UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Effects of flexible modes and vibration damping for scaled teleoperation Chen, Chia-Tung 1996

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1996-0303.pdf [ 8.78MB ]
Metadata
JSON: 831-1.0064869.json
JSON-LD: 831-1.0064869-ld.json
RDF/XML (Pretty): 831-1.0064869-rdf.xml
RDF/JSON: 831-1.0064869-rdf.json
Turtle: 831-1.0064869-turtle.txt
N-Triples: 831-1.0064869-rdf-ntriples.txt
Original Record: 831-1.0064869-source.json
Full Text
831-1.0064869-fulltext.txt
Citation
831-1.0064869.ris

Full Text

EFFECTS OF FLEXIBLE MODES AND VIBRATION DAMPING FOR SCALED TELEOPERATION B y Chia-Tung Chen B . A . S c . (Engineering Physics) University of Br i t i sh Columbia, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF A P P L I E D SCIENCE in T H E FACULTY OF GRADUATE STUDIES ELECTRICAL ENGINEERING We accept this thesis as conforming to the ^ q u i r e d standard T H E UNIVERSITY OF BRITISH COLUMBIA 1996 © Chia-Tung Chen, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Electrical Engineering The University of Br i t i sh Columbia 2075 Wesbrook Place Vancouver, Canada V 6 T 1W5 Date: Abstract During experiments with a dual-stage bilateral teleoperation system proposed in [1], it was observed that flexibilities in the coupling linkage of the fine manipulators and in the structure of the coarse manipulator produce vibrations that can induce instability. The effects of flexible modes on the stability robustness and performance of a bilateral motion scaling system are studied empirically. Performance is examined by comparing the environment impedance to that transmitted to the operator, while stability robustness is evaluated using the multivariable Nyquist test. Difficulties in implementing an effective active damping control wi th endpoint feedback are then discussed. They can be traced mostly to inaccuracies in modeling and/or sensing and actuation. The approach of passive damping is investigated in detail. The constrained layer damping technique is proposed to achieve intrinsic structural damping. The stress/strain analysis based on the Strain Energy Method first proposed by Ross, Ungar and Kerwin [2] is summarized and explained. The extension of this analysis to a hollow beam with mult i-ple viscoelastic laminates is developed and simulation results are presented. Experiments were performed wi th a hollow cantilever aluminum beam with multiple damping layers, and the computed damping coefficients were compared to the measured ones. i i Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgements xi 1 Introduction 1 1.1 Bilateral Mot ion Scaling System for Microsurgery 1 1.1.1 Structural Flexible Modes 2 1.1.2 Stabil i ty and Transparency Analysis 4 1.2 Vibra t ion Suppression 5 1.2.1 Act ive Damping 5 1.2.2 Passive Damping . 6 1.2.3 Constrained Layer Damping 9 1.3 Thesis Overview . 11 2 Stability and Performance Analysis of the Motion Scaling System 12 2.1 Overview 12 2.2 Analysis 13 2.2.1 Model ing 13 2.2.2 Formulation , •. . 15 2.2.3 Objectives 20 i i i 2.3 Results , 24 2.3.1 Case #1 : master and slave rigidly mounted and decoupled . . . . 25 2.3.2 Case #2 : master rigidly mounted and flexibly coupled to slave . 32 2.3.3 Case #3 : decoupled master and slave mounted to a flexible platform 37 2.3.4 Case #4 : flexibly coupled master and slave mounted to a flexible platform 40 2.4 Summary 43 3 Active/Passive Damping and Endpoint Vibration Control 46 3.1 Overview . 46 3.2 Model ing and Open-Loop Response . 47 3.3 Closed-loop Responses . 49 3.3.1 Jo in t /Endpoint Feedback 50 3.3.2 P D Control . 51 3.3.3 Command-Preshaping 52 3.3.4 State-space Controller 55 3.3.5 Q-Parameter Control 56 3.4 Summary 63 4 Vibration Damping with Constrained Layer Damping 65 4.1 Structural Design . . . 66 4.1.1 Implementation . 67 4.2 Beam Analysis 70 4.2.1 History 71 4.2.2 Preliminaries on Structural Analysis . . . 72 4.2.3 Analysis Assumptions and Notat ion 86 4.2.4 Single Layer Formulation 88 iv 4.2.5 Double Layer Formulation 93 4.2.6 Multi- layer Generalization 100 4.2.7 Issues of Implementation 105 4.3 Beam Design . : ; • • 108 4.3.1 Overview 108 4.3.2 Damping Layer Composit ion . • . 109 4.3.3 Target Vibra t ional Frequency 110 4.3.4 Mul t ip le Damping Layers : . . . I l l 4.3.5 Symmetric Layer Arrangement . . I l l 4.3.6 Beam Design Guidelines 112 5 Experimental Work 115 5.1 Introduction 115 5.2 Damping Measurements 115 5.2.1 Experimental Procedure 115 5.2.2 Results and Discussion . 117 6 Conclusions 125 6.1 Contributions 126 Bibliography 128 Appendex 132 A Motor Control 133 A . l A r m Actuat ion . . . : . . . 133 A.1.1 Motor Selection 133 A.1.2 Motor Actuat ion • • 135 v A. 1.3 Experiment Setup . . . 136 A. 1:4 Experiment Result 139 A. 1.5 Conclusion 140 B M A P L E Source Code for Beam Design 142 C M A T L A B Code for Beam Design 150 v i List of Tables 2.1 Input/output vectors for different cases . 15 2.2 Feedback matrices for proper motion coordination 17 2.3 Values for the base model parameters and their changes in each case. . . 24 4.4 Advantages of symmetrical layer configuration is evident 112 5.5 Experimental results with different beam constructions. . 122 v i i List of Figures 1.1 A proposed system configuration. (Drafted by M r . Joseph Yan.) : . . . . 3 2.2 1-DOF Bilateral motion scaling system models 14 2.3 Free-body diagram of the master system 22 2.4 Case #1: Position/force controls. 26 2.5 Case #1: Stability/transparency and position/force scalings 28 2.6 Case #1: Stability/transparency and hand/environment stiffness 29 2.7 Case #1: Stability/transparency and hand/environment damping 30 2.8 Case #1 Summary Plot : Ms vs kh/ke, bh/be and np/nf 32 2.9 Case #2: Stability/transparency and position/force controls. . . . . . . . 33 2.10 Case #2: Stability/transparency and position/force scaling wi th different passive linkage damping 34 2.11 Case #2: Stability/transparency and hand/environment impedance. . . . 36 2.12 Case #3: Stability/transparency and position/force scaling with passive platform damping 39 2.13 Case #3 : Stability/transparency and hand/environment impedance. . . 40 2.14 Case #4: Stability/transparency and position/force control with different flexible structure masses 41 2.15 Case #4: Effects of passive damping and position/force scaling on the stability margin 42 3.16 Robot arm modeling block diagram 48 3.17 Free structural vibration due to endpoint or base disturbances 48 v i i i 3.18 Root-locus plots for robot control 50 3.19 Nyquist plots for robot control 51 3.20 Endpoint vibration wi th P D control 52 3.21 Vibra t ion suppression wi th command pre-filtering 54 3.22 Robot control with full state-space controller. 55 3.23 Q-parameter control reference plots. . 57 3.24 Time responses of Q-parameter control with modeling errors 59 3.25 Block diagram of the endpoint feedback closed-loop control system. . . . 60 3.26 Nyquist plots for Q-parameter control with modeling uncertainties. . . . 61 3.27 Bode plots for Q-parameter control with modeling uncertainties 62 4.28 A hollow rectangular beam with damping layers 66 4.29 Viscoelastic materials have frequency dependent damping capacity 73 4.30 Stress-Strain plot defines loss factor r\ 74 4.31 Phasor diagrams representation of sinusoidal stress and strain 74 4.32 Total deformation is the sum of flexural and shear angles. . . . . . . . . . 80 4.33 Strain.diagram of a beam under centric bending 82 4.34 Strain diagram of a beam under eccentric bending 83 4.35 Bending moment of a multi-layer beam is additive 84 4.36 Shearing in a multi-layer beam 85 4.37 A single-layer constrained-layer damping model 88 4.38 A double-layer constrained-layer damping model. 94 4.39 A multi-layer constrained-layer damping model 101 4.40 Loss factor peaks at a specific configuration .109 4.41 Frequency response of the loss factor and multi-layer application. . . . . I l l ix 5.42 Experimental setup of the beam experiment with internal optical endpoint motion sensing 116 5.43 Free vibration of the bare aluminum beam 120 5.44 Design guidelines for one-side and two-side configurations 121 A.45 Electrical connection of the motor and mechanical setup of the experiment with optical position sensing. (Graphs are not drawn to scale.) 136 A.46 Motor torque and angle of rotation relationship 138 A.47 Frequency responses of the closed-loop motor control system 140 x Acknowledgements During this journey, I have encountered many people who have helped me graciously and I owe much gratitude to each of them. Firs t , I like to thank my mom and dad who have encouraged me wi th their trust and strength to take on this journey of graduate study. I am grateful to many friends who have always been part of my life, contributing much joy and meaning to my existence and offering sincere friendship which has enriched and fueled my soul; among them, I like to thank especially Marianne Hamberger, Roxanne Louie, Joseph Yan , Beatrice Mor i t z , N i a l l Parker, Angela Chou, Mor ie Chen, Ian Burke, Shyan K u and Stefanie Ebelt . I like to thank my supervisor, Dr . T i m Salcudean, for all my work. Over the years, T i m has been giving me all the support and patience, and provided me wi th opportunities in learning new knowledge; I have learned much from h im in al l disciplines and greatly indebted to h im. Final ly , I like to thank my loving parents again for their extraordinary effort in raising me, and their endless and unconditional love and support for everything I do. I would like to dedicate this work to my mom and my dad. XI Chapter 1 Introduction 1.1 Bilateral Motion Scaling System for Microsurgery A bilateral teleoperation motion scaling system consisting of magnetically levitated (ma-glev) master and slave systems was proposed for microsurgery experiments [1, 3, 4]. A prototype was designed and built , and preliminary evaluation experiments were per-formed [1]. A surgeon or an operator would handle the master to teleoperate on an environment that is in direct contact with the slave. The surgeon's hand motion and force are transmitted to the slave, and the motion and forces sensed by the slave while interacting wi th the environment are also transmitted back to the master to let the sur-geon feel the environment. The transmission of position and force between the master and the slave can be scaled independently to achieve a more suitable system for micro-surgery. For example, the motion from the master to the slave can be scaled down and the force reflected back from the slave can be scaled up in order to increase the operator's dexterity. During the evaluation experiments for the system mentioned above, instability was observed at the master and the slave. When the bilateral motion scaling system was mounted to a stationary surface, the vibrations at the master and the slave could be reduced by manually holding onto the flexible linkage between the wrist systems. This action effectively applied some damping to the linkage and the associated flexible mode. When it was mounted to the endpoint of a moving robot C R S - A 4 6 0 , the vibrational 1 Chapter 1. Introduction 2 problem at the wrists became even more severe. These observations suggested that the flexibilities in the physical structure of a bilateral motion scaling can affect the stability of the control system and prevent the successful operation of such a teleoperation system in microsurgery. In order to safely and effectively use such a system for microsurgery, it is necessary to analyze the stability of the control system under different operating conditions and to evaluate the transparency achievable between the master and the slave, both under the consideration of structural flexibilities. The stability and transparency of a special case of the bilateral motion scaling system described above were studied in [5], where the motion and force scalings between the master and the slave are of 1:1 relationship. Furthermore, the analysis assumed that no flexible modes are present in the structure. This may not be true in a realistic implementation. Since the existing robot structures mostly do not incorporate structural damping in the designs, it is useful to design a well damped moving structure as a coarse-motion stage suitable for the bilateral teleoperation system. Thus, the modeling in the mechanical structure and the flexible modes does not particularly conform to a specific existing robot structure, and is simplified to focus the analysis mainly on understanding the effects of structural flexible modes on the stability of a bilateral motion scaling system. 1.1.1 Structural Flexible Modes The master and the slave are two maglev robotic wrists each with a flotor and a sta-tor. The flotor is magnetically levitated above the stator wi thin a confined magnetic gap, which defines the operational workspace for the wrist system. The stators of the master and the slave of the motion scaling system proposed in [1] are coupled together by an arm linkage, the common stator, as shown in Figure 1.1. It was also proposed to mount the teleoperation system to the endpoint of the a robotic structure as shown in Chapter 1. Introduction 3. Figure 1.1: A proposed system configuration. (Drafted by M r . Joseph Yan.) Figure 1.1 [1, 3]. This serves the purpose of potentially extending the dynamic range of the teleoperation system by virtually enlarging master and slave workspace. Both the common stator linkage and the robotic structure can exhibit structural flexible modes, which translate into vibrational motions at the endpoint of the robot and unwanted disturbances at the common stator. The structural vibration can be excited both from actuating the robot and/or the motion scaling system, and from applying external forces to parts of the system such as contacting of the slave wi th the operating objects and/or contacting the robotic structure or command stator wi th the operator's hands. Since the controller design in the proposed motion scaling system assumes that the master and the slave are two completely decoupled and undisturbed rigid bodies, these structural vibrational modes and external excitations can potentially threaten the Chapter 1. Introduction 4 stability of the proposed control systems [1, 3]. In fact, instability has been observed in actual laboratory experiments; especially when the motion scaling system was mounted to the C R S - A 4 6 0 robot. It was "also observed when the position and/or.force scalings were changed to higher values. To verify these observations and understand the source of difficulties in the control tasks, a complete stability and transparency analysis is yet to be performed. The analysis should not only isolate the most problematic causes for the control system, but also suggest design guidelines for the motion scaling system and predict the performance. Such a tool is necessary in realizing a practical motion scaling system for microsurgery. 1.1.2 Stability and Transparency Analysis Since the bilateral motion system generally involves two coupled robotic systems with multiple inputs, i t is difficult to obtain necessary and sufficient conditions for stabil-ity [5]. Colgate proposed the following definition in [6]: a bilateral manipulator is said to be robust if, when coupled to any passive environment, it presents to the operator an impedance (admittance) which is passive. Under the assumption of passive hand and operator impedances, this definition describes a necessary and sufficient condition for stability, and was used in the analysis of a number of telemanipulation control architec-tures [6, 5, 7]. However, it is not clear from the passivity argument how the stability margin changes with varying system parameters; a system satisfying the theorem may st i l l have a small stability margin and the performance may not be satisfactory. Fur-thermore, to the author's best knowledge, none has considered the effects of structural flexibilities on the choice of controllers and position/force scaling factors. A different analytical approach is exercised in this thesis. Operator hand and envi-ronment impedances are assumed and M I M O eigenloci analysis on the control transfer Chapter 1. Introduction 5 matrix of the teleoperation system together with the generalized Nyquist stability cri-terion defines a precise quantitative measure of the stability margin of the closed-loop system. The changes in the measure of stability margin can be observed wi th the in-troduction of different flexible modes, hand/environment impedances and position/force scaling factors. Simulations over a range of values can reveal the relationship between the stability margin and a particular physical or control parameter. 1.2 Vibration Suppression Much study has been done on analyzing the dynamics of flexible robotic structures [8, 9]. Structural vibration is not a unique problem specific to robotics. Rather, it is a generic problem faced by many in various fields [10, 11, 12]. In the aerospace field, structural v i -brations from air turbulences or engine roars can damage the on-board electronic systems of a fighter jet or a satellite. In acoustic engineering, structural vibrations generate and transmit unwanted audible noise. In robotics, they act as un-intended excitation forces and disturb sensing and actuation systems, and seriously l imi t resolution and tracking response. Rap id dissipation of the structural vibratory energy is an important goal for many applications. In different fields, this goal is achieved with different approaches. 1.2.1 Active Damping In the field of robotics, active control mechanisms play a major role in most of the ap-plications. A s the bandwidth of actuation systems increases, structural vibrations could be compensated directly wi th active control, given precise knowledge of the dynamics of the flexible structure. However, this is yet to be demonstrated in applications involving stiff structures. In order to actively control endpoint vibrations, it is often required to feed back Chapter 1. Introduction 6 the endpoint position of the robot [13, 14].. Besides linear modern control methods such as state-space and Q-parameter controls [15], many other active control approaches and algorithms have been studied to achieve vibration suppression/control for flexible robotic structures. Some of them involve pre-filtering the command input to move the robot without exciting residual vibration [16, 17], constructing part of the structure with piezoelectric material in order to actively control the structural vibration [18], introducing additional active damping mechanisms between the joint link and the robot arm that bears the actual payload [19], and mounting a high performance robot wrist at the endpoint to compensate for any endpoint vibrations [13]. Many of these approaches require complex hardware additions or modifications, and the effectiveness of the linear active control algorithms depends heavily on the accuracy of the modeling of the actual physical system. This requirement on modeling precision is not practical in many cases and difficult to achieve in real applications wi th linear control methods. The main difficulties are due to the inevitable sensing errors and time varying dynamics of the robotic structure during motion. These difficulties may be overcome with non-linear control actions such as adaptive control methods. A partial knowledge of the physical structure can be used in a model-reference adaptive controller or a self-tuning regulator to properly change the control action such that more effective active damping can be achieved [20, 21, 22]. Regardless of the active control used, better results would be obtained i f the structure has a higher intrinsic damping achieved by passive means. 1.2.2 Passive Damping There are a number of ways to passively reduce the endpoint vibration in a structure. For a robot, the conventional approach is to increase the mass and stiffness of the arm, and consequently require bigger and more powerful actuators to move the whole system and Chapter 1. Introduction 7 possibly causing compromises on the desired frequency response. In addition, although the magnitude of the vibration is reduced, the actual vibration is s t i l l present in the system and this is intolerable for systems that require high position and force resolution. Therefore, increasing the internal structural damping seems to be a direct and sound solution to the realization of light-weight and high performance robots. Passive damping has had a long history in aerospace and acoustic engineering, but poor understanding in the implementation and design methods have prevented much effective practices unti l recent years [23]. In robotics, viscous damping is often applied only to the actuation joints. This is ineffective in reducing the endpoint vibrations because the flexibilities in the arms also contribute to the structural vibration. Intrinsic material damping has not been exercised widely in designing or constructing a robot. However, some research has started and was presented in [10, 24]. Part of the work presented in this thesis also addresses this problem. General Techniques A number of passive damping techniques for mechanical structures are surveyed in [25]. The most straightforward method is to use high damping materials in the construction. Conventional metals such as aluminum and steels have poor damping capacities in the range of ^ = 0.01% - 1% (or damping coefficients C = 8 x 10~ 6 - 8 x 10~ 4) 1 [27]. High damping metals such as nickel and cast irons have \& = 10% —70% (or £ = 0.008 — 0.056), but are not suitable because they are either costly and/or heavy [27, 28]. Al though poly-mer materials generally possess high damping capacities (\I/ > 100%), the Young's moduli are usually in the. range of 10 5 — 10 6 Pascal, .which are low comparing to conventional 1The damping capacity measures the percentage of the energy dissipation per cycle, and is related to the damping coefficient ( as * = 4TT£ when V& < 63% [26]. Chapter 1. Introduction 8 metals in the range of 10 9 — 10 1 1 Pascal and not stiff enough for constructing weight-bearing robotic structures. For example, for a 1 — m long 1" x 1" hollow square beam with wall thickness of 1/16", the endpoint deflection due to an endpoint load of mass m = 4 kg is expected to be 0.5 cm for steel, 1.3 cm for aluminum and 19 cm for a typical Nylon material. Composite Materials Recent advancements in the composite materials offer an attractive solution. Depending on the exact resin/fiber mixture and the orientation of the fiber, the epoxy/carbon-fiber composite materials can have flexural Young's modulus E = 20 x 10 9 — 200 x 10 9 Pascal and damping capacity * = 0.5% - 10% (or C = 0.0004 - 0.008) [29, 30]. The stiffness is comparable to steel Esteei = 200 x 10 9 Pascal and aluminum Eat = 72 x 10 9 Pascal [31]. This type of composite materials may be the suitable construction materials for they possess higher intrinsic damping capability while preserving the stiffness of conventional metals. However, the best damping capacity achievable with composite materials are st i l l much less than high damping metals such as nickel and cast iron. Moreover, these materials are not readily available commercially, and their damping performances are heavily dependent on the specific alignments of the fibers and the manufacturing pro-cesses. Therefore, they are not suitable for the investigative work of passive damping in robotic structures presented in this thesis, but their damping capacities can be used for comparison wi th any experimental results. Further exploration in another suitable alternative passive damping technique is preferable. Chapter 1. Introduction 9 1.2.3 Constrained Layer Damping If polymer or viscoelastic materials offer high dampings over ^ = 100% and metallic ma-terials offer the high stiffnesses, the combination of both materials may yield the desired damping performance in robotic applications. This is the idea behind constrained, layer damping. First patented in 1939 [23], it consists of a layer of high damping viscoelastic material sandwiched between two stiff layers of metal. The shearing between the two stiff layers deforms the viscoelastic layer resulting in the dissipation of the vibratory energy in the system [2]. Shen took this approach further by replacing one of the stiff layers with an actively controlled piezoelectric layer [32]. His aim was to manipulate the constraining layer such that more shearing is induced in the viscoelastic layer at a particular vibra-tional frequency. The work presented in this thesis follows the conventional approach without any active piezoelectric member. Literature Survey Analysis of such a sandwich structure is required due to the frequency dependent damp-ing property of the viscoelastic material [23, 33]. The geometrical configuration of the layers determines the final damping capacity achieved, which is also frequency depen-dent [2]. Ear ly works were mostly in solving for the closed-form solutions of the beam dynamics. Ross, Ungar, and Kerwin dealt wi th flexural vibration of beam/plate struc-tures [2]. DiTaranto [34] solved the complete equations of motion of a finite length beam with boundary conditions using a different set of coordinate system from the one used in [2]. L u , Douglas, and Thomas looked at the sandwiched rings with radial loads [35]. Pan [36], Wi lk ins [37] and Y u [12] addressed axis-symmetrical vibration damping of a sandwiched cylindrical shell. These closed-form solutions mostly involve high order dif-ferential equations wi th various boundary conditions. Chapter 1. Introduction 10 Due to the complexity of the closed-form solutions and their l imitations when applied to more complex structures, recent researchers have been tending toward the finite ele-ment method ( F E M ) approach. Johnson [38] did some ini t ia l work on damping prediction using the F E M approach. Hwang and Gibson followed and expanded the problem to con-sidering the generic multi-layer construction of sandwiched structures [39]. Ramesh and Ganesan [40, 41] recently revisited the sandwiched cylindrical shell problem and further investigated the harmonic responses using F E M based on Wi lk in s ' earlier work. Research Tasks In controlling the endpoint of a robot with flexible modes, it is necessary to know the end- • point position. For the purpose of maximizing the stiffness-to-weight ratio and possibly of accommodating an internal endpoint sensing system, it is advantageous to design robot arms with hollow cross-sections. In order to properly design such a system where multiple constraint-viscoelastic layers may be applied, an analytic tool must be developed to aid the design process. The F E M approach does not offer a quick answer due to the amount of work involved in setting up the necessary software system. No closed-form solutions developed earlier consider generic sandwiched structures with hollow cross-sectional area and multiple damping layers. Sandwiched cylindrical shell theories readily assume low shell thickness-to-radius ratio, which is not true for a usable beam capable of supporting any reasonable endpoint payload. Therefore, it is one of the tasks in this thesis to de-velop an analytical approach for estimating the structural damping of a hollow robotic arm sandwiched by multiple layers of constraint-viscoelastic layers. Chapter 1. Introduction 11 1.3 Thesis Overview This thesis answers a series of questions in the process of realizing a practical and usable bilateral motion scaling system for microsurgery experiments: How are the stability and transparency performance affected when there are flexible modes present in the system? Is passive damping a more practical and effective method than the active approach in suppressing the structural vibration caused by these flexible modes ? If so, what may be the appropriate technique and design methods for a well-damped structure? Chapter 2 presents a complete stability/performance analysis of the bilateral motion scaling system wi th different structural flexibilities. Effects of material damping, posi-tion/force scalings and hand/environment impedances on the stability and performance of the control system are studied. Chapter 3 addresses the effectiveness of active control in structural vibration suppression wi th endpoint position feedback. Several algorithms are investigated, and the results are evaluated against the ones obtained wi th just passive material damping. Chapter 4 develops a design tool for a hollow square tube robot arm damped wi th several constrained layers. The intrinsic structural damping capacity of an arm with certain physical configurations can be estimated prior to assembly. Chapter 5 reports some experimental results to verify the formulation developed in Chapter 4, and discusses some general design guidelines. Chapter 2 Stability and Performance Analysis of the Motion Scaling System Vibrations have been observed from time to time in operating the bilateral motion scal-ing system [1]. A full stability and performance analysis is useful to isolate and un-derstand the causes. This chapter details the analysis procedure and discusses how the stability and performance of the control system are affected by the changes in different control/mechanical parameters. 2.1 Overview The simplest bilateral motion scaling system consists of a master and a slave, where each system is controlled to follow the motion of the other. The stability and transparency of a special case of such a system is studied in [5] where a position/force scaling of 1:1 was considered and both master and slave have stationary bases. This is the ideal case. The system proposed in [1] not only allows motion scaling factors other than 1, but also exhibits some structural dynamics. For example, the mechanical design of the system requires the stators of master and slave be mounted together through a linkage arm that may have some flexibility. Furthermore, in order to position the system conveniently at different operating sites, the system needs to be mounted on a moving or transporting platform which may also exhibit flexible modes. It is often desirable to independently design and optimize the motion coordinating control schemes without the prior knowledge on any flexible structures where master and slave may be mounted. This allows the freedom to implement the bilateral motion .12 Chapter 2. Stability and Performance Analysis of the Motion Scaling System 13 system under different environments where different transporting structures are available. However, this often results in poor compensation for any flexural motions present in the system. O n the other hand, the difficulties in precise modeling and prediction of the flexural dynamics often prevent any effective compensation from succeeding. Thus, it is necessary to investigate whether it is possible to safely operate a bilateral motion scaling system mounted on a flexible structural without altering the design of the original control system. As wi l l be shown in this chapter, by increasing the mechanical damping of the flexible structure, this is possible and the threat to the stability and performance of the motion scaling system diminishes. Further addition to the work in [5] also includes the study of how position/force scaling factors can affect the stability. 2.2 , Analysis 2.2.1 Modeling Four different cases (shown in Figure 2.2) are considered: Case #1 The master and the slave are both on a stationary surface and rigidly mounted with respect to each other. Case #2 The master is on a stationary surface and is coupled to the slave by a flexible linkage. Case #3 The master and slave are rigidly mounted with respect to each other, but both are supported by a flexible arm. Case #4 Same as Case #3, but there exists some coupling flexibility between the master and the slave. Case #1 is the ideal case, Case #2 describes the motion scaling system proposed in [1] mounted securely to a stationary surface and Case #4 illustrates the scenario where the Chapter 2. Stability and Performance Analysis of the Motion Scaling System 14 m. Case #1 : Ideal Case Case #2 f, f. Case #3 Case #4 Figure 2.2: 1-DOF Bilateral motion scaling system models. system is mounted to a transporting robot or platform. For curiosity, Case #3 is to study how the ideal.motion system would behave if the master and the slave were both subject to exactly the same disturbance from any structural flexural motion. The analysis uses specific and simplified models to describe the bilateral motion scal-ing system in 1-degree-of-freedom ( D O F ) . It can be generalized to more complex models. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 15 2.2.2 Formulation Plant Descriptions State space representations can be used to describe the physical plants shown in F i g -ure 2.2. Each system is subject to a particular input force / , and assumes a state space realization x = Ax+Bifi ; y = Cx . (2.1) The nxn matr ix A describes the dynamics of the physical system, where n is the number of the states. The input force / ; can be one of fm and fs - actuation forces of master and slave, fh and Je - hand/environment forces experienced by master/slave and /j and fp - disturbance forces on the flexible link and platform (see Figure 2.2.) The matrices P>i are n x l . The system state vector x consists of the positions and velocities of the masses and the output vector y are just the positions. For each case studied, they are defined as follows: X y Case #1 Case #2 Case #3 Case #4 \xm Xm Xs Xs\ \Xp Xp Xjyi Xm Xs Xs] \Xp Xp X[ X{ XfYi XJYI Xs X s ] \X{ Xm Xg\ \Xp Xfyi X 5 j \Xp X[ XJJI xs] Table 2.1: Input/output vectors for different cases. Taking the Laplace Transform of (2.1), the plant transfer function Gi(s) due to a force fi can be expressed as: y = Gi(s)fi ; = C(sl - A)~1Bi , (2.2) where fi is the Laplace transform of a particular input force B y convention, the Laplace transform of a signal / is denoted by / . The transfer functions Gm, Gs, Gh, Ge, Chapter 2. Stability and Performance Analysis of the Motion Scaling System 16 Gi and Gp are denned as in (2.2) and each is subject to a input force of fm, fs, fh, fe, fi and fp, respectively. Hand/Environment Force As discussed in [1], the hand force fh and environment force fe consist of active and passive parts: fh = fha ~ H(s)xm ; fe = fea - E(s)xs , (2.3) where fha and fea are the active hand and environment forces, and xm and xs are the mas-ter and slave positions. H(s) and E(s) represent the hand and environment impedances and are defined by (2.3) when no active forces are present (i.e. fha — 0 and fea = 0) 1 : H(s) = - A ; E(s) = -k- • (2.4) xm xs Control Forces and Objectives As proposed in [1], the control actions are fm = -mmfc + nffe ; / , = -msfc + f h + { U f " n m U p ) f e , (2.5) nf nmnp where rif and np are force and position scaling factors, mm and ms are master and slave masses, nm = mm/ms is the mass ratio and fc is the position coordinating force between the master and slave. The control actions can also be separated into two parts: position control (terms wi th fc) and force control (terms with fh and fe). The coordinating force fc can be thought of as a physical spring/dash-pot type of connection between the master and the slave. This kind of force can be generated by a P D type of control action and the Laplace transform of it can be formulated as fc = Kc(s)e = Kc(s)Hc y ; Kc(s) = kvs + kp ' , (2.6) 1Note that (2.4) differs slightly from the usual definition of impedance Z = F/V, where Z is the impedance, F is the force and V is the velocity. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 17 where Kc(s) represents a high gain P D controller with the proportional term kp > 0 simulating the stiffness of the spring and the derivative term kv > 0 simulating the viscous damping of the dash-pot. Hc is the feedback matrix necessary to generate the error signal where ym and ys are the positions of the master and the slave wi th respect to their own mounting platforms and np is the coordinating scaling between their positions. The appropriate scalar matrices i f c ' s used to generate the correct error signal from the output vector y for each case studied are listed below : Case #1 Case #2 . Case #3 Case #4 [1 -np] rhp 1 rip [rip — 1 1 — rip] —1 Up 1 — np] y [xm xs Ipl %m %s. r" ~ - I T T* r1' T* l^p *^5. Xp X\ Xm X S J Table 2.2: Feedback matrices for proper motion coordination. The controller in (2.5) was designed to achieve the transparency objectives fm — Tlffe j -Em — ripXs , (2-7) where the force applied to the master is scaled by a factor of ny from the actual force experienced at the environment and the position of the master is scaled by nv from that of the slave. System Formulation Taking the Laplace Transform of the state space description in (2.1) and uti l izing the transfer function expression in (2.2), an open-loop system can be stated as y = Gmfm + Ggfg + Ghfh + Gefe + Gifi + Gpfp , Chapter 2. Stability and Performance Analysis of the Motion Scaling System 18 where fi = 0 and fp = 0 in Case #1, fp — 0 in Case #2 and / ( = 0 in Case #3 by default. Substituting in equations from (2.3) to (2.6), the above expression becomes V = -{mmGm- —msGs)KcHcy nf -[(Gh + ^—Gs)H (Ge + n f G m + n f ~ n m n p G s ) E ) y fimrip fimnp +(—Gs + Gh) fha + (nfGm + U f ~ nmUp.Gs + Ge) fea + Glfl + Gp fp \ TlmTlp TlmTlp This evolves into a closed-loop system description, and can be formulated and simplified as (I + Hp + Hf)y = Ghafha + Geafea + Gifi + Gpfp , (2.8) where we define the position control transfer matrix Hp(s) G s f t m x m & g Hp(s) = (mmGm-—msGs)KcHc , (2.9) "/ the force control transfer matrix Hf(s) G 3 ? m x m as Hf(s) = {{Gh + -^-Gs)H (Ge + nfGm + n f ~ n m n p G s ) E } (2.10) rim^p rimnp and the plant transfer matrices Gha(s) G 5 R m x l and Gea(s) G ! R m x l as Gha(s) = —-—Gs + Gh • Gea(s) = nfGm + n f ~ n m n p G s + Ge . (2.11) Properness of Hp(s) and Hf(s) In order to investigate the invertibility of the matrix (I + Hp + Hf) in (2.8), it is necessary to show that the control transfer matrices formulated above are proper. First , it is shown below that the open-loop physical plant transfer functions Gj(s) in (2.2) are stable and strictly proper. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 19 Proof: B y taking the Laplace transform of Newton's equations for the cases presented in Figure 2.2, the system output y d u e to an input force / ; can be expressed as A(s)y = Bji or y = A^^Bji = Gl{s)fi , where A(s) is the plant matrix and P>i is a scalar matrix associated wi th a particular input force From [42], the inverse of a matrix A is A'1 = . - ^ A 1 /icof det A where Acof is the cof actor matrix of A. For Case #4, the system dynamics matrix is Ms) = mps2+(bp+bi+bm)s+(kp+ki+km) -(fcjs+fc/) -(bms+km) 0 -(hs+ki) mis2+(bi+bs)s+(ki+ks) 0 bss+k3 -(bms+km) 0 mms2+bm,s+km 0 0 -(bss+ks) 0 mss2+bss+ks Assuming detA(s) ^ 0, the degree of the determinant of A(s) is always at least 2 greater than the degree of any of the cofactors. In Case #4, the determinant would have a degree of 8 while the cofactors can have maximum degree of 6. This is true for the other cases studied. Therefore, the plant transfer function Gi(s) = A~l(s)P>i is strictly proper and the degree of the denominator is always 2 greater than the one of the numerator. Furthermore, as defined in (2.4), the passive hand and environment impedances H(s) and E(s) can model the physical connections between the flotors of master and slave and the masses of hand and environment objects, respectively. Therefore, H(s) and E(s) have at most two more zeros than poles. To model just the spring-dashpot connection, H(s) and E(s) are first order systems in the form of (bs + fc); to include mass inertia effects, they are second order systems in the form of (ms2 + bs + fc). (In the analysis, H(s) and E(s) are modeled as first order systems.) Similarly, as defined in (2.6), the coordinating controller Kc(s) is a first order system with one zero but no poles. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 20 The position control transfer matrix Hp(s) in (2.9) is the product of 1. a linear combination of open-loop plant transfer functions (mmGm — msGs/rif) which has two more poles than zeros because each transfer function G{ has at least two more poles than zeros as shown in the proof above, 2. a first order coordinating controller Kc with.only one zero and 3. a feedback matr ix Hc wi th only scalar entries. Therefore, the number of poles is always greater than the number of zeros in Hp(s) and it is stable and strictly proper. Similarly, i f H(s) and E(s) are modeled as first order systems, the force control transfer matrix Hf(s) in (2.10) is stable and strictly proper. This is because the entries of Hf(s) consist of the product between the open plant transfer function Gj(s)'s and impedance transfer functions H(s) or E(s). Even if the impedances are modeled as second order systems, Hf(s) w i l l remain stable and proper, but not strictly proper. 2.2.3 Objectives A - Invertibility The closed-loop system expression (2.8) shows that the stability of the system output y depends on the invertibility of the matrix (I + Hp(s) + Hf(s)). Since both Hp(s) and Hf(s) are stable proper transfer matrices, the control transfer matrix (I + Hp + Hj)~l is stable if and only i f det(J + Hp + Hf) has a stable inverse [43]. Therefore, it is necessary to first find the transfer function det(J + Hp + Hf) and to determine whether it has any zeros in the right-half-plane of the s-domain, before proceeding any further. If the matrix is invertible, then the analysis interest turns to the robustness of the system. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 21 B - Stability Margin For a single-input-single-output (SISO) system, the Nyquist stability criterion is the classical technique for studying the stability of the control system. Since the formula-tion (2.8) results in a .multi-input-multi-output ( M I M O ) system, the Nyquist criterion used for the SISO case must be generalized for the M I M O case. There exist several ways of generalizing the classical Nyquist stability criterion and the approach proposed by Desoer and Wang [15, 44] is exercised here. A brief description of the theorem and analysis procedure are described below. Characteristic Loci of a Transfer Matrix and Nyquist Plot Let H(s) £ J R ( s ) m x m be a proper rational open-loop transfer matrix with m inputs and m outputs. From the characteristic equation \XI-H(s)\ = 0 , the eigenvalues Aj(s) can be found, with i = 1.. .m. Each eigenvalue ~\(s) forms an eigenlocus or characteristic locus ji on the real-imaginary plane when it is evaluated across the entire frequency spectrum, s = jcu where ui € ( — 0 0 , 0 0 ) . Each eigenlocus 7; may not form a closed curve, but there is always a combination of 7J'S that collectively forms a closed curve or a circuit- C [44]. Thus, the Nyquist plot of the open-loop transfer matrix H(s) is the circuit C formed by the characteristic loci 7;'s of H(s). Generalized Nyquist Stability Criterion Let H(s) € R(s)mxm be proper as defined above. The MIMO closed-loop system [I + i ^ ( s ) ] - 1 is stable if and only if the charac-teristic loci jt's of H(s) (taken together) or the circuit C encircles the critical point — 1 counter-clockwise as many times as the number of right-half-plane (RHP) or unstable poles of H(s). Chapter 2. Stability and Performance Analysis of the Motion Scaling System 22 m m , 'm m "-rn Figure 2.3: Free-body diagram of the master system From the closed-loop formulation (2.8), either H(s) = Hp(s) + Hf(s) or H(s) = Hp(s) can be used as the open-loop transfer matrix for studying the stability of the closed-loop system wi th position/force or position only control. Since both Hp(s) in (2.9) and Hf(s) in (2.10) consist of linear combinations of stable physical systems and controllers, H(s) cannot have any unstable poles. Therefore, no encirclement of the critical point by the eigenloci of H(s) or C is the suitable stability criterion and the relative distance between them can serve as a reasonable measure to the robustness or the stability margin. The stability margin measure Ms is then formally defined as Since the Nyquist plots of strictly proper rational systems always converge to the origin at high frequencies, Ms cannot be greater than 1. Thus, a system wi th a stability C - Transparency Measure Since the goal of the bilateral teleoperation is to achieve transparency between the op-erator (or master) and the environment (or slave), a measure of transparency achieved is useful to quantitatively assess the performance of the control system. The control in (2.5) has motion scaling transparency objectives (2.7) of fm = njfe and xm — npxs. Assuming no active environment force (i.e. fea = 0) and considering the impedances Ms = min |1 + Aj(jo;)|, i = l...m (2.12) margin Ms = 1 can be viewed as a robust system. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 23 defined in (2.4), the transparency objectives above can be translated to imply Xfyi TlpXg np 1 (2.13) L nffe nfE(s) From the free-body diagram of the master mass in Figure 2.3, Newton's equation can be written as ^"m^m — fm fh k>m%m ^m-^m Substituting in fh — fha — Hxm from (2.3) and taking the Laplace transform of Newton's equation above, the Laplace transform of the local master actuation force fm can also be expressed as fm = ~ fha + H{s)xm + {pimS + bmS + km)xm Combining wi th (2.13), an ideal input/output transfer function for transparency is arrived and can be used as a suitable reference for an ideal response to observe between the master position and the applied hand force: 1 Tref{s) fl ha fm = riffe Xjfi — TlpXs /ea = 0 (nf/np)E(s) + H(s) + (mms2 + bms + km) (2.14) Thus, achieving the transparency goals in (2.7) means the impedance between the master position and the active hand force is the summation of a scaled environment impedance, the passive hand impedance and the inertial dynamics of the master and its local stiffness and viscous damping. Final ly, a transparency measure Mt can be formally defined as Mt = 20 log 10 fi ]-ref ha (2.15) There are different ways of defining a transparency measure depending on the method of measurement [1]. Mt measures the difference in decibel scale between the actual Chapter 2. Stability and Performance Analysis of the Motion Scaling System 24 Base Case #1 Case #2 Case #3 Case #4 Unit np 1 1 - 20 1 - 20 1 - 16 1 - 40 -nf 1 1 - 20 1 - 20 1 - 2 0 1 - 4 0 -mm 0.670 - - - - kg ms 0.040 - • - - - kg k 26 - - - N/m bm 0.8 - - - 0.8 Ns/m ' ks 1.6 - - - - N/m bs 0.05 - - - 0.075 Ns/m mi 2 - 2,200 - 2,200 kg mp 2 - - 2,200 2,200 kg Ci 0.1 - 0.005 - 1 - 0.005 - 1 -Cp o . i - - 0.005 - 0.5 0.005 - 1 -Ul 10 - • - - - Hz up 10 - - - - Hz kh le3 Ie2 - le4 5e2 - 7e4 Ie2 - le4 5e4 N/m . bh 100 50 - 200 . 30 - 200 1 - 200 100 Ns/m ke le5 le4 - le6 le4 - le6 le4 - le6 le5 N/m . be 1 1 - 100 1 - 100 1 - 100 1 Ns/m Note : ki, kp, bi and bp in Figure 2.2 are defined be ow as 2nd order systems: k = micuf ; kp = mpu?v ; bi = 2miUJiQ , bp — *2mpUJp(^p Table 2.3: Values for the base model parameters and their changes in each case. input/output response of the closed-loop system and the ideal response (2.14). The ideal transfer function Tref is setup against a known hand and environment impedances H(s) and E(s), respectively. The purpose is to potentially benchmark the transparency performance of the bilateral motion scaling against some expected (or pre-determined) best performance from the measurable input fha and output xm. 2.3 Results The goal of the analysis is to look at how different control and mechanical parameters affect the stability and performance of the system. The variables of interest include -Chapter 2. Stability and Performance Analysis of the Motion Scaling System 25 flexible structures: masses, stiffness and damping; position/force control systems: po-sition and force scalings; and hand/environment systems: stiffness and damping. It is difficult to symbolically study the contribution of each parameter to the overall stability and performance of the control system. The approach used is to first define a reference numerical model based on the real bilateral motion scaling system. The values of the parameters under study are altered within specific ranges to observe their effects on the invertibility, stability margin and transparency measure of the base system. Conclusions are drawn from studying how the Generalized Nyquist plots and the transparency measure plots change according the changes in these parameters. The base model reflects a master/slave system much similar to the existing experiment setup in the laboratory [1]. The masses and local stiffnesses and dampings are chosen based on the real magnetically levitated (maglev) macro/micro wrists. The hand and environment impedances approximate human hand properties and hard contact surfaces. The weights of the linkage arm and moving platform are selected to reflect light weight robotic assemblies. Table 2.3 lists the values chosen for the base system parameters and the ranges that they were changed in the analysis for each case studied. The block diagram of each case studied below are shown in Figure 2.2. Furthermore, in the Nyquist plots shown below, a line is drawn to indicate the shortest distance from the eigenloci to the crit ical point. Due to the unsymmetrical aspect ratio of the horizontal and vertical axes, this line may not necessarily appear to be perpendicular to the tangent of the eigenloci curve at the nearest point, as it should be perceived intuitively. 2.3.1 Case #1 : master and slave rigidly mounted and decoupled The ideal case describes the system where both the master and the slave are mounted rigidly to a stationary surface. In order to establish a basis for the analysis in the rest of Chapter 2. Stability and Performance Analysis of the Motion Scaling System 26 T r a n s p a r e n c y : P o s i t i o n O n l y T r a n s p a r e n c y : F o r c e / P o s i t i o n - 1 . 5 - 1 - 0 . 5 O — 1 . 5 - 1 - 0 . 5 O ( c . ) R e a l o f H _ p ( d . ) R e a l o f H _ p f Figure 2.4: Coordination only control is much different from position/force control: Nom-inal values for the Base system listed in Table 2.3 (np=nj=l) are used. The transparency achieved wi th position/force control in (b.) is clearly better than wi th just position co-ordination in (a.). However, the control system is more robust for the latter than the former. W i t h just coordination between the master and the slave, the system is robust with Ms = 1 in (a) . W i t h the addition of force control, the stability margin reduces to Ms = 0.38 in (d.). (For this ideal system, Hp(s) G 2 x 2 and Hf(s) G 2 x 2. Both eigenloci and 72 can be seen in (d.) while only one can be seen in the range shown in sections, the stability/performance analysis results on the ideal case are discussed first in this section. The stability margin and transparency measure obtained for the base system here can be used as reference measures for other cases. Base System Figure 2.4 shows the stability and performance plots for the closed-loop systems with position only and position/force controls. F i g . 2.4(a.) and (b.) show that significant improvement in transparency is achieved when the additional force control scheme is Chapter 2. Stability and Performance Analysis of the Motion Scaling System 27 implemented. However, F ig . 2.4(c) and (d.) show that under the same plant configura-tions, the position only control scheme has better stability margin than the position/force control approach, while the transparency is much worse. Position Only Control Figure 2.4 demonstrates the classical tradeoff between sta-bil i ty and performance. The computed stability margin of a system wi th position only control scheme remains robust with parameter variations. There are changes observed in the eigenloci for position control due to parameter changes, but none threatens the stabil-ity of the system. Thus, unless specifically mentioned, the results (stability margins and transparency plots) presented in this chapter assume position/force control approach. The stability margin Ms = 0.3834 for the position/force control scheme in F i g . 2.4(d.) can be used as a reference measure to gauge the sensitivity of the stability of the control system to different system parameters. Position/Force Scaling Figure 2.5(a.) and (c.) show the transparency and stability measures when the force scaling is increased to rif = 10. Comparing wi th F i g . 2.4(b.), the transparency is worse at higher frequencies and the stability margin is reduced down to Ms = 0.1350. Increasing the force scaling effectively amplifies the force controller gain and results in a less robust system. O n the other hand, increasing the position scaling results in better transparency and a more stable system. F i g . 2.5(b.) and (d.) shows an improved performance at higher frequencies and a stability margin of Ms = 0.9507. This improvement can also be observed readily from the proposed control law (2.5). If np >> rif in (2.5), the local feedback term [rif — nmnp)/nmnp in the slave actuation force wi l l approach —1. A s predicted by Y a n in [1], this results in a more stable system. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 28 T r a n s p a r e n c y : n p = 1 ; n f = 1 0 T r a n s p a r e n c y : n p = 1 0 ; n f = 1 C O S - - 1 Q O ' - 2 0 0 - 3 0 0 2 1 . 5 1 1 O " 1 O " ( a . ) F r e q u e n c y ( H z ) N y q u i s t P l o t : n p = 1 ; n f — 1 O CO S - - 1 Q O - 2 0 0 - 3 0 0 °T 31 0 . 5 O - 1 . 5 M a r g i n = 0 . 1 3 5 0 p o o 3 o ooorarnJ 2 1 . 5 1 0 1 0 ( b . ) F r e q u e n c y ( H z ) N y q u i s t P l o t : n p = 1 0 ; n f = 1 0 . 5 - 1 - 0 . 5 ( c . ) R e a l o f H _ p f O - 1 . 5 M a r g i n = 0 . 9 5 0 7 o - 1 - 0 . 5 O ( d . ) R e a l o f H _ p f Figure 2,5: Posit ion and force scalings affect the transparency and stability of the po-sition/force control system: The stability margin drops to Ms = 0.13 in (c.) when the force scaling is increased to n / = 10. The transparency plot in (a.) is not changed much from F ig . 2.4(b.) except for some more heightened response in the high frequency region. When the position scaling is increased to np = 10, a better stability margin Ms = 0.95 is obtained in (d.) and better transparency is observed across the frequency spectrum. The increasing negative local feedback term effectively increases the slave mass inertia as the position scaling np increases against the force scaling rif. This additional effective mass inertia is achieved by counter balancing the sensed environment force through the local feedback in the actuation action. Thus, a more stable system is achieved. Al though this supports the desire for a motion-scaled bilateral teleoperation in micro-surgery due to the improved transparency and stability at high position scaling, the same results are not observed in the non-ideal cases studied. Those findings are discussed later in this chapter. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 29 N y q u i s t P l o t : k h = 1 e 2 ; k e = 1 e 5 M y q u i s t P l o t : k h = 1 e 4 ; k e = 1 e 5 _ 1 . 5 en 1 0 . 5 O - 1 . 5 o o o o M a r g i n = 0 . 3 8 5 1 o \ ° c y ° o < 2 1 . 5 r 1 0 . 5 — 1 — 0 . 5 ( a . ) R e a l o f H _ p f N y q u i s t P l o t : k h = 1 e 3 ; k e = 1 e 4 O - 1 . 5 o o o o o M a r g i n = 0 . 9 5 4 5 — 1 — 0 . 5 O ( c . ) R e a l o f H _ p f T r a n s p a r e n c y : k h = 1 e 2 ; k e = 1 e 5 - 1 0 0 So - 2 0 0 - 3 0 0 1 0 1 0 ( e . ) F r e q u e n c y ( H z ) T r a n s p a r e n c y : k h = 1 e 3 ; k e - 1 e 4 _ 1 . 5 * o 1 CO 0 . 5 0 — 1 2 1 . 5 1 0 . 5 O -1 . 5 c o o M a r g i n = 0 . 3 6 8 2 o \ o J o o 5 - 1 — 0 . 5 O ( b . ) R e a l o f H _ p f N y q u i s t P l o t : k h = 1 e 3 ; k e = 1 e 6 M a r g i n = 0 . 1 2 4 0 o o o o c 3 o o o m j - 1 — 0 . 5 O ( d . ) R e a l o f H _ p f T r a n s p a r e n c y : k h = 1 e 4 ; k e = 1 e 5 - - 1 0 0 § > - 2 0 0 - 3 0 0 i o " 1 cr ( f . ) F r e q u e n c y ( H z ) T r a n s p a r e n c y : k h = 1 e 3 ; k e = 1 e 6 0 3 S - - 1 0 0 f - 1 0 0 h -20CH -2oo^ ( g . ) F r e q u e n c y ( H z ) ( h . ) F r e q u e n c y ( H z ) Figure 2.6: The hand/environment stiffness can affect the stability margin and trans-parency of the position/force control system: The hand stiffness kh is ' adjusted by ± 1 decade from the nominal value kh = 1 x 10 3 N/m in (a.) and (b.), respectively. A l -though no significant changes are observed, the stability margin is slightly better when kh is decreased and worse when it is increased. The same ± 1 decade change is applied to the nominal stiffness ke = 1 x 105Ns/m in (c.) and (d.). More drastic changes in the stability margin are observed. A t ke = 1 x 1 0 4 i V / m , the system gains much robustness with Ms = 0.95 and at ke = 1 x 10 6 N/m, the stability margin drops rapidly to Ms = 0.12. Graphs (e.), (f.), (g.) and (h.) are the corresponding transparency plots to (a.), (b.), (c.) and (d.), respectively. Significant improvement in transparency is observed only when environment stiffness is decreased. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 30 2 1 . 5 N y q u i s t P l o t : b h = 5 0 ; b e = 1 N y q u i s t P l o t : b h = 2 0 0 ; b e = 1 o 1 0 . 5 M a r g i n o = 0 . 1 9 8 2 o o o o \ o c J o o < - 1 . 5 - 1 - 0 . 5 ( a . ) R e a l o f H _ p f N y q u i s t P l o t : b h = i O O ; b e = 1 0 _ 1 . 5 3 = " S 1 «3 e — 0 . 5 O - 1 . 5 o M a r g i n = 0 . 4 1 5 7 o o o o \ ° 6 / ° o c - 1 - 0 . 5 O ( c . ) R e a l o f H _ p f T r a n s p a r e n c y : b h = 5 0 ; b e = 1 C O S . - 1 0 0 - 2 0 0 - 3 0 0 1 O 1 O ( e . ) F r e q u e n c y ( H z ) T r a n s p a r e n c y : b h = 1 O O ; b e = " I O - - 1 0 0 - 2 0 0 - 3 0 0 1 0 1 o ( g . ) F r e q u e n c y ( H z ) 0 . 5 0 . 5 M a r g i n = 0 . 7 1 4 6 o o o . . o / o o o O . 5 - 1 - 0 . 5 C ( b . ) R e a l o f H _ p f N y q u i s t P l o t : b h = 1 0 0 ; b e = 1 0 0 M a r g i n = 0 . 7 1 1 8 o o o / o o o O -- 1 . 5 - 1 - 0 . 5 O ( d . ) R e a l o f H _ p f T r a n s p a r e n c y : b h = 2 0 0 ; b e = 1 - 1 0 0 - 2 0 0 - 3 0 0 i O " 1 0 " ( f . ) F r e q u e n c y ( H z ) T r a n s p a r e n c y : b h = 1 0 0 ; b e = 1 0 0 . - 1 0 0 - 2 0 0 - 3 0 0 1 O 1 O ( h . ) F r e q u e n c y ( H z ) Figure 2.7: The hand/environment damping has a direct influence on the stability margin of the system wi th force/position control: The hand damping is halfed from the nominal value bh = lOONs/m in (a.) and doubled in (b.). The stability margin clearly decreases or increases with bh from the reference stability margin Ms — 0.38. When the environment damping is increased by 1 and 2 decades from ke = INs/m in (c.) and (d.), consistent improvements in the stability margin can also be observed. Graphs (e.), (f.), (g.) and (h.) are the corresponding transparency plots to (a.), (b.), (c.) and (d.), respectively. The changes in the transparency plots are insignificant. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 31 Hand/Environment Stiffness and Damping The hand and environment impedances also affect the stability and performance of the bilateral motion scaling system. Figure 2.6 shows the stability and transparency plots due to the change in the hand/environment stiffness and Figure 2.7 reflects the changes in the damping. Bo th stiffnesses kh and ke are adjusted by one decade from their nominal values of kh = 1 x 103 N/m and ke = 1 x 10 5 N/m. Decreasing kh and ke generally improves stability margins (Fig. 2.6(a.) and (c.)) and increasing them brings the system closer to the crit ical point (Fig. 2.6(b.) and (d.)). As shown in Figure 2.6(e.)-(h.), this is also true for transparency where it is better wi th softer hand and environment impedances and worse when they are stiff. As can be observed from Figure 2.6, the stability margin is affected more drastically when the environment impedance was changed by 2 decades than when the hand environment was changed. Similar trends are also observed in changing the dampings constants, (see Figure 2.7.) More damping in the hand/environment impedance improves the stability margin sig-nificantly and vice versa. However, more damping has less effect on the performance as shown in F i g . 2.7(e.)-(h.). Summary Figure 2.8 summarizes the discussion above. The plots show the continuous shifts in the stability margin Ms due to parameter variations. A n increase in position scaling and hand/environment damping and a decrease in force scaling and hand/environment stiffness are generally associated with an improvement in the stability margin, although each of them exhibits different behavior in the plots. The analysis results obtained for the ideal case can be used as design guidelines. However, wi th the addition of structural flexural dynamics, these general guidelines may Chapter 2. Stability and Performance Analysis of the Motion Scaling System 32 M s v s k h ( a t k e = 1 e 5 ) M s v s k e ( a t k h = 1 e 3 ) O 5 0 1 0 0 1 5 0 2 0 0 O 5 1 0 1 5 2 0 ( c . ) H a n d / E n v . D a m p i n g s : b h & b e ( d . ) P o s i t i o n / F o r c e S c a l i n g s Figure 2.8: Changes in the stability margin due to changes in hand/environment stiffness and damping and position/force scaling: The units used are N/m for the spring constants (kh and ke) and Ns/m for the damping constant (bh and be): W i t h the environment stiffness fixed at the nominal value ke = 1 x 10 5 N/m, the stability margin is plotted against the hand stiffness kh in (a.). Li t t le change in Ms is observed. W i t h the hand stiffness fixed at ^ - 1 x 10sN/m in (b.), it is clear that Ms is strongly related to the environment stiffness ke. A s shown in (c ) , Ms can generally be improved by increasing the damping in the hand/environment impedance. In (d.), the dashed line shows that increasing the position scaling np improves Ms and the solid line shows otherwise for the force scaling rif. not hold true depending on the properties of the flexural motion. It is the goal of this analysis to determine whether these variables can threaten the stability of the system and whether better structural designs can eliminate the threat. This leads to the analysis of the non-ideal cases below. 2.3.2 Case #2 : master rigidly mounted and flexibly coupled to slave As shown in Figure 2.2, flexural motion is introduced to the base of the slave in this case. Figure 2.9(d.) shows that wi th heavy linkage mass of mi — 200 kg, the stability of the Chapter 2. Stability and Performance Analysis of the Motion Scaling System 33 T r a n s p a r e n c y : p o s i t i o n o n l y . - 1 0 0 S ? - 2 0 0 - 3 0 0 1 — 1 0 ° °T 2 1 . 5 o 1 C7) 0 . 5 z l = Q . 1 : 1 0 1 0 1 0 ( a . ) F r e q u e n c y ( H z ) N y q u i s t P l o t : p o s i t i o n o n l y UK ***** . O : r n l = = 2 C * : m l = 2 r > 0 ; M s = 1 , 0 M s = 1 . 0 y*-- 1 . 5 - 1 — 0 . 5 O ( c . ) R e a l o f H _ p - - 1 0 0 5 ^ - 2 0 0 — 3 0 0 ' — 1 O 2 1 . 5 0 . 5 T r a n s p a r e n c y : p o s i t i o n / f o r c e : : : m l = 2 : : - : r r V P = 2 0 0 : : z l = 0 . 1 : 1 0 1 0 ( b . ) F r e q u e n c y ( H z ) N y q u i s t P l o t : p o s i t i o n / f o r c e O - 1 . 5 : 34£ o : m l = 2 0 * : m l = 2 ; O ; M s — 0 . 3 8 1 6 c ^ c M s = 0 . 1 9 3 7 c3K - 1 - 0 . 5 O ( d . ) R e a l o f H _ p f Figure 2.9: Transparency and Nyquist plots for Case #2 with position only and posi-tion/force control modes: W i t h just the position coordination controller in place, the transparency and Nyquist plots in (a.) and (c.) do not look much different from F ig . 2.4(a.) and (c.) in the ideal Case #1. This is also true when a massive link-age mi — 200kg is used. More drastic changes can be observed for the position/force control case. First , the transparency (dashed line in (b.)) is much worse comparing to F ig . 2.4(b.) and the Nyquist plot (with symbol " in (d.)) shows a worse stabil-ity margin Ms = 0.19. When the mass of the linkage is increased by a hundred fold to mi = 200kg, the Nyquist plot in (d.) (with symbol 'o') shows the same stability mar-gin M ; = 0.38 as in the ideal system and the transparency (solid line in (b.)) is improved from the case where the lighter linkage mass mi = 2 kg is used. Al though not shown, further increase in the linkage mass further improves the transparency. system is.much the same as in the ideal case F ig . 2.4(d.); but when mi is decreased to model a light weight linkage arm, the stability margin Ms is reduced to 0.19 from 0.38. Comparing wi th F i g . 2.4(b.), F i g . 2.9(b.) reveals worse performance in transparency when flexible modes are added to the system. The performance also suffers when the mass of linkage is reduced. From F i g . 2.9(a.) and (c ) , these changes to the system stability and performance are not observed when force control is absent. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 34 T r a n s p a r e n c y P l o t .--too - 2 0 0 - 3 0 0 — i :: n p r ? 1 O ; r\1— 1 : : : :: : T : H ^ V . — : ^ : : : : : : - .- .•>—-^^^.v •. •. :.-. • : n p - 1 : : ! : n f = 1 l O T ^ s i u , 0 . 5 0 . 4 J o . 3 JE O . I S t a b i l i t y P l o t ( a t z l = 0 . 1 ) o : n p = 1 | ; n f = 1 0 M s = 0 . 1 3 4 4 ' * : n p = 1 p ; n f = 1 ; M s = 0 . 0 7 0 8 o : o • T t i m i ° O o , 1 0 " 1 0 1 o ( a . ) F r e q u e n c y ( H z ) M s v s n f ( — ) a n d n p ( ) 0 . 2 5 CO 0 . 2 cz CG 0 . 1 5 CO 0 . 1 C O 0 . 0 5 - 1 . 5 —1 — 0 . 5 O ( b . ) R e a l o f H _ p f M s v s z l ( a t n p = 1 0 ; n f = 1 ) \ \ ( a t n p — t ) \ \ \ V. : ( a t n f = 1 ) 5 1 0 1 5 2 0 ( c . ) P o s i t i o n / F o r c e S c a l i n g s Figure 2.10: Effects of position/force scaling with passive linkage damping on the trans-parency and stability in Case #2 : Increasing the scalings to np = 10 or rif = 10 reduces the stability margin to Ms = 0.07 or Ms = 0.13 in (b.) from the original response of Ms = 0.19 in F i g . 2.9(d.). In (a.), the transparency is better wi th higher force scal-ing rif = 10 than wi th nv — 10 or with rif — 1 in F i g . 2.9(b.). The stability margin is plotted against the position or force scaling in (c.) when the other one is fixed at the nominal value of rif = 1 or np = 1. Ms is more sensitive to the change in np (dash line) than in n / (solid line), but it is clear that increasing either of them wi l l reduce the robustness of the control system. This situation can be improved by increasing the base damping of the flexible structure and the result is shown in (d.). A t np — 10 and nj = 1, the stability margin can be improved from Ms = 0.07 at Q = 0.1 to Ms = 0.42 at Q = 1.0. Position/Force Scaling and Linkage Damping Contrary to the findings in the ideal case, an increase in both the position and force scalings can cause changes in the transparency and the stability margin. These are shown in F i g . 2.10(a.) and (b.). When rif = 10 or np = 10, the stability margin is reduced to Ms = 0.13 or Ms = 0.07 from Ms = 0.19 observed in F i g . 2.9(d.), where the linkage mass is m; = 2 kg. The transparency frequency response is less smooth when np = 10, and is improved when rif = 10. From observing how Ms changes in F i g . 2.10(c) when rif or np are varied from 1 to 20, it is evident that the position scaling is a more Chapter 2. Stability and Performance Analysis of the Motion Scaling System 35 important parameter affecting the stability of the system than is force scaling. Thus, it is clear that the stability margin is reduced due to the addition of the structural flexible mode. This situation can be improved by increasing the material damping of the linkage arm £/• A s shown in F ig . 2.10(d.), the stability margin is roughly proportional to the amount of damping Q in the arm. Conversely, if the arm consists of low-damping metals such a aluminum or steel whose damping is in the range of Q = 0.005, the stability of the motion scaling system is expected to be low, causing difficulties in its operation. Thus, the goal is to increase the base damping of the flexible structure as much as possible in order to accommodate for an expected decrease in the stability margin when the position and/or the force scalings are changed. For example, F i g . 2.10(c) shows that with a nominal setup, the decrease in the stability margin seems to converge to Ms = 0.07 from Ms = 0.2. Thus, in order to maintain the stability margin around Ms = 0.15, F i g . 2.10(d.) suggests that a material damping of Q — 0.3 should be sufficient in preventing the control system from approaching marginal instability. Hand/Environment Impedance In some cases, a decrease in the stability margin and performance is caused by the physical hand/environment configurations and cannot be prevented from by using more structural damping. The analysis on the stability and transparency due to the changes in hand and environment impedances is summarized in Figure 2.11. Comparing F i g . 2.11(a.) to F ig . 2.8(a.), the hand stiffness has a much more significant effect on the stability margin than in the ideal case. The stability margin falls rapidly after a certain hand stiffness where maximum stability occurs. A n increase in material damping from Q — 0.1 to Q = 0.3 does not have significant effect on improving the stability margin. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 36 Ms vs kh (at ke=1e5) Ms vs ke (at kh=1 e3) C D i— CO 0.5 CO CO 10 10 (a.) Hand Stiffness : kh Transparency Plots 10° CD 2 , - 1 0 0 CD T3 a ? - 2 0 0 -300 10' bh~30 ; bes=1 bh=100; be=1 bh=100 ; be= 100 0.6 .£ 0.4 CO i= 0.2 _ Q CO CO 10" 10" (b.) Environment Stiffness : ke Ms vs bh(-) and be(—) r 1 1 y-S / / , s i/ / / / f ( at bh=1i00 ) / : ( at be = 1 > 10' 10" (c.) Frequency (Hz) 10° 0 50 100 150 200 (d.) Hand/Env. Dampings : bh & be Figure 2.11: Plots of stability/transparency with changes in hand/environment impedance: The hand stiffness kh affects the stability margin differently in Case #2 than in Case #1. Due to the additional linkage flexible mode, (a.) shows there is a par-ticular hand stiffness where the stability margin peaks, then falls rapidly. Since normal operating hand stiffness is not expected to be so high, it can be said that kh does not sig-nificantly affect Ms as observed in the ideal case. Similar to the ideal case in F i g . 2.8(b.), (b.) shows that increasing the environment stiffness ke can reduce the stability mar-gin severely. The dashed line in (a.) and (b.) are the same plots but with d = 0.3. More material damping does not seem to significantly change the relationship between the stability margin and the hand/environment stiffness. The transparency plots in (c.) show that with more hand/environment damping, the performance is better. The solid line represents the base system with nominal values, the dotted line wi th hand damp-ing reduced to bh = 30Ns/m and the dashed line wi th environment damping increased to be = lOONs/m. Consistent improvements to the stability margin due to increases in bh and be are also observed in (d.). The dashed line represents the increase in be while bh = lOONs/m and the solid line in bh while be = INs/m. The system becomes unstable if the hand damping is below bh = 30Ns/m. This implies that when force con-trol is implemented and the slave is in contact with a stiff environment of low damping, the master should be in contact with a hand environment of sufficient damping to insure stability in the control system. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 37 F ig . 2.11(b.) behaves much the same way as in the ideal case (Fig. 2.8(b.)). The stability margin decreases steadily as the environment stiffness increases. Again , material damping does not change the picture much. Similar results are also observed for bh and be in F i g . 2.11(d.). Increasing the hand/environment damping generally improves the stability margin and from the transparency plot in F i g . 2.11(c), it tends to smooth out any resonant modes in the system. Summary B y placing the slave system on a flexible structure, a threat to stability and perfor-mance of the control system are observed. This is because the force/position controller is designed for the ideal system. Without sufficient damping in the additional flexible structure, the stability of the control system is in question. This problem can be avoided by increasing the structural damping and as a result, the motion scaling system can in fact operate quite well wi th the original control system. Decreases in the stability margin due to hand/environment impedance changes cannot be rooted to the lack of the struc-tural damping. Much the same to the ideal case studied earlier, the problem needs to be solved appropriately by adjusting the impedances. 2.3.3 Case #3 : decoupled master and slave mounted to a flexible platform In this case, both the master and slave, are mounted to the same platform and receive the same base excitation from the flexural motion of the structure (See F i g . 2.2). Fig.2.12(a.) and (b.) show the transparency and stability achieved when the platform mass is set to mp = 2 kg and mp = 200 kg. The case mp = 200 kg was chosen as a l imi t ing case to show that Case #3 reduces to Case #1 when the platform mass is large. When the mass is large, better transparency is achieved but the stability is not much affected. This finding Chapter 2. Stability and Performance Analysis of the Motion Scaling System 38 is different from the one in Case #2 and may be because the master and slave receive the same base disturbance from the flexural motion of the structure. Position and Force Scalings Fig . 2.12(c) and (d.) show that the stability margin is affected by changes in the position and force scalings. Similar to Case #1 and #2, an increase in the force scaling rif clearly decreases the stability margin. The damping in the moving platform does not seem to help the situation. A n increase in position scaling np does help the stability margin as shown in the ideal case, but Ms w i l l start to drop rapidly if insufficient platform damping £ p is present. From F ig . 2.12(c), a damping coefficient of (p = 0.1 is sufficient to eliminate this threat to the stability of the system. Hand/Environment Impedance Figures 2.13 shows how the stability margin Ms is affected by the hand/environment stiffness and damping. Consistent with the observations from the previous two cases, an increase in stiffness generally reduces the stability margin and an increase in damping improves it. The platform damping Qp does not affect these observations significantly. Summary When both the master and slave are subject to identical flexural motion disturbance at their bases, the threat to the stability margin due to the additional structural flexural dy-namics only surfaces when one increases the position scaling np. W i t h sufficient damping in the moving platform, the motion scaling system can be expected to be robust much the same as in the ideal case, although the transparency is not as good. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 39 T r a n s p a r e n c y P l o t S t a b i l i t y P l o t : M s = 0 . 3 8 2 7 C O S - - 1 0 0 - 2 0 0 - 3 0 0 : : m p = 2 k g : — : m p = 2 0 0 k g _ 1 I t as 0 . 5 1 0 1 CO S 0 . 8 c= S > 0 . 6 as 1 O 1 O ( a . ) F r e q u e n c y ( H z ) M s v s n p 1 0 O - 1 . 5 o : m p = 2 0 0 k g — : m p = 2 k g — - 4 - 1 — 0 . 5 O ( b . ) R e a l o f H _ p f M s v s n f 7 — - : 2 r P = Q . 1 \ • : z p = 0 . 0 1 - : z p = O . O O E 5 1 0 1 5 2 0 ( c . ) P o s i t i o n S c a l i n g : n p 0 . 4 CO d E ? 0 . 3 CO 0 . 2 "as C O 0 . 1 ° \ : \ • : z p = p . 1 &> o : z p - O . 5 O 5 1 0 1 5 2 0 ( d . ) F o r c e S c a l i n g : n f Figure 2.12: Transparency and stability of the position/force control system with different position/force scalings: Transparencies achieved in (a.) are closer to the ideal case shown in F ig . 2.4(b.) than then one studied in Case #2 as shown in F i g . 2.9(b.). Whi le a more massive platform wi th mp = 200kg (solid line in (a.)) shows a better transparency than a light weight one with mp = 2kg (dashed line in (a.)), there seems to be no significant gain to the stability margin in (b.). Notice that at mp = 200kg in (a.), the performance is roughly the same as the one in Case #1 except at lower frequency ranges. This is because although the mass is increased, the simulation continues to model a lOHz structural natural frequency at the base of the platform which is clearly picked by the bilateral motion scaling system as shown in (a.). Using the nominal value ( p = 0.1 for the platform damping (dashed line in (c.)), the stability margin increases as the position scaling increases. This observation is similar to the ideal case. However, for ( p = 0.01 in the dotted line and £ p = 0.005 in the solid line, the stability margin only partially increases wi th the position scaling before it starts to decrease rapidly wi th the continuing increase in np. This is not observed in (d.). The relationship between the stability margin and the force scaling stays consistent no matter how much passive damping there is in the platform. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 40 M s v s k h ( a t k e=1 e 5 ) M s v s k e ( a t k h = 1 e 3 ) ( c . ) H a n d D a m p i n g : b h ( d . ) E n v i r o n m e n t D a m p i n g : b e Figure 2.13: Relationships between stability/transparency and hand/environment impedance: Similar to the ideal Case #1, (a.) — hand stiffness kh does not affect the stability margin in any significant manner, (b.) — environment stiffness ke directly influ-ences the stability margin and (c.) & (d.) — increasing hand/environment damping im-proves the stability margin. As plotted in the 'o' symbol above, a poorly damped platform does not have significant effects on the relationships between Ms and hand/environment impedance. 2.3.4 Case #4 : flexibly coupled master and slave mounted to a flexible platform This case combines the previous two cases by introducing the flexible modes from both the moving platform and linkage arm, as shown in F i g . 2.2. Figure 2.14 shows the transparency and stability achieved by the base system under this configuration. As shown in F i g . 2.14(c), the stability margin stays at the ideal level of Ms — 0.38 when heavy masses of mp = mi = 200 kg are considered. When light weight arm is considered as in F i g . 2.14(d.), the stability margin drops severely to Ms = 0.07. The transparencies achieved in the light weight case wi th position and position/force controls are both worse than the ideal case (see F i g . 2.14(b.)). However, this is improved as the masses are Chapter 2. Stability and Performance Analysis of the Motion Scaling System 41 T r a n s p a r e n c y : m p = m l = 2 0 0 k g C D S . - 1 0 0 ^ - 2 0 0 - 3 0 0 2 ^ . 1 . 5 ~S 1 CT5 ca E — 0 . 5 ~* „ — : p o s i t i o n o n l y ~~ -— : p o s t i o n / f o r c e \ m S . - 1 0 0 - 2 0 0 - 3 0 0 T r a n s p a r e n c y : m p = m l = 2 k g ^ ^ \ ~ - ^ — : p o s i t i o n D n l y — : p o s i t i o n / f o r c e 1 O 1 o ( a . ) F r e q u e n c y ( H z ) 1 0 1 0 ( b . ) F r e q u e n c y ( H z ) O - 1 . 5 o M s — 0 o o : 3 7 8 9 o o o o ' o 0 . 5 - 1 - 0 . 5 O ( c . ) R e a l o f H _ p f 0 - 1 . 5 ! O o o M s - 0 . 0 7 2 2 o o o o r> i o o o o o o<aB - 1 - 0 . 5 ( d . ) R e a l o f H _ p f Figure 2.14: Transparency and stability plots for position only and position/force control schemes: W i t h two additional flexible modes introduced in the system, the transparency achieved by the position/force control in Case #4 is better in the heavy masses case (see the solid lines in (a.) and (b.)). Similar results can be observed wi th position only control as well (see the dashed line in (a.) and (b.)). W i t h massive platform mass mp = 200kg and linkage mass mt — 200kg, the ideal stability margin Ms — 0.38 is maintained for the position/force control. When the masses are decreased to reflect the light weight robotic structures of mp = 2kg and mt — 2kg, the transparency wi th position control only worsens and the stability margin with position/force control reduces to Ms = 0.07. increased as shown in F i g . 2.14(a.). Platform and Linkage Dampings F i g . 2.15(a.) shows that by increasing the platform and linkage dampings, the stability margin can be much improved. It is also observed that the linkage damping is more effective at improving the stability than the platform damping because sufficient linkage damping makes the system in Case #4 look more like the system in Case #3, which is more robust than the system in Case #2. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 42 M s v s z p ( - ) a n d z l ( ) M s v s n p / n f ( a t k h = 5 e 4 ) . : n p ( a t z l = z p = 0 . 2 ) ( a t z p = 0 . 1 ) , -^ 0 . 3 r - — : O P ( a t z l = z p = 0 . 1 ) — : n f ( a t z l = z p = 0 ; i ) : o : n f ( a t : z l = z p = O J 2 ) 0 . 0 5 O O 0 . 5 1 ( a . ) P l a t f o r m / L i n k . D a m p i n g : z p & z l O 1 0 2 0 3 0 4 0 ( b . ) P o s i t i o n / F o r c e S c a l i n g s : n p / n f Figure 2.15: Relationships between the stability margin and platform/linkage damping and position/force scaling: The passive damping of the whole flexible structure consists of both platform and linkage dampings. The contribution of each damping toward the improvement of the stability margin is plotted in (a.) while the other is held at the nominal value. The result shows that increasing damping in the linkage (the dashed line) has more significant effect in improving the stability margin than in the platform (the solid line). From (b.), it is shown again that the relationship between the stability margin and the force scaling (the solid line and the 'o' symbol) is independent of the amount of intrinsic dampings present in the system. However, increasing these intrinsic dampings £ p and Q can improve the stability margin as the position scaling increases (the dotted and dashed lines). Wi thout sufficient passive structural dampings, the control system can become unstable when high position scalings are implemented. Position and Force Scalings F i g . 2.15(b.) summarizes the effects of position and force scalings on the stability margin of the control system. Similar ly to Case #3, an increase in force scaling nj reduces Ms and neither the platform damping nor the linkage damping affect this. However, the behavior of the control system due to the increase in the position scaling np is much affected by the addition of the structural flexible modes. Increases in both the platform and linkage dampings are shown to be effective in improving the stability margin. Contrary to Case #3, much damping is required before the stability margin Ms starts to increase with position scaling Tip, cLS observed in the ideal case. Chapter 2. Stability and Performance Analysis of the Motion Scaling System 43 Summary Very poor stability is observed in this configuration even when high structural dampings °f Cp = 0 — 0-1 a r e considered and so as the transparency achieved. Increasing the structural dampings of the platform and the linkage clearly improves the stability margin. The range of position scaling one can use before the stability margin starts to decrease is severely l imited. 2.4 Summary The causes of instability and/or degradation in performance of a bilateral motion scaling system exhibiting structural flexible modes have been discussed. Together with realistic parameters, physical models and control system formulations were used to analyze the problem. The purpose was to describe the real laboratory system as closely as possible. Four representative cases were studied with one being the ideal case upon which the control for the bilateral motion scaling system was originally designed. It was found that with sufficient damping in the flexible structure, the stability of the control system can be maintained at the same level as in the ideal case without resorting to redesigning the controller to actively compensate for flexural modes in the system. This observation stays true even for light weight structures. However, depending on how the flexible modes are introduced to the ideal system, the transparency achieved is generally degraded. The degradation is most significant when two different flexible modes are present in the system, as simulated in Case #4. Several pairs of physical/control parameters were studied in detail in determining their relationships with the stability of the system and how these relationships are affected by structural vibrations. The goal of the latter study was to isolate the parameter(s) that have the most effect on threatening the stability of the closed-loop system due to the Chapter 2. Stability and Performance Analysis of the Motion Scaling System 44 introduction of flexible modes in the structure. For the ideal system, an increases in force scaling nj and hand/environment stiff-ness kh and ke were found to reduce the stability margin of the control system. Further-more, an increase in position scaling np and hand/environment damping bh and be tended to improve the stability margin. W i t h flexibilities in the physical system, it was found that structural vibrations generally do not significantly affect the relationships between the stability margin Ms and the passive parameters such as the hand and environment impedance H(s) and E(s); they continue.to influence the stability of the system in much the same way as they do in the ideal case. For the relationship between the stability margin Ms and the active control parameters such as the position and force scalings np and n / , the scenario is a lit t le different. A bilateral motion system performs both force reflection and motion scaling tasks. As observed in the cases studied, the relationship between the force scaling.n/ and the stability margin Ms was not changed significantly from one case to another. This is expected because the analysis assumed a perfect force reflection system with precise force sensing and actuation; therefore, all additional disturbance forces due to structural vibrations or other external factors are detected and reflected, perfectly. However, this is not the case for the motion scaling task. This perfect force-reflection system could not prevent the drastic reduction in stability margin when position scal-ing np was increased in the non-ideal cases studied. This ' is because the precise actuation forces are required to move the masses in order to achieve accurate motion scaling. In non-ideal cases, the actuation action from the motion scaling task also caused the move-ments of other masses not accounted for in the control design. Thus, if the master and slave were on the different platforms and received different un-controlled flexural motions at their bases in addition to the local active actuation forces, an accurate motion scaling task would have become difficult. A s the position scaling factor increased, the active Chapter 2. Stability and Performance Analysis of the Motion Scaling System 45 actuation force required to increase the slave inertia would consequently induce more excitation in the structural flexible modes, causing a loss of performance, a decrease in the stability margin. In the ideal case, increasing np tended to improve Ms. When the master and slave were subject to different flexural base disturbances as in Case #2, this was no longer true. The system moved rapidly towards marginal stability as the position scaling increased. W i t h sufficient damping in the system, this situation could be reverted and one could expect to observe the same relationship between Ms and np as in the ideal case. However, there is an upper l imi t on increasing the position scaling before the stability margin starts to drop; this l imi t is proportional to the amount of damping present in the system. A t last, there are other factors that also cause the degradation in the stability mar-gin and transparency. This can be observed in changing the hand and environment impedances. They pose threats to the stability of the control system independently from the introduction of structural flexible modes. These threats cannot be indiscriminately eliminated even if there is more structural damping. Chapter 3 Active/Passive Damping and Endpoint Vibration Control 3.1 Overview The previous chapter shows that the stability of the bilateral motion scaling system can be threatened i f the structural flexible modes present in the system are poorly damped. Therefore, in order to realize a practical and effective motion scaling system for micro-surgery, these flexible modes must be well damped. O n the linkage arm, this has to be achieved passively since no actuation is present in that part of the system. For the endpoint motion of the moving platform, it may be believed that active control of the transporting robot can provide a well damped endpoint motion. For a conventional serial link robot, this is a difficult task because the actuation of a link occurs at the base of the link and the endpoint is s t i l l allowed to vibrate freely about the actuator much like a cantilever beam held fixed at one end. Of course, one can also take the passive damping approach by constructing a transporting robot with high structural damping to dissipate any structural vibration that may be induced during operation. The goal of this chapter is to evaluate and compare the effectiveness of active and passive damping approaches to suppress the endpoint vibrations of a flexible robot arm. The passive scheme simply involves increasing the material damping of the beam. The active scheme involves actuating the base of the link with different control methods in order to avoid exciting the vibrational modes of the l ink while at the same time to achieve a desired endpoint motion. Linear active control schemes such as direct P D control, 46 Chapter 3. Active/Passive Damping and Endpoint Vibration Control 47 command pre-shaping, state-space control and Q-parameter control are investigated and the results are compared against the passive damping approach. Non-linear control schemes such as adaptive control is not evaluated in detail, but briefly discussed at the end of the chapter. It was found that the success of active damping depends heavily on the precision of the plant modeling and actuation. In addition, the time varying dynamics of the robot structure during operation makes active vibration suppression become even more difficult. Passive damping introduces fundamental changes to the design of a robotic structure, but clearly offers intrinsic advantages in effective vibration suppression. The same effects often cannot be achieved easily with complex control algorithms, which sometimes are forced to sacrifice system performance in order not to excite any structural resonant modes. 3.2 Modeling and Open-Loop Response A flexible robot arm can be modeled by considering only the fundamental mode of v i -bration as shown in Figure 3.16. The joint and arm stiffnesses are ki and k2, and the damping constants are b\ and b2, respectively. The endpoint vibration is the position dif-ference between masses mi and m2; i.e. &c — x2 — x\. Either the joint position x\ or the endpoint position x2 can be used as feedback signal to generate the actuation force / , which is applied at the joint. The plant input/output transfer functions in frequency domain are „ , x m2s2 + b2s + k2 x2 b2s + k2 P*AS) = j = • ; PM = j = - D ^ - - . ( 3 - 1 6 ) where D(s) = {m2s2 + b2s + k2)[m1s2 + {bl + b2)s + {k1 + k2)}-{b2s + k2)2 . (3.17) Chapter 3. Active/Passive Damping and Endpoint Vibration Control 48 Vibration with 1-N Impulse Excitation at Base Figure 3.16: A flexible robot arm link: x\ Figure 3.17: Open-loop response to a IN and x2 represent the joint and endpoint impulse force at (a.) the base and (b.) positions, ki and b\ are the joint stiffness the endpoint. '•': bx = 10 and b2 = 0.25 and-viscous damping and k2 and b2 are the material stiffness and damping. ; these are the nominal values. ' —•' : bi = 50 and b2 = 1. ' - ' : bx = 1 and b2 = 50. Uni t : [Ns/m]. Modeling Parameters The nominal numerical values used in the analysis and simulation are listed below. m i = 2 % m2 = 2 kg = 315 N/m k2 = 7900 N/m h = 10 Ns/m b2 = 0.25 Ns/m The masses m i and m2 are chosen to reflect a light weight robotic structure much like the one analyzed in Chapter 2. The joint stiffness ki corresponds to a low structural oscillation of 2 Hz about the base and linkage arm stiffness k2 models a stiff arm with fundamental vibration mode at 10 Hz. The damping constants b\ and b2 reflect a struc-ture wi th much damping at the joint and very little damping in the arm. They have the Chapter 3. Active/Passive Damping and Endpoint Vibration Control 49 equivalent damping coefficients (joint'= Ci = 0-2 and (material = (2 = 0.001 1. Impulse Responses Although both are part of the structural damping, the joint and material dampings shown in Figure 3.16 should be distinguished. Before engaging in any analysis, some simple simulations can readily show that the material damping is more effective in controlling the endpoint vibrat ion than the joint damping. Figure 3.17 displays the endpoint vibration responses due to a 1 N impulse excitation at either the base or the tip of the beam. In both cases, high material damping suppresses more vibration than high joint damping. This is intuitive as from Figure 3.16, even if the joint stiffness k\ and damping bi are both high, the endpoint can st i l l vibrate freely if the material damping b2 is low. Since the actuation occurs at the joint, extra joint damping additional to b\ can be conveniently introduced v ia some active local feedback control. Therefore, the rest of this chapter focuses on increasing the material part of the structural damping rather than the joint part. 3.3 Closed-loop Responses In the attempts to suppress the structural vibration v ia active means, it is often required to feed back some sensed position signals to generate appropriate actuating action - clos-ing the loop. This section addresses the philosophy in active damping and evaluates the effectiveness of some of the control approaches proposed and used today, in comparison with passive damping. 1For conventional metals such as aluminum, the material damping coefficient is actually in the range of ( = 2 x 10 - 5 ; the value £2 used is overestimated to reflect any additional damping not accounted for. Chapter 3. Active/Passive Damping and Endpoint Vibration Control 50 J o i n t F e e d b a c k E n d — p o i n t F e e d b a c k - 1 5 0 1 0 0 co S O C O O ^ - S O - 1 0 0 - 1 5 0 y-I t - 2 - 1 O ( b . ) R e a l A x i s Figure 3.18: Root-locus plots for open-loop plants with joint and endpoint feedback modes. 3.3.1 Joint/Endpoint Feedback The conventional practice in active robot control is to feed back the joint angles. This ap-proach produces errors in the actual endpoint position of the robot because of flexibilities in the link. In order to achieve precise endpoint positioning and/or effective active vibra-tion suppression for a light-weight flexible robotic structure, endpoint position feedback is essential. Even wi th the same control algorithm, the control dynamics wi th joint and endpoint feedback are quite different. The distinction between these two feedback modes and the potential effects on the stability of the resulting control systems are addressed here first. The active control algorithms evaluated in the rest of the chapters assume endpoint feedback approach. Root-locus Analysis From the root locus plots of the open-loop plant transfer functions (3.16) shown in Figure 3.18, a general picture on the control stabilities wi th joint and endpoint feedback can be illustrated. Al though both x\/f and x2/f are stable plants, the former guarantees . global stability independent of the controller gain while the latter can result in instability as the controller gain increases. Chapter 3. Active/Passive Damping and Endpoint Vibration Control 51 J o i n t : k p = 5 0 ; k v = 2 E n d — p o i n t . : k p = 5 0 ; k v = 2 - 2 —1 O 1 2 — 2 —1 O 1 2 ( c . ) R e a l A x i s ( d . ) R e a l A x i s Figure 3.19: Nyquist plots of the open-loop system wi th joint and endpoint feedback modes. Changes in control gains can cause instability in the latter, but not in the former case. Nyquist Stability Analysis Similar observation is also reflected using the Nyquist stability analysis. B y controlling the plants xi/f wi th a simple P D controller Kc(s) = kvs + kp, the open-loop Nyquist plots are shown in Figure 3.19. The change in controller gain from kv = 2 to kv = 4 while kp = 50 causes part of the Nyquist curve to expand. For the joint feedback case in F i g . 3.19(c), the expansion tends to move toward the right-half-plane without threatening the stability of the system. For the endpoint feedback case in F i g . 3.19(d.), it moves toward the other direction to encircle the crit ical point —1, indicating instability. 3.3.2 PD Control Using the P D controller in F i g . 3.19(b.) as an example, the endpoint vibration resulted from a 1 mm step command is shown in Figure 3.20. The vibration induced due to this operation is quite severe. The damping in the controller cannot be increased because Chapter 3. Active/Passive Damping and Endpoint Vibration Control 52 0 . 0 2 0 . 0 1 M. o cz o S - 0 . 0 1 ^ - 0 . 0 2 - 0 . 0 3 V i b r a t i o n d u r i n g 1 — m m S t e p 0 . 5 1 1 . 5 ( a . ) T i m e ( s e c ) 0 . 0 5 0 . 0 4 S 0 . 0 3 o 1-— 0 . 0 2 0 . 0 1 A c t u a t i o n d u r i n g 1 — m m S t e p 0 . 5 1 1 . 5 ( b . ) T i m e ( s e c ) Figure 3.20: (a.) Endpoint vibration due to a 1 mm step command. A P D control with endpoint feedback and gains of kp = 50 N/m and kv = 2 Ns/m is used. The dotted line represents vibration from the default system wi th = 0.2 and £2 = 0.001. Even increasing the joint damping to £1 = 0.7 (dashed line), there is st i l l vibration present. W i t h an increase in the material damping to C2 = 0.1 (solid line), the structural vibration is suppressed more effectively, (b.) Corresponding control efforts are also plotted. Increase in passive damping reduces the amount of control efforts required. it w i l l cause instability in the control.. system as shown in the previous section. The other alternative is to increase the passive damping. B y increasing the joint damping to (1 = 0.7, there is st i l l some high frequency structural oscillation left. Only after sufficient material damping is introduced, £ 2 = 0.1, then the result is satisfactory. Less control effort is observed when more passive damping is present in the system. 3.3.3 Command-Preshaping To actively reduce the endpoint vibration observed in F i g . 3.20(a.), a command, pre-shaping technique proposed by Singer and Seering [45, 46] was evaluated. Rather than avoiding exciting the structural vibration modes, they are purposely excited, but at different time intervals. If the t iming is precise, this wi l l induce structural vibrations which are out of phase to each other; thus, cancellation of the vibration is achieved. This excitation cancellation mechanism is embedded in the actuation pre-filtering the command prior to feeding it to the plant. The plant is s t i l l controlled by the same P D controller used in the previous section, Chapter 3. Active/Passive Damping and Endpoint Vibration Control 53 with the addition of the command pre-filter between the controller and the plant. As proposed in [45, 46], in order to cancel the fundamental mode of the structural vibration, the digital pre-filter would consist of two pulses with magnitudes P - 1 • P - Y which are apart by a time A T . The magnitude of Y and time separation can be found by Y = e ; A T = / , where £2 is the material damping and u>0 is the circular frequency of the fundamental mode of vibration. The model in F i g . 3.16 exhibits a structural vibration of 14 Hz wi th the material damping at C2 = 0.001. This results in Y - 0.7292 and A T = 0.0359 sec. W i t h a sampling period of T = 1 msec, the required series of pulses can be translated to an equivalent digital filter expressed in the z-transform H(z) = 0.5783 + 0.4217z~ 3 6 . This is basically a digital filter of two pulses with magnitudes Pi = 0.5783 and P2 = 0.4217. The first pulse P i is energized at T = 0 sec. to induce an in i t ia l structural vibration, and at a time of 36 sampling periods later, a second pulse P2 is energized to induce another structural vibration out of phase to the first one. Reduction in the magnitude of vibration is then achieved. Note that this implementation is in fact off by 0.1 msec, from the required A T found above. The resulting endpoint vibration is shown in F i g . 3.21 (a.). The high frequency vibration is mostly suppressed leaving the low frequency oscillation about the link joint, whose damping can be increased passively wi th more viscous damping in the Chapter 3. Active/Passive Damping and Endpoint Vibration Control 54 P r e — c o m m a n d S h a p i n g A c t u a t i o n R e q u i r e d 1 1 . 5 ( a . ) T i m e ( s e c ) M i s s b y — 0 . 0 0 5 s e c 0 . 5 1 1 . 5 ( b . ) T i m e ( s e c ) M i s s b y + 0 . 0 1 O O s e c 0 . 5 1 1 . 5 ( c . ) T i m e ( s e c ) 0 . 5 1 1 . 5 ( d . ) T i m e ( s e c ) Figure 3.21: A digital command-preshaping filter with ^-transform 0.5783 + 0A2l7z~36 is implemented to filter the P D control effort before applied to the plant. The sampling rate is at 1 kHz, and the fundamental vibration frequency of the arm is at 14Hz. In (a.), the solid line shows the successful vibration suppression, and the dotted line indicates the magnitude of the original vibration. The filtered control effort (solid line) is shown in (b.), wi th the original control effort in the dotted line, (c.) and (d.) show that if the implementation of the filter was off by as lit t le as —5 or +10 sampling periods, the structural vibrat ion would start to surface again. joint bearings or actively with additional local joint-feedback control. The control effort shown in F i g . 3.21(b.) remains much the same as in the original case, except some changes at the in i t ia l stage to appropriately excite the structure. Al though the method seems to work rather well, precision of actuation and t iming in the actual implementation is crucial to the success. Figure 3.21(c) and (d.) show the re-sulting endpoint vibration when the application of this method was off by 5 and 10 msec; the cancellation was imperfect and the structural vibration started to surface again. Er -rors in identifying the structural stiffness and damping can easily lead to miscalculation in the precise t iming required for effective vibration cancellation. Chapter 3. Active/Passive Damping and Endpoint Vibration Control 55 x 1 Q " 3 S t a t e S p a c e C o n t r o l A c t u a t i o n f o r 1 — m m S t e p O 0 . 5 1 1 . 5 2 O 0 . 5 1 1 . 5 2 ( c . ) T i m e ( s e c ) ( d . ) T i m e ( s e c ) Figure 3.22: A state-space controller is implemented. In (a.), the endpoint vibration re-sulted from a 1 m m step command is improved; the high frequency modes are suppressed. Al though the controller gains are high, the control effort is reasonable and achievable in real systems, as shown in (b.). (c.) and (d.) demonstrate as l i t t le as 3% of error in estimating joint and material stiffness can drastically affect the effectiveness of the active controller. 3.3.4 State-space Controller Instead of using the conventional P D controller, a full state feedback controller can be implemented to achieve the desired endpoint trajectory while avoiding inducing vibration. The model in F i g . 3.16 can have a state-space realization x = Ax+ Bf ; y = Cx , Chapter 3. Active/Passive Damping and Endpoint Vibration Control 56 where x = [xi X\ x2 x2]T is the state vector, / is the input and y = x2 is the output. The dynamics, input and output matrices are respectively 0 1 0 0 0 A = fci + fa mi 6i +b2 mi kz_ mi m i ; 5 = l m i 0 0 0 1 0 m2 bi_ m2 m2 A_ m 2 . 0 C = .[0 0 10] Using the nominal values for the variables above, a full state controller K = [-14932 20 14622 - 28] is computed to place the poles of the closed-loop system [si — (A — BK)]~l at —10, resulting in a system with a frequency response of approximately 1.6 Hz. As shown in Figure 3.22, li t t le endpoint vibration is observed during a 1 mm step command. Despite the high gains of the controller, the actuation effort required is achievable wi th practical actuators! As observed in the command pre-shaping technique, precise modeling of the physical system is crucial to the performance. F i g . 3.22(c) and (d.) show that wi th as little as 3% of deviation in estimating the joint or arm stiffness, the effectiveness in vibration suppression can suffer significantly. 3 .3 .5 Q - P a r a m e t e r C o n t r o l Many modern controller design methods formulate the plant dynamics into the controller transfer function in such a way that the overall closed-loop system would exhibit a desired response. This kind of approaches can be generalized by the Q-parameter control design. The design rationale is briefly discussed here. Let one specify a closed-loop system response in the form of PX2(s)Q(s), where PX2{s) is the stable input/output transfer function of the physical plant and Q(s) is the transfer Chapter 3. Active/Passive Damping and Endpoint Vibration Control 57 D e s i g n e d M a g n i t u d e R e s p o n s e - 5 0 - 1 0 0 -5 x 1 0 1 0 1 0 ( a . ) F r e q u e n c y ( H z ) -=> Q — P a r a m e t e r C o n t r o l 0 . 5 1 1 . 5 ( c . ) T i m e ( s e c ) 1 0 . 5 O - 0 . 5 —1 0 . 0 8 0 . 0 6 g 0 . 0 4 o u _ 0 . 0 2 O D e s i g n e d N y q u i s t P l o t 1 - 0 . 5 O ( b . ) R e a l A x i s A c t u a t i o n f o r 1 — m m S t e p 0 . 5 1 1 . 5 ( d . ) T i m e ( s e c ) Figure 3.23: Reference plots for the Q-parameter controller design. The designed mag-nitude response in (a.) shows a well damped 2 Hz frequency response. The Nyquist plot in (b.) shows the stability margin of the control system . Endpoint vibration in (c.) demonstrates the effectiveness of the controller in suppressing the structural flexural mode. Actuat ion effort in (d.) is achievable in real systems. function necessary to yield that desired closed-loop response. PX2(s) is proper because it represents a physical system. Thus, if Q(s) is also proper, then a proper controller C(s) Q(l - PX2Q)~l (3.18) can be found to satisfy the closed-loop transfer function PX2Q-To better compare wi th the state space control approach discussed in the previous section, a closed-loop system response with control bandwidth J0 = 2 Hz (or w t = 4-7T rad/sec) and damping £ c = 1.0 are specified. These specifications can be translated into a fourth order transfer function P„{s)Q(s) = {s2 + 2(CL0B S + CO2)2 ' where the plant transfer function PX2(s) = x2/f as defined in (3.16) is used to reflect the endpoint feedback control. Q(s) is obtained from above and substituted into (3.18) to Chapter 3. Active/Passive Damping and Endpoint Vibration Control 58 compute for the required controller. Using the nominal values of the physical parameters and substituting in the design specifications above, the appropriate controller is found to This controller gives the desired closed-loop system response shown in Figure 3.23(a.), and the Nyquist plot of the open-loop system in F ig . 3.23(b.) indicates the system is robust. G iv ing the same 1 mm step command, the endpoint vibration induced and the control effort required are shown in F ig . 3.23(c) and (d.). The resulting endpoint vibration is comparable to the case where a full state-space controller is implemented and similarly, the actuator force expected is achievable even though the controller gains are high. Modeling Error The implementation of the Q-parameter control also requires the precise transfer function of the physical plant in order to be effective in vibration suppression, but errors in parameter estimation are inevitable in any realistic application. Thus, the stability or performance of the control system can be greatly compromised when there are modeling uncertainties present. Figures 3.24(a.) and (b.) show the endpoint vibration during a 1 mm step command when there is a 20% error in estimating the joint and arm stiffnesses. It is evident that the structural vibration starts to appear again and less damping in the endpoint motion is achieved. In order to study whether a controller with high bandwidth can compensate for the modeling uncertainties, a second controller wi th the closed-loop frequency response of f0 = 15 Hz is computed be Ci(s) = 992 g 4 + 3 x 10 4 s 3 + 8 x 10 6 s2 + 2.16 x 10 7 s + 6.18 x 10; s(s 4 + 364 s 3 + 1.67 x 10 4 s 2 + 3.05 x 10 5 s + 2.50 x 10 6) C2(S) 3.14 x 10 6 s 4 + 9.47 x 10 7 s 3 + 2.55 x 10 1 0 s2 + 6.85 x 10 1 0 s + 1.95 x 10 s(s 4 + 690 s 3 + 1.72 x 10 5 s 2 + 2 x 10 7 s + 1.05 x 10 9) ~ Chapter 3. Active/Passive Damping and Endpoint Vibration Control 59 x 1 0 - 3 k 1 o f f b y - 2 0 % ( 2 H z ) x 1 0 - 3 k 2 o f f b y - 2 0 % ( 2 H z ) O . S 1 1 . 5 ( a . ) T i m e ( s e c ) x 1 0 " 3 k 1 o f f b V - 2 0 % ( 1 S H z ) 0 . 5 1 1 . 5 ( b . ) T i m e ( s e c ) k 2 o f f b y - 2 0 % ( 1 5 H z ) 0 . 5 1 1 . 5 ( c . ) T i m e ( s e c ) 0 . 5 1 1 . 5 ( d . ) T i m e ( s e c ) Figure 3.24: T ime responses of Q-parameter control with modeling errors: In (a.) and (b.), the modeling uncertainties in the joint and arm stiffnesses result in unwanted end-point vibrations. Similar results can be observed in (c.) and (d.) when higher control bandwidth (at 15 Hz) is implemented. G iv ing the same 1 mm step command and modeling errors in the joint and arm stiffnesses, the vibration induced are shown in F i g . 3.24(c) and (d.). It can be observed that the effect of modeling errors is more pronounced when the frequency response of the Q-parameter controller is higher. This may be due to the fact that the control bandwidth is closer to the natural resonant frequency of the structure. Further analysis from the Nyquist and Bode plots explains the observations made above. Nyquist and Bode Analysis The Nyquist and Bode plots for each case simulated in F i g . 3.24 are shown in F ig . 3.26 and F i g . 3.27. The Nyquist plots study the open-loop system Topen(s) — PX2 (s)C(s) , Chapter 3. Active/Passive Damping and Endpoint Vibration Control 60 Figure 3.25: Block diagram of the endpoint feedback closed-loop control system. where PX2(s) is the perturbed version of the original physical system transfer func-tion PX2(s) in (3.16) and C(s) is the controller C i ( s ) or C?(s) defined earlier, which is designed specificly for the unperturbed plant PX2(s). The Bode plots look at the closed-loop system response Tciose(s) of the endpoint vibration bx = x2 — x,\ due to the endpoint input command xref. The block diagram of the system is shown in Figure 3.25 and the closed-loop transfer function can be derived as ' close bx x. ref X2 - Xi Xref *2 PXl(s)C(sf-<-^ x. ref X ref = [I + Pxl(s)C{8)] *X£8]C{*] , -K(s)C(s) I + PX2(s)C(s) [PX2(s) - PXl(s)]C(s) (3.19) I + PX2(s)C(s) where PXl(s) is the perturbed version of the original physical system transfer func-tion PXi(s) in (3.16). These Bode plots indicate directly the magnitude of vibration one can expect when a certain required endpoint motion is commanded. Low Bandwidth Control When the controller C\(s), wi th a designed 2 Hz close-loop response, is implemented wi th 20% error in the joint stiffness estimation (Fig. 3.26(a.)), the Nyquist plot moves Chapter 3. Active/Passive Damping and Endpoint Vibration Control 61 k 1 o f f b y - 2 0 % ( 2 H z ) k 2 o f f b y - 2 0 % ( 2 H z ) 0 . 5 0 . 5 C D CO E - 0 . 5 — 0 . 5 ( ( a . ) R e a l A x i s k 1 o f f b y - 2 0 % ( 1 5 H z ) —0.5 O ( b . ) R e a l A x i s k 2 o f f b y - 2 0 % ( 1 5 H z ) - 0 . 5 - 1 - 0 . 5 ( c . ) R e a l A x i s - 0 . 5 ( d . ) R e a l A x i s Figure 3.26: Nyquist plots for Q-parameter control with modeling uncertainties: De-creases in the stability margin can occur with error in estimating the joint or arm stiff-ness. In (a.) and (b.), the 2 Hz controller C\(s) is used. Error in the arm stiffness does not affect the stability margin because the control bandwidth is far away from the arm flexible mode of 12 Hz. In (c.) and (d.), the control bandwidth of 15 Hz is closer to the flexible mode of the arm. Thus, an error in k2 affects stability significantly. closer to the cri t ical point —1; thus, the stability margin is decreased. This is also reflected in the Bode plot from F i g . 3.27(a.). Wi thout modeling error, a resonant peak at 2 Hz can be observed for the designed system response (dotted line). W i t h the modeling error in the joint stiffness, a new resonant mode appears and the peak overtakes the original maxima. When the modeling error is switched to the stiffness of the arm, the stability of the control system does not appear to be threatened, as shown in F i g . 3.26(b.). This , , however, does not prevent the vibration from surface. A n additional resonant response is observed in the Bode plot shown in F i g . 3.27(b.) and this heightened response can be translated into poorly damped endpoint motions. The stability margin remains the same because the maximum response of the system st i l l stays at 2 Hz and is part of the' Chapter 3. Active/Passive Damping and Endpoint Vibration Control 62 k1 o f f b y - 2 0 % ( 2 H z ) k 2 o f f b y - 2 0 % (2 H z ) 1 0 " 1 1 0 ° 1 0 1 1 0 2 1 Q - 1 1 0 ° 1 0 1 ( c . ) F r e q u e n c y ( H z ) ( d . ) F r e q u e n c y ( H z ) Figure 3.27: Bode plots for Q-parameter control wi th modeling uncertainties: Modeling errors in k\ or k2 induce heightened magnitude responses at the corresponding joint or arm natural frequency. If there is li t t le passive damping present in the joint or arm, poorly damped oscillatory motion can be observed at the endpoint of the arm. original design in the desired closed-loop system response. High Bandwidth Control When the 15 Hz controller C2(s) is used, similar results are observed; they are shown in F ig . 3.26(c) and (d.) and F ig . 3.27(c) and (d.). The maximum vibration occurs at 15 Hz and the magnitude is much larger than the one observed in the case of C i ( s ) . The 20% error in the joint stiffness does not affect the stability margin, but the Bode plot shows a heightened response at the lower frequency. This poorly compensated low frequency wave acts like a carrier wave which allows the high frequency structural vibration at 15 Hz to surface. ' The same amount of error in arm stiffness has much more drastic effects over both the stability margin and the endpoint vibration. The Nyquist plot shows the stability Chapter 3. Active/Passive Damping and Endpoint Vibration Control 63 margin is much decreased and the additional resonant peak in the Bode plot dominates the maximum vibrational response of the closed-loop system. 3.4 Summary A number of active control methods were investigated for endpoint vibration suppression of a flexible robot arm link. Due to the additional flexural dynamics from the joint to the end of the link, endpoint feedback method was used for these active controls. W i t h joint-feedback, the global stability of the control system can be guaranteed; but, with endpoint feedback, the stability can be threatened as the control gains increase. W i t h a regular P D controller, increasing the derivative gain decreased the stability margin of the control system introducing more vibration at the endpoint. Modern control methods such as command pre-shaping, state-space control and Q-parameter control were also investigated. Passive damping seemed to be the only viable way for consistent vibration control. It was found that the precision in modeling the flexural mode of the structure and in actuating the calculated control efforts was crucial in the success of the modern l in-ear control approaches. In addition, although the controller gains might be high, the required control efforts were not in proportion to the gains and in some cases achievable. Uncertainties in modeling and actuation not only directly resulted in the appearance of the flexural motions at the endpoint, but also affected the stability of the control system in some cases. Al though not evaluated here, non-linear adaptive control may be a suitable approach to overcome the difficulties in linear controls with modelling uncertainties. W i t h a model-reference adaptive system ( M R A S ) , partial priori knowledge for a robot manipulator can be utilized to control the manipulator appropriately [20]. It is important that as much a Chapter 3. Active/Passive Damping and Endpoint Vibration Control 64 priori knowledge should be used as possible. If the arm were poorly damped, the endpoint flexural motions due to modeling uncer-tainties would have been highly oscillatory and poorly damped, which had been shown in Chapter 2 to affect the stability and performance of the bilateral motion scaling sys-tem. However, wi th sufficient amount of passive damping in the arm structure, even if the plant modeling or actuation were not precise and caused the flexible mode of the structure to induce endpoint flexural motions, they would have been well damped and the stability and performance of the motion scaling system would have not been threat-ened. Therefore, the evidence showed that active damping methods cannot be effectively implemented and relied upon with confidence in many realistic applications considering there is little room for modeling errors. No matter what control methods are to be used, sufficient passive damping in the flexible robot structure for transporting the motion scaling system is absolutely necessary. Chapter 4 Vibration Damping with Constrained Layer Damping From the stability and performance analysis of the bilateral motion scaling system in Chapter 2 and the evaluation of active and passive damping schemes in Chapter 3, passive structural damping is clearly important to the operation of the motion scaling system. In order to provide a suitable supporting platform, a well damped physical structure is required. This chapter develops and proposes a methodology for designing such a structure. Section 4.1 discusses the rationale, the implementation considerations and the design procedures in realizing a robotic structure with links of high intrinsic damping. Sec-tion 4.2 presents the methodology for designing a hollow beam with multiple viscoelastic laminates. Specifically, a formulation first developed in 1959 [2] is first summarized in Section 4.2.4. The analysis rationale and basic physical concepts used in [2] are intro-duced and explained in detail in Section 4.2.2. Extension of the original formulation that considers a hollow structural layer and multiple viscoelastic laminates is presented in Section 4.2.5 and Section 4.2.6. The mult i-layer arrangement is a preferred configuration for designing a robot link wi th sufficient intrinsic damping. Section 4.3 discusses the design guidelines in achieving a well damped beam based on the formulation developed. 65 Chapter 4. Vibration Damping with Constrained Layer Damping 66 Figure 4.28: A hollow rectangular beam with damping layers. 4.1 Structural Design In a typical serial l ink robot, vibrations can be caused by both the joint and link flexibili-ties. Sufficient joint damping can be achieved by introducing viscous damping within the joint bearings, or if there is a joint actuator, direct local control can provide additional active damping. Linkage damping is more difficult to achieve through active means (See Chapter 3). The methodology first proposes that in order to achieve an overall well damped struc-ture via passive means, each link of the structure must exhibit high intrinsic damping. It . is then proposed to realize such a link construction through the constrained layer damping technique. This technique involves applying one or more damping layers to the surface of the structure. Each damping layer consists of one viscoelastic layer wi th low stiffness and high intrinsic damping, and one stiff layer constraining the viscoelastic layer between itself and the stiff layer on the other side of the viscoelastic layer, which can be either the structure or another constraint layer. The concept is illustrated in Figure 4.28, where one damping layer is applied to both sides of a hollow beam structure. Chapter 4. Vibration Damping with Constrained Layer Damping 67 4.1.1 Implementation Target Frequency for Effective Damping As wi l l be elaborated later in the chapter, the damping of such a composite beam (referred to as the beam hereafter) is frequency dependent [23, 2, 11]. It peaks at a particular frequency and eventually degrades down to the raw damping of the structural layer at frequencies far away from the peak frequency. Since the vibrational modes of a robot change as it moves through space, it is necessary to determine the flexible mode that needs the most suppression and use it as the target frequency where peak damping should occur. Once a damping layer configuration is decided, the frequency response of the beam damping can be obtained to show its damping capabilities wi thin the frequency range that the robot structural resonant mode is expected to span in a particular application. Beam Modeling First , by considering al l the shearing effects between layers and incorporating their con-tributions toward the overall flexural motion of the beam, it is assumed that such a beam can be analyzed just as a homogeneous beam where shearing does not exist within the material [31, 34]. Therefore, the flexural properties of the beam can be analyzed by finding the effective flexural rigidity, which includes the flexural and shearing properties of al l composing layers. To be become apparent later in the chapter, a complex valued flexural rigidity B = B0(l + jrj) describes the intrinsic damping in a beam, where n is the damping factor. Thus, the effective damping of a beam with constrained layer damping can be. determined by finding the ratio between the imaginary and real parts of the effective flexural rigidity. . Chapter 4. Vibration Damping with Constrained Layer Damping 68 Analysis Approach In an actual implementation, the beam is used as a link in the robotic structure. Both ends of the link are coupled to joints with actuators. The stiffnesses in the joints are actively controlled to produce certain robot motion and are constantly changing with time. This makes solving the effective complex flexural rigidity of a finite length beam with excitation forces at the beam boundaries difficult. This also requires one to solve the entire wave equation of the beam which often involves high order partial differential equations. For example, wi th only one damping layer applied, the solution of the wave equation is a 6th order partial differential equation [34]. As the number of the damping layers grows, the order of the final wave equation increases, too. From the design point of view, a practical design tool should provide sufficient guidelines for a good design without complicated problem setups and time-consuming solution approaches. Solving the high-order differential wave equation with the time-varying excitations at the link boundaries does not serve as a good candidate. Based on the analysis method developed by Ross, Ungar and Kerwin in [2] for estimat-ing the effective flexural rigidity of an infinitely long solid beam wi th one damping layer, a generalized formulation for an infinitely long hollow rectangular beam with multiple damping layers applied both inside and outside of the beam is developed and presented in this chapter. The formulation determines the effective damping that a beam has on a passing flexural vibratory wave with a certain wavelength. It is then used repetitively with different design parameters, such as the layer thicknesses and number of applied layers, to find the best combination of these parameters that yields the most damping for a particular vibrational mode. Chapter 4. Vibration Damping with Constrained Layer Damping 69 Boundary Conditions It has been found that for the same vibration modes, the effective damping achieved by clamping al l layers together at one end of the beam is consistently higher than leaving both ends free [47]. Therefore, the generalized formulation mentioned above can serve as a conservative estimation on the effective damping of a particular beam configuration, with the confidence that the actual damping achievable can be higher when the beam is used as a l ink in the robotic structure. This is because at one end of the link, all the layers are clamped together and coupled to the actuator while at the other end, only the structural layer is coupled to the next actuator leaving the viscoelastic and constraining layers free. Design Procedure Therefore, passive damping can be considered as part of the fundamental procedure in designing a robotic structure. A n outline of the design process for such a structure is proposed below: 1. The architecture of a robotic structure is designed and decided first. 2. Assuming al l members are elastic, simulations or structural analysis assuming a simplified model on the operation of the robot with expected payloads can be performed to determine the design parameters that may include: (a) structural vibration frequency that needs the most suppression (b) structural dimensions of each link (c) maximum mass allowed on each link (d) min imum stiffness required on each link Chapter 4. Vibration Damping with Constrained Layer Damping 70 As w i l l be shown in the analysis later, the above parameters are in fact inter-related. Therefore, a satisfactory design needs to find the best combination among these parameters that yields the most damping. 3. The desired frequency is used to determine the best damping layer designs for all links with no boundary conditions. They are designed considering the following variables: (a) Thicknesses of the viscoelastic and constraining layers (b) Number of the layers applied to the beam (c) How the layers are applied to the beam There may be several designs that can achieve sufficient damping for the required vibrational frequency, but the one that meets the physical design constrains wi l l be chosen. 4. Each designed link is assembled with al l the viscoelastic and constraining layers clamped to the structural layer at one end leaving the other end free. This is to achieve more damping than the one predicted from the formulation for an infinitely long beam. 4.2 Beam Analysis Lit t le explanation was presented in [2] regarding the physics and complex notations used in the original formulation of a solid beam with one damping layer. They are first explained in detail in the sections below. The formulation for a one-layer beam by Ross, Ungar and Kerwin [2] is then summarized in Section 4.2.4. The extensions to a double layer beam and a multi-layer formulation are presented in Section 4.2.5 and Section 4.2.6 Chapter 4. Vibration Damping with Constrained Layer Damping 71 The goal of the formulation is to find the effective damping of a multi-layer beam. It can be shown that the effective damping can be summarized into a frequency dependent linear system, which can be subjected to some type of optimization against a particular set of design constraints. A brief discussion is presented at the end of this section. 4.2.1 History The analysis method is based on the Strain Energy Method ( S E M ) first suggested by Ungar [2] and his colleagues in 1959. The basis of S E M is that damping of a material can be characterized by the ratio of the energy dissipated in the material to the energy stored in the material. The leads to the idea of expressing the Young's modulus as a complex number to describe the energy dissipation characteristic in a material. The method then focuses on the stress/strain analysis of a multi-layer beam structure, ut i l iz ing the complex moduli of al l composing materials. The original formulation developed was sometimes referred to as the R K U formulation [11]. S E M has become the accepted method for analyzing damping of this type of struc-ture, even for composite materials [39]. DiTaranto [34] solved the equations of motion with boundary conditions for a finite length 3-layer beam. Pan [36] and Wi lk ins [37] extended the concept to formulate the problem for cyl indrical /conical structures. Recent researchers util ized the power of the Fini te Element Method ( F E M ) to analyze complex structures of this nature, wi th distributed models also based on S E M [38, 39, 40, 41]. Al though the application of the R K U formulation is l imited to the analysis of one-dimensional flexural motion of an infinitely long or simply supported sandwiched-struc-ture, wi th the reasonable assumptions summarized in the next section, it does not involve solving the wave equations and thus offers an attractive and practical approach as the basis of beam analysis required here. Since it may be useful to design a robot arm link Chapter 4. Vibration Damping with Constrained Layer Damping 72 with multiple damping layers, the original formulation cannot be readily used in the de-sign process. Al though Ungar speculated that more damping wi l l result from using more damping layers [2], the precise amount of improvement is not known unless al l layers are considered in the formulation. These factors motivated the generalization of the formu-lation from a 3-layer beam to a beam with arbitrary number of layers and various layer arrangements. 4.2.2 Preliminaries on Structural Analysis The analysis on the viscoelastic behaviour of a multi-layer beam depends on the knowl-edge of the physical properties of the materials under bending and shearing, and the mathematical notations and equations that govern these properties. These discussions are presented here and they include the following: • nature of viscoelastic materials, • the use of complex moduli in representing intrinsic dampings in materials, • the approximation of damping coefficient £ from the material damping factor n, • the relationship between vibration frequency uo and vibratory wavelength A in an Euler-Bernoull i beam, and • the governing physics behind the stress/strain analysis. Viscoelastic Material The physical properties of a viscoelastic material are often temperature and frequency dependent. From observation, the stiffness of a polymer material lowers as the temper-ature rises. Effects of temperature or frequency on the damping capability of,a polymer are less intuitive. Fortunately, the manufacturer usually summarizes these properties Chapter 4. Vibration Damping with Constrained Layer Damping 73 I S D 1 1 2 1 Pascal = 1.45 x 10"' PSI Figure 4.29: The damping capacity of a viscoelastic material is often frequency and temperature dependent. This chart is provided by 3 M Scotchdamp for the ISD-112 material. in a chart. Figure 4.29 shows such a graph provided by 3M-Scotchdamp for the vis-coelastic material ISD-112 material. The material achieves peak damping under normal room temperatures (15° C - 20° C) for frequencies ranging from 10 Hz - 100 Hz. These properties best match the operating temperature and low structural frequencies in a typ-ical robotic application. Note that not all viscoelastic materials are suitable for a given application; the best material can be selected using the charts provided by the manufac-turer. 3M-Scotchdamp manufactures different types of viscoelastic materials, each has peak damping performance at different temperatures and frequencies. The manufacture produces three different thicknesses: l'mil, 5 mil and 10 mil. The material is adhesive Chapter 4. Vibration Damping with Constrained Layer Damping 74 Im Stress a AW ' / t i A / ' / Strain e 2 W = E e /2 1 AW ^ " 2 7 1 w 1 7 5 CO V V j c o t j ( c o t -Re Figure 4.30: Stress-Strain plot defines Figure 4.31: Phasor diagrams represen-loss factor n tation of sinusoidal stress and strain. in nature; thus, it can be applied to a structure as normal adhesive tapes. Depending on a particular damping layer design, multiple layers can be applied to achieve the required thickness. Complex Modulus and Complex Physical Variable The Young's modulus and shear modulus are defined by Hooke's law [31], E G e r 7 ° ~ * (4.20) where o and 7 are the longitudinal and shear stresses on the contact surface and e and tp are the longitudinal and shear strains, respectively. Hooke's law defines E and G to describe the physical property of a material when it is under stress. When the study of a material is l imited only to its elastic behavior, Young's modulus is a real value and is sometimes referred to as the modulus of elasticity (or elastic modulus). Chapter 4. Vibration Damping with Constrained Layer Damping 75 This is because of the assumption that no energy dissipation takes place during any stress loading, and if a sinusoidal stress is applied to the material with a circular frequency to, the strain observed will be in phase with the applied stress. This results in a real valued elastic modulus. In reality, it is observed that all materials exhibits hysteresis relationship between stress and strain when subject to a stress cycle (See Figure 4.30). This means that there is a phase lag between the applied stress and the resulting strain. This indicates that some energy dissipation is present within the material. The applied cyclic stress a(t) with an amplitude o0 and frequency ui, and the measured strain e(i) with a magnitude e0 and a phase delay of 5 can be visualized as phasors in Figure 4.31 and expressed as a(t) = a 0 e ^ f ; e(t) = e D e ^ ~ S ) . From Hooke's law in (4.20), Young's modulus is then p °° e J U J t a° iS a° r tx\ i • • / X M E = — — = r r — - — F T = — eJU = — \cos(d) + j sin(d) = E0(l+jrj) , where E is the complex Young's modulus, and the elastic modulus Ea and the damping factors n can be defined as E0 = — ; rj = tan(5) . It is then clear that the elastic modulus used in conventional structural analysis occurs when the phase lag in the stress/strain phasors is zero. A phase lag implies energy dissipation, and is used to define a dimensionless damping factor r), which is related directly to this phase lag 5. By analogy, the shear modulus G = GQ(l + jrj) is also a complex value when considering damping properties in real materials. Chapter 4. Vibration Damping with Constrained Layer Damping 76 Later in the analysis, it is found that the distance of the neutral bending plane of the composite beam from the central plane of a composing layer is a complex number. A c -cording to the governing equation (4.34), this distance describes the relationship between the longitudinal force in the material and the bending moment about an arbitrary axis a distance away from its central plane. A complex distance suggests that the bending moment observed wi l l not be in phase with the force applied, implying that the structure exhibits some intrinsic damping characteristic. Al though not presented in this chapter, simulations have shown that the imaginary part of the neutral plane location decreases as the effective damping achieved wi th the composite beam decreases. Damping Factor n and Damping Coefficient £ In control engineering, the damping coefficient £ is often used in place of the damping factor r\ and it is important to note that they are two different measures. The damping factor is specifically a measure of material damping property from Hooke's law while the damping coefficient describes the general damping property of a plant or a structure which is treated as a mass-spring system governed by Newton's equation and can be modeled as a second order system. Consider a cantilevered beam of length L wi th complex Young's modulus E and cross-sectional area A The spring constant of the beam due to a force at the free end can be approximated as . k = ^ = A;0(l + j'77) . This hints that i f one were to model the endpoint oscillation with a mass-spring system, a complex spring constant can be used to represent the stiffness and damping in the beam. This is demonstrated below. Assume one can model the endpoint deflection y(t) due a force f(t) as a mass-spring Chapter 4. Vibration Damping with Constrained Layer Damping 77 system and the internal material damping mechanism as a type of viscous damping, Newton's equation can be written my(t)+Cy(t) + k0y(t) = f(t) , (4.21) where m is the payload mass at the endpoint, C is the viscous damping constant and k0 is the real spring constant. When considering a sinusoidal force f(t) = Fe^^ and endpoint oscillation y(t) = Ye^^, the equation above becomes /(£) = (-mco2 + jCuo + k0) y(t) = (-mco2 + k) y(t) , where the viscous damping can be included directly in a complex spring constant k = k0(l+jn) = k0 + jCuo ; n = (4.22) Taking the Laplace transform of Newton's equation (4.21), the input/output transfer function T(s) of the mass-spring system is a second order system wi th the fundamental resonant frequency con and the corresponding damping coefficient £ T(s) = y^-J- = 2 + 2 ! C = Xunm . (4.23) Combining (4.22) and (4.23), the relationship between r\ and C is found to be C = ^ % • (4.24) This relationship suggests that at the structural resonant mode, i.e. u> = u)n, the damping coefficient of the modeled mass-spring system is related to the material damping factor as C = rj/2. Further, i f the beam is forced to oscillate at other frequencies, i.e. LO ^ con, the effective damping coefficient (as computed) from the intrinsic damping in the material is less at higher frequencies than at the lower frequencies. This further suggests that one should be careful wi th different damping measures because depending on the governing physical laws, some measure the structural damping Chapter 4. Vibration Damping with Constrained Layer Damping 78 and some measure the intrinsic material damping, al l under different contexts. In addi-tion, the conversions between them may not simply involve a constant factor, but may be a function of excitation frequency, as demonstrated in (4.24). Euler-Bernoulli Beam A lumped mass-spring model such as the one illustrated above can only serve as a rough conceptual approximation to the endpoint oscillation of a beam. A complete analysis of a multi-layer beam is beyond the scope of this thesis. It w i l l be assumed that the length of the beam is much greater than the width and height; so, the longitudinal vibrational modes are well separated from the transverse ones and the coupling terms such as dy(x,y, z,t)/'dz and dy(x,y, z,t)/dy can be neglected from the analysis, (the coordinate system is as shown in Figure 4.28.) In addition, only bending is considered in the beam. Therefore, the analysis presented in this chapter assumes an Euler-Bernoull i beam with the Euler-Bernoull i wave equation rtd4y d2y B a ? + " a f = 0 • ( 4 ' 2 5 ) describing the free vibration of such a beam, JJL is the mass per unit area, while B = EI is the flexural rigidity where / is the area moment of inertia of the cross-sectional area. There has been experimental work performed to analyze bending and shear damping in a beam. In [48], it was reported that Euler-Bernoull i beam theory works well in the low frequency range (less than 128 Hz), but becomes inadequate when the excitation frequencies gets higher (up to 434 Hz in one of the experiments) or when the ratio of the thickness of the beam to the wavelength of the vibration increases; Timoshenko beam theory works better at higher frequencies. As discussed in the previous section, for practical design purposes, it is neither desir-able nor necessary to solve for the solution of a complete wave equation to approximate Chapter 4. Vibration Damping with Constrained Layer Damping 79 the effective damping in a multi-layer beam. However, for the implementation purpose, it is necessary to determine the relationship between the vibratory wavelength A and the vibration frequency u of the beam. Euler-Bernoull i beam theory is used to approximate this relationship. A general sinusoidal wave solution for the Euler-Bernoull i equation in (4.25) is y(x,t) = Yefox-ut) , (4.26) where p = 2n/\ is the wave number [49] and w is the vibration frequency in rad/s, can be substituted in (4.25) leading to the following relationship between the wave number and the vibration frequency for an Euler-Bernoull i beam: P = f = ( § " 2 ) * • (4-27) Step 2 in the robotic structure design procedures proposed in the previous section esti-mates the structural vibrational frequency UJ of the design robotic structure, assuming elastic behaviour in the constructing material. W i t h (4.27), this value can be used to approximate the equivalent wave number of the vibratory wave traveling through a link i with a certain flexural rigidity Considering a composite beam with intrinsic material damping, B is the effective flexural rigidity and is complex. Equation (4.27) suggests that i f one specifies a real valued wavelength A c in space traveling along the longitudinal axis of the beam, one can expect a complex vibrational frequency implying its damping in time. Vice versa, if one instead concentrates on a real valued oscillation in time, one wi l l find a complex wave number indicating its damping in space. This makes one speculate that a flexural vibratory wave is actually damped both in space and time for a freely vibrating Euler-Bernoull i beam. Indeed, if a force term is introduced to the right-hand-side of the equation in (4.25), the.vibration frequency UJ can be forced to be real and is no longer an inherent physical Chapter 4. Vibration Damping with Constrained Layer Damping 80 y dy/dx= c|> y = F / A ( | ) - \ ( / + flexural angle shear (or shear angle) Figure 4.32: Total deformation is the sum of flexural and shear angles. property of the beam. This leaves the wave number to be complex hinting intrinsic damping in material. Governing Equations The mechanics of bending is covered in detail in [31], but the relevant ideas and equations are summarized below. Bending analysis of a beam involves mainly strain e, stress a, radius of curvature p, flexural angle 4>{x,t) = dy/dx and bending moment M. They are explained below. Bending and Shearing A s shown in Figure 4.32, an object can be under both bend-ing and shearing at the same time. Bo th shearing and bending contribute toward the deformation in material. From the geometry of bending, the flexural angle is the same as the slope of the curvature: shear (4.28) shear Chapter 4. Vibration Damping with Constrained Layer Damping 81 where the added minus sign means that while the force FShear is s t i l l defined positive in the positive x direction, the shear angle is defined in the counter-clockwise fashion as shown in Figure 4.32. Further coupling between bending and shearing can be demonstrated through express-ing the shear modulus G as a function of Young's modulus E. The detailed derivation is not included here. A bar subject to an axial tensile load F directly along the longitu-dinal x-axis w i l l elongate in the x direction and contract in both the transverse y and z directions due to lateral strains where ex is the longitudinal strain and v is Poisson's ratio. It can be shown (in [31]) that if ex << 1, a small cubic volume dx • dy • dz wi th side area dA in the material experiences an equivalent shear strain ib due to this longitudinal strain 4>, I/J = (1 + v)tx . From Hooke's law (4.20), this expression becomes I - <'+">§• Again, it can be shown that the same longitudinal force F generates a shear stress 7 = F/2A and a longitudinal stress a — F/A; thus, 7/a = 1/2, and the shear modulus G is related to Young's modulus from the above equation as: E T E G = t ^ - 1 = 7-77——r • (4.29) l + i^ cT • 2(1 + v) v ' Usually, Young's modulus E and Poisson's ratio v are supplied by the manufacturer of viscoelastic materials; the shear modulus can be computed from (4.29). Chapter 4. Vibration Damping with Constrained Layer Damping 82 (a) (b) Figure 4.33: Strain diagram of a beam under centric bending. Bending Analysis A s shown in Figure 4.33, a segment of beam wi th length L — dx under pure bending has a neutral bending plane where the length of the beam is preserved under loading implying zero strain; with a homogeneous material, the bending plane divides the beam into two equal parts. A local coordinate system can be defined such that y = 0 at the bending plane. The strain of the beam at y ^ 0 is defined as A L (p-y)0-p0 e = y p (4.30) L p9 where 9 is the bending angle and p is the radius of curvature. A s shown in [31], the radius of curvature is related to slope of the bending curve dy/dx (or the bending curvature </>) as (4.31) where the approximation 1/p = d(f)/dx assumes that the flexural angle is small. Substituting the expression of strain (4.30) in Hooke's law (4.20), the longitudinal stress can be expressed as a = Ee = - E - (4.32) Chapter 4. Vibration Damping with Constrained Layer Damping 83 0 F M = Figure 4.34: Strain diagram of a beam under eccentric bending. For pure bending of a beam about its own neutral axis (or centric loading), the bending moment M about the cross-sectional area A is defined as M = / —yadA J A Using the stress expression in (4.32), the above expression can be simplified as M j I -y2dy Jx Jy p dx E I -P where / is the second moment of inertia (or area moment of inertia). Defining the flexural rigidity as B = EI and substituting in the expression for radius of curvature in (4.31), the bending moment of a beam can be expressed as M EI d2y B dcj> (4.33) dx2 dx This is the governing equation of bending in a homogeneous beam. For a multi-layer beam, it is assumed that this equation st i l l holds provided the reasonable assumptions given in the next section are satisfied. Therefore, an effective flexural rigidity of the composite beam can be found to enable one to analyze it as i f it were a homogeneous entity. Eccentric Loading A s a multi-layer beam is deflected, al l composing layers no longer bend about their own bending planes, but about a neutral plane that is specific to the Chapter 4. Vibration Damping with Constrained Layer Damping 84 D + + z D M = Mi  M 2 Figure 4.35: Bending moment of a multi-layer beam is additive. composite beam. The layers are said to be under eccentric loading. Figure 4.34 shows the strain diagram of a homogeneous layer under eccentric loading. Comparing with the pure bending case shown in Figure 4.33, there is an additional constant strain (with respect to the y axis) due to a longitudinal force F at the neutral plane of that layer. The bending moment experienced by an individual layer is then dcf) M = F-d + B dx (4.34) where the central plane of the layer is, a distance d away from the supposed bending plane. The bending moment for the eccentric case in (4.34) converges to the centric case in (4.33) when d — 0. As shown in Figure 4.35, a beam can be sliced into layers and the total bending moment of the multi-layer beam can be expressed as the sum of individual bending moments of each composing layer about the neutral bending plane of the beam. For example, applying (4.34) to the two layers in Figure 4.35, the total bending moment is: M = M i + M 2 F-D + B d(j)/2 dx = - B ^ + - B ^ 2 dx 2 dx F-D + B d(f>/2 dx = B 30 dx This shows that the sum is actually the bending moment of the whole beam using (4.33). For a composite beam with different layers, the superposition principle wi l l s t i l l apply, Chapter 4. Vibration Damping with Constrained Layer Damping 85 no shearing in material F, up shearing in between layers F d o w n F, shear = F -up F, d o w n Figure 4.36: Shearing in a multi-layer beam. but the location for the neutral bending plane for the beam is not as obvious as the above example. Shearing A s shown in Figure 4.36, a multi-layer beam under flexural motion experi-ences shearing in between the composing layers while there is no shearing in a homoge-neous beam. Therefore, in order to justify treating a multi-layer beam as a homogeneous one, al l shearings must be accounted for and be incorporated into the overall flexural motion of the beam. Let shear be denoted by ip(x,t). For a sinusoidal wave solution y(x,t) in (4.26), the shear angle ip(x,t) w i l l also be proportional to a sinusoidal wave as Together wi th Hooke's law in (4.28) where the shear or shear angle is defined in the opposite direction as the shear force, the following relationship can be established ip(x,t) = -1 Fshear = 1 d27p GAshear p2.dx2 To account for the energy dissipation property in materials, a complex shear modulus G = G Q (1 + jrj) is used. Since the shearing force Fshear is due to the longitudinal strains from a flexural deflection, this equation can be used to couple shearing action and flexural motion. ijj(x:t) oc e j(px-wt) Chapter 4. Vibration Damping with Constrained Layer Damping 86 4.2.3 Analysis Assumptions and Notation The following assumptions are exercised in the analysis of a composite beam: 1. A composite beam (hereafter as beam) consists of one structural layer and a number of damping layers. Each damping layer consists of one viscoelastic layer and one constraint layer. 2. The longitudinal axis of a beam is defined as x-axis. y-axis is defined in the same direction as the transverse force applied to the endpoint of a beam, and the centroid plane of the structural layer is aligned with y = 0. The z-axis is orthogonal to both x and y axes; together they form a right-handed coordinate system. 3. There exists a neutral bending plane in a beam where the strain e = 0. It is defined a distance D along the y-axis from the neutral plane of the structural layer, and its location is dependent on the vibrational frequency. A s discussed above in the preliminaries on complex modulus and complex physical variables, D is a complex value implying intrinsic damping in the material causing constant phase lag between a strain force phasor and a bending moment phasor. 4. A l l composing layers in a beam are bounded such that no slipping exists in between any two layers. This ensures continuing of strain and displacement at the boundary between layers. 5. A l l layers are assumed incompressible. Thus, the thickness of each layer does not change under deflection. 6. Unless otherwise specified, al l constraint layers are identical in dimension and ma-terial, as well as al l viscoelastic layers. Chapter 4. Vibration Damping with Constrained Layer Damping 87 7. The Young's modulus of the viscoelastic material is assumed to be negligible relative to that of the constraining or structural material. Therefore, a stiff layer being sandwiched by two viscoelastic layers experience no significant shearing; i.e. the actual shearing angle is too small to be significant. 8. A l l layers partake the same flexural angle 4>(x,t) as the neutral bending plane of the beam. 9. As shown in Figure 4.32, al l viscoelastic layers, sandwiched in between two stiff lay-ers, receive additional shear deformations on top of flexural deformation; however, each viscoelastic layer i experiences a different shear ipi(x,t) satisfying (4.35): 1 _ \d2i) ~GJi ~ 'p^dx2 ' 10. The principle of equilibrium applies to the total longitudinal force on a cross-sectional area anywhere along the x-axis; i.e. the net longitudinal force is zero = o . i 11. A beam can be analyzed as a homogeneous beam with an effective complex flexural rigidity B = B0(l + jrj) and conforms to the Euler-Bernoull i beam theory d4y d2y B a ? + "a? = 0 • that has a solution of the form y = Ye-fV-"*) , where the wave number p is related to a vibration frequency UJ p = {-UJ )4 • . Chapter 4. Vibration Damping with Constrained Layer Damping 88 Figure 4.37: single-layer constraint-layer damping model: a segment of the beam dx is shown both in an unloaded position and loaded position. Under loading, there exists a neutral bending plane for the beam a distance D away from the central plane of the structural layer. The location of D is dependent on the vibrational frequency UJ of interest, which is coupled to the wave number p v ia (4.27), assuming the Euler-Bernoull i beam theory. The wave number p is then used in (4.43) below to determine the effects from shearing. The central plane of the structural layer is displaced by Hns(j), viscoelastic layer by a total displacement of Hnv(j) — Hvip and constraint layer by Hnc(f) — 2Hvip. The viscoelastic layer experiences shearing forces Fvc and Fsv. 4.2.4 Single Layer Formulation The R K U formulation for a beam with one damping layer is derived below as a precursor to the multi-layer generalization. A model of this configuration is shown in Figure 4.37. The thicknesses of the structural, viscoelastic and constraint layers are denoted as 2HS, 2HV and 2HC, respectively. The subscripts s, v and c represent each of the three layers composing the beam. This convention is used throughout this section. A neutral plane N is a distance D away from the central plane of the structural layer, and the distances between it and the viscoelastic and the constraint layers are denoted as Hsv and Hsc, respectively. The model shows a segment dx of the beam along the longitudinal x-axis under both the unloaded and loaded conditions. Under loading, the flexural angle <j){x, t) is defined clockwise while the shear angle ip(x, t) is defined counter-clockwise. It is caused by the shear forces Fvc between the viscoelastic and constraint layers and Fsv between the structural and viscoelastic layers. Chapter 4. Vibration Damping with Constrained Layer Damping 89 When the overall flexural motion of the composite beam is considered, the governing equation for centric loading in (4.33) applies ox ox or the effective flexural rigidity can be expressed as The total centric bending moment M is the sum of al l the eccentric bending moments of individual layers about the neutral bending plane of the beam. According to (4.34) and Assumptions 8 and 9, this can be expressed as M = Ms + Mv + Mc . • (4.36) = -BSTT -^ + FsHns + Bv{-^- — —^ ) + FvHnv + Bc-^- + FcHnc . ox ox ox ox The individual flexural rigidities are Bs = ESIS ; Bv — EVIV ; Bc = ECIC . B y assumption 5, the distances between the layers and the neutral bending plane are constants and can be expressed as scalars Hns = —D ; Hnv = Hs + Hy — D ; Hnc = Hs + 2HV + Hc — D . The longitudinal forces can be obtained by integrating the longitudinal stresses over the solid cross-sectional areas and simplified as Ft = f OidAi = f Ei6i dAi = EiA^ = , (4.37) JAi JAi where i is the index for each layer and K{ is the effective stiffness per unit length. These forces are acting on the central planes of the layers, and they are specifically Fs — K S 6 S , Fv — K V 6 V , Fc — Kc€c Chapter 4. Vibration Damping with Constrained Layer Damping 90 From Assumption 7, Kv is negligible against Ks and Kc\ however, Fv is s t i l l used here, as later in the next section when the double layer formulation is derived, Fv needs to be included to show that D = 0, exactly. Thus, Assumption 7 is mainly used for neglecting shearings on stiff layers. Therefore, the longitudinal strains are the strains experienced at the central plane of each individual layer. From the model diagram in Figure 4.37, the longitudinal strain of the structural layer is given by Hns((/>+^dx-(f>) dl dx OX where the sign is changed comparing to (4.30) because the y-axis in Figure 4.37 is defined in a opposite direction as defined in Figure 4.33 - away from the center of curvature. A s discussed in the preliminaries on the total deformation of a viscoelastic layer due to bending and shearing, the change in length for it considers both the flexural deformation and the shear angle (defined in the opposite direction) and the strain is Hnv((f> + ^dx -</>)-Hv{tl} + l^ cfe - t/>) _ d(f>. d^Jj 6V - ; — " n » 7 j fly— , dx ox ox where the total amount of deformation at the central plane of the viscoelastic layer due to shear angle ip is just Hv • dtp/dx. Similarly, the change in length for the constraint layer is Hnc(<f> + ^dx - <j>) - 2HV(^ + ^dx - VQ _ _ dl_ w dx ox ox where because the constraint layer is directly in contact with the viscoelastic layer (from Assumption 4), it receives an additional displacement from the shear ip, resulting in an additional deformation of 2HV • dip/dx. The longitudinal forces can now be expressed fully Fs = -KJ)d4- (4.38) Chapter 4. Vibration Damping with Constrained Layer Damping 91 Fv = KV[(HS + Hv - D)^ - Hv^} (4.39) Fc = KC[(HS + 2HV + HC-D)^-2HV^} . (4.40) The total bending moment can now be assembled by substituting (4.38), (4.39) and (4.40) in (4.36); it becomes M = [Bs + Bv + Bt '<v(Hs + h -KsDHns + KV(HS HV- D)Hnv + KC(HS + 2HV + HC- D)Hnc] ^ By + (Ky + 2KC)Hy , - R W + R ^ — -Drf, - h D j , — OX OX dcf) ^dx and the complex effective flexural rigidity is formally given by B = B0(l+jv) = B 0 + ^ | | , (4.41) where Bcf, — Bs + Bv + Bc — (KsHns + KvHnv + KcHnc) D +KV(HS + Hv)Hnv + KC(HS + 2HV + Hc)Hnc B^ = -[By + (KV + 2KC)HV]^ B^ can be viewed as the effective flexural rigidity due to the flexural motion in each layer, and B^p as the effective flexural rigidity from the shearing. The damping factor of the beam is then the ratio between the imaginary and the real parts of the effective flexural rigidity B. In order to compute B, D and dip/dcj) need to be computed first. Two equations are required for this purpose. From Assumption 10 and equations (4.38), (4.39) and (4.40), 0 = Fs + Fv + Fc Chapter 4. Vibration Damping with Constrained Layer Damping 92 •= [-(KS + KV + KC)D}^-(KV + 2KC)HV^ +[KV{HS + 'HV)-+KC(HS + 2HV + HC)]^ . Mul t ip ly ing the above expression by dx/d(p on both sides, D can be expressed as = Kv(H> + Hv) + Kc(H* + 2Hv + Hc) _ KV + 2KC diP Ks + Kv + Kc Ks + Kv + Kc v d(P '. l " ' From Assumption 9 and Figure 4.37, the shear strain satisfies - 5 5 T S S « - * * " " ? 5 E ' ( 4 ' 4 3 ) where b is the width of the beam and Ashear = 2b • dx is the shearing area. The shear modulus G can be obtained from (4.29). Expanding Fc in (4.40) and Fs in (4.38) and simplifying the expression, the second equation becomes Ks-K KCHV I ,0^ KS(HS + Hv) - KC(HS + 2HV + Hc) 2bG 1 bG p2)d<P 2bG ' 1 j This concludes the derivation of the original R K U formulation. Equations (4.42) and (4.44) form a linear system that can be put into the form D dip i n = bn where AQ and bo are matrices wi th complex physical variables. From (4.41), the effective flexural rigidity can have the form B = A ^ b o , (4.45)' where A\ is another system dynamics matrix with associating physical parameters. This expression can be used to design of beam structure with just one damping layer. U n -fortunately, for different number of damping layers, the entries and dimensions of the matrices A\, A0 and 6 0 w i l l be changed as the number of unknowns in the system grow. The motivates the continuing development of the R K U formulation to consider mult i -layer cases. Chapter 4. Vibration Damping with Constrained Layer Damping 93 Numerical Example For example, a L — 1 m long robot arm is made of 1/16 i n thick 1 in x 1 in hollow aluminum square tube. In order to suppress the fundamental vibration of the beam with wavelength A = AL = 4 m [49], a damping layer consisting 100 mils = 1/10 in thick 3M-Scotchdamp ISD-112 viscoelastic material and 1/16 in thick solid aluminum plate is applied to the aluminum tube. The material loss factors are nai = 2.7 x 10~ 5 [29] and r/sM = 1 [33]. The elastic Young's moduli for aluminum and ISD-112 material are Eal = 72 x 10 9 N/m2 and E3M = 5 x 10 5 N/m2 [33], respectively. The Poison's ratio for the 3 M material is v = 0.5 and the elastic shear modulus for &• homogeneous mate-rial is related to the elastic Young's modulus as in (4.29); thus G3M = 1.67 x 10 5 N/m2. The complex elastic and shear moduli for all the layers are Es = Ec = Eai(l + jrjai), Ev = E3M(1 + jr]3M) and Gv = G3M(1 + jn3M)-Solving the linear system comprised of (4.42) and (4.44) wi th the numerical values above, the location of the neutral plane N is found and so is the coupling factor between the flexural motion of the beam and the shear motion of the viscoelastic layer: D = 3.04 + J0.16 mm ; ^ = 3.9 - jl.2 rad/m . Substituting both D and dip/d(f) into (4.41), the complex rigidity and loss factor of the composite beam are found B = B0(l+jrj) = 1176 + J73.5 Nm2 ; r] = 0.063 . This is a tremendous increase in damping from that of a homogeneous aluminum beam. 4.2.5 Double Layer Formulation This section demonstrates the procedure in developing beam formulation when more than one layer is applied to the structural layer. It also shows that when identical number of Chapter 4. Vibration Damping with Constrained Layer Damping 94 Figure 4.38: A double-layer constraint-layer damping model: Similar to the single layer case in Figure 4.37, a segment beam of length dx is shown both in the unloaded and loaded positions. Following the same convention, notice the shear angle ipi contributes positive displacement while ip2 contributes negative displacement. These has to be re-flected correctly in the strain expressions for each layer. In addition, the shearing force on viscoelastic layer 1 is Fsvi — FvcX while for layer 2, it is Fvc2 — Fsv2. The neutral plane of the composite beam is st i l l assumed to be a distance of D away from the structural layer. As wi l l be shown later, it actually coincides with the central plane of the structural layer. damping layers are used on each side of the beam, the neutral plane iV coincides with the neutral plane of the structural layer, i.e. D = 0. It wi l l aid in the simplification of the generalized formulation by reducing the number of unknowns by one and simplifying the expression for the closed form solution. Further, the numerical example at the end of the section shows that with the same damping layer configuration, although the number of damping layers are doubled from the previous example in the single-layer case, the effective damping factor achieved is more than doubled. This suggests a possible design guideline, that distributing a number of layers symmetrically about both sides of the structural layer is more effective in achieving more damping than pil ing them all on just the one side. Further verification on this idea by finding the average increase in damping per damping layer motivates the development of a raw algorithm that enables systematic computation of the effective damping of a beam with a number of damping layers. The derivation presented in this chapter w i l l lead to that development. Chapter 4. Vibration Damping with Constrained Layer Damping 95 The model for a symmetrically sandwiched double-layer beam is shown in Figure 4.38. The bottom damping layer is designated as layer 1 and the top as layer 2, and two shear angles ipi and tp2 are specified and both defined in the counter-clockwise fashion. W i t h Assumption 8, al l layers partake the same flexural motion; thus, only one flexural angle 4> is specified. The neutral plane is st i l l defined a distance D away from the central plane of the structural layer. Variables associated with the structural layer are st i l l identified with a subscript s, the constraint layers with c l and c2, and the viscoelastic layers with vl and v2. The analysis would start wi th the same governing equation for the whole beam where pure bending is assumed: B = MU • <446) The goal is the find the total bending moment of the beam in order to compute the effective flexural rigidity B. Similar to the rationale presented in the previous case, the total centric bending moment of the composite beam is the sum of a l l the eccentric bending moments of individual composing layers about the neutral bending plane. This total bending moment is M = M c i + M„i + Ms + Mv2 + Mc2 (4.47) = Bc —^ 4- FciHnci + Bv{-^- —^) + FviHnvi + Bs-^- + FsHns + ox ox OX ox Bv{-^ ^) + FV2Hnv2 + Bc——h Fc2Hnc2 ox ox ox FsHns + FciHnci + FviHnvi + Fc2Hnc2 + Fv2Hnv2 Assumption 6 enables al l constraint layers to use the same flexural rigidity parameter Bc and likewise for the viscoelastic layers; they are BS ESIS , By Eyly , B£ • EQ!C Chapter 4. Vibration Damping with Constrained Layer Damping 96 Incompressibility in the material from Assumption 5 allows one to specify the distrances between the neutral bending plane and the central planes of individual layers as constants Hns = —D ; Hnvi = —(Hs + Hv) — D ; Hnci = —(Hs + 2HV + Hc) — D Notice that the distances from the bottom layers to the neutral bending plane, Hnvi and Hnci, are of negative quantities. From the model defined in Figure 4.38, this sign change wi l l reflect compressive strains for the bottom layers instead of tensile strains as for the top layers. Again , from the equation (4.37), longitudinal forces are products of effective stiffnesses per unit length and the longitudinal strains Fs = Ksts ; Fci = Kceci ; Fc2 = Kcec2 ; Fv\ = Kvtv\ ; Fv2 = Kvev2 . As shown in the modeling figure, al l layers in a beam segment of length dx experience different strains at the central planes; they are Similar to the single layer case, a sign change from (4.30) is due to the definition of the transverse y-axis in Figure 4.38 being opposite from the one defined in Figure 4.33. As defined in Figure 4.38, shear angle tpi generates positive displacements while ip2 is with negative displacements. Thus, the coefficients associated wi th ipi and ip2 are of opposite signs. In addition, although layers vl and c l also experience a flexural angle 4> a s the other layers, as discussed earlier, they experience compressive strains and Hnvi and Hnci are negative values. The longitudinal forces can now be expressed clearly as Hnv2 = (Hs + Hv) —D Hnc2 = (Hs + 2HV + Hc) — D Fs (4.48) Chapter 4. Vibration Damping with Constrained Layer Damping 97 Fvl = KV{[-(HS + HV)-D}^ + HV^-} (4.49) Fel = KC{[-(HS + 2HV + HC)-D]^ + 2HV^-} . (4.50) Fv2 = KV{[(HS + H V ) - D ] ^ - H V ^ } (4.51) Fc2 = KC{[(HS + 2HV + HC)~D]^-2HV^-} (4.52) As shown previously, these force expressions (4.48)-(4.52) can be substituted into the total bending moment expression (4.47) and used with the governing equation (4.46) to find the final expression for B. However, this wi l l lead to a lengthy and complex algebraic expression which is not useful to be displayed here. But , similar to the. previous case, the final expression for the effective flexural rigidity wi l l assume the form dipi dip2 B = Bt + B ^ + B * - ^ . Again, as shown clearly above, B can be summarized as a linear system in (4.45) which can be used wi th some optimization procedure for this particular damping layer arrange-ment. Thus, the derivation wi l l proceed to find the unknowns in this system. There are three unknowns need to be solved: D, dipi/dx and dip2/dx. Thus, three equations are required. Using (4.48)-(4.52) and Assumption 10, we obtain 0 = Fs + Fvl + Fcl + Fv2 + Fc2 (4.53) = -(KS + 2KV + 2KC)D + (KV + 2 K C ) H V ( ^ - ^ ) . Assumption 9 on the sinusoidal properties of shear angles tpi and ip2 w i l l yield one equa-tion per shear angle. A s shown in Figure 4.38, the total shearing force on viscoelastic layer 1 is W W - F ^ J L W - d F c l r ! "shear! — "svl "vcl ' ^ o *^ • 1 ox OX Apply ing equation (4.35), the second equation is obtained KC(HS + 2HV + He) - (Ks - KC)D. KJU 1 djh u ,^ 2bG 1 bG p2' d<f> ' 1 j Chapter 4. Vibration Damping with Constrained Layer Damping 98 Similarly, the total shearing force on viscoelastic layer 2 is: F - W - F - ^Lrl-r - d F s J r r shear2 — rvc2 r sv2 — ^ U J / ^, ""•<' j OX OX and the thi rd equation is ' 2~bG = { ^ G - + p ^ W • ( 5 5 ) B y solving the linear system comprised of (4.54), (4.54) and (4.55), three unknowns are found: '8(f) d<j> d<\> 2KCHV + 2bG/pi ' [ ' ' Usually these solutions are solved numerically. Because of the damping arrangement, they are much simplified and can be shown here in entirety. The simplification is largely due to the fact that D = 0. This does not mean the physical variable D stops demonstrating the energy dissipation property, just that its magnitude is zero forcing both the real and imaginary parts to be zero as well. This results is obtained from purely symbolic manipulation of the formulation from Maple. If one were to solve this linear system numerically, small errors from numerical manipulations may not yield a perfect zero for the distance D. It w i l l be shown in the next section on the generalized formulation that this fact holds whenever the beam is applied with identical number of damping layers on each side of the beam, no matter inside or outside of the hollow structure. Intuitively, this means that the neutral bending plane iV of a composite beam is the same as the neutral plane of the structural layer. A s a consequence, both shear angles contribute the same effects toward the overall flexural motion, which can be stated as dipi/dcp = dfa/dfi. Therefore, the linear system solved above could be effectively reduced to be one equation, (4.54) or (4.55), wi th one unknown, dtp/dcf). Chapter 4. Vibration Damping with Constrained Layer Damping 99 W i t h some algebraic manipulations and letting dip/d(p = dipi/dcp = dip2/d(p, the complex effective flexural rigidity of the beam is formally B = B0(l+jV) = B<f + 2 B ^ (4-57) . = [Bs + 2Bc + 2Bv + 2Kv(Hs-r-Hv)2 + 2Kc(Hs + 2Hv + Hc)2} -2[BV + KVHV(HS + Hv) + 2KCHV(HS + 2HV + Hc)&-- . (4.58) Notice with D = 0 and dip±/dx = dip2/dx, the linear system in (4.45) reduces down to a first order system with only one unknown dip/dx. Solutions in (4.56) can be substituted into (4.58) and the loss factor rj is simply the ratio between the imaginary and real parts of B. Numerical Example In this example, the same beam and damping layer in the numerical example for the single-layer case are used. The damping layer is applied to both sides of the square aluminum tube as shown in Figure 4.38. The unknowns for this configuration are solved numerically wi th (4.56) and they are ^ dipi dip2 dip D = Omm ; —— = —— = — = 3.9 - jl.2 rad/m ocp d(p 8(p From (4.58), the complex rigidity and loss factor of the beam are B = 1602 + j294 Nm2 ; rj = 0.18 . Al though only one additional layer is added to the structure, the effective damping achieved is more than doubled from the single layer case. This is because the neutral bending plane is shifted to an optimal position which coincides with the central plane of the structural layer. A t this position, the neutral bending plane is positioned half way Chapter 4. Vibration Damping with Constrained Layer Damping 100 in between the two damping layers, causing the most shearing simultaneously in both viscoelastic layers. This property can be used as a design guideline, and is discussed with further detail in Section 4.3. 4.2.6 Multi-layer Generalization As demonstrated in the two examples above, the effective damping of a multi-layer beam cannot be approximated directly from the one achieved wi th only one damping layer. The damping factor depends on a complex set of variables that can significantly alter the amount shearing created in the beam structure. Following the same analysis ratio-nale exercised in the above two sections, the section develops and presents a method of formulating a generic hollow multi-layer beam with damping layers applied both inside and outside the structural layer. Modeling and Notation Figure 4.39 shows the model for an arbitrary multi-layer configuration. In the figure, there are two damping layers on top of the beam, one below and one inside the hollow structure. The beam can be divided into four regions: t-\— top outside, t top inside, b+ - bot tom inside and b bottom outside. These are the four regions where damping layers can be applied and they can be collectively expressed as (t, b)±. The structure layer is referred to by the letter index s and/or an unique numerical index {0} as layer 0. A n y viscoelastic and constraining layers are accompanied with the letter indices v and c. The damping layers in each one of the four regions are assigned wi th numerical values {1, 2, . . .} in an ascending order where layer 1 being immediately next to the structural layer. Variables that are associated wi th a particular layer are superscripted wi th the region it is located and subscripted wi th its type (s, v, or c) and layer rank (0, 1, or 2 . . . ) . Chapter 4. Vibration Damping with Constrained Layer Damping 101 top 1-+ unloaded p o s i t i o n s loaded p o s i t i o n s dx damping l a y e r 2 | damping l a y e r 1 n e u t r a l plane N D s t r u c t u r e S v i s c o e l a s t i c V c o n s t r a i n i n g C dx bottom Figure 4.39: A multi-layer constrained-layer damping model: Add ing more layers to the beam results wi l l more tedious and repetitive algebra, which can be much simplified with a unified notation system. Four regions where damping layers can be applied are specified: top+, top—, bottom+ and bottom—. The displacement due to the shearing angles are precise opposite to the sign of each region. Total displacement of a layer due to any shearing action is the sum of all the contributions from the viscoelastic layers between itself and the structural layer; i.e. the outside layer would be displaced the most. For example, the shear experienced by the viscoelastic layer of the first damping layer located on the top outside surface of the beam can be expressed as and its distance to the beam's neutral plane is ff^ti- Notice ip\\ can also be abbreviated as ip\+ since only the viscoelastic layers have shears ip associated with them. Other simplifications to the notation system can also be exercised. For instance, the variables associated with the structural layer requires only a subscript of numerical index {0} or letter index {s} for an unique identification; i.e. ~{s,v,c)0 — €o — es Chapter 4. Vibration Damping with Constrained Layer Damping 102 Bending Moments The total centric bending moment of the composite beam can also be expressed the sum of al l eccentric bending moments as M _ V - M ( t ' 6 ) ± -4- V M[t'b)± - R ^ 4- V R ( * ' h ) ± (A eq\ M - 2^ M{s,v,c)ii + M{s,v,c)in ~ ^ 0 , 0 + l^^in o _ ' l 4 - b y i i>0 i>0 , U J j i>l u x where M^Jf^ is the bending moment of layer i about its own neutral plane, M^s'^in about the beam's neutral plane N and and' B^f^ are the effective flexural rigidities of the composite beam associated wi th the overall beam flexural angle <f> and each of the shear strains ipi m the viscoelastic layers. From (4.33) and (4.34), the bending moments can be defined as M(t,b)± = B(t,b)± (d(f) dV>f'&)±, . M(t,b)± = F(t,b)± ff(t,b)± (s,v,c)ii X J ( s , i i , c ) i ^ dx dx ' (s>v>c)in. (s,v,c)i n{s,v,c)i The flexural rigidity of each individual layer i is R(*.*)± _ F V T^'^ ^(s^rfi — ^ ( s ^ c ) x 1(s,v,c)i » where E^s^v^ assumes that the same materials are used for each viscoelastic and constraint layers (from Assumption 6). A s discussed earlier in the notation system, the partial derivative di/jf ' 6 ^ /dx = 0 when i = 0 because there is no shear associated with the structural layer. F^'^{ is the longitudinal force experienced by each layer i. H^^c^ is the distance from the neutral axis iV to the layer's own neutral plane and can be expressed as rj{t,b)± . _ ir(t,b)± _ D n(s,v,c)i s(s,v,c)i Note that the sign of H^'^c^ is positive for the top regions to reflect a strain of tensile nature, and negative for the bottom reflecting a compressive strain. Chapter 4. Vibration Damping with Constrained Layer Damping 103 Longitudinal Forces For simplicity, the formulation presented below assumes that al l damping layers consist of the same viscoelastic and constraining layer geometries. It can be further generalized to consider otherwise. The longitudinal strains present in each layer can be summarized in the expression below „ o / (t,b)± p, i (t,b)± (t,b)± _ rr(t,b)± °(p ^ °V(-,v,v)j , rr aV{-,v,-)l (Am\ e(s,v,c)i ~ Hn{stv,c)ifa = F 2 / / « 2^ . " d~X ' ( ' 3— When layer i is a viscoelastic layer, all three terms are included in the equation above; otherwise, only the first two terms are included for constraining layers and the first term for structural layer. Due to the geometrical symmetry in each layer, the integration in (4.37) can be solved symbolically and the longitudinal forces can be readily defined as F(t,b)± _ K{t,b)± Xf(t,b)± . K(t,b)± _ F .(t,b)± M f i n where the notation E^s^v^ implies that the same materials are used for each viscoelastic and constraint layers. The equil ibrium principle from Assumption 10 requires the net longitudinal force be zero. This is expressed as £^,S = 0 . . (4.62) Shearing Forces Equation (4.35) reflects the sinusoidal property i n the shear angles (as discussed in As -sumption 9), a l l shearing angles can be summarized in the expression below ,{t,b)± _ +)j_]_jL(F{t,b)±_F(t,b)±) _ _ i _ d 2 4 , b ) ± f 4 6 3 1 Chapter 4. Vibration Damping with Constrained Layer Damping 104 where (—, +) indicates the minus sign is used if layer i is in the top region, and plus sign if it in the bottom region. Equations (4.62) and (4.63) provide the complete linear system required to solve for D - the position of the neutral plane' N and al l the other partial derivatives dipf '^/dtp. From the bending moment expression in (4.59), the complex effective flexural rigidity wi l l assume the form B = ^ + 1 ^ % - > (4-64) i>i ocP which can define the appropriate system structural matrices A0, A\ and bo as in the expression (4.45), and the loss factor of the composite beam is n = 9 m ( 5 ) / S e ( f l ) . (4.65) For the configuration shown in Figure 4.39, the solution of the complex flexural rigid-ity B in (4.64) requires one to solve the linear system (4.62) and (4.63) of five equations and five unknowns: D, ip{+, ip^'> i'l'j a n d ip\~ • If more damping layers are added in the t+ region, the number of unknowns that need to be solved is st i l l five. This is because the additional viscoelastic layers are al l sandwiched in between two constraint layers and (4.63) would suggest that S T = ~ 8 f ~ • • ( 4 - 6 6 ) This suggests that for a beam with arbitrary number of damping layer, one only needs to solve nine unknowns; i.e. these are D, dipi'^/dx and di^^/dx. O f course, this is only true wi th Assumption 6, when each damping layer consists of identical viscoelastic and constraint layers. Chapter 4. Vibration Damping with Constrained Layer Damping 105 Symmetrical Configuration In addition, i f the damping layers applied in the top region are symmetrical to the ones in the bottom region, one can use (4.62) and (4.66) to show that the location of the neutral plane N indeed coincides with the neutral plane of the structural layer; i.e. D = 0. This w i l l not only reduce the number of unknowns to solve by one, but also simplify the symbolic formulation significantly as demonstrated earlier in Section 4.2.5. Later in Section 4.3, it is shown that a good design of the beam tends to be this k ind of symmetrical configuration. 4.2.7 Issues of Implementation The above discussion implies the following for a symmetrically arranged beam: 1. by programming in the expressions for the general formulation presented, one can solve for the effective complex flexural rigidity of a beam with arbitrary number of layers. 2. Let it be the vector of unknowns one needs to find to solve for the effective complex flexural rigidity B. From the Principle of Equi l ibr ium in (4.62) and the assumption of sinusoidal shearing motion in (4.63), a linear system can be expressed to solve for u A0u = b0 , where both A0 and bo are dependent on the physical configuration of the beam and the wave number p of the vibratory wave, which can be converted into a corresponding vibrational frequency to as shown in (4.27) using the Euler-Bernoull i beam theory. Chapter 4. Vibration Damping with Constrained Layer Damping 106 3. one can express B as a solution to the linear system B = Aiu — AiA^bo , where due to the symmetry, the maximum number of unknowns drops to 8 (because D — 0). So, solving B involves three matrices of Ax E K l x 8 , A0 G !f t 8 x 8 and b0 € 5 R 8 x l . 4. wi th the known numerical values, one can compute the numerical representations of the system matrices Ai, A0 and b0 above directly. 5. following from the expression in (4.65), the effective damping of the multi-layer beam is then, 3 m ( £ ) _ S m ^ i A ^ & o ) V ~ fte(B) ~ ^e{AxAolbo) ' This expression can be used as the basis of an optimization problem where the ge-ometry of the damping layer and the number of the damping layer can be optimized to yield the maximum effective damping factor 77. Numerical Example Equations The beam shown in Figure 4.39 is used as an example in implementing the formulation given in the previous section. For simplicity, it is assumed that all the damping layers consist of the same constrained and viscoelastic layers, i.e. K^c'bJ^f = K(c>vy The first task is to specify the longitudinal force in each layer. From (4.60) Chapter 4. Vibration Damping with Constrained Layer Damping 107 and (4.61), they are: d F£ = Kv(Hl-vl^ + H v ^ ) ; ft = K C ( H ^ + 2 H V ^ ) Ffr = Kv{H%i% + Hvyjt) ;-^r = ^(^11 + 2 ^ ^ 1 ) : Fit = K V { H ^ - 2 H V ^ - Hvd-§-) ^ .= K c ( H ^ - 2 H v d - ^ - 2 H v ^ ) Apply ing (4.62) and (4.63), five equations are produced to solve for five unknowns: Fs + Flt + F^ + F^ + F^ + F^ + F^+F^ + F^ = 0 , . _ i < _ ' (F£-F.) . _xdtf+ ._ ( i f f -Fjt) p2^x ~ 2bGv ' P2~8X ~ 2bU~v i d$- _ (Fjr 7FS) . _ (FJAZ-FS) p2~dx ~ + 2bGv ' P2~8X ~ 2bGv Numerical Simulation Use the same structural and damping layer configurations as in the numerical examples of Section 4.2.4 and Section 4.2.5, where Hc = 1/16 in and Hv — 1/10 in. The five unknowns are found to be D = 2.1 +j 1.1 mm = 3.9 — j 1.24 rad/m ; = 1.0 -j 0.32 rad/m = - 1 . 9 + j 0.60 r a d / m ; = 3 . 9 - j 1.24 r a d / m . Substitute the numerical values into (4.64), the flexural rigidity is B = 1946 + j471 Nm2 and the loss factor is then 77 = 0.24. Chapter 4. Vibration Damping with Constrained Layer Damping 108 4.3 Beam Design Using the procedure developed in the previous section, the damping of a hollow beam sandwiched by constrained damping layers can be analyzed. The results are presented in this section and used to suggest appropriate design approaches; they focus mainly on suppressing the fundamental vibration mode of the structure. A l l the analysis and simulations are based on the structural layer of a 1 — m long and 1/16 in thick 1 in x 1 in hollow aluminum square tube. As discussed in [23], effective passive damping cannot be achieved without careful and specific design processes. A s demonstrated in this section, the performance can be poor or drop rapidly when the actual application of passive damping deviates from the original design. 4.3.1 Overview A damping layer consists of both the viscoelastic and constraint layers. Since the function of the constraint layer is to restrict the viscoelastic layer in order to create the required, deformation for effective dissipation of the vibrational energy in the system, it is necessary to address how different configurations between the viscoelastic layer thickness and that of the constraint layer affect the damping capacity of the damping layer. In real robotic applications, the structural vibration of the robot changes as the arm extends or shortens to perform a task. Therefore, it is essential to see how the peak damping performance shifts in the frequency domain wi th different design parameters. A t last, wi th multiple damping layers, there can be a number of ways of applying them to the structure. It is also interesting to investigate whether there is a best way to construct the composite beam when only a specific number of damping layers are available. Chapter 4. Vibration Damping with Constrained Layer Damping 109 L o s s F a c t o r a t H c = 1 / 2 i n L o s s F a c t o r a t H v - O 0 5 i n O.I S 1 s t m o d e 1 s t m o d e O : 3 r d m o d e CT3 "33 o O.OS 0 . 1 O O 0 . 0 5 ( a . ) V i s c o e l a s t i c " T h i c k n e s s ( i n ) O O.S ( b . ) C o n s t r a i n t T h i c k n e s s ( i n ) Figure 4.40: Loss factor is not a simple variable proportional to the amount of viscoelastic material used. Depending on the targeting vibrational modes, the loss factor peaks at a specific damping layer configuration. 4.3.2 Damping Layer Composition Viscoelastic Layer It is intuitive to expect the damping of the sandwiched beam to increase when more viscoelastic material is used in its construction. In aerospace or acoustic engineering where constrained layer damping has been exercised, it was found that more viscoelastic material does not imply more damping [11, 2, 50]. In robotic applications, the structural frequencies that need to be suppressed are lower than the ones in aerospace or acoustic fields. Thus, it was found that when designing constrained layer damping for low frequen-cies, it is safe to use much viscoelastic material without degradation in the final damping capacity of the composite beam. Figure 4.40(a.) shows how the loss factor of the beam is affected by adjusting the thickness of the viscoelastic layer for the 1st and 3rd modes of vibration. The 1st mode has endpoint vibration at about 20 Hz and the 3rd mode at 175 Hz. For the 1st mode, the beam loss factor increases and quickly converges to an upper l imi t as the thickness of the viscoelastic layer increases. For the 3rd mode, the loss factor peaks at a specific viscoelastic layer thickness and starts to decrease beyond that. Thus, one should first check whether more viscoelastic material means more damping for the vibrational frequency interested. Chapter 4. Vibration Damping with Constrained Layer Damping 110 Constraint Layer Figure 4.40(b.) shows that for a fixed thickness in the viscoelastic layer, there is an optimal constraint layer thickness where maximum damping occurs. Too much flexibility in the constraint layer does not produce enough shear in the viscoelastic layer. O n the other hand, the composite becomes stiffer as the thickness of the constraint layer grows, causing less deflection in the beam which results in less shear deformation in the viscoelastic material. Only a specific thickness balances the trade-off between the rigidity of the beam and the shear deformation of the polymer material. This is observed in both cases for the 1st and 3rd modes of vibration. 4.3.3 Target Vibrational Frequency As already shown in Figure 4.40, the loss factor is a frequency dependent function: it behaves differently for the 1st and 3rd vibrational modes. For example, wi th the same damping layer configurations Hc = 1/2 in and Hv = 0.05 m , the loss factor is r\ = 0.12 and n = 0.04 for the 1st mode and the 3rd mode vibration, respectively. The solid line in Figure 4.41 (a.) plots the frequency response of the loss factor of the composite beam with constraint layer thickness Hc = 1/16 in and viscoelastic layer thick-ness Hv = 0.01 in. W i t h this configuration, the peak damping occurs at around 100 Hz and it drops by about 60%/decade as the frequency increases and 80%/decade as it de-creases. This observation emphasizes the importance of careful design of passive damping for a particular structural resonant mode. The dashed line in Figure 4.41 (a.) shows the increase and shift of the damping peak toward the lower frequency region as the amount of the viscoelastic material is increased to Hv = 0.03 in. The decaying rate of the loss factor with respect to the frequency is unchanged. Therefore, it is useful to estimate the range that the structural vibration mode may change during the intended operation, and Chapter 4. Vibration Damping with Constrained Layer Damping 111 L o s s F a c t o r a t H c = 1 / 1 6 i r « 0 . 0 2 H v = 3 0 m i l / N / / . . . . /. \ \ H v = 1 O m i l / / \ \ / / \ \ / / \ \ / / \ \ / / \ \ . O . O S c b -H c = 1 / 1 6 " a n d H v = 1 0 m i l : 3 r d m o d e s t m o d e 10 10 10 ( a . ) V i b r a t i o n a l F r e q u e n c y ( H z ) 2 3 4 ( b . ) # o f d a m p i n g l a y e r Figure 4.41: Frequency response of the loss factor and multi-layer application, then design the beam such that sufficient damping is provided within that range. 4.3.4 Multiple Damping Layers The solid line in Figure 4.41 (b.) shows the loss factor of the beam for the 1st mode vibration (20 Hz) when multiple damping layers are assembled together and applied to one side of the structure layer. It is evident that the damping increases as the number of layers goes up. The damping layer configuration is Hc = 1/16 in and Hv = 0.01 in. From F i g . 4.41 (a.), this configuration yields more damping at the 3rd mode (175 Hz). The dashed line in Figure 4.41(b.) shows the loss factor for the 3rd mode as number of layers grow. The composite beam starts at a base loss factor wi th just one damping layer; wi th each additional layer, a constant increase is contributed toward the overall beam loss factor. However, it is evident that the increase is not exactly 100% although the same damping layer is used. This is because the additional viscoelastic layers are sandwiched between two constraint layers and the shear deformation due to structural vibration is lower, in the layers that are not immediately next to the structural layer. 4.3.5 Symmetric Layer Arrangement One can apply a fixed number of damping layers to a structure in a number of ways. It was found that much greater damping can be achieved per each additional layer when al l Chapter 4. Vibration Damping with Constrained Layer Damping 112 # 1st mode 3rd mode one-side symmetric one-side symmetric V . A?7i V Am Ar/2 V Am A772 A771 A772 1 0.020 0.02 - - - - 0.050 0.05 - - - -2 0.026 - 0.007 0.05 0.025 0.070 - 0.02 0.15 0.075 -3 0.031. - 0.006 - - - 0.088 - 0.02 - - -4 0.036 - 0.006 0.07 - 0.01 0.104 - 0.02 0.25 - 0.05 5 0.041 - 0.005 -_ - - 0.122 - 0.02 - - -Table 4.4: Advantages of symmetrical layer configuration is evident. the layers are symmetrically distributed on both sides of the structural layer. Table 4.4 lists the numerical values obtained in Figure 4.41 (b.), the loss factors achieved when two and four layers are used and are distributed equally on both sides of the beam and the average increase in loss factor per damping layer in each case. Two types of increases can be distinguished: A771 - when the additional layer is right next to the structural layer and Ar)2 - when it is not. Arji measures the average increase wi th respect to the raw damping capacity of the aluminum structural layer r)s = 2.7 x 1 0 - 5 and A772 wi th respect to the damping of the composite beam with only damping layer(s) immediate to; the structural layer. The results for both the 1st and 3rd modes are tabulated. In both cases, Arji and A772 in the symmetric configurations are clearly larger than the ones in the one-side configurations. 4.3.6 Beam Design Guidelines Based on the observation discussed above, the following procedures are proposed to design a multi-layer beam as a link in a robot structure: 1. A particular vibration frequency UJ is determined from the structural simulation of a robot as the most problematic vibration mode. A l l beams are to be designed to have peak damping capacity at this frequency. Chapter 4. Vibration Damping with Constrained Layer Damping 113 2. For a particular link in the robot structure, the equivalent vibratory wavelength can be approximated for the structural layer using the Euler-Bernoull i beam theory and equation (4.27). This is the targeted wave number for the damping layer design for this particular link. Other links with different physical dimensions wi l l have different wavelengths. 3. Try a double layer design first. Using the computed wave number and the formula-tion presented in Section 4.2.5 for symmetric damping layer arrangement, determine the best combination between the viscoelastic layer thickness and the constrained layer thickness. A range of values can be simulated according to the physical pa-rameters, choose the pair yielding the highest damping factor. 4. Check the obtained thicknesses on the viscoelastic and constraint layers. If the overall dimensions or mass of the link violate the design constraints decided for a particular robot structural, repeat the simulation with four damping layers, i.e. two layers on each side of the beam. Repeat this step unti l the best damping layer design satisfying the design constraints of this link is arrived. 5. The damping coefficient £ of this link with the effective damping factor r\ can be approximated wi th equation (4.24). The equation models a link as a second order system. If the robot vibration frequency u coincides with the link natural frequency con, the relationship 2£ = 77 holds. However, it is likely the robot structure vibrates at a lower frequency than the link's natural frequency. The effective damping coefficient can be higher than n/2. 6. In implementation, al l layers at one end of the link are clamped while the other ends are left free; only the structural layer is coupled to the next link. Thus, higher intrinsic damping than the one estimated wi th the formulation is likely to Chapter 4. Vibration Damping with Constrained Layer Damping 114 be achieved in the actual robotic structure. 7. W i t h a chosen damping layer design, re-run the simulation wi th different wave-lengths to determine the damping frequency response of this particular design for this link. (Refer to Figure 4.41 as an example.) 8. Repeat the design process starting from Step 2 above for the other links which have different physical dimensions resulting in different vibratory wavelengths and damping layer designs. Chapter 5 Experimental Work 5.1 Introduction Experimental work in support of the analysis presented in Chapter 4 is described next. The task is to measure the structural damping of a hollow aluminum beam with/without constrained layer damping. The experiment attempts to achieve the following: 1. To verify the generic formulation developed in Chapter 4. 2. To suggest the above mentioned formulation as a suitable design tool in constructing hollow beam structures with constrained layer damping. 3. To prove that constrained layer damping is effective in achieving passive damping. 5.2 Damping Measurements 5.2.1 Experimental Procedure The experimental setup is shown in Figure 5.42. It consists of a 1 m long aluminum square tube with width of 1 in and wall thickness of 1/16 in and an optical sensor. The beam length simulates the full reach range of many industrial robots such as CRS-A460 or PUMA560. One end of the beam is clamped firmly to a massive bronze table sitting on the floor while the other end is freely hanging in the air. The purpose of the bronze table is to 115 Chapter 5. Experimental Work 116 photo sensing diode (PSD) Laser Diode Constraint Layer \ Viscoelastic Layer \ Structural Layer Composite Beam C-clamp Bronze Basi Figure 5.42: Experimental setup of the beam experiment with internal optical endpoint motion sensing. eliminate any base vibration or damping that can be introduced to the beam if it was mounted to a light weight base or to a wooden table. Endpoint Vibration Sensing System The endpoint vibration of the beam is detected with an internal optical system (See Figure 5.42). A laser diode is mounted to the free end of the beam and a photo sensing diode (PSD) to the fixed end. The laser beam from the laser diode is first calibrated to focus 1 meter away at the center of the P S D when there is no load on the beam. Any endpoint movement relative to the base is then registered as the movement of the laser beam spot on the P S D . Due to the sensitivity and resolution of the optical system, the setup is capable of measuring very fine movements at the endpoint. Furthermore, since the damping of the beam measures merely the rate of decay of vibrational magnitude, the endpoint displacement measured on P S D need not be mapped to any standard physical measuring units for the purpose of finding the corresponding damping coefficient. Chapter 5. Experimental Work 117 Data Acquisition System A Sun Sparc V M E board running the real-time operating system V x W o r k s was the center of the data acquisition system. The optical sensing system was connected to a X V M E 5 4 5 board, a 16-bit analog-to-digital converter. The control software written in C language running under VxWorks sampled the input from X V M E 5 4 5 at a frequency of F = 500 Hz. The experimental results were recorded wi th the software Stethoscope. Mat lab was used to compile and analyze the recorded experimental data. Procedure Free vibration was first introduced to the structure either by striking the beam lightly once (impulse response) or releasing it quickly with a certain amount of in i t ia l deflection (step response). The endpoint vibration was then measured by the internal optical system and recorded wi th the data acquisition system. The sampling frequency used (at F = 500 Hz) was much higher than the expected fundamental mode of the beam which is at about f0 = 20 Hz. A 1 sec long segment was then extracted from the recorded vibration history for each free vibration induced and normalized such that the maximum oscillation amplitude in the 1 sec history is 1. 5.2.2 Results and Discussion Data Interpretation The fundamental mode of vibration of the beam in Figure 5.42 can be treated as a second order system where the spring constant is the stiffness of the beam and the viscous damping is the intrinsic material damping in the structure. For an under-damped second Chapter 5. Experimental Work 118 order system, the normalized endpoint vibration follows the equation [51] 1 7^= ^dt _ (-2 y(t) = -y—^e s i n ^ - t a n - 1 V ^ ) , where y(t) is the endpoint displacement, ( < 1 is the damping coefficient and cud is the damped natural circular frequency. The envelope of the decaying sinusoidal function is then This can be linearized to a first order system as shown below l n £ ( C , t ) = - I n ^ -^udt + ln^L^ . (5.67) This shows the linear relationship between the natural logarithm of the decaying enve-lope In £•(£,£) and the time t. Therefore, the equivalent damping coefficient of the beam can be obtained by first extracting the envelope E{kT) from the recorded endpoint displacement data y(kT), and then fit it to the linear equation (5.67). The fitting is performed wi th the standard least square fit algorithm wi th the error function ferr = Y,[^F(kT)-\nE(C,kT)}2 , (5.68) k where k = 0 , 1 , . . . is an index and T = 1/F — 2 msec is the sampling period. The damp-ing coefficient ( that minimizes the error function ferr is the interpreted damping of the beam. A detail example of this analysis and interpretation process for the bare aluminum beam is shown in Figure 5.43. Verification Approach The raw structural damping capacity of the aluminum beam £ 0 was determined first to establish a base line for comparing to the later measurements wi th constrained layer Chapter 5. Experimental Work 119 dampings. Instead of using the documented value, this measured value of raw aluminum material damping was used in the formulation presented in Chapter 4 to predict the structural damping of the beam with different damping layer configurations constructed during the experiment. Several damping layer geometries were tested and the results were compared wi th the predicted values. Recall from Chapter 4 that the formulation basically treats the composite beam as one homogeneous beam and assumes identical flexural motion in each layer. As the thickness of the constraint layer increases, this assumption falls apart because the cohesion between the structural and constraint layers from the adhesive viscoelastic material is not strong enough to sustain the assumed synchronous flexural motion. Therefore, a formulation verification was first performed on a number of different configurations wi th thin constraint layers. Damping layers wi th thick constraint layers were also constructed as an attempt to assemble a composite beam with the most damping. The results are also presented and compared wi th the predicted values based on the assumption discussed above. Experimental Results Raw Damping Capacity The damping of a bare aluminum beam was measured first. The detailed experimental analysis procedure is presented in Figure 5.43. The recorded endpoint vibration is first normalized, then the spectral analysis is performed to find the damped natural frequency to be ood = 2n x 17 Hz. The envelope of the oscillatory endpoint motion was extracted and together with the known cod, it was fitted to (5.67) wi th the least square fit algorithm (5.68). The damping coefficient was found to be Co = 0.005. Since the loss factor is related to the damping coefficient as rj = 2£ in (4.24) when considering the resonant modes, the value of nai = 0.01 was used in the damping beam Chapter 5. Experimental Work 120 O 0.5 1 O . 0.5 1 (c.) Time (sec) (d.) Time (sec) Figure 5.43: Free vibration of the bare aluminum beam: The normalized endpoint dis-placement of the beam with respect to time is shown in (a.). The data was recorded at a sampling frequency of F = 500 Hz. The power spectral density analysis in (b.) shows that the fundamental structural resonant mode is at around cud = 17 Hz. The envelope of the decaying endpoint oscillation is extracted in (c ) . It is then fitted in (d.) according to the reference function in (5.67) with the minimizing error function in (5.68). The damping coefficient of the bare aluminum beam with the fundamental resonant mode was found to be £ 0 = 0.005. This value was used later in the formulation to predict the damping coefficient of the beam with constrained layer damping. formulation to predict the structural damping of beams wi th different damping layers later in this chapter. Damping Layer Designs W i t h the knowledge of the structural layer and the cor-responding raw damping capacity, different constraint and viscoelastic layer thicknesses can be used to compute the optimal combination between Hv and Hc which results in the most structural damping £ for the fundamental vibrational mode. Figure 5.44 display the plots that indicate where the maximum damping occurs. If only one damping layer is applied to the structural layer, maximum damping of C = 0.065 can be achieved with Chapter 5. Experimental Work 121 One—Side Configuration Both—Side Configurati ion O 0.5 (a.) Constraint Layer : Hc (in) O 0.5 (b.) Constraint Layer : Hc (in) 1 Figure 5.44: Design guideline plots for one-side and both-side configurations: (a.) If only one damping layer is applied, the structural damping increases as the viscoelastic layer thickens. It peaks to a value of £ = 0.065 at Hv = 60mils and Hc = 0 .5m before starting to decrease, (b.) If both sides of the structural layer are applied wi th one damping layer, the optimal configuration is at Hv — 20 mils and Hc — 0.75 in, yielding a damping of C = 0.25. the geometry of the damping layer being Hv = 60 mils and Hc — 1/2 in. The maximum damping achievable when both sides of the structural layer are applied wi th a damping layer is much higher. From F i g . 5.44(b.), a damping layer geometry at Hv = 20 mils and Hc = 3/4 in can result in a total beam damping of £ = 0.25. Beams wi th both of these optimal configurations were constructed and tested. Be-cause these designs call for thick constraint layers, whose flexural motions may not syn-chronize wi th the flexural motion of the structural layer during vibration, they may not serve well as the candidates for any meaningful verification on the formulation devel-oped in Chapter 4, but merely for experimental interest. Therefore, for the purpose of more credible verification, some composite beams with thinner constraint layers were also constructed and tested. The experimental results are presented below. Chapter 5. Experimental Work 122 Structural Free Vibra t ion Different Configurations geometry Hv (mils) Hc(in) Damping Coeff. £ Predict Expt 2 x 10 -5 0.005 10 1/16 0.013 0.017 10 1/16 0.025 0.035 100 1/16 0.036 0.055 100 1/16 0.095 0.100 60 1/2 0.065 0.050 20 3/4 0.045 0.055 20 3/4 0.250 0.100 Table 5.5: Experimental results with different beam constructions. W i t h the thin con-straint layers at Hc = 1/16 in, the experimental values are either close or better than the ones predicted. When the thick constraint layers are applied only to one side of the hol-low beam, the experimental values are within the neighborhood of the predicted values. The prediction gets worse when thick constraint layers are applied to both sides of the structural layer. (The viscoelastic material was kindly supplied by 3M-Scotchdamp V i -bration Control Systems. The model is ISD-112. One can contact M r . Tom Rubbelleke at 3 M , 612-733-3760 for ordering information.) Chapter 5. Experimental Work 123 Discussion Table 5.5 summarizes the experimental results. The structural vibrat ion decays away faster when more damping is achieved. The experimental results obtained for beams with thin constraint layers Hc = 1/16 in are well in agreedment wi th the predicted values. In some cases, more damping was measured. The results for the beams with optimal designs discussed above are also shown in Table 5.5. A s suspected, the predicted values start to diverge wi th the experimental values as the thickness of the constraint layer increases. For one-side applications, the results were close. More drastic differences are observed when both sides of the structural layer were applied with these thick constraint layers. The experimental results also demonstrate that increasing the amount of viscoelastic material used in constructing the composite beam is not necessarily the only route to improve structural damping. More damping can be achieved wi th stiffer constraint layers resulting in more deformation in the viscoelastic layer. For example, for a constraint layer with thickness Hc = 1 /16in, as much as Hv = lOOmils of viscoelastic material was required to achieve a structural damping of C = 0.055. However, the same amount of damping was achieved wi th much less viscoelastic material at Hv = 20 mils but wi th a stiffer constraint layer at Hc = 3/4 in. In fact, by replacing the constraint layer material from aluminum to steel, the formulation predicted that the thickness of the constraint layer can be reduced to half at Hc = 3/8 in without any significant loss in the achieved damping capacity. Notice that the ratio of reduction in the thickness of the constraint layer was not necessarily proportional to the ratio between the material stiffnesses of aluminum and steel. The increase in viscoelastic material deformation per unit endpoint deflection is partially traded off wi th the increase in the stiffness of the composite beam, which results in less deflection and/or less overall deformation in the viscoelastic layer. Furthermore, the mass of the composite beam in Chapter 5. Experimental Work 124 the example above increased to 2.25 kg from 1.90%. This may also be undesirable. W i t h a reliable formulation as an effective design tool, the optimal damping layer design with various design constrains can be determined prior to the actual assembly. As supported by the analysis and many researchers in aerospace or acoustic engi-neering, bl indly applying viscoelastic material to structures and constraining it wi th any stiffening layer is not likely to yield the desired damping successfully. However, from both the analysis and experimental work presented in this thesis, it seems that i f damp-ing of low vibrational frequencies is of interest, it is not as crucial to match the physical damping layer construction exactly to the one designed. For example, as shown in F ig -ure 4.40(a.), one can afford to increase the viscoelastic layer thickness without worrying a significant drop in the damping performance for the first mode vibration, while the room for error for the thi rd mode is much less. Therefore, considering that most robotic structural resonant modes tend to be in the low frequency region, one can almost apply a certain amount of viscoelastic material constrained by a stiff layer to a robot structure and expect to observe some damping in the structural flexible mode. Chapter 6 Conclusions A complete stability and performance analysis were performed on the bilateral motion scaling system proposed in [1, 3, 4] using eigenloci and generalized Nyquist stability criteria. In particular, the analysis considered the fundamental flexible modes in the system and studied the effects on the stability margin of the proposed position/force control system due to these flexible modes. The analysis provided evidence that damping in the structural flexural motions is essential in keeping the motion scaling system robust. In achieving the required damping in the endpoint motion of a transporting robot, two approaches are considered: active and passive dampings. It was shown that active vibration control with endpoint position feedback not only poses many implementation challenges, but also threatens the robustness of the control system. Vibra t ion control with passive damping, on the other hand, does not share these difficulties and offers an intrinsically well damped structure. This suggests the use of well damped materials in constructing a robotic structure rather than the conventional materials. No homogeneous engineering material possesses sufficient intrinsic damping required to ensure a comfortable stability margin in the moving bilateral motion scaling system. A different method in constructing a robot structure must be found. Cross-field research into the aerospace and acoustic engineering has revealed constrained layer damping as being a suitable solution to achieve sufficient passive damping. The dynamic analysis on the viscoelastic behaviour of a hollow rectangular robot arm applied wi th a number of damping layers were performed. The generalized governing equations in estimating 125 Chapter 6. Conclusions 126 the effective damping capacity of such a composite beam was formulated. The work was a multi-layer extension of the original R K U formulation [2] developed for a solid beam structure with just one damping layer. This set of formulation was proposed to be used as a practical design tool in constructing a structure with certain vibration modes. A design procedure based on iteratively (trial-and-error) evaluating the above mentioned formulation against a set of known design constrains was developed. A n optimization approach was also suggested. Beams wi th different damping layer configurations were assembled, and experimental results were recorded and compared with the predicted ones to verify the correctness and effectiveness of the formulation developed. The measured beam damping coefficients were mostly close and slightly higher than the predicted ones. The actual and predicted coefficients started to diverge as the thickness of the constraint layer in a damping layer started to increase. The best result obtained from the experiments showed a damping coefficient of ( = 0.1. This is 2 orders of magnitude better than the raw aluminum material, and is better than any homogeneous engineering material suitable for structural construction. Such a beam structure can be mounted to a motor to test the damping capability of the endpoint vibration when it is actuated by a motor. Prel iminary work has been performed on the operation of D C brushless motor, and the results are presented in Appendix A . 6.1 Contributions The main contributions are summarized below: • The stability and transparency of a bilateral motion scaling system were analyzed v ia M I M O eigenloci and transmitted impedance plots. Structural flexibilities were considered in the analysis. Quantitative stability and performance measures were Chapter 6. Conclusions 127 defined. • Difficulties in achieving effective active damping via linear control methods with fixed controller gains and endpoint feedback were discussed and illustrated. A passive damping approach was proposed. • Constrained layer damping technique was suggested as an effective method to achieve passive damping. Extension of the original R K U formulation [2] to a hollow " beam structure wi th multiple damping layers were developed. Design guidelines for such a composite beam and eventually a structure consisted of a number of these beams were proposed. Experimental work was performed to verify this extended formulation for a single composite beam with different multiple viscoelastic lami-nates. Bibliography [1] J . Yan , "Design and control of a bilateral motion system using magnetic levitation," Master's thesis, University of Br i t i sh Columbia, March 1994. [2] Donald Ross, Er ic E . Ungar, and E . M . Kerwin , "Damping of plate flexural vibra-tions by means of viscoelastic laminae," in Structural Damping (J. E . Ruzicka, ed.), pp. 49-88, The American Society of Mechanical Engineers, 1959. [3] S.E. Salcudean and J . Yan , "Towards a force-reflecting motion-scaling system for microsurgery," in Proc. IEEE Conf. Robotics Automat., (San Diego, California), pp. 2296-2301, M a y 9-12 1994. [4] S.E. Salcudean and J . Yan , "Towards a Mot ion Scaling System for Microsurgery," June 8-11 1993. The T h i r d Annual I R I S - P R E C A R N Conference, Ottawa, Canada . [5] D . A . Lawrence, "Stabili ty and transparence in bilateral teleoperation," IEEE Trans-actions on Robotics and Automation, vol. 9, pp. 624-637, October 1993. [6] J . E . Colgate, "Robust Impedance Shaping Telemanipulation," IEEE Transactions on Robotics and Automation, vol. 9, pp. 374-384, August 1993. [7] Kazuhiro Kosuge, Tomotaka Itoh, Toshio Fukuda, and Manabu Otsuka, "Scaled telemanipulation system using semi-autonomous task-oriented vi r tual tool," IEEE International Conference on Intelligent Robots and Systems, vol. 2, pp. 124-129, 1995. [8] Linag-Wey Chang and J . F . Hamil ton, "Dynamics of robotic manipulators with flexible links," Transaction of the ASME, vol. 113, pp. 54-59, March 1991. [9] B . S. Yuan , W . J . Book, and J . D . Huggins, "Dynamics of flexible manipulator arms: Alternative derivation, verification, and characteristics for control," Transaction of the ASME, vol. 115, pp. 394-404, September 1993. [10] Eugene I. R i v i n , Mechanical Design of Robots. M c G r a w - H i l l Book Company, 1987. [11] A . D . Nashif, D . I. G . Jones, and J . P. Henderson, Vibration Damping. John Wiley, Inc., 1985. [12] Y . - Y . Y u , "Vibrations of elastic sandwich cylindrical shells," Transactions of the ASME : Journal of Applied Mechanics, pp. 653-662, December 1960. 128 Bibliography 129 Andre Sharon and David Hardt, "Enhancement of robot accuracy using endpoint feedback and a macro-micro manipulator system," in American Control Conference proceedings, (San Diego, California), pp. 1836-1842, June 6-8 1984. M . W . Spong and M . Vidyasagar, Robot Dynamics and Control. John Wi ley and Sons, 1989. J . M . Maciejowski, Multivariable Feedback Design. Addison-Wesley, 1989. N . C. Singer and W . P. Seering, "Preshaping command inputs to reduce system vibration," Transactions of the ASME, vol. 112, pp. 76-82, March 1990. James M . Hyde and Warren P. Seering, "Using input command pre-shaping to sup-press multiple mode vibration," Proceedings of the IEEE International Conference on Robotics and Automation, Sacramentao, California, vol. 112, pp. 2604-2609, A p r i l 1991. Donald J . Leo and Daniel J . Inman, "Pointing control and vibrat ion suppression of a slewing flexible frame," Journal of Guidance, Control, and Dynamics, vol. 17, pp. 529-535, May-June 1994. A . Zalucky and D . E . Hardt, "Active control of robot structure deflections," Journal of Dynamic Systems, Measurement, and Control, vol. 106, pp. 63-69, March 1984. K a r l Johan Ast rom and Bjorn Wittenmark, Adaptive Control. Addison Wesley, 1989. M . C . Tsai , "Model reference adaptive control for robot manipulators - continuous time theory and digital implementation," American Society of Mechanical Engineers, Dynamic Systems and Control Division, vol. 11, pp. 199-209, 1988. H . M . Schwartz, "Model reference adaptive control for robotic manipulators with-out velocity measurements," International Journal of Adaptive Control and Signal Processing, vol. 8, pp. 279-285, May 1994. M . L . Drake, "Passive damping: Has its time finally come?," Mechanical Engineer-ing,^. 20-24, December 1985. C . K . Sung and B . S. Thompson, " A methodology for synthesizing high-performance robots fabricated with optimally tailored composite laminates," ASME : Journal of Mechanisms, Transmissions, and Automation in Design, vol. 109, pp. 74-86, March 1987. Bibliography 130 [25] E . J . Richards and A . Lenzi , "On the prediction of impact noise, v i i : The structural damping of machinery," Journal of Sound and Vibration, vol. 97, no. 4, pp. 549-586, 1984. [26] V i k r a m K . K i n r a and Carlos L . Yapura, " A fundamental connection between in-trinsic material damping and structural damping," in M3D : Mechanics and Mech-anisms of Material Damping, ASTM STP 1169 (V. K . K i n r a and A . Wolfenden, eds.), pp. 396-420, American Society for Testing and Materials, Philadephia, 1992. [27] R. D . Adams, "The damping characteristics of certain steels, cast irons, and other metals," Journal of Sound and Vibration, vol. 23, no. 2, pp. 199-216, 1972. [28] R. D . Adams and M . A . O. Fox, "Measurement of the damping capacity and dynamic modulus of high-damping metals under direct cyclic stresses," Journal of Physics, vol. 5, pp. 1274-1283, 1972. [29] R. D . Adams, "Damping properties analysis of composites," in Composite Materials Analysis and Design, pp. 206-217. [30] R . F . Gibson, "Damping characteristics of composite materials and structures," Journal of Materials Engineering and Performance, vol. 1, pp. 11-20, February 1992. [31] Ferdinand P. Beer and E . Russell Johnston, Jr., Mechanics of Materials. McGraw-H i l l Book Company, 1981. [32] I. Y . Shen, "Hybr id damping through intelligent constrained layer treatments," Jour-nal of Vibration and Acoustics, vol. 116, pp. 341-349, July 1994. [33] 3 M , "Scotchdamp vibration control systems - product information and performance data," tech. rep., 3 M Industrial Tape and Specialties Divis ion, 1993. [34] R. A . DiTaranto, "Theory of vibratory bending for elastic and viscoelastic layered finite-length beams," Journal of Applied Mechanics, pp. 881-886, December 1965. [35] Y . P. L u , B . ' E . Douglas, and E . V . Thomas, "Mechanical impedance of damped three-layered sandwich rings," AIAA Journal, vol. 11, pp. 300-304, March 1973. [36] H . H . Pan, "Axisymmetr ical vibrations of a circular sandwich shell wi th a viscoelastic core layer," Journal of Sound and Vibration, vol. 9, no. 2, pp. 338-348, 1969. [37] J . D . J . Wi lk ins , "Free vibrations of orthotropic sandwich conical shells with various boundary conditions," Journal of Sound and Vibration, vol. 13, no. 2, pp. 211-228, 1970. Bibliography 131 [38] Conor D . Johnson and David A . Kienholz, "Finite element prediction of damping in structures wi th constrained viscoelastic layers," AIAA Jounral, vol . 20, pp. 1284-1290, September 1982. [39] S. J immy Hwang and Ronald F . Gibson, "The use of strain engergy-based finite element techniques in the analysis of various aspects of damping of composite mate-rials and structures," Jounral of Composite Materials, vol. 26, no. 17, pp. 2585-2605, 1992. [40] T . C . Ramesh and N . Ganesan, "Finite element analysis of cylindrical shells with a constrained viscoelastic layer," Journal of Sound and Vibration, vol. 172, no. 3, pp. 359-370, 1994. [41] T . C . Ramesh and N . Ganesan, "The harmonic response of cylindrical shells with constrained damping treatment," Journal of Sound Vibration, vol. 180, no. 5, pp. 745-756, 1995. [42] G . Strang, Linear Algebra and Its Applications, 3rd Ed. Harcourt Brace Jovanovich, 1988. [43] M . Vidyasagar, Control Systems Synthesis: A Factorization Approach. Cambridge, M A : M I T Press, 1985. [44] Charles A . Desoer and Yung-Terng Wang, "On the generalized nyquist stability criterion," IEEE Trans. Automat. Cont., vol. A C - 2 5 , pp. 187-196, A p r i l 1980. [45] N . C . Singer and W . P. Seering, "Preshaping command inputs to reduce system v i -bration," Journal of Dynamic Systems, Measurement and Control, vol. 112, pp. 76-82, March 1990. [46] Nei l C . Singer and Warren P. Seering, "Design and comparison of command shap-ing methods for controlling residual vibration," IEEE International Conference on Robotics and Automation, pp. 888-893, May 1989. [47] P. Raju Mantena, Ronald F . Gibson, and Shwilong J . Hwang, "Opt imal constrained viscoelastic tape lengths for maximizing damping in laminated composites," AIAA, pp. 1000-1008, September 1990. [48] H . T . Banks, Y . Wang, and D . J . Inman, "Bending and shear damping in beams: Frequency domain estimation techniques," Transactions of the ASME, vol. 116, pp. 188-197, A p r i l 1994. [49] David Hall iday, Robert Resnick, and Jearl Walker, Fundamentals of Physics, J^th ed. John Wiley, Inc., 1993. Bibliography 132 [50] E . E . Ungar, "Damping of panels," in Noise and Vibration Control (L. L . Beranek, ed.), pp. 434-475, M c G r a w - H i l l Book Company, 1988.. [51] A . F . D'Souza, Design of Control Systems. Prentice-Hall, 1988. Appendix A Motor Control A . l Arm Actuation There are a number of factors to consider when selecting an appropriate actuator to drive the transporting structure. These factors are discussed in this section and a motor was selected according to the requirements. Feedback control of the motor wi thin small angles was experimented and the method, procedures and results are presented. A. 1.1 Motor Selection In order to effectively follow the motion of the master held by the surgeon, the endpoint motion of the transporting structure should have the frequency response of the normal human hand motion, approximately 7 Hz to 10 Hz. Thus, the actuator should have the positioning frequency response within that frequency range. A direct drive robot with electric motor actuators would be a suitable approach. Conventionally, most motors powerful enough to drive a robotic structure have been brush type motors which generate electric sparks when rotating. This is highly dangerous in an operating room where pure oxygen is present. W i t h recent advancements in motor designs and constructions, brushless type of D C motors are capable of producing large amount of torque necessary for robotic applications. 133 Appendix A. Motor Control 134 Specifications In order to find the maximum torque requirement for the motor, a numerical model of the moving bilateral motion scaling system is constructed to aid the design process. Numerical Model Weight of the bilateral motion scaling system mi = 4 kg Weight of the transporting robot : (consider a light weight robot) m2 = 14 kg Fu l l reach of the robot : L = 1.2 m Diameter of the robot arm : d = 6 cm M a x i m u m angular acceleration : a = 1 rad/sec2 The mass moment of inertia of a cylindrical arm is [31] J = ^ - m ( 3 d 2 + L2) = 307 oz -in- s2 , where m = m i + m 2 is the total mass. Therefore, the torque required for the D C brushless motor is [31] r = Ja = 307 oz • in . A D C brushless motor by Pacific Scientific Model No. BL7008-24-0-S-007 was selected and purchased. The specifications are shown in the table below. Appendix A. Motor Control 135 Motor Specification M a x i m u m continuous torque at stall r c = 540 oz • in M a x i m u m continuous current at stall Ic = 4.8 amp. M a x i m u m peak current Ip = 38.2 A Torque Constant (or Sensitivity) KT = 114.1 oz • in/amp. Voltage Constant (or Back E M F ) Ke = 84.4 V/KRPM Input Impedance to = 5 ohm Consider the mass moment of inertia of the robotic arm approximated above, the mechanical time constant of the motor can be estimated as: TM = x 10" 3 = 17 msec . . Thus, the selected motor meets the specified torque requirement, and wi th the mechanical time constants calculated above, it also should be able to achieve the required positioning frequency response under the full loading condition. A. 1.2 Motor Actuation For D C brushless motors, phase switchings in the currents applied to the motor stator coils (or commutation) are necessary to complete one revolution. A P W M (pulse width modulation) brushless servo amplifier with resolver feedback, Copley Controls Model No. 526R, was experimented to control the motor rotation. It was found that P W M type of actuation mechanism is not suited for high-torque/low-speed applications. The motion of rotation wi th in small angles (±10°) was not smooth, particular when the direction of rotation was switched. Since the full reach of the arm is 1-meter long, only small angles of rotation are neces-sary to give the necessary motion range required at the robot endpoint for expanding the Appendix A. Motor Control 136 Mounting Bracket V+ I •o - B +45 o \ Photo Sensing Diode (PSD) Motor Shaft V -(a.) (b.) Figure A.45: Electr ical connection of the motor and mechanical setup of the experiment with optical position sensing. (Graphs are not drawn to scale.) dynamic workspace of the the motion scaling system. Therefore, without commutation, one can potentially achieve a maximum of 90° of rotation with a 4-pole D C brushless mo-tor. The coil current can then be supplied by a linear stereo amplifier, without resorting to any expensive brushless servo amplifiers that do not perform well for low-speed/non-continuous-rotation applications. This was the approach used in this experiment. The actual setup and the results of feedback control are presented in the sections below. A. 1.3 Experiment Setup The motor is electrically connected to a Techron 7520 Stereo Amplif ier as shown in Figure A.45(a.). The magnet motor rotor can rotate in a ± 4 5 ° space. A t 0°, the current is perpendicular to the magnetic field. The linearity between the coil current and the generated torque is expected to fall rapidly beyond ± 2 0 ° . Appendix A. Motor Control 137 Position Sensing A potentiometer type of joint angle sensor was used.first to measure the rotation of the' motor. It was found un-suitable because the electrical contact noise in the sensor was in the range of 0.2°. This translates to an endpoint positioning repeatability of 3.5 mm, which was not acceptable. In addition, due to the high torque constant of the motor and the fact of diminishing in control linearity as the motor shaft rotates beyond ± 2 0 ° , the sensor noise severely l imited the choices in controller gains that can keep the closed-loop system stable. As shown in Figure A.45(b.), an optical sensing system was proposed. A laser diode was mounted to the motor shaft and a large photo sensor diode (PSD) was mounted to the stator of the motor. The laser diode is a distance away from the center of the shaft. As the shaft rotates a full revolution, the laser beam draws a complete circle on the P S D . B y knowing the position of the laser beam at a particular moment with respect to the center of the circle, the angle of rotation can be computed from simple trigonometry. The sensor calibration procedure simply involves extrapolating the center of the circle while the experimenter manually rotates the shaft by complete revolutions. O f course, in order to maximize the usable sensing surface of the P S D , it should be mounted such that the center of the P S D is as close to the center of the circle as possible. This method of sensing improves the sensing resolution from 0.2° to 0.02°. Ultimately, the endpoint of a 1-meter long arm can be positioned within the range of 0.35 mm. •Consequently, higher controller gains were permitted for a better performed closed-loop system. Appendix A. Motor Control 138 cos 0 -90 20 20° 9rf Figure A.46: Torque per ampere generated decreases rapidly as the angle of rotation leave the opt imal 0° position. Control effort must be compensated by the appropriate factor. Digital Control and Data Acquisition A Sun Sparc V M E board running the real-time operating system VxWorks was the center of the control and data acquisition systems. The optical sensing system was connected to a X V M E 5 4 5 board, a 16-bit analog-to-digital converter. The motor control command output was generated by a 12-bit on-board digital-to-analog converter on the X V M E 5 4 5 board. The Techron 7520 stereo amplifier was calibrated to drive 0.5 amp of current per voltage of input command. A control software written in C language running under VxWorks sampled the in-put from X V M E 5 4 5 at a frequency of F — 500 Hz, and updated the actuation com-mand at the same frequency. The experimental results were recorded wi th the software Stethoscope. Mat lab was used to compile and perform spectral analyze on the recorded experimental data. Appendix A. Motor Control 139 Torque Compensation As discussed above in Figure A.45(a.) and illustrated in Figure A.46, as the rotor rotates away from the 0° position, the magnetic filed is less perpendicular to the stator coil current supplied by the stereo amplifier and the torque generated per ampere of coil current input is reduced. Therefore, the control effort generated by the digital control must be compensated by a factor of 1/ cos 9, where 9 is the angle of rotation. A t 9 = ± 9 0 ° , this compensation factor grows to infinity; thus, the control output was switched off at ± 2 0 ° in order to avoid generating control efforts exceeding the capacity of the stereo amplifier or causing computational errors as the angle of rotation approaches ± 9 0 ° . Filtering Although the optical sensing system yielded a much higher sensing resolution at ±0 .02° , it had been attempted to further improve the quality of control by low-pass filtering the sensor signals at either 25 Hz or 50 Hz before computing the control effort. However, this filtering process worsened the control of the motor and resulted instability. Thus, no filtering was practiced in controlling the D C brushless motor. For the purpose of evaluating the performance of the motor and the control system, no load was applied to the motor shaft. The results of the experiment are discussed below. A. 1.4 Experiment Result The spectral analysis was performed to determine the magnitude and phase transfer functions of the closed-loop motor control system, as shown in Figure A.47. Whi te noises wi th magnitudes ± 1 ° and ± 5 ° were used as the reference input of the motor. The output of the motor was measured and recorded to determine the input/output transfer function. A P I D controller was used to control the motor. Taking into account both the gain of Appendix A. Motor Control 140 Transfer Function(-+-/— 1 deg) Transfer Function(-*-/— 5 deg) - 2 0 1 O Frequency (Hz) (+ / - 1 deg) 10 10 Frequency (Hz) ( + / - 5 deg) o> -100 -1 50 10 % -SO a> -100 10 Frequency (Hz) 10 10 Frequency (Hz) Figure A.47: Frequency responses of the closed-loop motor control system. the Techron amplifier at 2 volt/amp and the motor torque constant at 114.1 oz • in/amp, the proportional, derivative and integral controller gains used are respectively k„ 28.525 oz • in k,. 0.4 oz • in - s 1.14 oz • in degree degree ' s • degree The frequency responses with respect to both ± 1 ° and ± 5 ° input noises are similar. The phase frequency responses behaved much like a second order system decading at 180°/decade. A structural resonant mode of the motor can be observed at about 35 Hz. The P I D controller design allowed a 3 dB resonant peak at around 20 Hz, and the bandwidth of the closed-loop system shown is approximately 40 Hz. A.1.5 Conclusion The results showed the control of the D C brushless motor can be achieved by a linear stereo amplifier, and the rotation jit ter with the P W M motor amplifier was completely eliminated. The optical sensing system worked well and improved the sensing resolution ten times from using the conventional joint angle sensor. In addition, since there is no Appendix A. Motor Control 141 mechanical contacts both in the sensing system and in the rotor/stator assembly of the motor, it is an attractive system setup for a light weight and high performance robot designed to follow human hand motions. Appendix B M A P L E Source Code for Beam Design This M A P L E code wi l l compute the effective flexural rigidity and damping for a beam upto ten layers on one side, six layers on the other side and two layers on the inside of the beam. One can change the physical parameters below to reflect different damping layer designs. Initialization ' > readlib(unassign): > r e a d l i b ( i s o l a t e ) : > unassignOEs', 'Ev' , 'Gv','Ec', 'Hs' , 'Hv', 'He','Ts', 'b', 'L' , 'lambda', 'p', > 'As','Av','Ac','Ks','Kv','Kc','Is','Iv','Ic','Bs','Bv','Be','Dist','dphi', > 'dpsi','dpsil','dpsi2','dpsi3','dpsi4','dpsim','dpsilm','dpsi2m','dpsii', > 'dpsidphi','dpsildphi','dpsi2dphi','dpsi3dphi','dpsi4dphi','dpsimdphi', > 'dpsilmdphi','dpsi2mdphi','dpsiidphi','BEQ1','BEQ2','BEQ3','BEQ4', > ' BECJ5' ,' BEQ6',' BEQ7' ,' BEQ8'): Model l ing Physical Parameter Definition Es : Young's Modulus for the structural layer E v : Young's Modulus for the viscoelastic layer G v : Shear modulus for the vis-coelastic layer Ec : Young's Modulus for the constraint layer Hs : Thickness of the structural layer H v : Thickness of the viscoelastic layer Hc : Thickness of the constraint layer Ts : thickness of the structural layer b : width of the structural layer L : length of the structural layer Units : Es [N /m/m] ; Hs [in]; Hc [mils]; Ts [in]; b [in]; L [m] > Es:=72000000000*(l+l/100*I) : > Ev:=500000*(l+I): 142 Appendix B. MAPLE Source Code for Beam Design 143 > Gv:=Ev/2/(1+5/10): > Ec:=Es: > Hs:=l: > Hv:=60: > Hc:=l/16: > Ts:=l/16: > b:=l: > L:=l: Adjusting Physical Parameters > Hs:=(l/2)*Hs*(254/100)*(l/100): > Hv: = (l/2)*Hv*(l/1000)*(254/100)*(l/100) : > Hc:=(l/2)*Hc*(254/100)*(l/100): > b:=b*(254/100)*(l/100): > Ts:=Ts*(254/100)*(l/100): Calculate Model l ing Parameters > lambda:=4*L: > p:=(2*Pi/lambda): > As:=b*Ts: > Av:=b*(2*Hv): > Ac:=b*(2*Hc): > Ks:=Es*As: > Kv:=Ev*Av: > Kc:=Ec*Ac: > Is:=(l/12)*b*(2*Hs)"3-(l/12)*(b-2*Ts)*(2*Hs-2*Ts)~3: > Iv:=(l/12)*b*(2*Hv)"3: > Ic:=(l/12)*b*(2*Hc)-3: > Bs:=Es*Is: > Bv:=Ev*Iv: > Bc:=Ec*Ic: Define Thicknesses > H11:=0: > H21:=Hs+Hv: Appendix B. MAPLE Source Code for Beam Design > H21i:=(Hs-Ts-Hv): > H31:=Hs+2*Hv+Hc: > H31i:=(Hs-Ts-2*Hv-Hc): > H41:=Hs+2*(Hv+Hc)+Hv: > H51:=Hs+2*(Hv+Hc)+2*Hv+Hc: > H61:=Hs+4*(Hv+Hc)+Hv: > H71:=Hs+4*(Hv+Hc)+2*Hv+Hc: > H81:=Hs+6*(Hv+Hc)+Hv: > H91:=Hs+6*(Hv+Hc)+2*Hv+Hc: > H101:=Hs+8*(Hv+Hc)+Hv: > Hlll:=Hs+8*(Hv+Hc)+2*Hv+Hc: > H10:=H11-Dist: > H20:=H21-Dist: > H30:=H31-Dist: > H40:=H41-Dist: > H50:=H51-Dist: > H60:=H61-Dist: > H70:=H71-Dist: > H80:=H81-Dist: > H90:=H91-Dist: > H100:=H101-Dist: > H110:=H111-Dist: > H20m:=-H21-Dist: > H30m:=-H31-Dist: > H40m:=-H41-Dist: > H50m:=-H51-Dist: > H60m:=-H61-Dist: > H70m:=-H71-Dist: > H20i:=H21i-Dist: > H30i:=H31i-Dist: Define Strains and Forces > el:=H10*dphi/dx: > e2:=H20*dphi/dx-Hv*dpsi/dx: Appendix B. MAPLE Source Code for Beam Design 145 > > > > > e3:=H30*dphi/dx-2*Hv*dpsi/dx: e4:=H40*dphi/dx-2*Hv*dpsi/dx-Hv*dpsil/dx: e5:=H50*dphi/dx-2*Hv*dpsi/dx-2*Hv*dpsil/dx: e6:=H60*dphi/dx-2*Hv*(dpsi/dx+dpsil/dx)-Hv*dpsi2/dx: • e7:=H70*dphi/dx-2*Hv*(dpsi/dx+dpsil/dx)-2*Hv*dpsi2/dx: > e8:=H80*dphi/dx-2*Hv*(dpsi/dx+dpsil/dx+dpsi2/dx)-Hv*dpsi3/dx: > e9:=H90*dphi/dx-2*Hv*(dpsi/dx+dpsil/dx+dpsi2/dx)-2*Hv*dpsi3/dx: > elO:=H100*dphi/dx-2*Hv*(dpsi/dx+dpsil/dx+dpsi2/dx+dpsi3/dx)-Hv*dpsi4/dx: > ell:=H110*dphi/dx-2*Hv*(dpsi/dx+dpsil/dx+dpsi2/dx+dpsi3/dx)-2*Hv*dpsi4/dx > e2m:=H20m*dphi/dx+Hv*dpsim/dx: > e3m:=H30m*dphi/dx+2*Hv*dpsim/dx: > e4m:=H40m*dphi/dx+2*Hv*dpsim/dx+Hv*dpsilm/dx: > e5m:=H50m*dphi/dx+2*Hv*dpsim/dx+2*Hv*dpsilm/dx: > e6m:=H60ra*dphi/dx+2*Hv*(dpsim/dx+dpsilin/dx)+Hv*dpsi2m/dx: > e7m:=H70m*dphi/dx+2*Hv*(dpsim/dx+dpsilm/dx)+2*Hv*dpsi2ni/dx: > e2i:=H20i*dphi/dx+Hv*dpsii/dx: > e3i:=H30i*dphi/dx+2*Hv*dpsii/dx: > Fl:=Ks*el: > F2:=Kv*e2: > F3:=Kc*e3: > F4:=Kv*e4: > F5:=Kc*e5: > F6:=Kv*e6: > F7:=Kc*e7: > F8:=Kv*e8: > F9:=Kc*e9: > F10:=Kv*elO: > F l l : = K c * e l l : > F2m:=Kv*e2m: > F3m:=Kc*e3m: > F4m:=Kv*e4m: > F5m:=Kc*e5m: > F6m:=Kv*e6m: > F7m:=Kc*e7m: Appendix B. MAPLE Source Code for Beam Design 146 > F2i:=Kv*e2i: > F3i:=Kc*e3i: Solving Model l ing Parameters C H A N G E two expressions: 1. E Q 1 : to include al l the logitudinal forces 2. solution : to include all the equations and unknowns > EQ1:=F1+F2+F3+F2m+F3m+F4+F5+F4m+F5m=0: > EQ2:=-(F3-Fl)/(Gv*2*b)=-dpsi/(dx*p~2): > EQ3:=-(F5-F3)/(Gv*2*b)=-dpsil/(dx*p~2): > EQ4:=-(F7-F5)/(Gv*2*b)=-dpsi2/(dx*p~2): > EQ5:=-(F9-F7)/(Gv*2*b)=-dpsi3/(dx*p"2): > EQ6:=-(Fll-F9)/(Gv*2*b)=-dpsi4/(dx*p~2): > EQ7:=-(Fl-F3m)/(Gv*2*b)=-dpsim/(dx*p~2): > EQ8:=-(F3m-F5m)/(Gv*2*b)=-dpsilm/(dx*p~2): > EQ9:=-(F5m-F7m)/(Gv*2*b)=-dpsi2m/(dx*p~2): > EQ10:=-(Fl-F3i)/(Gv*2*b)=-dpsii/(dx*p~2): > solution:=solve({EQl,EQ2,EQ7 )EQ3,EQ8} ){Dist,dpsi,dpsim,dpsil,dpsilm}): > assign(solution): > e v a l f ( D i s t ) : > dpsidphi:=dpsi/dphi: > dpsildphi:=dpsil/dphi: > dpsi2dphi:=dpsi2/dphi: > dpsi3dphi:=dpsi3/dphi: > dpsi4dphi:=dpsi4/dphi: > dpsimdphi:=dpsim/dphi: > dpsilmdphi:=dpsilm/dphi: > dpsi2mdphi:=dpsi2m/dphi: > dpsiidphi:=dpsii/dphi: > unassign('dphi','dpsi','dpsil','dpsi2','dpsi3','dpsi4','dpsim','dpsilm', > 'dpsi2m','dpsii'): Calculat ing Total Bending Moment > Mll:=Bs*dphi/dx: > M22:=Bv*(dphi/dx-dpsi/dx): > M33:=Bc*dphi/dx: Appendix B. MAPLE Source Code for Beam Design 147 > M44 =Bv*(dphi/dx-dpsil/dx): > M55 =Bc*dphi/dx: > M66 =Bv*(dphi/dx-dpsi2/dx): > M77 =Bc*dphi/dx: > M88 =Bv*(dphi/dx-dpsi3/dx): > M99 =Bc*dphi/dx: > M1010:=Bv*(dphi/d x-dpsi4/dx) > Mllll:=Bc*dphi/dx > M22m =Bv*(dphi/dx -dpsim/dx): > M33m =Bc*dphi/dx: > M44m =Bv*(dphi/dx -dpsilm/dx) > M55m =Bc*dphi/dx: > M66m =Bv*(dphi/dx -dpsi2m/dx) > M77m =Bc*dphi/dx: > M22] =Bv*(dphi/dx -dpsii/dx): > M33] =Bc*dphi/dx: > M10 =F1*H10: > M20 =F2*H20: > M30 =F3*H30: > M40 =F4*H40: > M50 =F5*H50: > M60 =F6*H60:. > M70 =F7*H70: > M80 =F8*H80: > M90 =F9*H90: > M100 =F10*H100: > MHO =F11*H110: > M20m =F2m*H20m: > M30m =F3m*H30m: > M40m =F4m*H40m: > M50m =F5m*H50m: > M60m =F6m*H60m: > M70m =F7m*H70m: > M20i =F2i*H20i: Appendix B. MAPLE Source Code for Beam Design 148 > M30i:=F3i*H30i: C H A N G E the total bending moment expression below to include al l the individual bending moments. > M:=M11+M22+M33+M22m+M33m+M44+M55+M44m+M55m+ \ > M10+M20+M30+M20m+M30m+M40+M50+M40m+M50m: > M _ s i m : = c o l l e c t ( c o l l e c t ( c o l l e c t ( c o l l e c t ( c o l l e c t ( c o l l e c t ( c o l l e c t ( c o l l e c t ( c o l l e c t ( \ > M,{dphi,dpsi}),dpsil),dpsi2),dpsi3),dpsi4),dpsim).dpsilm),dpsi2m),dpsii): > Calculate Effective Flexural Rigidi ty > BEQl:=collect(op(l,M_sim),dx)*dx/dphi; > BE n2:=collect(op(2,M_sim),dx)*dx/dphi; > BEQ3:=collect(op(3,M_sim),dx)*dx/dphi; > BEQ4:=collect(op(4,M_sim),dx)*dx/dphi; > BEQ5:=collect(op(5,M_sim),dx)*dx/dphi; > BE n6:=collect(op(6,M_sim),dx)*dx/dphi; > BEQ7:=collect(op(7,M_sim),dx)*dx/dphi; > BE n8:=collect(op(8,M_sim),dx)*dx/dphi; > Bl:=evalf((BEQl*dphi/dpsilm)*dpsilmdphi); > B2:=evalf((BEQ2*dphi/dpsim)*dpsimdphi); > B3:=evalf((BEQ3*dphi/dpsil)*dpsildphi); > B4:=evalf((BEQ4)); > B5:=evalf((BEQ5*dphi/dpsi)*dpsidphi); > B6:=evalf((BEQ6)); > B7:=evalf((BEQ7*dphi/dpsi)*dpsidphi); > B8:=evalf((BEQ8)); > B:=evalf(B1+B2+B3+B4+B5); Calculate Effective Damping Appendix B. MAPLE Source Code for Beam Design > eta:=op(l,op(2,B))/op(l,B); Display Some Unknowns > e v a l f ( D i s t ) ; > evalf(dpsidphi); > evalf(dpsildphi); > evalf(dpsi2dphi); > evalf(dpsi3dphi); > evalf(dpsi4dphi); > evalf(dpsimdphi); > evalf(dpsilmdphi); > evalf(dpsi2mdphi); > evalf(dpsiidphi); Appendix C M A T L A B Code for Beam Design function out=finddamp(H_v,H_c,horizontal,method,material); %Conversion Factor: 1 pound = 4.44 N ; 1 in = 0.0254 m; 1 m i l = 1/1000 in % % H_v : thickness of viscoelastic layer in [mil] % H_c : thickness of constraint layer in [in] % horizontal : 0 = no gravity (ie. horizontal position) % 1 = with gravity (ie. vertical position) % method : 1 = single layer % 2 = two layers stack together % 3 = one layer on both sides % material : 1 = aluminum % 2 = steel % Defining conversion factors mil2m=(l/1000)*(2.54 , )*(l /100); in2m=2.54*(l/100); % Defining structural layer geometry % Units : L[m]; b[in]; H_s[in]; T_s[in] 150 Appendix C. MATLAB Code for Beam Design L = l ; b = l ; H_s=l ; T_s=l /16; % convert to SI units H_s=(l /2)*H-s*in2m; H_v=( l /2)*H_v*mi l2m; H_c=(l /2)*H_c*in2m; T_s=Tj3*in2m; b=b*in2m; % Define material properties % Modulus of Elast ici ty [N/m/m] E_steel=200e9; E_al = 72e9; E_3M = 5e5; % mass density [kg/m/m/m] rho_al = 2.8e3; rho^steel = 7.2e3; rho_3M = 960; % damping factor [dimensionless] eta_al=0.01; Appendix C. MATLAB Code for Beam Design eta_steel=0.01; eta_3M=1.0; % Define structural layer parameters E_s= E_al*(l+i*eta^al); rho_s = rho_al; % Define viscoelastic layer parameters E_v= E_3M*(l+i*eta_3M); G_v = E_v /2 / ( l+0 .5 ) ; rho_v = rho_3M; % Define constraint layer parameters if mate r ia l==l , E_c = E_al*(l+i*eta_al); rho_c = rho_al; elseif material==2, E_c = Ej3teel*(l+i*eta_al); rho.c = rho_steel; else sprintf( ' Unknown M a t e r i a l ! ! \ n ' ) ; end % Define vibrational modes % 1st mode 3rd mode 5th mode 7th mode Appendix C. MATLAB Code for Beam Design % 4L (4/3)L (4/5)L (4/7)L . %lambda=4*L; % wavelength of the fundamental mode, vibration lambda=(4)*L; p = 2*pi/ lambda; % calculate geometrical parameters if hor izonta l==l , if (method==l) , A ^ l = ( b + H _ c ) * ( 2 * H ^ ) - ( b - 2 * T _ s ) * ( 2 * H _ s - 2 * T _ s ) ; elseif (method==2 | method==3), A_sl=(b+2*H_c)*(2*H_s)-(b-2*T_s)*(2*H_s-2*T_s); else sprintf( 1 Unknown Method! ! \ n 1 ) ; end else A ^ l = ( b * 2 * H _ s ) - ( b - 2 * T _ s ) * ( 2 * H ^ - 2 * T _ s ) ; end A _ s = b * T j 3 ; A_v=b*(2*H_v); A_c=b*(2*H_c); if hor izonta l==l , i f m e t h o d = = l , . I ^=( l /12 )* (b+H_c)* (2*H_s ) ' 3 - ( l / 12 )* (b -2*T^)* (2*H_s -2*T_ Appendix C. MATLAB Code for Beam Design elseif (method==2 | method==3), I^=( l /12 )* (b+2*H_c)*(2*H^)^3- ( l /12 )* (b -2*T_s)* (2*H^-2*T_s) else sprintf( ' Unknown Method! ! \ n ' ) ; end else , , I_s=(l /12)*b*(2*H_s)"3-(l /12)*(b-2*T_s)*(2*H_s-2*T_s)^3; end Lv=( l /12)*b*(2*H_v)^3; Lc=(l /12)*b*(2*H_c)"3; BJ3=E_S*I_S; B _ v =E _ v *I _ v ; B_c=E_c*I_c; K_S=E_S*AJS; K _ v = E _ v * A _ v ; K _ c = E _ c * A _ c ; % complete solution % solve for the unknowns D , and dpsi /dphi if me thod==l , H_sv=(H_s+H_v); H_sc=(H_s+2*H_v+H_c); g=b*G_v/(K_c*H_v*p*p) ; D = ( K _ c * H _ s c * ( K _ v + 2 * K _ c ) - 2 * K _ c * ( K _ v * H _ s v + K _ c * H ^ c ) * ( l + g ) ) / ( ( K _ c - K _ s ) * ( K _ v + 2 * K _ c ) - 2 * K _ c * ( K _ s + K _ v + K _ c ) * ( l + g ) ) ; d p s i d p h i = ( K _ c * H _ s c - ( K _ c - K _ s ) * D ) / ( 2 * K _ c * H _ v * ( l + g ) ) ; Appendix C. MATLAB Code for Beam Design 155 B=B_s + B_v + B_c + K _ s * D * D + K _ V * ( H J S V - D ) ~ 2 + K _ c * ( H _ s c - D ) ~ 2 - ... (B_v + K _ v * H _ v * ( H _ s v - D ) + 2*K_c*H_v*(H_sc-D ) )*dpsidphi ; mu=A_sl*rho_al+A_v*rho_v+A_c*rho_c; elseif method==2, H11=0; H21=HJ3+H_V; H31=H_S+2*H_V+H_C; H41=H_s+2*H_v+2*H_c+H_v; H51=H_s+2*H_v+2*H_c+2*H_v+H_c; K 1 = K ^ ; K 2 = K _ v ; K3=K_c ; K4=K_v ; K5=K_c ; g=b*G_v/(K_c*H_v*p*p); D=(K_c*H31*(3*K_v+4*K_c) -2*K_c*( l+g)* (K2*H21+K3*H31+K4*H41+K5*H51)+ . . 2*K_c*(K4+2*K5)*(H_v+H_c) ) / ( (K_c-K_s)*(3*K_v+4*K_c) - ... 2*KjC* ( l + g ) * ( K ^ + 2 * K _ v + 2 * K j c ) ) ; d p s i d p h i = ( K 3 * H 3 1 - ( K 3 - K l ) * D ) / ( 2 * K 3 * H _ v * ( l + g ) ) ; dpsi ldphi=((H_v+H_c) /H_v) / ( l+g) ; H 1 0 = H 1 1 - D ; H 2 0 = H 2 1 - D ; H 3 0 = H 3 1 - D ; H 4 0 = H 4 1 - D ; H 5 0 = H 5 1 - D ; B = B ^ + 2 * B _ v + 2 * B _ c + K l * H 1 0 ^ 2 + K 2 * H 2 0 ' 2 + K 3 * H 3 0 ' 2 + K 4 * H 4 0 ~ 2 + K 5 * H 5 0 ' 2 ... - (B_v+(K2*H20+2*K3*H30+2*K4*H40+2*K5*H50)*H_v)*dps idphi ... - (B_v+(K4*H40+2*K5*H50) *H_v) *dpsi ldphi ; mu=A_sl*rho_s+2*A_v*rho_v+2*A_c*rho_c; elseif method==3, g=b*G_v/(K_c*H_v*p*p); dpsidphi=(H_s+2*H_v+H_c)/(2*H_v*(l+g)); Appendix C. MATLAB Code for Beam Design B=B^+2*B_v+2*B_c+2*K_v*(H_s+H_v)~2+2*K_c*(H^+2*H_v+H_c) '2 ... - 2*(B_v + K_v*H_v*(Hjs+H_v)+2*Kx*H_v*(Hjs+2*H_v+H_c))*dpsidph mu=A_sl*rho_s+2*A_v*rho_v-|-2*A_c*rho_c; else sprintf( 1 Unknown Method! ! \ n ' ) ; B=0; mu=0; end % find the effective damping eta=imag(B)/real(B); % using Euler—Bernoulli beam theory to find the corresponding vibrational % frequency mass=mu*L; omega=real(p*p*sqrt(B/mu)); freq=omega/2/pi; % output results out=[eta;freq]; 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0064869/manifest

Comment

Related Items