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UBC Theses and Dissertations

Active-passive motion compensation systems for marine towing Stricker, Peter Andrew 1975-01-29

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ACTIVE - PASSIVE MOTION COMPENSATION SYSTEMS FOR MARINE TO WING by PETER ANDREW STRICKER B. Eng. McGill University Montreal, 1971. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER CF APPLIED SCIENCE in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH CCLUMEIA VANCOUVER CANADA January 1975 In presenting this thesis in partial fulfilment of the requirements ft an advanced degree at the University of British Columbia, I agree tha 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MEGUAmCAL Bd£ldEEg.^& The University of British Columbia Vancouver 8, Canada Date APg.lL 2, l<?7S~-ii ABSTRACT The dynamic behaviour of an active-passive motion compensation system for handling towed marine vehicles is examined, and a mathematical model developed. In the analysis, the passive system considered is pneumatic, while the active system is electro-hydraulic. The towed body is assumed to be a point mass subjected to uydrodynamic drag, and attached to the motion compensator by means of a linear spring representing the cable. It is not intended, in this project, to mcdel the tewed body in greater detail. The equations of the passive, active, and towed tody systems are derived, and linearized tc permit a relatively simple frequency-domain solution. A time simulation based on the nonlinear equations, including Coulomb friction in the compensator, is developed for use on an IBM Systeis/370 computer. A laboratory model is used to conduct experiments at three frequencies, and the results indicate good agreement ketween the linear, simulation, and real models. Extension of the equations to cover multi-frequency inputs, two-dircensiona1 tewing cables, and slow-acting servovalves is also discussed to facilitate application to marine systems. iii TABLE OF CONTENTS Chapter I - Introduction ............................. 1 1.1 Problem Description ....................... 1 1.2 State of the Art .......................... 6 1.3 Objectives and Scope of Project ........... 13 Chapter II - Theoretical Analysis 14 2.1 Typical System 12.2 Equivalent Model 7 2.3 The Passive System 20 2.4 The Active System ......................... 30 2.5 The Active-Passive System 37 2.6 The Control System ........................ 42 2.7 Computer Simulation 46 Chapter III - Linear Analysis 7 3.1 Linearized Passive System 43.2 Linearized Active System 55 3.3 Linearized Active-Passive System .......... 58 3.4 Performance Analysis and optimization ..... 61 Chapter IV - The Laboratory Model 70 4.1 General Description4.2 Performance Prediction and Evaluation ..... 77 Chapter V - Application 81 5.1 Input Conditions .......................... 81 5.2 Two-Dimensional Cable Model 85 5.3 Servovalve Model Extension ................ 87 5.4 Control System Considerations 90 Chapter VI - Conclusions ............................. 93 References 94 Appendices 6 iv LIST OF ILLUSTRATIONS Figure Page 1.1.1 Motion Compensation Systems 4 1.2.1 Isolation and Absorption 7 1.2.2 Performance Characteristics1.2.3 Passive Pneumatic System 9 1.2.4 Tuned Bam Tensioner 12 2.1.1 Active/Passive Bam Tensioner 15 2.2.1 Equivalent System 8 2.3.1 Passive System 21 2.3.2 Block Diagram of Tank Dynamics ....................... 23 2.3.3 Block Diagram of Cylinder Dynamics 22.3.4 Block Diagram of Valve Dynamics ...................... 27 2.3.5 Block Diagram of Passive System Dynamics ............. 29 2.4.1 Active System 31 2.4.2 Servovalve Characteristics , 32.4.3 Block Diagram of Active System ....................... 36 2.5.1 Cable/Body Model 8 2.5.2 Block Diagram of Cable/Body Dynamics 32.5.3 Block Diagram of Active/Passive System ............... 41 2.6.1 Active System with Control Blocks 43 3.1.1 Pressure-Flow Curve for Throttling Valves ............ 51 3.1.2 Linearized Passive System Transfer Function .......... 51 3.2.1 Linearized Servovalve Characteristics 57 3.2.2 Linearized Active System Transfer Function ........... 57 3.3.1 Linearized Cable/Body Transfer Function 60 3.3.2 B. D. of Linearized Active/Passive System ............ 60 3.4.1 Bam Centering Network 65 4.1.1 Laboratory Apparatus 71 4.1.1 Laboratory Apparatus - Schematic 72 4.1.3 Motion Generator Arrangement 3 4.1.4 Control System 5 4.2.1 Theoretical and Experimental Besults 80 5.1.1 Sea State Spectral Density Function ... 83 5.1.2 Ship Heave Response .................................. 83 5.1.3 Ship Heave Spectral Density Function ................. 83 5.2.1 Two-Dimensional Cable Model 86 5.3.1 Typical Servovalve Response 9 5.4.1 Motion Compensation Transfer Function ................ 91 5.4.2 Typical Feedback Network 9V LIST OF SYMBOLS SYMBOL MEANING A area of piston AA area of active cylinder piston AP area of passive cylinder piston Co throttling valve coefficient CR capillary coefficient CH hydrodynamic drag factor CL linear drag factor CSV servovalve flow coefficient D differential operator d/dt f friction force in ram FA active cylinder force FMBT cable tension FP passive cylinder force FBAM total ram force FB(S) body transfer function FC (S) cable transfer function GP(S) passive system transfer function H (s) open loop transfer function feedback loop transfer function H^tS) feedforward loop transfer function Hsv(s) servovalve transfer function Hy (S) ram centering loop transfer function K, displacement feedback gain K2 velocity feedback gain K3 acceleration feedback gain KFF feedforward static gain KMA mechanical advantage KP passive system static gain Ks gas spring stiffness Ksv servo-amplifier gain m mass of gas in passive cylinder H mass of towed body N passive system volume ratio P pressure Po initial pressure in passive system P* . P* passive cylinder pressures P. pressure downstream throttling valve PM pressure upstream throttling valve Ps supply pressure in active system AP pressure drop in active cylinder QA oil flow into active cylinder QL leakage flow Qv servovalve flow r low level servo signal R gas constant for nitrogen 5 Laplace variable t time T absolute gas temperature u disturbance input displacement V volume Vc,, VC2 passive cylinder volumes passive tank volumes vi W power consumption x body displacement xt tow point displacement y piston displacement Yc tow pt displacement relative to ship r servovalve actuating current V ratio of specific heats 8 small piston displacement 5 passive system critical damping ratio 5 servovalve critical damping ratio \, servovalve flow gain "Xz servovalve flow-pressure coefficient X3 bypass valve flow coefficient tr, t, passive system time constants •cw servovalve time constant u30 design frequency oin passive system natural frequency co<,v servovalve natural frequency vii ACKNOWLEDGEMENT I am most grateful for the patient help and encouragement of my supervisor, Dr. R. BcKechnie. Special thanks is due to Dr. Keefer of B. C. Research Council who initially gave me the idea for this project, and for his subsequent assistance. Thanks are also due to Mr. Johnson of Fleck Brothers for donating a servovalve and amplifier; to.Dr. Vickers for making the valve work; to Messrs. Hoar and Hurren and their crew for building the apparatus and lending equipment; to the Department of Electrical Engineering for the use of their analogue computer; and to Miss Wendy Allen for keypunching the manuscript. All computing was done at the University of British Columbia Computing Centre. The project was funded by the National Research Council of Canada under Grant #67-8183. 1 CHAPTER I I8TB0DUCTI0H 1.1 Problem Description The safety and performance of towed submersible vehicles depend, to a large extent, on the ability of the handling gear to decouple wave induced motions of the surface ship from the towing cable. This decoupling is usually accomplished by a motion compensation system — essentially a special class of a vibration isolator. Vibration isolating devices are employed in systems where a mass is to be isolated from an external force cr motion disturbance. Some common examples are automobile suspensions, earthquake absorbers, and rocket-borne instrumentation cushions. The marine towing isolation problem has three distinguishing features. First, the frequency of surface ship motion is low, in the order of 0.1 Hz, while the associated amplitude may be in the order of fifteen feet. Second, the mass of the submersible, including the water it entrains, is large — typically in excess of 30, 000 pounds-mass. Finally, the vertical displacement of the stern of the ship consists of the superposition of several harmonic functions of different amplitudes and frequencies as defined by a spectral density function. Thus, the input disturbance is somewhat more 2 predictable than the forms of vibration present in the previous examples. The marine towing motion compensation system is usually designed to maintain constant towing cable tension. Once this requirement is met, it then follows that the acceleration of the submersible will be zero, and all undesirable motion will be eliminated. To provide this constant cable tension, all systems attempt to pay out or haul in cable as the ship moves up or down, thus decoupling the motion of the ship from the cable. The index of performance of such a motion compensation system is the ratio of the amplitude of towed body to ship displacement. It is generally stipulated that the performance index must be less than a given value at the frequency which contains the greatest energy as taken from the sea state spectral density function. This is then considered the primary design frequency. In addition, the minimum acceptable index of performance for other frequencies within the spectrum may be specified. Once the overall physical constraints are met (e.g.. weights, geometries, etc., over which the designer has little or no control) the problem then becomes that of determining the physical characteristics of the system. Some of the design considerations include simplicity, reliability, initial cost, and power consumption. Three of the more popular motion compensation systems are 3 shewn in Fig. 1.1.1. although very different in appearance, each employs a pneumatic spring in the form of a gas accumulator to operate a hydraulically actuated positioner. With systems <a) and (b), the positioner is a cylinder which controls the vertical displacement of a sheave over which the cable is reeved. System (c) uses a hydraulic winch to haul in and pay out cable. In each case, the cable tension is balanced by a passively-acting pneumatic spring1. As will be seen in Section 1.2, such systems, under certain conditions, can be tuned to perform adequately over a narrow frequency range. It is possible to design a purely active system, in which a significant amount of energy is expended to achieve the stabilization effect. In such a system, a transducer monitors the motion of the load and after suitable signal processing, controls the flow of oil to a hydraulic actuator. Active systems are superior to passive ones in that they are capable of good motion isolation over a wider frequency range. However, because they require a bulky power source and consume a large amount of energy when controlling a massive load, they are not suitable for marine applications. To improve the performance of a passive system without a large expenditure of power, a hybrid active-passive system is 1 In a passive system, the sum of the potential energy in the spring and kinetic energy of the load is conserved, apart from some dissipation due to damping and friction. Thus, no external energy is required to operate the system. 4 2U MUTATOR ACTUATOR TOWING CABLE (o) RAM TENSIONER ACCUMULATOR (b) BOOM BOBBER ACCUM. HYDRAULIC MOTOR REEL Cc) TENSIONING WlklCM FIG. 1.1.1 MOTION COMPENSATION SYSTEMS 5 proposed. Such systems have been successfully used to isolate small components from an environment of severe shock and vibration. However, as outlined in the next section, no work was found related to the application of such systems to the marine towing field. 6 1.2 State of the Art Vibration isolation systems, both active and passive, have been widely investigated in the past decade. In general, it appears that there were two distinct methods of dealing with the problem; one being highly theoretical, and the ether being the analysis of a particular problem. The latter method, especially in the field of ocean engineering, has been very empirical in nature, with little or no mathematical justification of the ideas presented. This section will discuss some of the relevant work that has been done, first, in vibration isolation, and second, in the field of ocean engineering applications. In general, there are two distinct vibration reduction methods available: absorption and isolation.1 Isolation involves placing a resilient material between the disturbance source and the receiver (the system to be protected), whereas absorption involves the attachment of an energy absorbing device to either the source or receiver (Fig. .1.2.1). Isolation can be achieved either actively or passively, and can be made effective over a wide range of frequencies. Absorption is generally achieved passively using a spring-mass system which is in resonance with the source and receiver at one particular frequency. At that frequency, the receiver experiences no input at all, but this 1 See Ref. (15) a RECEIV£R RECEIVER ISOLATOR SOURCE 4— Cc) ISOLATION! ft>) ABSORPTION! ABSORBER RECEIVER SOURCE ^ FIG. 1.2.1 VIBRATION ISOLATION 6, ABSORPTION! (a) ISOLATION (b) ABSORPTION FIG- 1.2-2 PERFORMANCE CHARACTERISTICS 8 effect is confined to a very narrow frequency band. Also, undesirable resonant peaks occur at two frequencies, corresponding to the separate natural frequencies of the receiver and absorber. Fig. 1.2.2 illustrates the performance of isolators and absorbers. A passive pneumatic isolator as shown in Fig. 1.2.3 (a) has been examined by Cavanaughl. He solved the linearized third order system equations in the freguency domain, and found the optimum critical damping ratio in terms of the tank to cylinder volume ratio. Fig. 1.2.3(b) shows the frequency response of the system, and Fig 1.2.3(c) shows the critical damping ratio function which yields the smallest maximum amplitude ratio. Another passive isolation system directly applicable to automobile suspensions has been examined by Thompson2. He considered a two-dimensional linear system with four degrees of freedom, and developed an optimum performance index based cn ride comfort and road-holding ability. A more general approach to optimizing passive suspensions has been presented by Hedrick for use in the design of high speed tracked vehicles3. An optimum passive shock isolator, which uses a variable friction element to dissipate energy has 1 See Ref. (5) 2 See Refs. (20) and (22) 3 See Ref. (7) * See Ref. (11) 9 LOAD G.AS TANKS PNEUMATIC-CYLINDER A -U Ca) CONFIGURATION! o i FREQUENCY (b) PERFORMANCE (c) OPTIMUM CRITICAL DAMPING RATIO N = TANK VOLUME CYLINDER VOLUME FIG. 7.23 PASSIVE PNEUMATIC SYSTGLM 10 been proposed by Mercer and Rees*. Active systems for shock and vibration isolation have also been examined. Soliman proposed a servovalve controlled pneumatic system using displacement and velocity feedback to control the servovalve1. Thompson considered active systems for automobile suspensions2. Porter, Athans, and Karnop all presented highly mathematical methods for dealing with linear active systems3, Kriebel developed an active system for shock isolation*. None of the above-mentioned work is in a form which is readily applicable to the problem of motion compensation in the marine environment. The theoretical solutions relate mostly to linear systems, while the more practical solutions are too specific and require much modification tc make them useful for other design purposes. There are a number of marine motion compensation systems (mostly passive) operational around the world, but little documentation exists to help predict a system's performance before it is built. Most systems consist of a pneumatic spring, as described in Section 1.1, and are classed as vibration isolators. Keefer proposed a simple manner in which an isolator 1 See Refs. (17) and (18) 2 see Ref. (21) a See Refs. (13), (1) and (8) * see Ref. (10) 11 can be made into an absorber by making the direction of moticn of the compensator mass orthogonal to the input disturbance1. Fig. 1.2.4(a) illustrates a ram tensioner in such a configuration. Such a system is tuned so that the anti-resonance occurs at the freguency which contains the dominant amplitude of vibration. Fig 1.2.4(b) shows the typical performance of a tuned system. Note that damping increases bandwidth at the expense of the system's attenuation. Buck2 and Sutherland3 suggested the use of active systems, but neither has developed a complete analysis of such a system, nor suggested a method of predicting the performance of a real, nonlinear system. In addition, they have not recognized the fact that power consumption can be reduced by incorporating a passive system to support the static weight of the load while the active system is used solely for motion compensation. 1 See Bef. (9) 2 See Ref. (3) 3 See Ref. (19) 12 (cO GENERAL ARRANGEMENT FREQUENCY (b) PERF ORMANCE FIG. 1.2.4 TUNED RAM TENSIONER 13 1.3 Objectives and Scope of Project The objectives of this project are first, to study the dynamics of an active-passive motion compensation system for marine towing, and second, to use the results of this study to develop guidelines for use in designing real systems. These objectives are accomplished by proceeding in six steps: 1. Representing a typical system in a form which closely approximates reality, yet which lends itself to mathematical analysis and simulation. 2. Developing the mathematical model, including such nonlinearities as hydrodynamic drag and dry friction. 3. Linearizing the mathematical equations and conducting a frequency-domain analysis to obtain a first approximation of the important system parameters. 4. Developing a digital computer simulation program which will validate and optimize the parameters derived in step #3, 5. Constructing a small working model to check the validity of the mathematical and simulation models. 6. Relating the results of the foregoing to the design of real systems. 14 CHAPTER II THEORETICAL ANALYSIS 2 . 1 Typical System Some of the commonly used passive motion compensation systems were shown in Fig. 1.1.1. It is possible to devise an active-passive system by adding an active actuator parallel to the passive one. In the case of the ram tensioner and boom bobber, this means adding a second cylinder, while for the tensionig winch, adding a second hydraulic motor. The three configurations shewn in Fig 1.1.1 are' very similar mathematically. The similarity between (a) and (b) is obvious — only the mechanical advantages of the reeving, in the case of (a), or the boom, in the case of (b), are different. In the case of the winch, the hydraulic motor is equivalent to a number of cylinders connected in parallel, and hence can be modelled as a single cylinder. Thus, for the purpose of this project the ram tensioner is selected as a typical system. The overall configuration of the typical system is shown in Fig. 2.1.1. The passive subsystem is the same as before, while the active subsystem consists cf a hydraulic cylinder controlled by an electrohydraulic servovalve. The control system consists of accelerometers mounted on the towing and ram sheaves, whose signals are processed and fed to the servovalve. The signal processing network is the most vital component of the system, FIG. 2,1.1 ACTIVE/PASSIVE RAM TENSIONER 01 16 and will be the subject of thorough analysis. As pointed out earlier, the load can be a diving bell suspended from a stationary ship, a submerged body towed at high speed (in the order of 10 knots), or a surface vessel such as a barge. For the case of a submersible supported from a ship which is not moving horizontally with respect to the water, the cable can be represented as a one-dimensional elastic link whose longitudinal axis is vertical. In the case of a moving ship towing a barge or submersible, the cable will assume a complex three-dimensional curve. Since this project is concerned primarily with the behaviour of the motion compensation system, the typical system considered will include the one-dimensicnal cable. Application of the approach to the case of three-dimensional cable, as developed by Walton and Polachek, is discussed in Chapter V. 17 2.2 The Equivalent Model To facilitate the analysis of the ram tensioner described in Section 2.1, the following simplifications will be made: 1. The static tension in the cable due to the submersible's weight is not considered. This simplification does not affect the dynamics of the motion compensator. 2. The cable is considered to be a one-dimensional elastic link for the reasons set forth in Section 2.1. 3. The passive subsystem is considered to be purely pneumatic. This restriction actually increases the complexity of the problem, but is included to demonstrate the method of application of the compressible fluid flow eguations. In many applications, the passive system would actually be an "air-over-oil", or hydropneumatic system, as shown in Figure 1.1.1. Using these simplifications, it is possible to model the chosen system as shown in Figure 2.2,1. The form shown in Fig. 2.2.1 was devised to facilitate mathematical analysis and test-model construction. The active and passive motion compensation SERVO VALVE GAS BOTTLES X DAMPING VALVES on ill X BYPASS VALVE MM $•3 CONTROL SYSTEM r —f-y-j l_ ACCELEROMETERS CABLE PASSIVE CYLINDER ACTIVE CYLINDER U FIG. 2.2.1 EQUIVALENT SYSTEM 19 cylinders are placed horizontally on a carriage, with their piston rods connected sc as tc function in parallel. The carriage is driven horizontally in a sinusoidal manner, with the desired frequency and amplitude, simulating the vertical motion of the ship. The load is modelled by a second carriage containing the desired mass, connected to the motion compensation piston rod by means of a spring which is assumed to model the cable. The entire system is similar tc the real case of Fig. 2.1.1 except that all motion is horizontal instead of vertical. As a result, the static weight of the towed body is not considered. Therefore, in modelling the spring characteristic of the passive system, it is necessary to pressurize both sides of the passive cylinder, such that the net static force at the piston red is zero. In general, this equivalent system accurately models the motion compensation system, but does not fully consider the dynamics of the cable and towed body. However, the design method developed here is flexible enough to accomodate these additions if the necessary parameters are available to the designer. 20 2. 3 The Passive System The passive side of the system under consideration consists of a pneumatic ram, with each end connected via a throttling valve to a receiving tank (Fig. 2.3.1). The throttling, valves are used to introduce damping into the system. The mass flow to and from a tank or cylinder is derived in Appendix A, and is given by R is the gas constant for the particular gas used, T is the absolute temperature, V is the tank or cylinder volume, P is the absolute pressure, and i is the ratio of specific heats. Because the receiving tanks have fixed volumes, and the mass flows are proportional to the negative of the pressure changes, we can write (2. 3. 1) where m is the mass flow rate, (2.3. 2) 22 and m oc - Pt (2.3.3) where Vt is the tank volume and Pt is the tank pressure. Substituting (2.3.2) and (2.3.3) into (2.3.1) gives , VM p " 7RTT7 *' (2.3.4) ™ - - Vt* P where the subscripts 1 and 2 refer to the left and right hand sides of the passive system, respectively. In general, the volume of the receiving tanks, and hence the volume of fluid contained, is large compared to the mass flow in and out of the tanks; thus the temperature of the gas can be considered constant,1 i.e. 1 For an actual system having an effective piston area of 13.46 square inches, a displacement of five feet as measured at the tow point causes a change in absolute temperature of only 1.9%, assuming adiabatic compression or expansion (worst case). 23 r R To • i ng. 2.3.2  BLOCK DIAGRAM of TANK DYNAMICS FIG. 2.3.3  BLOCK DIAGRAM of CYLINDER DYNAMICS 24 Ti = = T„ (2.3.5) Fig. 2.3.2 is a block diagram of the tank flow equations. The mass flow to and from the cylinders is also expressed by (2.3.1), except that: Vz - VC2 - A2^/ (2.3.6) V, = Atj V2 = -A2j (2. 3.7) VV1, oC P, (2.3.8) where Vn and V2 are the volumes of the left and right sides of the cylinder, VC| and VC2 are the initial values of Vr and Vz, flt and A2 are the effective piston areas, and y is the piston displacement. 25 Substituting (2.3.6), (2.3.7) and (2.3.8) into (2.3.1) gives I Z ' 1 1 J J K-l, (2. 3.9) I £T2 Due to the relatively small volume of the cylinder and the large variation in pressure, temperature change is nc longer negligible. Using the equation of state for an ideal gas, p. y. T; = —1—L (i*l,2) (2.3.10) where m-{ is the mass of gas in the i-th side of the cylinder. Substituting (2.3.10) into (2.3.9) gives r ± A + A , i ' _ m f I Pz Az . "j (2.3.11) Fig.. 2.3.3 shows the block diagram of the the cylinder equations. The mass flow through each throttling valve is derived in 26 Appendix A and is given by (2.3.12) where C0 is the valve constant, Pu is the upstream pressure, Pd is the downstream pressure, and Tu is the upstream temperature. It is observed that the direction of mass flow is from high to low pressure; that is, Pu, the upstream pressure, is the greater of PttrVA . By the convention shown in Fig. 2.3.1, if P^ > Pt then ml> 0 ; conversely, when Pti < Pt , < 0. Consequently, in solving (2.3.12) the upstream end must first be determined. Then the correct algebraic sign can be assigned to m-. Fig. 2.3.4 shows the block diagram of the valve eguaticns. The equations (2.3.4), (2.3.9) and (2.3.12) must be solved simultaneously to yield P, and Pz given a piston displacement y. The force generated in the ram can then be found: (2.3. 13) where Fp is the ram force. FIG. 2.3.4. BLOCK DIAGRAM of VALVE DYNAMICS 28 The block diagram of the passive system dynamics is shown in Fig. 2.3.5. Since it is difficult to sclve this system of equations analytically, a numerical solution is now developed. The strategy used in the numerical solution is as follows: 1. Calculate Pti as shown in Fig. 2.3.2 2. Calculate P-, as shown in Fig. 2.3.3 3. Calculate m;as shown in Fig. 2.3.4. 4. Repeat steps 1 to 3 with the new value of m; until Pt. and P-, no longer change from iteration to iteration. 5. Calculate the ram force from (2.3.13). Mi TANK. * 1 Pt, CYUMDER. SIDE * I FIG. 2.3.5 BLOCK DIAGRAM of PASSIVE SYSTEM DYNAMICS 30 2.4 The Active System The active side of the system under consideration consists of a positive displacement hydraulic pump, relief valve, gas accumulator, servo-valve, and hydraulic cylinder. (Fig. 2.4.1) It is assumed that the pump discharge flow rate always exceeds the system requirement, hence maintaining the pressure in the accumulator equal to the relief valve setting. In fact, the accumulator pressure will vary slightly with changes in flow rate due to friction losses in the hydraulic lines, but this fluctution is negligible compared to the working pressure. Therefore, the supply pressure is considered constant. The compressibility effect of the hydraulic fluid is examined in Appendix C, and,is found tc contribute an error in flow of only 2.6% for a typical full-scale system. Therefore, the compressibility of the fluid is not considered. The flow-pressure relationship for the servo-valve is given by the manufacturer for selected values of actuating signal, z (Fig. 2.4.2). As shown in Appendix B, this relationship, for a zero-lapped valve1, can be accurately modelled by 1 See Ref. (6) for the equations of under- and over-lapped servo-valves. GAS OIL FROM PUMP -> ACCUMULATOR TO TANK C ->— 11 ACTUATING SERVO VALVE. CURRENT" FA FIG. 2.4.1 ACTIVE SYSTEM AP F1 Gr 2.4. 2 SERVOVALVE CHAR ACT ERISTICS 32 (2. 4. 1) w her 6 Qv is the volume flow through the valve. AP is the pressure drop across the load. Csv is the characteristic constant of the servo valve , Ps is the supply pressure, assumed constant, and z is the actuating signal. Leakage across the cylinder is often useful in stabilizing a servo-system, and is therefore included in the analysis. Leakage is provided by means of an auxiliary path around the piston, and controlled by means of a valve. The leakage flow, QL, is given by Cv is the characteristic constant of the valve. The total flow into the ram, QA, is the difference between the flow through the servo-valve and the leakage flow: (2.4. 2) where 33 The sign convention is such that QA is positive when it causes the piston to move to the right. Substituting (2.4.1) and (2.4.2) into (2.4.3) gives The velocity of the piston with respect to the cylinder, y, can now be expressed as where aAis the effective area of the piston. The force available to do work at the end of the piston rod, FA, is given by FA - AAAP (2.4.6) Equations (2.4.5) and (2.4.6) can now be combined to yield 34 y directly as a function of F : (2.4.7) In computing (2,4.7) it is necessary to introduce an artificial sign assignment to avoid negative values within the surds. This is done by noting that 1 . In the case of flow through the servo-valve, Ps must take the sign of z. The surd then becomes 2. In the case of the leakage flow, the surd takes the sign of FA, which is actually the direction of pressure drop. The surd thus becomes S;9n(FA)y|F*/^| (2-"-9) Substituting the modified expressions (2.4.8) and (2.4.9) into (2.4,7) gives the equation for computing y from FA consistent with the sign convention: 35 a- ct "/Ms-*/* I - ,2-<-,o) Fig. 2.4.3 shows the block diagram of the active system. 2 >• SICM (5 ABS ABS —3*- AA x: x < SI6M FIG. 2.4.3 BLOCK DIAGRAM of ACTIVE SYSTEM 37 2.5 Active-Passive System The total ram force is the sum of the forces exerted by the passive and active cylinders, less friction: FtoB- FP + F.-f ' (2.5.D where FR*M is tne ram force, FA is the active cylinder force, Fp is the passive cylinder force, and f is the friction fcrce, as discussed in Appendix E. The force felt by the cable is directly proportional to FRAM , where the constant of proportionality is the reciprocal of the mechanical advantage of the reeving: F = — Fo <2'5* 2) "NET J/ »RAM where KMfl is the mechanical advantage of the reeving, and F^ is the force acting on the cable. The towed body can be represented by a mass K, subjected to hydrodynamic drag and towing cable tension. (Fig. 2.5.1) The cable is assumed to be a massless linear spring. The compensator force causes an elongation of the cable 38 ==D>| k/VA BODY CABLE —*- Xi > % DRAG FIG. 2.5.1 CABLE /BODY MODEL FIG. 2.5.2 BLOCK DIAGRAM of CABLE. /BODY DYNAMICS 39 according to the relation fwT = ICC-X, - 7c) (2.5.3) The cable then applies the same force to the body, whose motion can be described by: FKET = M X + C %2 (2.5.4) where M is the mass of the towed body, and CH is the hydrodynamic drag factor. Equating (2.5.3) and (2.5.4) and rearranging, gives the nonlinear differential equation of motion of the body: * + % + ^ X = W'X' (2.5.5) where 60c is the cable-mass natural frequency, ^ Ke/M" . Once (2.5.5) is solved, it is possible to find FM61 by direct application of (2.5.4) or (2.5.3). The block diagram of the towed body and cable system is shown in Fig. 2.5.2. The absolute displacement of the shipboard end of the 40 cable, x„ is the sum of the input, u, and the displacement of the end of the cable with respect to the input, yc: X, = U + vjc (2.5.6) For the case where the actuator acts on the cable through a mechanical advantage (e.g., the ram tensioner of Fig. 2.1.1), the motion of the cable with respect to the ship's stern can be expressed as: yc = I^A y <2-5-7> w here y is the extension of the actuator. Equations (2.5.6) and (2.5.7) are combined with the block diagrams of the passive, active, and cable-mass systems (Figs. 2.3.5, 2.4.3, and 2.5.2) tc give the block diagram of the entire system, as shown in Fig. 2.5.3. CABLE/BODY FIG. 2.5-2 K MA PASSIVE SYST. FIG. 2.3.5 ACTIVE SYST-FIG. 2-4.3 CONTROL SYST. FIG- 2-6 2 -— FIG. 2.5-3 BLOCK DIAGRAM of ACTIVE-PASSIVE SYSTEM 42 2.6 The Control System The control system generates the signal tc operate the active actuator by means of monitoring and processing certain variables. The controlled variable can be considered as either x OR F»IET r and Figure 2.5.2 suggests that both can be controlled simultaneously because they are linearly dependent. This means that the index of performance, as discussed in Chapter I, can te either FHET /u or x/u, and both must be made tc fall below certain specified limits for acceptable operation. In addition, the control system must ensure that the long term average motion of the actuator piston does not drift from the centre of the cylinder. The controlled variable (FNCT or x) can be used to generate the primary actuating signal for the servovalve. This constitutes a simple feedback control system where the reference input is,zero, as shown in Fig. 2.6.1. (The passive system is omitted from Fig. 2.6.1 for clarity.) The "Control Elements" block may contain filters, integrators, etc., as required for best operation. In addition tc feedback, it may be desirable tc include a portion of the disturbance input, u(t), as indicated by the feedforward loop in Fig. 2.6.1. The piston centering control is a miner loop used to restore the piston to the centre of the actuator slowly with 43 FEEDFORWARD CONTROL ELEMENTS TRANSDUCER SERVO VALVE ACTUATOR SHIP MOTION U(i) CONTROL ELEMENTS TRANSDUCER LOAD CENTERING CONTROL ELEMENTS TRANSDUCER FEEDBACK FIG. 2.6.1 ACTIVE SYSTEM WITH CONTROL BLOCKS U(s) HFFfe) 1 V(s) X(s) K SY 2(s) FIG. 2.6.2 BLOCK DIAGRAM of CONTROL SYSTEM 44 respect to the forcing freguency. That is, the time constant of the loop is at least one order of magnitude greater than the reciprocal of the input freguency. The choice between using FNtT and x as the controlled variable is made based on the availability and suitability of transducers. In either case, however, it is necessary to provide a physical path in the towing cable for the feedback signal to reach the surface vessel. If this is not possible, then the absolute motion of the shipboard end of the cable, xt, can be used as the feedback signal. Such a strategy is easily feasible in the case of the ram tensioner and boom-bobber (Fig. 1.1.1 (a) , (b)) , but net so for the constant tension winch (Fig. 1.1.1(c)). Fcr demonstration purposes, it is assumed that x is the feedback variable. The control system elements are now lumped into three blocks, as shown in Fig. 2.6.2. The total control voltage, B (s) is given by (in Laplace notation): R(Y)= MfBCs)X(s) + WFFCs)\J(s) + U,(s)Y(s) (2.6.1) where Hpj,(s) represents the feedback element, Hpp(s) represents the feedforward element, and (s) represents the piston centering element. The servovalve actuating current is obtained by passing the control voltage through a power amplifier Z(s) = J<svR(s) where Z (s) is the actuator current, and Kw is the amplifier gain. 46 2.7 Computer Simulation The active-passive system as depicted in Fig. 2.5.3 has been modelled by means of a "Continuous Systems Modelling Program" (CSMP), an IBM product for use on their System/370. The programming language consists of a number of functional blocks such as integration, differentiation, etc., in addition to all the usual mathematical functions available in Fortran. These blocks are assembled in much the same manner as an analogue computer network, but without the inconvenience of scaling variables. Integration can be performed using any of five different built-in routines; the one employed in this project is fourth order Bunge-Kutta with fixed integration interval. This method was selected because it is found to be the least expensive for the degree of accuracy required. The logic flow of the program is exactly as depicted in Fig. 2.5.3. The listing is shown in Appendix F.1. 47 CHAPTER III LINEAR ANALYSIS The equations of the active/passive motion compensation system developed in Chapter II are difficult to handle without extensive use of computers. In order . that the designer can quickly gain a feel for the problem and thereby establish first approximations for important parameters, I have simplified the equations to permit a fast approximate solution. The simplified approach uses linearized equations and a frequency-domain solution. 3.1 Linearized Passive System In linearizing the passive pneumatic system, I have assumed that the changes in pressure within the cylinder and tanks are linear with respect to the piston displacement, y, and that y is too small to affect the temperature in the system. This leads to the following: 1. S=Ay: i.e., the perturbation in piston displacement is small, and denoted by S. 2. P1+P2=2P0; i.e., the average pressure in the cylinder is constant, and equal to the quiescent pressure P0 . 3. P1:=-Pa; i.e., the rate of increase of pressure on one side of the cylinder is equal to the rate of 48 decrease on the other. 4. The temperature throughout the passive system is constant and equal to T<>. Furthermore, the geometry of the system is assumed to be symmetric, which leads to the following: 5. At+AZ=2AP; i.e., the effective piston areas are averaged to a constant Ap. 6. Vc, + VCi=2Vc; i.e., the tank volumes are averaged tc a constant Vc . The gas flow equations are linearized by first considering the mass flow through the throttling valve as given in (2.2.12), and restated here: where (3.1.1) Considering the upstream pressure, Pw , as constant and equal to the guiescent pressure Pc, and small variations in pressure drop (Pe -Pa ), equation (3.1.1) can be linearized into 49 the form: A w -# Cr A (P.-Pa) (3. 1. 2) where 9w- 5>v^ 3Cfl/P.) S>CPJ/PO) 3(Po-^) (3. 1.3) The flow through the valve will be positive for the half cycle when the piston moves one way, and negative for the other half. Therefore, (P0"P<i) can assume positive or negative values. It is thus necessary to select m0= (Pe-Pd ) =0 as the equilibrium point about which perturbations are considered. However, this will yield infinite value for Cr, since the slope of the m vs (P0 -Pa ) curve, as shown in Fig. 3.1.1, is vertical at Po-Pd=0. It is therefore more reasonable to select values of Va, Pj which coincide with some average operating condition, for example the root-mean-sguare value. The equilibrium point, however, is still the origin. Renaming Pe and Pd to correspond to the tank and cylinder pressures, the linearized mass flow equations are: 50 = Cr f Pt,- P.) (3. 1.4) Equation (3.1.4.) is plotted together with its nonlinear form in Fig. 3.1.1. The cylinder flow equations are considered next. Incorporating the simplifications of geometry as discussed above, equation (2.2.9) becomes: v'+ A's p, + P. (3. 1. 5) The net mass flow into the cylinder is given by (3. 1. 6) Incorporating the approximations of pressure as described abcve, and introducing the differential operator B= ~t, equation (3.1.6) can be rewritten as: FIG 3,1.1  PRESSURE - FLOW CURVE {or TMROTTLIKJG VALVES 1 + 2 s/u)A 5 FP N+l L1 + 'N+t SJ FIG. 3.1.2 LINEARIZED PASSIVE SYSTEM TRANSFER FUNCTION! 52 (3. 1.7) Finally, the flow out of the receiving tanks is considered. Equation (2.2.4) can be rewritten, in differential operator notation, as: DP*. Vi (3.1.8) Solving (3.1.4.) for Ptl and pti and substituting into equation (3.1.8), the net flow into the cylinder can be expressed as: /. , x • vt r PCP.-PQ 1 (3. 1.9) Equating (3.1.9) and (3.1.6) and solving for P-P gives P. - P 2Y APP» (3.1.10) I + Vt/Vc  vt— D The force exerted by the ram is then given by 53 FP = Ap (p, _ Pz) (3.1.11) and the overall transfer function of ram force tc piston displacement is ^ _ 2 y Po AP (3.1.12) L 1 T TCR RT. U J Lumping parameters and introducing the Laplace operator, s, in place of the differential operator, D, the overall transfer function becomes N + 1 1 + 2 s 1 + (3. 1. 13) where K*= 2VyfcA' = Static stiffness, N=Vt/Vt= tank to cylinder volume ratio, AND 2S/CJN= YCVRT. The parameters w„and 3 are the natural freguency and critical damping ratio, respectively, and are ultimately a function of the mass which the system must control. In particular, 5% [W7 W„ - 7 H , J "" 2rcVET0 (3.1.14) The linearized transfer function Gp(s) is now extended to cover the full range of piston displacement, y, such that FP = GpGOi, (3.1.15) The transfer function is shown in block diagram form in Fig. 3.1.2. Introducing the time constants f,= 2%* ; Ta » — (3.1.16) the transfer function may be restated as Cf (i) (3.1.17, 55 3.2 Linearized Active System The equation of the active system, as presented in Section 2.3, is restated here: 9 = ^z v^ - ?a/aa ~ x V^/A* (3.2.D Equation (3.2.1) is to be linearized about some operating point (Yo »z0 ,FA(>) . Perturbations about that point can be represented, by A-notation, as Ay = (A., A* T- A^Ap;) + A3 AFA (3.2.2) where (3. 2. 3) ^* Sf )SETV6 ~ ^A^rTT 7=—71—1 (3.2.4) ..Cy ^S-UFJ^ ~ ZAZ JKJA* (3,2'5) The constants A, and X2 are termed the "flew gain" and > "flow-presure coefficient" of the servovalve, respectively, and 56 ^3 the "flow coefficient" of the bypass valve. Friction is net considered in the linear analysis because of the discontinuity a t y= 0. Equation (3.2.2) is valid only for perturbations about the operating point (ye ,z0 ,FAo) . However, during normal operation of the valve the operating point can travel in a band spanning both the negative and positive regions of j , z , and FA. The linearized equation then becomes inappropriate in its present form. It is proposed that the perturbations y, z, and FA be centred about the origin, i.e., y, =z„ =FAo=0, but that ">3 be calculated about a root-mean-square point using equations (3.2.3) to (3.2.5.). Equation (3.2.2) can thus be rewritten as y = + FA (3.2.6) and (Yo#zofFAo) can ^e considered as the BHS operating point. The effect of linearizing the servovalve equation is to change the family of parabolae to one of straight lines, as indicated in Fig. 3.2.1. The linear system thus developed is found to model the active system adeguately over the entire operating range. The transfer function is shown in block diagram form in Fig. 3.2.2. 57 ~FI6.. 3.2.1 LINEARIZED SERVOVALVE CHARACTERISTICS BYPASS VALVE r- CYLINDER • A —————— A FORCE -* SERVO-ACTUATING SWJWAL A, SERVOVALVE PISTON VELOCITY FIG, 3.2,2 LINEARIZED ACTIVE SYSTEM TRANSFER FUNCTION! 58 3.3 Linearized Active-Passive System The linear transfer functions of the active and passive systems are combined in the same manner as in Section 2.4. It will be necessary, however, to first linearize the body dynamics transfer function. Recalling equation (2.5.4) FN&T * M X + Cw X (3.3.1) it is seen that only the damping term is nonlinear. This can be linearized about i=0, giving FNe1 - M* + Ct X (3.3.2) where CL=Crt ||7=2CHie and x, is taken as the root-mean-sguare velocity. CL is then the linearized drag coefficient. The differential equation of (2.6,4.) can now fee linearized to give -X + ^ "X + C0c X = X, (3.3.3) The transfer function of the compensator motion x1# tc body motion x can then be expressed, in Laplace notation, as 59 *. 1 + 2%». s + s7«2 <3-3-'" where ic 2-MtOt and Fc (s) is the cable transfer function. The transfer function of body displacement, x, tc cable tension, FHfrt , is derived from (3.3.2) FB(s) = % = MsSCLS 0.3.5) The block diagram of the linearized cable and body dynamics is shown in Fig. 3.3.1. The block diagram of the linearized motion compensation system is shown in Fig 3.3.2. F 16, 3.3.1 LINEARIZED CABLE/BODY TRANSFER F UMCTION Ny(s) + Ksv z Xi FIG. 3.3-2 BLOCK DIAGRAM of LINEARIZED ACTIVE- PASSIVE SYSTEM 61 3.4 Performance Analysis and Optimization As a first approximation to system performance, the linear model derived in Section 3.3 will be examined in the freguency domain. It will be most convenient to use the ratio of body to surface ship displacement, x/u, as the criterion of performance. Due to the linearity of the system, a minimum x/u is equivalent to minimum variation in FNE.T /U. Solving the block diagram of Fig 3.3.2 yields the overall transfer function: X6) _ FcCs) Uts) | + W(s) (3'a*1) where H (s) is the open-loop transfer function, given by: In designing an active-passive system, the absolute value of the closed loop transfer function, |TJ(S)|, is tc be minimized within the range of operating frequencies. This in turn yields H (s), the closed loop transfer function, a maximum. In general, the designer has no control over the cable and body dynamics, which are represented by Fc (s) and F6(s). Furthermore, he has very little control over the passive system. 62 Gp.(s) , since the primary design criterion for that system is to be able to carry the static weight of the towed body.1 Thus, in maximizing H (s) , it is necessary to design HF8(s), HFF (s) , Hy(s), and to select a suitable servo-valve and hydraulic cylinder as represented by A| and 3.3.1 The Feedforward Element The first step in maximizing H{s) is to minimize its denominator. Ideally, it is set to zero, which yields = " ^ u S ^^X^Wl^V Uy£0l (3.4.3) As pointed out earlier, Hy (s) is the ram centering network. Because it is relatively slow-acting it has little or no effect at operating frequencies. Thus, it can fce deleted from (3.4.3). The feedforward compensator can now be given as Hf^) - - rrj—, [s + (VWo] (3-'-'" 1 The stiffness of the passive system is usually specified in order that the active-passive system he capable of operating in a purely passive mode when working in a low sea state, or in case of a power failure in the active system. 63 3.4.2 The Feedback Element The second step in maximizing H (s) is tc maximize the numerator. An examination of (3.4.2) reveals that this can be done by setting the gain of Hra(£) as large as possible; however, the servovalve saturates when the actuating current, z, exceeds a critical value, z0. Therefore, HFB (s) must be optimized with respect to the above constraint. As is customary in position servos, the feedback element HFB (s) is assumed to be a combination of displacement, velocity, and acceleration feedback. Thus, K, + I4S + SZ (3.4.5) where K j , Kz and K3 are constants. This yields a servo-actuating current z, given by ZM = KSV[HFF6)U(S)T- WpB(OX(sT>] (3.4.6) (The ram centering network, Hy(s), is again ignored because it has negligible effect at operating frequencies). The amplitude of the actuating current, |z<s)|, is now constrained to be less than or equal to the critical value, ze: 64 |Z(s)| - Ksv| HFF(s)U(s) + WFB(s)X(s)| ^ (3.4.7) For convenience, (3.4.7) is rewritten in terms of ratios: Z(s) - 1/ where HFF(s)+ H, Is (3. 4. 8) u„ is the amplitude of the input, ju (s)^ , and — (s) is the closed loop transfer function as given by (3.4.1). The optimum values of K,, Kz and K3 are fcund using a digital computer program listed in Appendix T.3. The optimization process is carried out at one frequency only, that being the "design frequency" which contains the greatest energy. 3.4.3 Ram Centering The ram centering network is required to maintain the long term average ram displacement zero. Dnder static conditions, this is equivalent to returning the ram tc centre position in a given (long) time after receiving a step input. Considering the centering loop as shown in Fig 3.4.1, the active ram force FA can be set to zero under static conditions. Q ^ r Z + ——* "FIG. 3.4.1 RAM CENTERING NETWORK 66 This gives a closed loop transfer function of Y(s) Letting Hy(*) = Ky = constant gives (3.4.9) (3.4. 10) where It now remains to select Kv such that where 0)„ is the design frequency. 3.4.4 System Stability The stability of the system is determined by solving the characteristic equation 1 + U(s) = O (3.4.11) and examining the root locus as the feedback gains K, , Kj_ and K3 are varied. In general, the characteristic function is cumbersome to deal with, and the following simplifications are therefore assumed: 67 1. Let Gp (s) = -Kp i.e., assume the passive system behaves as a simple spring (where Hccke's Law applies). 2. Let Fc (s)=1 — i.e., neglect the cable dynamics. This is reasonable because the effect of the cable cn the motion compensation system is relatively small, except at the natural frequency of the tewed system. This frequency should lie outside the range cf cperaticn. 3. Neglect Hy(s) as explained earlier. With the above assumptions, (3.4.11) can be expressed as: (3.4.12) The application of (3.4.12) to the laboratory rcodel is discussed in Chapter IV. 3.4.5 Power Consumption The power consumed by the active system is the SUIT cf the power dissipated in the servovalve and the power required to drive the load. This is equal tc the product of the volume rate of oil flow into the valve and the supply pressure: 68 W = AA ?s\i) (3.4.13) where W is the instantaneous power consumption. Note that although y changes in direction, W is always positive. This is due to the fact that the direction switching occurs in the valve itself, but the oil flow to the valve is unidirectional. Thus, the average power consumed is the roct-mean-sguare of the amplitude of W: and W is the average power consumption. It is convenient to express "w in the freguency domain as a ratio to input displacement U(s): (3.4.14) w her e 0.707 is the rms factor for a sine wave, Ps is the supply pressure, AA is the ram area. onoi AA R I6,A FcCs) uCs) (3.4.15) 69 where v 2-(s) is the closed loop transfer function given in (3.4.1.) The reason for the inclusion of an accumulator in the active system now becomes obvious: the hydraulic power supply needs only to provide power at the steady rate W, while the accumulator supplies (or absorbs) the difference between the average and instantaneous values. To avoid wasting power when the accumulator is fully charged, it is necessary to use either an unloading valve or a pressure compensated pump. This way, the pump does not discharge through the relief valve (at high pressure) when there is no flow demand. 70 CHAPTER IV THE LABORATORY MODEL 4,1 General Description A laboratory model, closely resembling the equivalent system discussed in Chapter II was constructed to validate the mathematical and simulation models developed, and as a prototype small scale mechanical simulator for designing and evaluating full scale systems. The apparatus consists of a hydraulic power supply, active and passive cylinders, a variable frequency displacement generator, and various pieces cf electronic monitoring, processing, and display equipment. See Figures 4.1.1 and 4.1.2. The power supply consists of a variable displacement pump of five gallons per minute capacity driven by a 5 BP electric motor. A pressure relief valve is provided on the discharge of the pump, which can be set to between 100 and 2000 psi. The entire unit is mounted on top of a 15 gallon oil reservoir. The motion compensator consists of a pair of cylinders mounted in tandem on a carriage, Fig. 4.1.3. The carriage is given an approximate sinusoidal displacement cf variable frequency and amplitude by means of a crank mechanism driven ty a 1-1/2 HP DC motor. The piston rods cf the two cylinders are pinned together, so that they act in parallel, rather than in 0 GAS STORAGE <2 5) X ^4 ACCUMULATOR 9 iXl-SERVO VALVE ® 2 PASSIVE ACTUATOR ACTIVE ACTUATOE I RELIEF a_VALVE PUMP FIG: 4.1.2 RYDR.AU LlC SUPPLY LABORATORY APPARATUS - SCHEMATIC PASSIVE. CYLINDER ACTIVE CYLIMPER 7a series as the arrangement might first suggest. The other end of the double-ended piston rod is pinned to a second carriage which contains weights to represent the mass whose motion is tc be isolated. The passive system is purely pneumatic, consisting of a pair of gas bottles, one connected to each end of the larger of the two cylinders described above. A flow control valve on each gas bottle is used to adjust the damping of the passive system. The active system consists of a gas/oil accumulator which feeds oil to a servovalve through a filter. The servovalve controls the flow of oil into the two ports of the smaller of the two cylinders mounted on the carriage... A gas line tc the top of the accumulator controls the charge pressure of the system. The control system monitors body and ram motions by a seismic velocity transducer mounted on the mass carriage, and a displacement potentiometer cn the ram, carriage. The signal processing function is achieved by an analogue computer which integrates, amplifies, and sums the two signals. A filter is also included to remove high frequency noise, such as the type generated by the wheels of the carriage. The processed signal is fed to a power amplifier which drives the servovalve. An oscilloscope and chart recorder are used to monitor any twc of mass velocity,mass displacement, input displacement, or low level servo-actuating voltage. (Fig. 4.1.4) DISPLACEMENT TRANSDUCER OSCILLOSCOPE VELOCITY TRANSDUCER BAMP- PASS FILTER DIFFERENTIATOR Ttt) TWO-CHANNEL CHART RECORDER SE.RYO-AMPLIFIER SIGNAL TO SERVOVALVE FIG.. 4. 1.4 CONTROL SYSTEM 76 The apparatus as constructed does not model the spring (i.e. cable) nor the hydrodynamic drag of the body. all other aspects of the equivalent system are included. The exclusion of these two parameters does not affect the model of the motion compensation system because they are both properties of the towed system. 77 4.2 Performance Prediction and Evaluation The physical properties of the laboratory model are given in Appendix D. The design is not based on any particular reguirement, but employs hardware which was readily available. The input amplitude was set to 1.5 inches, and the design frequency to 1 Hz. The velocity transducer used to measure the motion of the body was guite unsuitable at the low frequencies at which the system could operate, and as a result some electronic signal processing was necessary tc obtain a useable signal. As a result of this, only acceleration feedback was available for use in the control system. The stability equation (3.4.12) can now be expressed using acceleration feedback: As2 + BS + C = O ,n o -n where B = Oz +^3) Cu - 1 C ^ C ^ + ^3) kip By making the acceleration feedback K3 negative, the coefficients A, B and C are always negative (note that and >3 are negative, while Kp is positive). Therefore, the system is 78 stable for all negative values of K3. A point to note is that the feedback constant, K3, has the effect of increasing the apparent mass of the system, as shown in (4.2. 1). Because only one feedback variable is considered, it is not necessary to use the optimization technique outlined in Section 3.4. a feedback constant of K5VK3= -5 ma/(ft/sec2) was used in the experiment. The experiment was conducted at three frequencies: 0.5, 1.0, and 2.0 Hz. In each case, the system was first run in the passive mode (K3 = 0) , then with Ksv K5=-5. The computer simulation was then conducted under the same conditions, and the results of both are shown in Appendix G. The time-domain records are then transformed tc the frequency domain, and plotted as distinct points on a Eode plot in Fig. 4.2.1. The linear model response is also plotted on the same graph for comparison, over a frequency range of 0.1 to 10 Hz. There appears to be good agreement between the mathematical models and the real system. Any discrepancies are due tc the uncertainties involved in estimating hydraulic and pneumatic throttling coefficients and mechanical friction. However, by 79 designing a real system with variable throttling valves, the former uncertainty can be removed since the real system can then be matched to the model. Friction, on the other hand, can neither be easily predicted nor altered once the system is operational. OVERALL RESPONSE X/U 80 LIMEAfc - PASSIVE (K»=o) LINEAR-ACTIVE A EXPERIMENTAL -SIMULATION 0.10 0.20 0.50 1.00 2.00 FREQUENCY RRTI0 5.00 10.00 FIG. 4.2-t THEORETICAL 6s EXPERIMENTAL RESULTS 81 CHAPTER V APPLICATION This chapter is intended as a guide tc the application of the mathematical and computer simulation models tc the design of real systems. 5.1 Input Conditions The input, as stated in Chapter I, can be approximated by a Fourier series representing the vertical displacement cf the surface of the water at a given point. The Eretschneider eguation1 can be used to obtain an estimate of the spectral energy density from the average height and period cf the seaway for a given Sea State: SCT)= ^LL'T-e^-^T*] where 2 S (T) is the spectral energy densxty in ft /sec, T is the wave period in seconds, "f is the average wave period in seconds, and h is the average peak-to-trough wave height in feet. i See Ref. (12) 82 Values for h and T can be readily found in most nautical handbooks. Figure 5.1.1 shows a typical plot of the spectral density function for Sea State 4, having mean wave height and period of 4.9 feet and 5.4 seconds, respectively. A ship subjected to a multi-frequency displacement input will behave as a low pass filter, and will not respond significantly to waves whose length is less than one-half the ship's length, fi typical response curve is shown in Fig. 5.1.2. The motion of the ship is then the product of the sea state spectral density function and ship response, as shown in Fig. 5.1.3. The wave component which contains the most energy is found to have a period T0, and 0)o=27r/To is used as the primary design frequency. Figure 5.1.3 can now be used to obtain the coefficients A-t of the Fourier Series which can represent the motion of the ship in the time domain. This is done by dividing the Ship Motion Spectral Tensity Function into n cells spanning the entire range of period, and calculating the energy associated with each cell: <5. 1.2) StT) 20 -\ FIG, S.LI  5£A STATE "4"  SPECTRAL DENSITY 10  FUNCTION o 10 F16. 5,1.2  SHIP HEAVE RESPONSE SHIP ME AVE SbCT) ft'Aec FIG. 5.1.3 4oH SPECTRAL DENSITY FUNCTION 20 o -i o 84 AS •• J S(T)«IT H Z (5.1.3) where &St is the energy associated with the i-th cell, T is the central period of the i-th cell, and AT is the cell width. The coefficients A^ can be expressed as The approximate ship displacement as given by (5.1.2) can then be used in the simulation model to give a realistic input condition. It is not necessary to use more than three to five terms in the series to give a good wave profile. (5.1.4) 85 5.2 Two-Dimensional Cable Model when dealing with long cable lengths (over 2000 feet) or horizontal motion with respect to the water, the cable assumes a catenary shape which can no longer be assumed one-dimensicna1. Walton and Polachek1 developed a program tc compute the shape and tension of a cable subject to a displacement boundary condition at one end and hydrodynamic drag along its entire length. In essence, the continuous cable is modelled as a number of elastic links pinned together end-tc-end. Each link has masses concentrated at its two ends, as well as longitudinal and transverse drag coefficients. (Fig. 5.2.1) The towed body is represented as the last link of the cable, given the appropriate values for mass and drag coefficient. The program uses a displacement input at the top (i.e. surface) end of the cable, and calculates the displacements and axial elongations of all the links. This in turn yields the cable tension in each link. The model of the motion compensation system developed here can use this cable/body model by supplying the variable as the boundary condition, and receiving FNET, the cable tension at the surface tow point. If desired, the compensator motion can be resolved into horizontal and vertical components to increase the model's realism. i See Ref. (23) 86 HEAVE SUR&E t (b) CABLE SEGMENT 87 5.3 Servo-valve Hodel Extension The servo-valve considered here is assumed to operate instantly upon application of a control signal. However, in the case of large valves which are usually multi-stage, there is considerable time lag even at low frequencies. The dynamic response of such valves can be considered as a first cr second order system, depending on the accuracy desired. The valve equation then becomes (5.3.1) where Hsv(s) is the dynamic characteristic cf the valve. In the case cf a first-order valve, Hsv(s) = (5.3. 2) 1 + tsvS and for a second-order. c (5.3. 3) where Csv is the valve constant (as before) , T$y is the first-order time constant, 0)W is the second-order natural freguency, and 5*v is the second-order damping ratio. 88 The parameters t„Vf o)w and Ssw are estimated from the response curve supplied by the valve manufacturer, Fig. 5.3.1, They are chosen such that the phase lags of the real valve and the model coincide over the freguency range of interest. When operating near the resonance of the valve (which is generally not recommended since servo-valves are usually underdamped) a higher order model may be necessary. 89 FIG, 5.3.1 SERVOVALVE RESPONSE 90 5.4 Control System Considerations In setting down the performance reguirements of a real system, the frequency response must be carefully considered. In particular, long period.waves generally have larger.amplitudes than short waves, hence it is not always possible to compensate for them as effectively due to the limited travel of the compensator. Therefore, it is desirable to design for 2ero compensation at high wave periods (in the order of 20 tc 50 seconds), increasing to maximum compensation at the design frequency. An acceleration feedback system will inherently behave in this manner. At frequencies above the ship's natural frequency it is desirable to decrease compensation since the ship does not respond to such waves. Furthermore, shipboard vibrations due to the engine and propellors may be significant above one Hz. A typical frequency response which would give acceptable performance is shown in Fig. 5.4.1. The low frequency cut-off can be moved to the left by either decreasing the stiffness of the passive system, increasing amplifier gain, or increasing the time constant of the ram centering loop. The point of maximum compensation is set by introducing a second-order low-pass filter, Fig. 5.4.2. The corner frequency coincides with the design frequency, where motion compensation is maximum. The critical damping ratio determines the bandwidth 91 CO o < UJ 2. -2o4 FREQUENCY FIG. 5.4. I MOTION COMPENSATION TRANSFER FUNCTION 1 Slfi-KJAL TO f ACCELERATION 1 + 2 S/Us + S*A>2 SERVO-AMPUFlfcTR. ui0 SETS DESl&M FREQUENCY 5 CONTROLS BANDWIDTH FIG. 5.4.2 TYPICAL FEEDBACK NETWORK 92 of the response curve. Such a system to virtually any sea state system is designed to handle the feature can be used to improve an adding on an active one. filter can be used tc tune the condition, provided that the corresponding amplitudes. This existing passive system by 93 CHAPTER VI CONCLUSIONS The dynamic behaviour of an active-passive motion compensation system has been analysed and a mathematical model developed. Experiments performed, on a laboratory apparatus indicate that the system is adequately described by the equations derived. The mathematical model has been simplified by linearizing the equations, and computer programs have been developed which can assist in the initial design of real systems. In addition, a program which solves the nonlinear equations by simulation has been written, and can be used to refine the initial design. The programs are flexible enough to accomodate a variety of system configurations. This project, in essence, has provided a design tool, based on mathematical analysis, to an area which has traditionally relied on seat-of-the-pants engineering. 94 REFERENCES 1. Athans, M.: On the Design of a Digital Computer Program for the Design of Feedback Compensators in Transfer Function"Form NTIS~AccT~#AE-700-4 31 2. Blackburn, J. F.: Fluid Power Control MIT Press, Cambridge, Mass,, 1960. ~ ~ ~ 3. Buck, J. R. 6 Stall, H. W.: Investigation of a Method to Provide Motion Synchronization During Submersible Retrieval laval Eng. J., Dec. 1969. 4.. Burrows, C. R. : Fluid Power Seryomechanisms Van Nostrand Reinhold Company, London, 1972. 5. Cavanaugh, R. D.: Air Suspension and Servo Controlled Isolation Systems Shock &. Vibration Handbook, Ch. 33, McGraw Hill, 1961. 6. Guillon, M.: Hydraulic Servosystems Analysis and Design Butterworth and Company, 1969. 7. Hedrick, J. K.: A Summary of The Optimization Technigues that Can Be Applied to Suspension Systems Design Ariz."stateT u7 Report~#PB-2205537 8. Karnop, D.: Vibration Control Using Semi-Active Force Generators ASME~Paper # 73-DET-1227 ' ~~ 9. Keefer, I. G.: Improved Hydropneumatic Tensioning Systems for -Marine Applications B, C. Research Council Report, T9727~~~~ 10. Kriebel, H.: A Study of the Feasibility of Active Shock Inginieur Archiv Berlin, Vol. 36 #6, ^968. 11. Mercer, C. A, & Rees, P. L. : An Optimum Shock Isolator J. Sound S Vibr., 18(4) 1971." 12. Myers, J. J.: Handbook of Ocean.and Underwater Engineering McGraw Hill,~19697 ~ . 13. Porter, B. & Bradshaw, A.: Synthesis of Active Controllers for Vibratory Systems J. of M. E. Sci., V. 14~#5, 1972. 14. Raven, F. H. Automatic Control Engineering McGraw Rill, 1968. ' ~ . " . ~~~ '• 15. Ruzicka,.J. E.: Fundamental Concepts of Vibration Control Tech. Inf. Service, AIAA~Doc7 #172-295557 16. Shinners, S. M.: Modern Control Systems Theory and A£Eiication Addison Wesley, 1972. 17. Soliman, J. I. 8 Tajer-Ardabili: Active Isolation Systems Using a Nozzle Flapper Valve Inst. M. E. Proc., V. ^82 #30, 1967." " 18. Soliman, J. I., & Tajer-Ardabili: Servovalve Controlled Isolation Systems Inst. M. E. Proc. ,~v7~ 185~#107"970. 19. Sutherland, A.: Mechanical Systems for Ocean Engineering Naval Eng. j77~Oct."1970." ~ ~ 20. Thompson, A. G.: Quadratic Performance Indices and Optimum Suspension Design Inst. M. E. Proc, V. 187, 1973. 21. Thompson, A. G.: Design of Active Suspensions Inst. M. E. Proc., V. 185,~1970. 95 22. Thompson, A. G.: Optimum Damping in a. Bandojl^ Excited Nonlinear Suspension Inst. M. E. Proc, V. 184, 1969. 23. Walton S Polachek: Calculation of Transient Motion cf Submerged Cables Math. Tables 8 lids to Computation, V. 14, 1960. 24. Yeaple, F. D.: Hydraulic and Pneumatic Power and Control McGraw HillT 1966?" APPENDIX A & AS FLOW SOU A TlOhJS /• FLOIAI ll^TO A CLOSES VOLUME FIG- A-i ..RATS OFCU-AAJG E OF^EMTN-ALPY IA/ITM*) COUTfLOLVOL UM£ .OFA-i : 6J4 THE: /zA-re OFcm+JQE. OF I^JTEY^JAL En/ee&Y IS TW= FLOlAi HATS OF GAS MULTIPLIED BY ITS UM/T Ik/TE(LtJPrL g^eZC^: . : 7V£ RATE OF CH-AtJ&E OF GtJER^Y IS WE WOfM DOAJZ OS) THE GAS BY EXPAfiJSiDkJ d(Z CJDHPHE SS(OtJ OF TI4C COKfTR-Ol VOLUWC •• ~ • r dt •• • , SU&STi TUTltJG, (A-2) /WD CA'3) /A/TO (A>I ) S0L\JtA/6 FoQ. m AtiD Su&STFTUi IfJG, ± + J- - ± IAJTO (A-4) ., I r dt + " aj CA'2) (At) (4*0 2. FUHAJ THROUGH A tJeeo>te VALVG FIG. A-2 7?/£ KIEEOLC vAL\J£, pl&- A-2. , Co*JSlS>TS OF A VARIABLE A&tA ArJMUUJS, Av , WH-iCH /S C0AJT&OLL&D BY RMSI/J6, O& LoujeZ-uOCj A TA-Pe&.&0 h)&&DL£. 77V/S C&rJ BE MODELLGb 8/ A CPAJVS^^F^T tJoZZLZ. APPLYIK]& THB B*J&££Y &2UAT/O/J ~* Liu. = H/ + fZ 'J CA-l) \rifr9&£ .sir- i$ THE A<JEGAG& VBUoCllY ACROSS Ay. A-SSVtftiUG- TH-£ FLUID lS A PEIZFFCT GAS CpT* =• CpTd + £<r-*. (A-?!) JUS MASS FtOuJ dATE IS, 6(\JFtJ 3Y y * = f A„v- . (A-lo) Su6ST(rUr/iJ6 (A-Z), (A-I) MTO CA'/OJ Givei: SU&STiTVT 1^6, TH£ F$UA7(0AJ or srA-rE (oP _ _JL ' • •• • IK/TO (A-(0 Gives--Sl/JCe 7V€ VALVE IS (JoT A+J IDEAL Coi<J\/BRiS,lfJCj A/OZZLE" , . ir is TH-edGFc&G juerc^ssAieY ro //JT&ODOCE AA/ £MPlQiCAL 0lSa4A£$ £~ COEFFICIENT , C# I/0 TO LTQUArnotO (A-lZ) . 7H(S GD&FF^ia-G+JT C~A*J TU&O BET CDtigM>E-D IAJ iTR Av, SUCH 7UAT j Co = CzAv (A 14) : Wtt£(2£ Q IS A FVK)CTIosJ OF fJEEbLE POSITIVES-Q is pEredHifjeb EXPF&IMSAJTALLYJ A*ir> USUALLY PUBLISUFD BY MAdUFACTUHZas FOP. WBIR. VALVES. .. .£&UATIOAJ (A-13) CAU JUVS 6€ ^/P/ZFSSFO AS; i_ Nore 'THAT. - (A-IS) IS OIJLV VALID FO/Z i.£.} Pod poiA/Aisr/ZEAM pzesruze &z&ATFe. TUAA) THAT ae&uieep FOR. CFOK^O FLOU). Foe. hjirnoze/O AA/O Ate , Pc = O.S2g Pu APPENDIX £ UYDgAULIC SErZxlO-VALVES  TYPES QF VALVES: VALUES c&fJSipeizeD tfE&e AUB *FOOQ.-WAY SPOOL VALV^ GDA)St£Tl>J6 OF ..^ H&T£tllh}$ ORlFlC££, FI6 B.I. . . SUPPLY BMAOST MZDM/To LOAD . TO/FROM LOAD FIG. B. I SPOOL VALVE 'THIS A£&filJ6&M&JT CAfiJ B€ MODFLteb BY A NYDZAULlC WHEATS TotiE Mlb&E, F-fe. E>1. :~-f=l4. 5.2 WHEATSTUME 0ftD$E TUB HYDRAULIC ZESlSTAiJCeS . A&G ; OF <JZ>U/2£b, hJOfJLIMBAe. SC.R.VOVALVES WAY BE UAJPE FLAPPED, OVE^LAPPED , °& "ZERO - LAPPED AS Sht-ovJ^ /A) FlC-j. S3 . rl\ t ZERO - LAPPED 1— 1 n 0V£.g- LAPPED . r i .utJbefL-LAPfen F\<k. 8-3 VALVE SPOOL DE£f6fiJS 0VF2LAPPED VALUES EXf+lBiT A DEAD Zc?A/£ ABOUT . THBIP. cEUTPE POSIT'OMj X = 0, WH-EtZEAS UMDEHLAPPEb VALVGS AHE FAi&LY LtHEA/P. ItJ THAT HEGIOhJ. UUDE/Z-LAPPltJS, TEMPS TO E^UAtiCE STABILITY. OF WE SEMD SYSTEM AT raF expense OF PDMEQ. LOSS AT X-O. /M wis ANALYSIS , 2ee.o-LAfpeO VALVES AgE LEAF 101 OMITTED IN PAGE NUMBERING. \ o \ does w?T~ ex i if 2. PERlVAVON OF PZESSUZa-F-LOvJ EQUATIONS _A ZBHO - LAPPED VAL\lE MAY BE MODELLED SY THE SlMf>L(Fl£D Cl&CUtT OF Fl<*. B-4-s o- LOAD P ~ O FIG. 84 2ERO-LAPPED VALVE IF COMPRESSIBILITY h/IT(Jl*J THE SYSTEM /5 AJFej LECTEO, TUEM I....- -• ' • • • - . ASSuHthJG THAT OdiFtcE AQEA IS PgoPoZT ION A L To DISPLACEMENT, X,- ME. WAVE THE FAMILIAR ORIFICE FLOvS .EQUATIONS - .- — ;~. -• • .>' • -i WHEZE P&„ KX /.Fs - P, ..= QL) .. ICY Tr\ = QL (B.2) SUPPLY. peessueE A COKlSTA-fJj $ QL .» '. LOAD FLOW COM&/Nlid<$ 77-/£ TvJQ EQUATIONS OF (&-2) A^D R.EAR.ZAN^IM&,, YIELDS[..:..!. VlUE&E QL ~- | X /P&-.AP AR = P, - P, FOR. THAT NEGATIVE SfOOL DISPLACEMENT (X^-O), NOTE 0, QZ - O (6-4) 103 WIS Y/BLDS . QL = J X Jps +&p' CVHBll/llJG, (B.3) Arhib , AND LETTltJG Wi+ichi is Thit C&HZIZAL S&MO VAL\J€. EQUATION)'. . APPENDIX C COMPRESSIBILITY EFFECT OF MDRAULIL FLU ID CoUSWtd TUB CYLlMPStZ. SPorili! iN F/G- CJ • 0; k Q0 -1 . . 1 , p, p2 1 1 — FIG,. C-l TH-e F-LOti' OoT., QQ t. IS. . ErQAf&L TO THE FLOIAI IhJ j Q{ f L6SS THE IZATE OF ....FLUID COMPRESSION IHSlPE, THE "COMPRESSIBILITY FLOW'',.. 6?c. 4 A tfc Y... IS THE \/OLUHE OF TUB CYLlriOEZ, 3 IS THE SULU hiODOLOS OF THE . FLuib, dP/Jt THE RATE OF CHA+IG, E OF X. .„ pa ESS UZE .... isJ Tti€ CiLWlDER • .' .. .'FOB. THE FOUJVALEKJT MODEL DISCUSSED IAJ THE TEXT, V ~ 4-10 m3 . & - 3>5ooo psi 5 P. 2oo(l..+ sin oji) , IAJU-ERE CJ^ ^ 4 Hz. 25" raJ/sec. _ . mis Gfives ft = 3oo u> coso)t 3 dt VJUEA£ TH15 Gives (Qc)mx = $J^00 « 75-00 = 0,37/ /V/«c £0£ MAXIMUM F-LON CONDITION, - Q i = Ay wH-E-zE A = PISTON AREA -. 0.393 />2 4. y = P(ST6>N VELOCITY. COUSIPERHJG FULL 12-iNcu STROKE • y =• <S ?w £jt Cf -& cu cos cot 6j)i*c-=V ^ 'ISO m/scc 7-/^/S 6/i/<5S ..... (5/ ....= . 0<393 x /s~£> : _: *=..59. in3/sec. THIS <s,iVES THE HATiO - 0.6°Jo T14E COMPRESSIBILITY FLOU) IS THUS NEGLIGIBLE. CONSIDER. uoiN A FULL-SCALE SYSTEM, TYPICALLY, V - 5O00 in3 (8" DIA- * 8 FT LoNC P " IOOO (I + Sin Ot) WHERE comA = 0.2S' Uz = /.£\7 taj/sec. mis Gives C£)»*X = 1000* /• 57 * isio psi So 00 NENCE Oc - /S-70 = 83 }/73/s^c 106 mz. iKiter FLON RAI& IS Noui Of = -Aijnw = SO x 48 x },SI (In*/sec) - 3110 in3/sec-miS Cwes, THG RATIO Qc _ _B3_ Q; ZTlO 2.2% THIS IS ALSO iJe&LiG I8LE. APPENDIX 0 LABORATORY MODEL 'SPECIFICATIONS'  /• INPUT • ju0 .- .. i. \{ " . = i O.I2Sfh - 2. LP AD  M = So «sLu^-s *=* 97 !i>-iwass. . tfc = <*> ...;  3- PASSiy/E ''SYSTEM' Ll -P0 - 0 psig - X5psia. Aj ~ 314 inz ...... /42 = 2.95 in* ...... : L: . ; ... Av. AREA •• Ap - • 3-C4 i/i* ' . : 1: Vc = ZO /n* ' • ,! .: ...... j •v .4 : • V* = . 2.SO m3 ^ A/ = £ = /4 . . /i:'r. .. STIFFNESS: KP - ^777 2r^4p " = ..T-yXjk/\'»\=\W- Ih/iL 4- ACTIVE SYSTEM £E(Z\/0VALVE- VlOlERS SC4--03 3 US&Pfi] @> EOO psi Jrep Csv = OrOU8 Lin/sex)/ff?) A, Ps = j-7s- foe)/fj^i Ll*l£A£ . APPeoxiMATi'OtJ :  LET FAo - /a* . . . ...... -\. y0y^ns = C7o7^9-4-2= /"«/s<zo -Z„...= A- • • - • ! • Id-* AA _ _ * 0 . 3 93 (0 ma .•.,. ... c"sv. r ^ _ o-o a8 r-—1 I.""....'. = w 0.055. (^b/uc)/^. _.J ... ' =--o,on. (j*/*«-)//t- ,-.4, ... : 4 -0.OO14 (fr/s*c)/lh . 2 A^/^7AA •:'*"" ., ;,f ; g_?^ = -2.4- Qh/<*t)/ ik 2+(o.5°>3,y{2<t .... - -<D.2 0+/scc.)/ll> /)PP£A/D/X E FlZICTioN PARAMETER .fRiCTiOti .IS-. CONSIDERED AS AN E?)CTER.hiAL FOQLCS d>W THE SYSTEM , AS SHotJtJ >N £1$. £.[ • 109 FA FIG- E.I RAM THE . MET FORCE IS GiVEti BY ^AV* '==- I~A + Fp - f WUERE '.. f IS THE FZICTioN FO&CE i IS A FUNCTION OF t/ (a) y = 0 : f /5 'EQUAL A7v7> OPPOSITE TO (FA +FP) UNTIL MOTION ...„ BEGitJS r 4 (b) • y * O • Jr. IS COslSTAMT AMD OPPOSJTE IN- SE-tJS£ TO j •(FP + FA) L = 0 5 f ° IIC APPEMDIX F. 1 ONLIN 1 »^»aji»»»ainii»^3»ajt» REMCiNONLIN ******** 2 * 3 ***** NONLINEAR MODEL SIMULATION PROGRAM *** 4 * 5 INITIAL 6 CONSTANT CSV=0.0118, UG=0.125, AA=0.393, AP=3.0, ... 7 V0=20., VT = 280., M=3.G, GAM=1.2, PS=50G. ,.«,. 8 K0=6.28 9 PARAMETER K2 = G., K3 = C.O, KSV=1.0, P0=15.0,... 10 ZETA=1.00, CV=3.75, FF0=1., YDCR=0.CCOGGG1, CX=1. 10.25 PARAMETER K1=(0.,5.) 11 PARAMETER RW= 1.0, TCV=0. 025 12 N=VT/VO 13 KS=2.*GAM*P0*AP**2/VT*12. 14 WN=SQRT(KS/M) 15 TC1=2.*ZETA/WN 16 TC2=TC1/(N+1.) 17 KP=N*KS/(N+1.) 18 W' = Rfo#WO 19 DYNAMIC 20 Ul=UO*SINE(0.fW,0.) 21 U2=RA,MP( 0.0) -RAMP(l.O) 22 U=U1*U2 23 UD=DERIV<0.,U) 24 UDD=DERIV(0.»UD) 25 X=U+Y 25.25 X1=CMPXPL<Q. ,0. ,0.5,W,XD) 25.5 X0UT=W*X1 26 F1=LEDLAG(TC1,TC2,Y) 27 FP=-KP*F1 28 YD=IMPL( 0.t0.05iFYO) 31 XD=UO+YD 32 YDD=DERIV(0.,YD) 33 XCD=UDD+YDD 34 FNET=M*XDD+CX*XD 35 PROCEDURE FFR = FRIC(YD,FFG ,FNET,YDCR> 36 IF(ABS(YD)-YDCR)10,10,11 37 11 IF(YD)1,1,3 38 1 FFR=-FFG 39 GO TO 4 40 3 FFR=FF0 41 GO TO 4 42 10 FFR=LIMITi-FF0,FF0,FNET) 42.25 4 CONTINUE 43.25 ENDPROCEDURE 44 FA=FNET-(FP-FFR) 45 XDDD=-W**2*XD 46 R1=-K1*XDDD*W 47 R=CMPXPL(0.,0.,0.5,W,R1) 48 Z1=KSV*R 49 Z=LIMIT(-40. ,40. ,ZD 50 SGN=FCNSW (Z, -1 .0,0.0, 1.0 ) 51 SGN2=FCNSK(FA,-1.0,0.0,1.0) 52 YD1=CSV*Z*SQRT(PS-LIMIT(-PS,PS,SGN*FA/AA)) 53 YD2=SGN2*CV*SQRT(ABS(FA/AA)) /// 54 FVD={ YUi-VD2)/12./^A 55 V=1MGRL (0.,YC) 55. 25 NOSORT 55.5 * GO TO 30 55.6 51 IF(KfcfcP.NE.l ) CC TC 30 55.7 TX=TIME+C.CC1 55.8 IF{AMOD(TX » .0 5 ).GT#0.002) GO TO 30 55.81 WRITE(8,31) TIM£,XCuT,U 55.82 31 FORMAT(3E14.6) 55.83 30 CONTINUE 56 PRINT I, X, Y, XCLT, P, FNET, FP, F L 57 TITLE ACTIVE/PASSIVE MQTICN COMPENSATION SYSTEM 58 TIMER PRCEL=0.05, FINTIM=iO.» DELT = 0.05 59 METHOD RKSFX 60 END 60.7 PARAMETER RW=0.5 60.8 TIMER DELT = 0.1 60.81 ENC 61 STOP 62 ENDJOB :N0 OF FILE EC *SKIP 112 APPENDIX F. 2. LINSYS 1 c******************** REMC !L INSYS *************************** 2 C 3 C LINEAPIZEC MUCEL OF MOTICN CCKPtNSAT ION SYSTcM 4 C 5 £ ************«*$********* ******** **>!« ** v***i|-** *i;***j,'.j;t w*4«r.< :*****.• 6 COMPLEX G,F,T1,T2,FFF,FFB,Z1,S,H,PWR .7 REAL KS,Kl,K2,K3,KP,M,N,LCGRh,KFF 8 REA0(5,1,END=99)AA,A,PQ,VC,V T, G AM , M , Z , CV 1 ,C V2,CX,DELAY 9 KS=2.*GAM*PG*A**2/VT*12. 10 l»N = SCRT( KS/M) 11 N=VT/VC 12 IF<Z.EG.G.) Z = SQRT((N+l.)*(N + 2.)/(8.*N)) 13 TC1=2.*Z/WN 14 TC2=TC1/(N+1.) 15 KP=N-KS/(N+1.18 100 REAC(5,1,END=99) WQ,K1,K2,K3,KFF,CBP 18.1 HCV1=1./(CV2+CBP) 18.2 HCV2=CV1*HCV1 18.25 RWN=WN/WO 19 IFtKFF.EC.C.} KFF = KP*(CV2 + CEP )/CV1 20 WRITE(7,4> P0»VC,VT,N»KS,Z,kN,CX 21 WRITE(7,5) CVl,CV2,CbP,Kl,K2,K3,KFF 21.25 fcRITE(7,6) kG,RhN 22 L0GRW=-1. 22.2 5 TCV=T AM DEL AY/180 .* 3 . 14 159 )/W0 22.5 ALPH=SQRT(!.+(TCV*WC)**2I 23 DO 20 1=1,81 24 R^=10.0**L0GRW 25 W=kW*WO 26 S = CMPLX(C, I* ) 27 G=-KP*(l.+TCl*S)/(l.+TC2*S) 28 F=M*S**2+CX*S 29 HFB = ( K1+K2*S+K3*S**2)*ALPH/ ll.*TCV*S )*I«*S / ( W* *2<-k »S+S**2) 30 HFF=KFF 31 h=-(F/HCV1 + CV1*(HFE + HFF))/(G/FCVl+CV 1*FFF + D 32 Tl=l./(1.+H) 33 PHIH=ATAN2(AIMAG(H)»R E A L ( H)M:180./2.14_59 34 Z1=HF6*T1+FFF 35 T2=T1-CMPLX(1.,0.) 36 T1A=CABS(T1) 37 T2A=CA6S(T238 HA=CA6S(H) 39 PHIl=|ATAN2(AIMAG(Tl»tREAL(Tl)))*18C./3. 14159 40 PHI2=(ATAN2(A I MAG(T2 ).FEAL(f2)))* 180•/2.14159 41 DB1=20.*AL0G10{T1A) 42 CB2 = 20 .*AL CG10 (T 2 A ) 43 DBH=20.*ALGG10(HA) 44 Z2=CA6S(Z1) 44.25 PWR=500.*AA*S*T2 44.5 PWRA=CABS(PfcR) 45 IF (MUD{I,2).EQ.1) WRITE(7,S> Rh ,DB1 , PH 11 ,062 , PHI 2, 46 * CBH,PHIF,Z2,PWRA 46.25 DBZ=2C.*ALCG10(Z2) 46.5 PH I Z= { AT AN 2 { A IMAGt ZD, REAL (ZD ) I * 18u. / 2. 141 59 47 WRITE(8,3) LCGRW,DE1,C82,PH11,Ph12 113 48 20 LOGRW=LOGRW+0.025 49 GO TO 100 50 1 FORMAT(12E12.0) 51 3 FORMAT(10F13.4) 52 4 FORMAT {• 1PASSIVE SIDE' / 'OPRESS » = 1 , F6. 0, 5X, »C YL • VOL . =*, 53' *F6.0,5X,«TANK VOL. = • , F6.0»5X,«VOL RATIO =',F6.3/ 54 *• STAT.STIFFNESS = •,F8.2,5X, «CRIT.DAMP.RATIO = «,F7.3, 55 *5X, • NATo FREQ • =•,F602/, BODY OR0G COEFF. =«,F6.0) 56 5 FORMAT <'OACTIVE S IDE V CLI NEAR VALVE CCEFFS. CV1 ='» 57 *F9.5,5X,*CV2 =',F9.5,5X,,CBP=»,F9.5/« FEEDBACK CONSTS. Kl =«, 57.25 *F6.0, 58 *3Xt 'K2 = SF6.0,3X, «K3 = »,F6.1/' FEEDFWD CONST. KFF =«,F6.3/) 58.25 6 FORMAT(•OOPERATING FRECUENCY=•f F7.2/ 58. 5 * 5X,'NAT.FREQ./OP.FREQ.=•,F6.2/*0*) 59 99 STOP 60 END OF FILE SKI P i lit APPENDIX r.3 PTIM 1 C ******************* REMC :0PTIM **************** ********** 2 C 3 c **** PROGRAM TO OPTIMIZE PARAMETERS OF CCNTROL SYSTEM *** 4 c 4.5 c ********************************************************* 5 DIMENSION VAR(3,6) 6 EXTERNAL TRANSF,FLO,FH I,FMPL 7 COMPLEX S,HFF,F,G 8 REAL M,N,KP,KS 9 COMMON/PARAM/S,HFF,F,G,KCV1,HCV2 10 READ(5,1) AA, A , PO , VC , V T , G AM, Mt, Z , CV1 , CV2 , CBP 10.25 READ(5,1)W 11 1 FORMAT(12E10.0 ) 13 KS = 2.0*GAM,*P0*A**2/VT*12.0 14 WN=SQRT(KS/M) 15 N=VT/VC 16 IFiZ.EQ.O.) Z=SGRT((N+l.)*(N+2.)/(8.*N )) 17 KP=N*KS/(N+1. ) 18 S=CMPLX(0.,W) 18. 25 TC1=2.*Z/WN 18.5 TC2=TC1/(N + 1. J 18. 6 G=KP*(l.+TCl*S)/(l.+TC2*S) 18.7 F=M*S**2 18.8 HCV1=1./CCV2+CBPJ 18. 81 HCV2=CV1*HCV1 18.82 KFF=0. 18.83 H FF = KF F 19 VAR(1,1)=0.0 20 VAR(2, 1 )=0. 21 VAR(3,1)=0. 22 CALL C0MPLX(X,VAR,3,3,6,4,9.9,50,150,2 50,10,0.001,TRANSF,FLO, 23 *FHI,FMPL,£999,£777) 24 STOP 25 999 STOP 9 26 777 STOP 7 27 END 28 FUNCTION TRANSF(T,NN) 29 DIMENSION T(1) 30 COMPLEX S,G,F,HF8,T1,HFF 31 REAL K1,K2,K3 32 COMMON/PARAM/S,HFF,F,G,HCV1,HCV2 34 K1 = TU) 35 K2=T(2) 36 K3=T(3) 41 HFB=K1+K2*S+K3*S**2 42 T1=(G-HCV2*HFF-HCV1*S) /{F+G+HCV2*HFB-HCV1*S ) 43 TRANS F=CABS (T 1 ) 44 RETURN 45 END 46 FUNCTION FLO(T,N,J) 47 DIMENSION T(l) 48 GO TO (1,2,3,4),J 49 1 FLC=0. 50 RETURN 51 2 FLO=0 . us 52 RETURN 53 3 FL0=-10. 5 4 RETURN 54.25 4 FL0=0. 54.5 RETURN 55 END 56 FUNCTION FHI(T,N,J) 57 DIMENSION T{1) 58 GO TO (1 ,2 ,3 ,4) , J 59 1 FHI=.0. 60 PETURN 61 2 FHI=0. 62 RETURN 63 3 FHI=10. 64 RETURN 64.25 4 FHI=128. 64.5 RETURN 65 END 66 FUNCTION FMPL<T,N,J) 67 DIMENSION TU) 68 COMPLEX S,T1,HFF ,HF8,F,G 69 COMMON/PARAM/S•HFF t F,G,HC VI,HCV2 69.25 K1=T(1) 69. 5 K2=T(2) 69.6 K2=T( 3) 69.7 HF8=K1+K2*S+K3*S**2 69.8 Tl=(G-HCV2*HFF-HCV1*S)/(F+G+HCV2*HFB-HCV1*S) 69.81 FMPL=CABS(HFB*T1+HFF) 71 PETURN 72 END OF FILE SKI P APPENDIX G EtPER. I MENTAL j SIMULATION R.ESULTS Hz. *3 AM PL. RATIO PHASE LAC FIGURE EY.PT SWUL. &XP 'T 0-5 0 Odb -14 di> 0° 18° 6./ 0.5 ...5 -3db -z.i<u> 41° 27° 1-0 ... .0 -Adh -4T5Ji 54° 36° 6.3. 1.0 5 - 7.3 Ah -i&Jb . 72°. 54° . 1 , : 64-2.0 0 -ISAh ----- 11° 100° 2-0 5 -/hldl - JOSdh 30° 108° THE7 ABOVE ARE PloTTErD OfJ A BODE DIAGRAM Ik) FIG. 4.2-1- •-- -MOTES, ON . FI&UZCS G.I -..£.4 • - LOWE a 2 CURVES WERE GENERATED BY WE COMPUTER. USING THE SIMULATION PROGRAM OF .. APPENDIX Fj- L -I ; .. •• ; - UPPER 2 CURVES ARE FROM CHART RECORDER. FULL SCALE OEFLECTION IS + i-5 INCHES. . .1 - AMPLITUDE RATIOS WD PHASE SN/FTS IN ERE ESTIMATED ... FROM THESE. CVZVES, REMC PLOT*.00720098. 6)1 n-7 i 


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