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An experimental study of nonlinear oscillations in railroad friction control systems Talebi Bidhendi, Mohammad Reza 2018

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AN EXPERIMENTAL STUDY OF NONLINEAR OSCILLATIONS IN RAILROAD FRICTION CONTROL SYSTEMS byMohammad Reza Talebi Bidhendi B.Sc., Amirkabir University of Technology, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE inTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)September 2018 © Mohammad Reza Talebi Bidhendi, 2018 iiThe following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, a thesis/dissertation entitled:AN EXPERIMENTAL STUDY OF NONLINEAR OSCILLATIONS IN RAILROAD FRICTION CONTROL SYSTEMSExamining Committee: Dr. A. Srikantha Phani SupervisorDr. Jin Xiaoliang Supervisory Committee Member  Darren Loo, PENG, Engineering lead, LBFoster representative Supervisory Committee Member submitted by Mohammad Reza Talebi Bidhendi  in partial fulfillment of the requirements for the degree of Master Of Applied Science in Mechanical Engineering iiiAbstractFriction control at the wheel-rail interface has been an outstanding challenge in front of the rail road engineers throughout the world. On-board solid stick friction modifier system, simply named stick-applicator assembly, has proved to be one of the simple and efficient ways to tackle the excessive wear and rail corrugation. Interlocking solid sticks are applied to the wheel flange and tread by means of a mechanical applicator mounted on a bracket, which is connected to the bogie. Relative sliding motion in the stick-wheel interface provokes gradual transfer of solid lubricant film to the wheel-rail interface through the wheel’s motion. Consequently, friction control at wheel-rail interface could be achieved. Instability and failure of stick-applicator assembly due to stick-wheel interaction destabilize its performance. The present study uses a lab-scale setup to produce consistent instability, which helps examine the behavior of the stick-applicator assembly during instability. The lab-scale setup incorporates a mock-wheel connected to the stick-applicator assembly. Mock-wheel is used to simulate up - down and transverse motion based on the concept of parametric excitation in the presence of internal resonance. Dynamics of each substructure is investigated to gain better understanding of the behavior of the coupled system. Having known the characteristics of each substructure, the dynamics of the coupled system is studied. It is found that period doubling bifurcation occurs consistently in certain ranges of excitation frequencies and voltages. Lateral stiffness is identified as one of the design parameters of the lab-scale setup that governs the vibration level. Clearances in the stick-applicator assembly and looseness between each interlocking sticks are found to be parameters which weaken the lateral stiffness of the coupled system. Some modifications in the design of the main contributing parts to eliminate instability and suppress the vibration of the coupled system in the lab-scale setup are also addressed in this thesis. Furthermore, full-wheel rig experiments are carried out to check the practicality of the design modifications. ivLay Summary The idea of using lubricants to reduce the wear and friction at the wheel-rail interface in Vancouver mass transit systems dates back to 1980’s when the rails were replaced shortly after the establishment of the mass transit systems due to the huge amount of wear and rail corrugation. Engineers came up with a new idea of exploiting on-board solid stick friction modifier systems suspended from the bogie and connected to the wheel through frictional contact. Relative sliding motion in the stick-wheel interface leads to gradual transfer of solid lubricant film to the wheel-rail interface through the wheel’s motion and consequently friction control at wheel-rail interface could be achieved. Problems associated with the stability and failure of friction modifier systems due to stick-wheel interaction undermines their performance. The present dissertation scrutinizesthe instability mechanisms and investigates the ways to lower vibration in friction modifier systems by introducing design modification of their components through a lab-scale setup.vPrefaceThe present thesis entitled “An experimental study of nonlinear oscillations in railroad friction control systems” is original and unpublished work conducted by the author, Mohammad Reza Talebi Bidhendi, under supervision of Dr. A. Srikantha Phani. This project was financially supported by LBFoster Company and Natural Science and Engineering Research Council of Canada (NSERC). Portion of Chapter 2 will be submitted to a pertinent journal in the field. Parts of the same work about insert design modifications presented in Chapter 2 can be patented as a joint contribution with LBFoster Company.   viTable of Content Abstract ......................................................................................................................................... iiiLay Summary ............................................................................................................................... ivPreface .............................................................................................................................................vList of Tables .............................................................................................................................. viiiList of Figures ............................................................................................................................... ixGlossary ...................................................................................................................................... xivAcknowledgements ......................................................................................................................xvDedication ................................................................................................................................... xviChapter 1: Introduction ............................................................................................................... 11.1 General overview ........................................................................................................ 11.2 Problem statement ....................................................................................................... 21.3 Literature review ......................................................................................................... 41.3.1 Friction – induced vibration and instability ............................................................ 41.3.1.1 Negative slope of friction coefficient ............................................................. 41.3.1.2 Stick-slip ......................................................................................................... 51.3.1.3 Mode coupling ................................................................................................ 51.3.2 Vibro- impact phenomena ....................................................................................... 61.3.3 Non-linear mechanisms route to instability ............................................................ 71.3.4 Internal resonances .................................................................................................. 81.4 Objectives and outline ................................................................................................. 9Chapter 2: Lab-scale Experiments and Modelling .................................................................... 10vii2.1 Description of substructures in the coupled system .................................................. 102.1.1 Mock-wheel .......................................................................................................... 102.1.1.1 Linear natural frequencies and mode shapes ................................................ 112.1.1.2 Nonlinear behavior of the mock-wheel ......................................................... 132.1.1.3 Summary of the tests on mock-wheel ........................................................... 192.1.2 Stick- applicator assembly .................................................................................... 202.1.2.1 Axial test ....................................................................................................... 212.1.2.2 Mode shapes of stick- insert assembly .......................................................... 232.2 Stick- mock-wheel assembly (coupled system) ........................................................ 262.2.1 Evolution of nonlinear motion from linear motion ............................................... 272.2.1.1 Experimental stability plots .......................................................................... 292.3 Vibration reduction in stick-applicator assembly ..................................................... 332.3.1 Stick’s impacts on vibration suppression .............................................................. 352.3.2 Insert’s impacts on vibration suppression ............................................................. 372.3.3 Conclusion ............................................................................................................ 41Chapter 3: Full-wheel rig experiments ..................................................................................... 423.1 Introduction ............................................................................................................... 423.2 Experimental results.................................................................................................. 433.3 Discussion ................................................................................................................. 48Chapter 4: Conclusions and Future works ................................................................................ 49Bibliography .................................................................................................................................52Appendices ....................................................................................................................................57viiiList of Tables Table 2.1: Modelling parameter for Mock-wheel’s FEA ............................................................. 12Table 2.2 : Natural frequencies of the mock-wheel in Hz ............................................................ 13Table 2.3 : Geometrical properties of different sticks .................................................................. 20Table 2.4 : Linear natural frequencies (Hz) of the stick-applicator assembly measured by the axial test ................................................................................................................................................. 22Table 2.5 : Comparisons between the linear natural frequencies extracted by FEA analysis of stick-insert assembly, impulse test of stick-insert assembly, and axial test of stick-applicator assembly with two short sticks ..................................................................................................................... 25Table 2.6: Linear natural frequencies (Hz) of the coupled system with different type and number of sticks inside the applicator. ....................................................................................................... 28Table 2.7 : Comparison of static stiffness and damping ratio for different inserts ....................... 39List of Figures Figure 1.1: A schematic of operation of an on-board solid stick friction modifier system; a) wheel flange lubrication, b) thin transferred solid lubricant film from the wheel to the rail .................... 1Figure 1.2 : Schematic of the applicator-stick assembly; a) exploded view of the applicator, b) cross section of a loaded applicator showing interlocking sticks inside the tube, c) front view showing clearance between sticks and insert. Note that free-play exist in X and Z direction ........ 2Figure 1.3 : UBC lab- scale experimental setup; a) coupled system, b) uncoupled system ........... 3Figure 1.4 : A vibration model of drill with impact damper; note that in some cases impact damper can be equipped outside of the main oscillatory system as well [27] ............................................. 7Figure 1.5 : Adjusted two-degree of freedom model for a 2:1 autoparametric resonance investigated by Nayfeh [35] ............................................................................................................ 8Figure 2.1 : Schematic of the mock-wheel and its components ................................................... 11Figure 2.2 : Frequency spectrum of the mock-wheel with respect to the different excitation inputs; a) response of vertical low-level sine sweep excitation signal, b) response due to impact appliedon the lateral side of the tip mass. Note that response amplitudes are different since input excitations and the point they are applied are different. Va and La are vertical and lateral acceleration respectively. .............................................................................................................. 12Figure 2.3: Mock-wheel; a) FE model in ABAQUS, b) first lateral bending mode (17.41 Hz), c) first vertical bending mode (24.52 Hz) ......................................................................................... 13Figure 2.4 : Detection of nonlinearity in mock-wheel’s vertical motion by increasing the amplitude of forward sine-sweep excitation (20 Hz to 30 Hz), (b) nonlinear response (acceleration)at 800 ixxmv excitation voltage (before jump), (c) linear response (acceleration) at 800 mv excitation signal (after jump) ................................................................................................................................... 15Figure 2.5 : Detection of hysteresis in mock-wheel’s vertical motion by comparing the forward and backward sine-sweep excitation (20 Hz to 30 Hz) with 800 mv amplitude .......................... 16Figure 2.6 : Vertical (first row) and lateral (second row) acceleration of the mock-wheel measured by the tri-axial accelerometer mounted on top of the mock-wheel at different excitation amplitude at 24 Hz. ........................................................................................................................................ 17Figure 2.7 : Rms values of the mock-wheel acceleration at different excitation voltages at 24 Hz....................................................................................................................................................... 18Figure 2.8 : Using laser Vibrometer to capture the lateral displacement and velocity of the mock-wheel. Excitation signal is a sinusoid of frequency 24 Hz and amplitude 800 mv; (a) schematic of the experiment, (b) lateral displacement in time domain, (c) FFT of the lateral displacement, (d) lateral velocity in time domain, (e) FFT of the lateral velocity .................................................... 19Figure 2.9 : Geometrical properties of the sticks tested in the current work ................................ 20Figure 2.10 : Axial test of stick – applicator assembly, (a) experimental setup, (b) schematic of the axial test. ....................................................................................................................................... 21Figure 2.11 : Normalized response (acceleration) of the stick with respect to the force provided by the force transducer for stick-applicator assembly with two short sticks. Note that below 60 Hz, the response is noisy and low coherence was observed. ............................................................... 22Figure 2.12 : Stick - insert assembly clamped at the end .............................................................. 23Figure 2.13 : Direct and cross accelerance of the stick - insert assembly clamped at the end; (a) lateral impact, (b) vertical impact ................................................................................................. 24xiFigure 2.14 : (a) Stick-insert assembly with its geometrical properties in ABAQUS, (b) Top view of first mode (14.63 Hz), (c) second mode shape (54.02 Hz), (d) third mode shape (220.28Hz). 24Figure 2.15 : Schematic of the lab-scale setup designed to examine the dynamics of stick- applicator assembly ....................................................................................................................... 26Figure 2.16 : Experimental frequency spectrum of the stick in the coupled system with increasing excitation level (from 20 mv to 300 mv); (a) vertical acceleration of the stick, (b) lateral acceleration of the stick ................................................................................................................ 27Figure 2.17 : Appearance of nonlinear resonances between 10 to 60 Hz by increasing the excitation voltage of the swept-sine signal in the coupled system with two short sticks; (a) vertical acceleration of the stick, (b) lateral acceleration of the stick ........................................................ 28Figure 2.18 : Experimental stability plot for the coupled system including two short sticks inside the applicator. 37 Hz is chosen as an example to demonstrate the lateral nonlinear motion of the system. La stick stands for lateral acceleration of the stick.  FFT of the stick lateral acceleration at (a) 1200 mv, (b) 800 mv, (c) 400 mv and (d) 100 mv excitation voltage..................................... 29Figure 2.19 : :  Verification of the onset of PD at excitation frequency of 37 Hz for the coupled system comprised of two short interlocking sticks; (a) Rms values at different separate excitation amplitudes, (b) lateral acceleration of stick at 37 Hz excitation frequency and sweep amplitude signal, (c) stick’s acceleration in time domain due to sweep amplitude signal, (d) sweep amplitude signal. ............................................................................................................................................ 30Figure 2.20 : Measuring displacement and velocity of the lateral motion by the scanning Vibrometer (PSV 400) .................................................................................................................. 31Figure 2.21: Frequency spectrum of the lateral response measured on the stick due to the excitation voltage and frequency of 1200 mv and 37 Hz; (a) lateral displacement, (b) lateral velocity ....... 32xiiFigure 2.22 : Experimental setup utilizing external lateral supports connected to the stick ........ 32Figure 2.23: Comparisons between Rms values of the coupled system having two short sticks with and without adding external lateral supports ................................................................................ 33Figure 2.24: comparing the Rms values of stick and applicator (tube) at 37 Hz and different excitation level s............................................................................................................................. 34Figure 2.25 : Vibro-impact phenomenon by roving the microphone around the coupled system; (a) noise signal during system’s operation at high excitation voltages; (b) microphone close to the stick-mock wheel interface, (c) microphone close to the entrance of the applicator .................... 34Figure 2.26 : Rms values of (a) 4 short sticks (c) 2 short sticks (e) 1 long stick at 37 Hz excitation frequency and different separate amplitudes; STFT of lateral acceleration of (b) 4 short sticks (d) 2 short sticks (f) 1long stick at 37 Hz excitation frequency and continuously varying amplitude 36Figure 2.27 : (a) Regular insert, (b) Regular curved insert, (c) Short bend insert, (d) Top-bend insert .............................................................................................................................................. 37Figure 2.28 : a) Front view of the stick-applicator assembly, b) qualitative model of stick-applicator assembly showing clearances and stiffness discontinuity ........................................... 38Figure 2.29 :  (a) Insert under a point load applied at the entrance, (b) Insert under a distributed load applied on the sidewall .......................................................................................................... 39Figure 2.30 : Evaluating the inserts’ performance in terms suppressing the stick’s vibration level ; (a) Rms values of vertical, lateral and axial acceleration of the stick due to various excitationfrequencies and constant excitation amplitude of 1000 mv,(b) Rms values of vertical, lateral and axial acceleration of the stick due to various excitation frequencies and constant excitation amplitude of 800 mv ..................................................................................................................... 40xiiiFigure 3.1: Full-wheel test rig experiment using freight wheel; (a) general view of the setup, (b) top view of the wheel’s flange and stick-applicator assembly ..................................................... 43Figure 3.2: Stick acceleration at 10 rpm and regular insert in use; (a) total measurement, (b) rapid growth in the response (occurrence of instability), (c) FFT of response in b ............................... 44Figure 3.3 : Stick acceleration at 10 rpm and regular insert in use; (a) total measurement, (b) rapid growth in the response (occurrence of instability), (c) FFT of response in b ............................... 45Figure 3.4 : Stick’s acceleration at 600 rpm and top-bend insert in use; (a) total measurement, (b) detail of the signal in time domain, (c) FFT of response of b (Observation of wheel speed harmonics)..................................................................................................................................... 47Figure 3.5 : (a) Stick’s Rms (g) of different inserts at different wheel speeds; (b) tube’s Rms (g) of different inserts at different wheel speeds. ................................................................................... 48Figure 4.1 : Proposed lab-scale setup to study the effect of suspension motion of the stick-applicator assembly ....................................................................................................................... 50xivGlossaryAbbreviations FE Finite Element FFT Fast Fourier Transform FRF Frequency Response Function Aa Axial Acceleration La Lateral acceleration Va Vertical acceleration STFT Short Time Fourier transform PD Period doubling PSD Power Spectral Density Symbolsf FrequencyxvAcknowledgements “It is never a question as to whether it can be done - it is only whether one cares to spend the time and effort” Clarence Walton Musser I would like to express my deepest gratitude to my supervisor, Dr. Srikantha Phani, for boosting the sense of being a gladiator in research in me. This project would not have been possible without his encouragement.  I would also like to thank faculty members at UBC, particularly Professor Yousef Altintas and Clarence De Silva, who kindly answered my questions and help me achieve broader view in the field of dynamical systems. I am grateful to my lab-mates at DAL for creating a friendly research atmosphere. I would also like to extend my appreciation to our partners in LBFoster, Mr. Ron Hui, Mr. Darren Loo, Mr. David Elvidge for their important discussions and critical reviews. I want to take this opportunity to thank Andrew Mckay from university of Cambridge for his suggestions and comments about this project.Last but not the least, my especial thanks should be given to my parents and sister for supporting me throughout my life.   xviDedicationTo my parents.1Chapter 1: Introduction1.1 General overview Rail road engineers always seek the most effective, inexpensive and secured way to overcome the wheel-rail transportation related problems. These include track life increase, improving fuel efficiency by optimizing traction and train weight, wear decrease and excessive noise suppression during wheel-rail interaction. Friction management at the wheel rail interface has been recognized to be crucial. Several methods of friction modification have been developed and designed by the railway industries including liquid and spray based top of rail friction modifiers by which the top of rail is lubricated, and carbon-based solid stick technology by which wheel flange and tread are lubricated. More details about friction modification in railway transportation can be found in [1,2,3]. The idea of using solid lubricants to reduce the wear and friction at the wheel-rail interface in Vancouver mass transit systems dates back to 1980’s when rails were replaced shortly after the establishment due to the huge amount of wear and corrugation [4]. On-board solid stick friction modifier systems suspended from the bogie and connected to the wheel through frictional contact are used to tackle the aforementioned issues (Figure 1.1). Moreover, other features of solid lubricants such as environmentally friendly, fire resistance, non-toxic and low wear rates provoked them to be eminently suitable in freight, urban transit and heavy haul locomotives.  Bracket (connected to the bogie)WheelTreadFlangeTube (Applicator)Bolted connectionSolid lubricant (stick)WheelRailLayer of solid lubricanta) b)Figure 1.1: A schematic of operation of an on-board solid stick friction modifier system; a) wheel flange lubrication, b) thin transferred solid lubricant film from the wheel to the rail2Although using applicator-stick assembly leads to effective reduction of wear and friction at the wheel-rail interface, stability and failure of the solid stick friction control systems remain a challenging issue that can drop the efficacy of that technology. Stick-applicator assemblies are prone to excessive vibration and noise due to stick-wheel and stick-applicator interaction. Hence eliminating the instability and preventing failure of the stick -applicator assembly by experimental identification of instability mechanisms are the focus of this project.1.2 Problem statementAs depicted in Figure 1.1, interlocking solid sticks are applied to the wheel flange/ tread by means of a mechanical applicator mounted on a bracket, which is connected to the bogie. As shown in Figure 1.2, applicator is a hollow tube utilizing a folded and removable part named insert with a constant force spring. Sticks are placed inside the applicator and pushed from one side by a constant force spring to be continually in contact with the wheel flange at the other side.GapConstant ForceConstant force springInterlocking carbon-based solid lubricants (sticks)Tube and insert (applicator)ProtrusionTubeInsertConstant force springLock-bolta)b) c)ZYXFigure 1.2 : Schematic of the applicator-stick assembly; a) exploded view of the applicator ,b) cross section of a loaded applicator showing interlocking sticks inside the tube, c) front view showing clearance between sticks and insert. Note that free-play exists in  X and Z direction.400 microns3Excessive amount of stick’s vibration, which is the result of the relative motion between the wheel and stick-applicator assembly, is one of the main concerns in dealing with the failure of the mechanical applicators. As shown in Figure 1.2(c), sticks can freely move and hit the internal sides of the applicator because of less constraints in X and Z directions due to the presence of the clearance which exists inevitably in the system due to manufacturing tolerances. Consequently, stick-wheel and stick-applicator interactions may lead to the failure of the applicators through the friction –induced vibration and vibro-impact mechanisms.   In this thesis, practical ways of suppressing the sticks’ vibration are explored by performing experiments on a lab-scale setup, which includes a mock-wheel to simulate the relative motion between the wheel and the stick-applicator assembly (Figure 1.4), and available full-scale wheel test-rig at LBFoster Company. ApplicatorStickBracketAccelerometersMock-wheelShaker ApplicatorShakerStingerBeam Tip massa) b) Figure 1.3 : UBC lab- scale experimental setup; a) coupled system, b) uncoupled system41.3 Literature reviewSince the current project deals with three important phenomena including friction-induced vibration due to stick-wheel interaction, vibro-impact oscillation because of stick-applicator interaction and nonlinear mechanisms route to instability due to the mock-wheel’s dynamics, abrief pertinent literature review is given in the following subsections. 1.3.1 Friction – induced vibration and instabilityFriction-induced vibration and instability is usually known as the root cause of undesired noises and instabilities in many engineering applications such as disc-brake interaction in automotive industry, machining processes, wheel-rail interactions, ceramic-on-ceramic hip arthroplasty, etc. Numerous studies have been conducted and various models have been proposed to describe the behavior of the systems oscillating due to friction. Friction modeling is a key task for the mathematical models to successfully predict the system’s behaviors with frictional contact [5]. The underlying mechanisms of friction generated instability and chaos have been comprehensively reviewed in [6,7,8,9]. Some of the mechanisms are briefly presented here.1.3.1.1 Negative slope of friction coefficientWhen friction coefficient is a decreasing function of relative velocity of two sliding systems in contact, friction can be interpreted as a negative damper feeding energy to the system instead of dissipating it. This characteristic was recognized to be an essential destabilizing mechanism for a long time [7]. However, Jarvis and Mills [10] showed that decreasing of friction coefficient with sliding speed could not alone amount for the squeal, based on their setup consisting of a cantilevered beam on a unidirectional rotating disc. Geometry of the coupling of the motions was identified as the main cause of instability in their setup. Moreover, Chen et al [11] looked at the instabilities arising from the frictional interaction of a reciprocally driven pin-on-plate apparatus and observed squeal occurrence in both regions of negative and positive friction-velocity gradientsbut no clear explanation was given.51.3.1.2 Stick-slipStick-slip motion, a non-smooth behavior, usually occurs at a low sliding speed between two surfaces due to the difference between the static friction coefficient and the kinetic frictioncoefficient or transition between elastically deformed asperities during sticking phase followed by plastically deformed asperities during sliding phase. Stick-slip vibration occurs in many systems such as drill strings in oil and gas well drilling [12], brake system, and bowed music instruments. In some cases, stick-slip vibration alone cannot be the source of brake squeal noise, but it can lead to the coupling of various modes in the system by releasing an impact-wise energy during slipping phase and enhance the squeal propensity [13, 14]. Hence, finding its causes and avoiding its formation by changing system parameters including interface frictional properties are pursued inindustrial applications.1.3.1.3 Mode couplingFlutter is a common example of mode coupling when at least two closely spaced modes of a system, get coupled and merge into one due to certain conditions such as having an asymmetric stiffness and/or damping coefficient matrices arising from follower loads or frictional contact conditions. Akay et al [15] demonstrated mode coupling as one of the most important mechanisms in the disc brake squeal and the effect of damping in the squeal noise through a simplified designed test rig. F.Chen [16], by using the concept of aligned frequencies, demonstrated that the squeal frequency is close to the aligned frequency. He addressed that if one in-plane mode of the rotorfalls into the less than one-third frequency distance between the two adjacent out-of-plane rotor’s modes, in-plane and out-of-plane modes are said to be aligned and there is a high possibility of squeal to arise if a proper friction, pressure and temperature exist to couple those modes. Hoffman [17] studied a minimal two degree of freedom model exhibiting mode coupling instability. Flutterroute to instability has also been observed by several researchers in follower loading structures,both experimentally and theoretically [18,19].The above mechanisms are the accepted and identified sources of the instability in engineering systems. However, nonlinear mechanisms route to instability including modal interactions (internal resonances), parametric oscillations, subharmonic or super harmonic6resonances are also known to be involved in dynamical systems and some of them are discussed briefly in section 1.3.3.There is a strong analogy between the wheel-stick-applicator assembly and disc-brake system,widely discussed in the literature. Train wheel, applicator and sticks can be treated as disc, caliper and pads respectively. Although controlling friction at the wheel-rail interface is done by the stick-applicator assembly, the physics behind the chatter/squeal occurrence in wheel-stick-applicator assembly remains unresolved. Inspired by the linear stability theory in frequency domain used by Duffor and Woodhouse [20] in studying the disc brake squeal and Altintas [21] in constructing the stability lobes for milling process, the first rudimentary work in the context was done by Sharma et al [22]. He studied the stability of an applicator mounted on three different brackets in contact with the wheel in a virtual environment. Further, he did modal experiments to measure the required transfer functions for the linear stability theory. No chatter/squeal was observed during his field experiments and he mentioned the wheel-stick interaction as a forced vibration process.Robustness of his model is under question since the stick’s influence, clearances as one of the main sources of the nonlinearities and vibro-impact phenomena were neglected in his model. The effect of sticks and clearances are studied in the present work by the lab-scale setup (Figure 1.4).1.3.2 Vibro- impact phenomenaWoodpecker toy [23], ground moling [24], impact dampers [25] and various loosely connected structures have been studied as typical examples of vibro-impact phenomena. Clearances in loosely connected systems similar to the stick-applicator apparatus can prompt the vibro-impact effects which results in noise level increase during operation and fatigue intensification of components. Impact dampers used in machining process to passively suppress the unwanted vibration (chatter) [26, 27], however, are one of the desired application of vibro-impact phenomena. A single mass can act as an impact damper in systems with clearances (Figure 1.5) by absorbing part of the main oscillating system’s energy during impact. 7In general, vibro-impact phenomenon is a non-smooth process and can be a nonlinear mechanism leading to instability. Many complex behaviors such as drift in response, subharmonic response and chaotic motion might potentially be observed in certain conditions and due to system’s parameters in an impact oscillator [28, 29].Stick-applicator assembly can be treated as an impact oscillator due to the existence of the clearance between the stick and the applicator. To prevent the failure of the applicators caused bythe stick motion, inserts may be designed to act as an impact damper to absorb stick’s energy. Different types of inserts are studied in this work.1.3.3 Non-linear mechanisms route to instabilityNonlinearity is an ubiquitous characteristic in most practical engineering systems. Large deformation of structural elements such as beams, shells and plates, impact and backlash and elasto-plastic behavior of materials exemplify the geometric, non-smooth and material nonlinearities in the systems respectively. Identification of nonlinearities and their influence on system operation is of great importance in achieving proper and simple models to predict the system’s behaviors adequately. Furthermore, there are involved phenomena exclusive to nonlinear systems and using linear models for that systems lead to spurious results. Modal interaction (internal resonance), saturation, amplitude dependent frequency of oscillation, jump (hysteresis) ,subharmonic or super harmonic resonances due to certain circumstances can arise only in nonlinear systems. A brief review of one of the known features of nonlinear structures associated XImpact damper (single mass)ClearanceMain oscillatory structureFigure 1.4 : A vibration model of drill with impact damper; note that in some cases impact damper can be equipped outside of the main oscillatory system as well [27] 8Figure 1.5 : Adjusted two-degree of freedom model for a 2:1 autoparametric resonance investigated by Nayfeh [35]with the current project, in particular the mock-wheel used in the lab scale setup, are provided here. More details can be found in [30]. 1.3.4 Internal resonances Modal interactions occur in nonlinear multiple degree of freedom systems as a result of the presence of the internal coupling and energy exchange between the modes due to nonlinearities. Autoparametric or internal resonances, which is one type of modal interaction in nonlinear systems, is said to exist when two or more of a system’s linear natural frequencies are commensurate or nearly commensurate, i.e. Z Z2 12 , Z3 3Z r1 Z2 . Internal resonances depend on the order of the nonlinearities. For instance, when a structure has a quadratic damping or an asymmetrical geometry caused quadratic nonlinearity, internal resonances may be anticipated if k2Z Zm n or Zm Z rn Z , while for cubic nonlinearities such as cubic stiffness, internal resonances arise when m n m n kZ Z or Z3 2Z rZ . Internal resonances, which can be a nonlinear mechanism route to instability, when combined with external resonance can cause hazardous large responses in modes and enhance fatigue-related problems. However, they have been purposefully exploited in applications including vibration absorption [31] and energy harvesting [32, 33]. A two-DOF model (Figure 1.6) was widely used as a good approximation to explain the observed behaviors including saddle-node bifurcation (jump phenomenon), Hopf bifurcation, Period doubling (PD) bifurcation, saturation and chaotic motion during experiments on the autoparametric systems [31,34,35]. Fourier spectra, time-frequency analysis, time histories, autocorrelation for detection of chaotic motion, Poincare sections and maps are useful tools in characterizing the responses of nonlinear systems [36, 37]. m1m2ShakerUp and down motion9The mock-wheel designed in the current work draws on the approach of a 2:1 autoparametric resonance with new boundary conditions. The mock-wheel shows some fascinating responses such as jump phenomenon (Hysteresis) and period doubling bifurcation in certain frequency ranges by increasing the excitation level which are discussed in Chapter 2 of this dissertation.  1.4 Objectives and outline As mentioned earlier, chatter/squeal was not observed during the stick-wheel interaction by Sharma [22]. Moreover, it was reported to be an inconsistent phenomenon in the field experiments by the industry. Therefore, the lab scale setup (Figure 1.4), which uses mock-wheel as a parametric system in the presence of internal resonance to simulate up-down and transverse motion, is opted for producing consistent instability. The objective of this work is to understand how stick-applicator assembly performs at instability. Next, how to eliminate the instability of the applicator-mock-wheel assembly and suppress the vibration by identifying the contributing design factors and hence modifying the main contributing parts such as sticks, clearances and inserts are addressed based on the lab scale setup. Furthermore, full scale test rig is also used to check the usefulness of conclusions made according to the lab-scale setup findings.  The current dissertation is organized as follows: in Chapter 2, the components of the lab-scale setup is described along with modal experiments and Finite Element (FE) simulations to extract the linear natural frequencies and investigation of mock-wheel nonlinear behavior. After that, the dynamics of the applicator-mock-wheel assembly is analyzed experimentally to highlight the main contributing parts of stick-applicator assembly. In Chapter 3, the results of experiments conducted on the full-wheel test rig are presented and practicality of conclusions made based on the lab-scale setup findings is assessed. 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ORQJVWLFNDW+]H[FLWDWLRQIUHTXHQF\DQGGLIIHUHQWVHSDUDWHDPSOLWXGHV67)7RIODWHUDODFFHOHUDWLRQRIEVKRUWVWLFNVGVKRUWVWLFNVIORQJVWLFNDW+]H[FLWDWLRQIUHTXHQF\DQGFRQWLQXRXVO\YDU\LQJDPSOLWXGH   ([FLWDWLRQYROWDJHPY5PVJ([FLWDWLRQYROWDJHPY5PVJORQJVWLFN        ([FLWDWLRQYROWDJHPY5PVJVKRUWVWLFNV30 60 90 120 150 180)UHTXHQF\+]PSDG%+]7LPHsec20406080)UHTXHQF\+]PSDG%+]30 60 90 120 150 1807LPHsec)UHTXHQF\+]PSDG%+]/D VKRUWVWLFNV7LPHsec/DORQJVWLFN100EFH I3'ELIXUFDWLRQP 9D6WLFNP /D6WLFNP $D6WLFN2QVHWRI3'2QVHWRI3'3'ELIXUFDWLRQ1R3'ELIXUFDWLRQ1R3'ELIXUFDWLRQ60 120 180Intermittent20406080100d204060801009HUWLFDOVORWD EF G(QWUDQFH)LJXUH D5HJXODULQVHUWE5HJXODUFXUYHG LQVHUWF6KRUWEHQGLQVHUWG7RSEHQGLQVHUW2.3.2 Insert’s impacts on vibration suppressionInsert also has contribution in providing lateral stiffness in the coupled system. As mentioned earlier, clearances exist in stick-applicator assembly due to manufacturing process. The loose connection between the sticks and insert prevents the insert to be efficient in providing lateral stiffness in the coupled system. To overcome the aforementioned issue, insert’s designs are important. It is noteworthy to mention that insert can be treated as a passive compensator for the applicator (tube), so it is possible to reduce the tube’s vibration originating from stick motion by compensating the insert. Two insert’s designs named regular insert and regular curved insert (known as damped insert in industry) have been used earlier but they were not successful in reducing the vibration and noise level emanated by stick-train wheel interaction. Note that the measurements presented so far were for stick-applicator assembly with regular insert only.  Having identified the role of lateral stiffness in the present lab-scale setup at UBC, two more insert’s designs called short-bend insert and top-bend insert were asked to be made to investigate the efficacy of insert designs in decreasing the vibration level in the lab-scale setup. The schematic of four different insert designs LV illustrated in Figure 2.27. =݉6WLFN,QVHUW$SSOLFDWRU&OHDUDQFHG&OHDUDQFHGGG$SSOLFDWRU...DQG.DUHLQVHUWVWLIIQHVV;)RUFH'LVSODFHPHQW*DS.VWLIIQHVVRILQVHUWD E)LJXUHD)URQWYLHZRIWKHVWLFNDSSOLFDWRUDVVHPEO\ETXDOLWDWLYHPRGHORIVWLFNDSSOLFDWRUDVVHPEO\VKRZLQJFOHDUDQFHVDQGVWLIIQHVVGLVFRQWLQXLW\,QWKDWUHJDUGDKRUL]RQWDOVORWLVFUHDWHGLQWKHVLGHZDOORIERWKVKRUWEHQGDQGUHJXODUFXUYHGLQVHUWDQGEDVHGRQWKHOHQJWKRIWKHVORW WKHORZHUVHFWLRQLVGHIOHFWHG)LJXUHF7KDWGHIOHFWHGSDUWILOOVWKHJDSEHWZHHQWKHVWLFNDQGLQVHUWLQWKHHQWUDQFHRIWKHDSSOLFDWRU,QWRSEHQGLQVHUWDYHUWLFDOVORWQHDUWKHHQGRIWKHLQVHUWLVILUVWFUHDWHGDQGDFHUWDLQGHIOHFWLRQLVJLYHQWRWKHVHFWLRQIURPWKHHQWUDQFHWRWKHVORW)LJXUHGPDNLQJWKHLQVHUWLQFRQWDFWZLWKVWLFNVWKURXJKDOLQH,QVHUWVDUHPDGHRIWKHVDPHPDWHULDOThe main idea behind new insert designs is to increase lateral stiffness in the coupled system by reducing the gap between sticks and insert. Interaction between sticks and insert can be treated as a nonlinear joint due to the presence of the free-play. As shown in Figure 2.28, by assuming sticks, insert and applicator as a discrete mass, springs and rigid stops respectively, stiffness discontinuity may exist due to the presence of the clearances in the stick-applicator assembly. The presence of discontinuity in the system’s parameter can be the cause of nonlinear behaviors such as period-doubling bifurcation and chaotic motion [29,37,44,45]. Therefore, eliminating the gaps in the stick-applicator assembly could help eliminate the PD bifurcation and decrease the motion transfer from the sticks to the applicator.   6WDWLFVWLIIQHVVDQDO\VLVDVDQDSSUR[LPDWHPHWKRGIRUWKHILUVWPRGHFDQEHXVHGWRFRPSDUHWKHVWLIIQHVVRIHDFKLQVHUWZLWKRXWVWLFNV7RWKLVHQG WKHLQVHUWVZHUHPRGHOOHGLQ$%$486DQGWZRGLIIHUHQWORDGLQJFRQGLWLRQVZHUHDVVXPHG)LJXUH9,QWKLVDQDO\VLVVWDWLFVWLIIQHVVLVGHILQHGDVWKHUDWLRRIWKHWRWDODSSOLHGIRUFHWRWKHGHIOHFWLRQRIDUHGSRLQWLQWKHPLGGOHRIWKHVLGHZDOO RI WKH LQVHUWV¶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 ,QVHUW XQGHU D SRLQW ORDG DSSOLHG DW WKH HQWUDQFH E ,QVHUW XQGHU DGLVWULEXWHGORDGDSSOLHGRQWKHVLGHZDOOClamped boundary conditionPoint loadMiddle of the sidewall Clamped boundary conditionDistributed load on the whole a) b)5HJXODULQVHUWZDVUHSODFHGE\WKHRWKHUWKUHHLQVHUWVWRVHHZKHWKHULQVWDELOLW\RFFXUVRUQRWZKHQWZRVKRUWVWLFNVDUHLQVLGHWKHWXEH3'ZDVQRWREVHUYHGGXULQJWKHRSHUDWLRQRIWKHRWKHULQVHUWVLQWKHRSHUDWLQJIUHTXHQF\UDQJHRIWR+])LJXUH30VKRZVWKH5PVYDOXHVRIWKHFRXSOHGV\VWHPLQFOXGLQJWZRVKRUWLQWHUORFNLQJVWLFNVDQGGLIIHUHQWLQVHUWV9LEUDWLRQUHGXFWLRQRIVWLFNVFDXVHGE\WRSEHQGLQVHUWDQGVKRUWEHQGLQVHUWLVQRWLFHDEOHFRPSDUHGZLWKUHJXODUFXUYHGLQVHUWDQGUHJXODULQVHUW2QFHWKHVWLFN¶VYLEUDWLRQZDVVXSSUHVVHGWKHYLEUDWLRQRIWKHFRXSOHGV\VWHPGHFUHDVHGEHFDXVHRISURYLGLQJODWHUDOVWLIIQHVVWRWKHVWUXFWXUH     )UHTXHQF\+]9HUWLFDOVWLFNUPVJ     )UHTXHQF\+]/DWHUDOVWLFNUPVJ     )UHTXHQF\+]$[LDOVWLFNUPVJ([FLWDWLRQYROWDJHP9D     )UHTXHQF\+] 9HUWLF DOVWLFNUP VJ     )UHTXHQF\+]/DWHUDOVWLFNUPVJ     )UHTXHQF\+]$[LDOVWLFNUPVJ([FLWDWLRQYROWDJHP9E)LJXUH(YDOXDWLQJWKHLQVHUWV¶SHUIRUPDQFHLQWHUPVVXSSUHVVLQJWKHVWLFN¶VYLEUDWLRQOHYHOD5PVYDOXHVRIYHUWLFDOODWHUDODQGD[LDODFFHOHUDWLRQRIWKHVWLFNGXHWRYDULRXVH[FLWDWLRQ IUHTXHQFLHV DQG FRQVWDQW H[FLWDWLRQ DPSOLWXGH RI PYE 5PV YDOXHV RIYHUWLFDOODWHUDODQGD[LDODFFHOHUDWLRQRIWKHVWLFNGXHWRYDULRXVH[FLWDWLRQIUHTXHQFLHVDQGFRQVWDQWH[FLWDWLRQDPSOLWXGHRIPY5HJXODULQVHUW5HJXODUFXUYHGLQVHUW7RSEHQGLQVHUW6KRUWEHQGLQVHUWIt can be deduced from the above figures that filling the gaps between the sticks and insert can result in having effective lateral stiffness provided by the insert in the coupled system and lessvibration of sticks/applicator. The efficacy of different insert designs in the full-wheel test rig system is discussed in the next chapter.2.3.3 ConclusionThis chapter was concerned with identifying the dynamics of the lab-scale setup and recognizingthe practical and accessible ways to suppress the vibration in stick-applicator assembly. To better understand the dynamics of the coupled system, each subsystem dynamics wDV studied. It was shown that both mock-wheel and stick-applicator assembly are nonlinear systems and when these two subsystems were coupled to each other, interesting nonlinear behaviors such as internal resonance and period doubling bifurcation might occur. Having produced a consistent instability in the coupled system in certain excitation frequencies and amplitudes, the role of stick designs and insert designs were given attention to suppress the vibration level and eliminate the instability.It was shown that looseness between interlocking sticks and clearance between the sticks and the insert are significant parameters for preventing instability and the fatigue-related problems in the stick-applicator assembly. To that end, new inserts were designed to act as passive compensators to suppress the stick motion and consequently tube’s vibration. The effectiveness of different insert designs in the full-wheel test rig system is discussed in the next chapter.42Chapter 3: Full-wheel rig experiments3.1 IntroductionIt was explained in the last chapter that modifications in insert designs to fill the gap between thestick and the applicator were effective in reducing vibration of stick-applicator assembly in thelab-scale setup. Lab-scale setup was designed to produce consistent instability in a controllable condition. Furthermore, lab-scale setup helped the study of stick-applicator assembly behavior during instability. However, some limitations exist in the lab-scale setup which are not present inthe Full-wheel test rig (FWTR). First of all, the amount of input energy provided by the mock-wheel is less than the one afforded by the wheel in FWTR. Therefore, nonlinearities in the stick-applicator assembly in FWTR can be more intensified. Intensification of nonlinearities may leadto complex nonlinear behaviors. FWTR includes rotational motion of the wheel, which is thedominant motion, and unsubstantial reciprocal motion due to slight misalignments in the wheel and rotor connections. Whereas, lab-scale setup was targeted to provide reciprocal motion. Sticksare in contact with the wheel flange in field experiments. Flange effects were neglected in the lab-scale setup. Due to higher relative motion present in the stick-wheel interaction of FWTR, sticksget worn faster. Both the stick profile at the sliding interface and material of the tip mass used inthe lab-scale setup differ from the FWTR. Thereby, contact properties of FWTR is morecomplicated than the lab-scale setup.Considering the aforementioned restrictions on the lab-scale setup, FWTR experiments are carried out to evaluate the effectiveness of the insert designs in suppressing the vibration level in stick-applicator assembly. Figure 3.1 presents the test rig designed to investigate applicator-train wheel interaction using a freight wheel and without considering the rail-wheel interaction. Two tri-axial accelerometers are used to measure the acceleration of the stick and the applicator (Figure 3.1 (b))during the operational tests. Proximity sensor is also used to measure the wheel runout which was found to be negligible. The maximum speed provided by the driving motor is 600 rpm. Four short interlocking sticks were placed inside the applicator and four different inserts, whichwere introduced in section 2.3.2, were tested accordingly at different wheel speedV.433.2 Experimental results It was noticed that lower wheel speed (less than 10 rpm) produces instabilities and gradual growth in the response of the stick-applicator assembly. However, most of the vibration in the FWTR setup is a stable forced vibration. In reality, wheel operates higher than 10 rpm, therefore having instabilities in lower speeds is not as important as the fatigue-related issues of forced vibration in the higher speeds.Studying the response of the stick-applicator assembly in lower speeds can be helpful to gain an insight of the contributing factors in the dynamics of the coupled system. To achieve that, the responses of the stick-applicator assembly with regular-insert and top-bend insert in contact with the freight wheel are depicted in Figure 3.2 and Figure 3.3 respectively for wheel speed of 10 rpm. Figure 3.2 and Figure 3.3 show representatives record of friction-induced vibration. As shown in Figure 3.2 (a), instability occurs several times within 30 seconds. Figure 3.2 (b) demonstrates portion of the response which includes the vibration initiation (rapid growth), bounded vibration and gradual decay. Figure 3.2 (c) illustrates a dominant peak at 847.5 Hz. It can be a representative of a mode of the structure around 850 Hz becoming unstable due to certain conditions provided by the frictional contact. Furthermore, attachment and detachment status were observed between stick Figure 3.1: Full-wheel test rig experiment using freight wheel; (a) general view of the setup, (b)top view of the wheel’s flange and stick-applicator assemblyFreight wheel Wheel’s guide ApplicatorDriving motor Proximity Sensor ApplicatorWheel Accelerometers a) b)Stickprofile44and wheel flange. Attachment status was in coincidence with the rapid growth in response and annoying noise which are the indications of an unstable oscillation [7, 11,46]. Contact properties at the stick-wheel interface and the mechanism of instability (i.e. mode coupling, stick-slip, etc.) need to be carefully investigated by future studies.Using top-bend insert led to an unstable oscillation with the dominant frequency of 397 Hz (Figure 3.3 (c)) in vertical direction. Figure 3.3 (a) depicts instabilities within 30 seconds. Figure 15.15 15.2 15.25 15.3 15.35 15.4-40040Va (m/s2 )15.15 15.2 15.25 15.3 15.35 15.4-40040La (m/s2 )15.15 15.2 15.25 15.3 15.35 15.4-40040Time (sec)Aa (m/s2 )0 5 10 15 20 25 30-100-50050Va (m/s2 )0 5 10 15 20 25 30-100-50050La (m/s2 )0 5 10 15 20 25 30-100-50050Time (sec)Aa (m/s2 )500 1000 1500 2000 2500 3000012500 1000 1500 2000 2500 3000012500 1000 1500 2000 2500 3000012Frequency (Hz)Instabilitya) b)c)Va (m/s2 )La (m/s2 )Aa (m/s2 )I +]$PSOLWXGHI +]$PSOLWXGHI +]$PSOLWXGHFigure 3.2: Stick acceleration at 10 rpm and regular insert in use; (a) total measurement, E rapid growth in the response (occurreQFHof instability), (c) FFT of response in b453.3 (b) demonstrates portion of the response which includes the vibration initiation (rapid growth), bounded vibration and gradual decay.  300 350 400 450 500 550360 380 400 420 440 460Instabilitya) b)c)-100-50050-100-50050-100-500505 10 15 20 25 3005 10 15 20 25 3005 10 15 20 25 300Time (sec) Time (sec)Frequency (Hz)               040040040I +]$PSOLWXGHVa (m/s2 )La (m/s2 )Aa (m/s2 )Va (m/s2 )La (m/s2 )Aa (m/s2 )Va (m/s2 )La (m/s2 )Aa (m/s2 ) 0      0      0     I +]$PSOLWXGHI +]$PSOLWXGHFigure 3.3 : Stick acceleration at 10 rpm and WRSEHQGinsert in use; (a) total measurement,(b) rapid growth in the response (occurrence of instability), (c) FFT of response LQ bTwo other inserts (i.e. short-bend insert and regular curved insert) were also tested in lower speeds. The data pertaining to these tests are not shown in this section since repeatability was not observed in the response. Based on the above results, it can be understood that insert is an important part in stick-applicator assembly. It can govern the stick motion and motion transfer from the stick to the applicator. It was shown in chapter 2 that instability in the lab-scale setup can be eliminated by using top-bend 46insert. Whereas, using top-bend insert in the FWTR cannot prevent instability in the lower speeds of the wheel. However, it can reduce the unstable frequency of oscillation from 847.5 Hz to 397.2 Hz under the same operational conditions. Furthermore, it can make more delay in the recurrence of the instability. As it was shown in Figure 3.3 (a), instability occurs two times within 30 seconds by using top-bend insert in the stick-applicator assembly. Whereas the recurrence of instability was more than two when the regular insert was in use (Figure 3.2 (a)). Increasing the wheel speed causes the stable forced vibration. Harmonics of the wheel speed (600 rpm or 10 Hz) in the stick response illustrated in Figure 3.4 may have to do with the nonlinear friction force in the interface and intrinsic nonlinear behavior of the stick-applicator assembly. It is worth mentioning that instability was not observed in higher speeds but high level of forced vibration existed in the stick-applicator assembly due to motion transfer from the wheel to the stick through friction coupling. 0 5 10 15 20 25 30-200002000Va (m/s2 )0 5 10 15 20 25 30-200002000La (m/s2 )0 5 10 15 20 25 30-200002000Time (sec)Aa (m/s2 )15.15 15.2 15.25 15.3 15.35 15.4-200002000Va (m/s2 )15.15 15.2 15.25 15.3 15.35 15.4-200002000La (m/s2 )15.15 15.2 15.25 15.3 15.35 15.4-200002000Time (sec)Aa (m/s2 )0 500 1000 1500 2000 2500 3000010200 500 1000 1500 2000 2500 3000010200 500 1000 1500 2000 2500 300001020Frequency (Hz)a) b)c)        I  I  I  II I I I  I I I IIIVa (m/s2 )La (m/s2 )Va (m/s 2)La (m/s 2)Aa (m/s 2)Frequency (Hz)I +]ZKHHOVSHHG47As pointed out earlier, occurrence of instability in lower speeds (less than 10 rpm) is not of great  importance since the wheel rotates higher than 10 rpm most of the time and stable forced vibration is the common type of motion in the applicator- wheel assembly. Therefore, comparison between Rms values of stick-applicator assembly incorporating four different inserts at wheel speeds higher than 10 rpm may illuminate the efficacy of the insert designs in suppressing the forced vibration arising from applicator- wheel interaction. Figure 3.5 shows the variation of the Rms values of stick and the applicator at different speeds by using different inserts. Each experiment at every speed was repeated three times to make sure the trend is consistent. In the figure below, the average Rms values of accelerations are reported.  Figure 3.4 : Stick’s acceleration at 600 rpm and UHJXODUinsert in use; (a) total measurement,(b)detail of the signal in time domain, (c) FFT of response of b (Observation of wheel speedharmonics)a)b)50100 300 60005101520Speed (rpm)Stick's vertical rms (g)50100 300 60005101520Speed (rpm)Stick's lateral rms (g)50100 300 6000246810Speed (rpm)Tube's vertical rms (g)50100 300 6000246810Speed (rpm)Tube's lateral rms (g)50100 300 6000246810Speed (rpm)Tube's axial rms (g)50100 300 60005101520Speed (rpm)Stick's axial rms (g)5HJXODULQVHUW5HJXODUFXUYHGLQVHUW7RSEHQGLQVHUW6KRUWEHQGLQVHUW48Figure 3.5 : (a) Stick’s Rms (g) of different inserts at different wheel speeds; (b) tube’s Rms (g) ofdifferent inserts at different wheel speeds.3.3 DiscussionAs mentioned earlier, lab-scale setup has some limitations which do not exist in the FWTR. Low-level of input energy and lack of a flange, which may act as a lateral support in the FWTR, are the main restrictions of the lab-scale setup. Therefore, evaluation of findings based on the lab-scale setup needV to be done on the FWTR. Experiments conducted on the lab-scale setup and FWTR demonstrated that portion of the stick motion transfers to the applicator. According to the lab-scale setup, insert was shown to be aneffective part in eliminating the instability and suppressing the vibration. To check the efficacy of the insert designs, FWTR was exploited. Instability occurred in the FWTR at the lower speeds. It was shown that insert modification can reduce the frequency of the unstable oscillation and makemore delay in the recurrence of instability. However, insert modification could not eliminate theinstability.  Furthermore, it was observed that increasing the wheel speed results in the forced vibration. Therefore, fatigue-related problems due to forced vibration are the main concern in the stick-applicator assembly. In that regard, four different inserts were tested to assess their efficacyin suppressing the vibration. It can be understood from Figure 3.5 that when wheel speed is lessthan 600 rpm, top-bend insert is more capable of suppressing the vibration in the stick-applicatorassembly and compensating itself for stick motion compared with the other insert designs. Short-bend insert seems not to be as effective as it was in the lab-scale setup in decreasing the vibration in the stick-applicator assembly. Although FWTR and lab-scale setup experiments confirm the inappropriate performance of the regular insert to lower the vibration, to understand accurately theusefulness of the insert designs, fatigue tests should be conducted and S-N diagrams are requiredto be found [47,48]. For instance, the tube’s Rms value (g) in vertical direction at 600 rpm forregular insert, regular curved insert, top-bend insert and short –bend insert is 6.1, 1.88, 1.088 and 3.61 respectively. Fatigue tests answer how the differences between the Rms values influence on the induced stress of the tube and eventually failure of the tube. Conducting the fatigue tests and finding the S-N curves are proposed topics for future studies. 49x First, the dynamics of each subsystem used in the lab-scale setup was studied and it wasshown by various experiments that both substructures are able to represent nonlinearbehavior when they are subject to certain level of input force.x Having known the characteristics of each substructure, the dynamics of the coupled systemwas investigated. It was found thato Period doubling bifurcation occurs consistently in certain ranges of excitationfrequencies and above certain excitation voltages for the interlocking sticks insidethe applicator.o Lateral stiffness could be one of the controlling design parameters of the lab-scalesetup to decrease the vibration level.o More numbers of short interlocking sticks inside the applicator, the less therequired threshold for instability is in the coupled system. Moreover, vibrationreduction by using one long stick in the coupled system, experimentally proved theChapter 4: Conclusions and Future works Chatter/Squeal was recognized as an inconsistent and random phenomenon in the field experiments (i.e. when bogie, wheel and rail are in connection to each other). Experiments conducted on the full-wheel rig in Chapter 4 confirmed the claim made by Sharma [22], saying that most of the vibration in the stick-wheel interaction is forced vibration.  Therefore, it is reasonable to conclude that rotational motion of the wheel, as a source of energy in stick-wheel interaction, may not be the main reason of producing instability in the field experiments. Wheel suspension motion as another source of energy in the stick-wheel interaction could be responsible of producing instability in the field experiments. In this regard, the lab scale setup, which uses mock-wheel to simulate up and down and transverse motion was designed to produce consistent instability. Lab-scale setup helps examine the behavior of the stick-applicator assembly during instability. Furthermore, understanding the ways to lower vibration in stick-applicator assembly by identifying the main contributing parts can be achieved through the lab-scale setup. To better analyze the dynamics of the lab-scale setup:50necessity of thinking about new interlocking stick designs for future studies to minimize the impact of looseness between sticks on the lateral stiffness of the sticks.o The regular insert could have been the best insert design if no gap existed in thestick-applicator assembly. Furthermore, comparison made between the vibrationlevel of stick-applicator assembly using four different inserts revealed thesignificance of insert modifications in suppressing the vibration of the coupledsystem.Finding a mathematical model for the lab-scale setup to substantiate the observed nonlinearbehaviors needs to be done by future studies. Low-level of input energy and lack of having a flange, which may act as a lateral support in the FWTR, are the main restrictions of the lab-scale setup. Therefore, the efficacy of the new insert designs was tested on the full wheel test rig.  It was found that top-bend insert, which was requested to be designed during the present project, is more capable of suppressing the vibration in stick-applicator assembly and compensating itself for stick motion compared with the other insert designs. However, to clarify accurately the practicality of the insert designs, fatigue tests should be conducted and S-N curves are required to be found for the stick-applicator assembly with different inserts by the future studies. Furthermore, the disparities between tube and stick acceleration levels among different insert designs need to be carefully understood by investigating the transmissibility plots in future studies. Figure 4.1 : Proposed lab-scale setup to study the effect of suspension motion ofthe stick-applicator assembly  Shaker’stableShaker (up and down motion)ApplicatorTri axial accelerometersFixed body51As mentioned earlier, the other source of producing vibration in the stick-applicator assembly is the suspension motion of the stick-applicator assembly which was neglected during this project. The experiment depicted in Figure 4.1 can be carried out to gain more understanding of stick-applicator assembly and the significance of the suspension motion. As shown in the Figure 4.1, the stick-applicator assembly is mounted on the shaker capable of providing up and down motion and sticks are in contact with the fixed structure. Thereby, tube is prone to failure due to motion transfer of the shaker, which acts as the suspension motion, and stick-applicator interaction. Considering an appropriate suspension for the FWR experiments may also illuminate the impact of suspension motion on the instability which was observed inconsistently in the field experiments. The long term plan for this research is to have structurally robust solid stick friction control systems by optimizing the required design changes. In that regard, the findings of this research could be used to improve the stick-applicator assembly performance in the wheel-rail industry. 52Bibliography[1] Lewis, R. and Olofsson, U. eds., 2009. Wheel-rail interface handbook. Elsevier.[2] Sharma, C.P., 2011. Friction induced vibrations in railway transportation (MASc thesis,University of British Columbia) [3]Stock, R., Stanlake, L., Hardwick, C., Yu, M., Eadie, D. and Lewis, R., 2016. Materialconcepts for top of rail friction management–Classification, characterisation and application. Wear, 366, pp.225-232. [4]Kalousek, J. and Johnson, K.L., 1992. An investigation of short pitch wheel and railcorrugations on the Vancouver mass transit system. Proceedings of the Institution of MechanicalEngineers, Part F: Journal of Rail and Rapid Transit, 206(2), pp.127-135. [5]Berger, E.J., 2002. Friction modeling for dynamic system simulation. Applied MechanicsReviews, 55(6), pp.535-577. [6]Ibrahim, R.A., 1994. Friction-induced vibration, chatter, squeal, and chaos—part II:dynamics and modeling. Applied Mechanics Reviews, 47(7), pp.227-253. [7]Kinkaid, N.M., O'reilly, O.M. and Papadopoulos, P., 2003. Automotive disc brake squeal.Journal of sound and vibration, 267(1), pp.105-166. [8]Sheng, G., 2007. Friction-induced vibrations and sound: principles and applications. CRCpress.[9]Oberst, S. and Lai, J.C.S., 2011. Chaos in brake squeal noise. Journal of Sound andVibration, 330(5), pp.955-975. [10]Jarvis, R.P. and Mills, B., 1963. Vibrations induced by dry friction. 3roceedings of theInstitution of mechanical Engineers, 178(1), pp.847-857. 53[11] Chen, G.X., Zhou, Z.R., Kapsa, P. and Vincent, L., 2003. Experimental investigation into squeal under reciprocating sliding. Tribology International, 36(12), pp.961-971. [12] Lin, Y.Q. and Wang, Y.H., 1991. Stick-slip vibration of drill strings. Journal of Engineering for Industry, 113(1), pp.38-43. [13] Zhu, Xiaohua, Liping Tang, and Qiming Yang, "A literature review of approaches for stick-slip vibration suppression in oilwell drillstring," Advances in Mechanical Engineering, pp. 952-967[14] Murakami, Hideto, Naomasa Tsunada, and Terukiyo Kitamura., "A study concerned with a mechanism of disc-brake squeal.," SAE Technical Paper,, 1984[15] Akay, A., Giannini, O., Massi, F. and Sestieri, A., 2009. Disc brake squeal characterization through simplified test rigs. Mechanical systems and signal processing, 23(8), pp.2590-2607.[16] Chen, F., Chern, J. and Swayze, J., 2002. Modal coupling and its effect on brake squeal(No. 2002-01-0922). SAE Technical Paper. [17] Hoffmann, N. and Gaul, L., 2003. Effects of damping on modeǦcoupling instability in friction induced oscillations. ZAMMǦJournal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics, 83(8),pp.524-534.[18] Bigoni, D. and Noselli, G., 2011. Experimental evidence of flutter and divergence instabilities induced by dry friction. Journal of the Mechanics and Physics of Solids, 59(10),pp.2208-2226[19] Ziegler, H., 1953. Linear elastic stability. Zeitschrift für angewandte Mathematik und Physik ZAMP, 4(2), pp.89-121. [20] Duffour, P. and Woodhouse, J., 2004. Instability of systems with a frictional point contact. Part 1: basic modelling. Journal of Sound and Vibration, 271(1-2), pp.365-390. 54[21] Altintaş, Y. and Budak, E., 1995. Analytical prediction of stability lobes in milling. CIRPAnnals-Manufacturing Technology, 44(1), pp.357-362. [22] Sharma, C.P. and Phani, A.S., 2015. Stability analysis of on-board friction modifier systems at the wheel–rail interface. Journal of Vibration and Acoustics, 137(5), p.051007. [23] Leine, R.I., Glocker, C. and Van Campen, D.H., 2001, September. Nonlinear dynamics of the woodpecker toy. In Proceedings of DETC 2001 ASME Symposium on Nonlinear Dynamics and Control In Engineering Systems (pp. CDROM-9). [24] Pavlovskaia, E., Wiercigroch, M., Woo, K.C. and Rodger, A.A., 2003. Modelling of ground moling dynamics by an impact oscillator with a frictional slider. Meccanica, 38(1), pp.85-97. [25] Ibrahim, R.A., 2009. Vibro-impact dynamics: modeling, mapping and applications (Vol. 43). Springer Science & Business Media. [26] Ema, S. and Marui, E., 2000. Suppression of chatter vibration of boring tools using impact dampers. International Journal of Machine Tools and Manufacture, 40(8), pp.1141-1156. [27] Ema, S. and Marui, E., 1998. Suppression of chatter vibration in drilling. Journal of manufacturing science and engineering, 120(1), pp.200-202. [28] Shaw, S.W., 1985. Forced vibrations of a beam with one-sided amplitude constraint: theory and experiment. Journal of Sound and Vibration, 99(2), pp.199-212. [29] Moon, F.C. and Shaw, S.W., 1983. Chaotic vibrations of a beam with non-linear boundary conditions. International Journal of non-linear Mechanics, 18(6), pp.465-477. [30] Nayfeh, A.H. and Mook, D.T., 2008. Nonlinear oscillations. John Wiley & Sons. [31] Cartmell, M., 1990. Introduction to linear, parametric, and nonlinear vibrations. Chapman and Hall 55[32] Jia, Y. and Seshia, A.A., 2014. An auto-parametrically excited vibration energy harvester. Sensors and Actuators A: Physical, 220, pp.69-75 [33] Daqaq, M.F., Masana, R., Erturk, A. and Quinn, D.D., 2014. On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion. Applied Mechanics Reviews, 66(4),p.040801.[34] Haddow, A.G., Barr, A.D.S. and Mook, D.T., 1984. Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure. Journal of Sound and Vibration, 97(3),pp.451-473[35] Balachandran, B. and Nayfeh, A.H., 1991. Observations of modal interactions in resonantly forced beam-mass structures. Nonlinear Dynamics, 2(2), pp.77-117. [36] Nayfeh, A.H. and Balachandran, B., 2008. Applied nonlinear dynamics: analytical, computational, and experimental methods. John Wiley & Sons. [37] Virgin, L.N., 2000. Introduction to experimental nonlinear dynamics: a case study in mechanical vibration. Cambridge University Press. [38] Ewins, D.J., 1984. Modal testing: theory and practice (Vol. 15). Letchworth: Research studies press. [39] Kerschen, G., Worden, K., Vakakis, A.F. and Golinval, J.C., 2006. Past, present and future of nonlinear system identification in structural dynamics. Mechanical systems and signal processing, 20(3), pp.505-592. [40] Londoño, J.M., Neild, S.A. and Cooper, J.E., 2015. Identification of backbone curves of nonlinear systems from resonance decay responses. Journal of Sound and Vibration, 348,pp.224-238.56[41] Feldman, M., 1994. Non-linear system vibration analysis using Hilbert transform--I. Free vibration analysis method'Freevib'. Mechanical systems and signal processing, 8(2), pp.119-127.  [42] Noël, J.P., Renson, L. and Kerschen, G., 2014. Complex dynamics of a nonlinear aerospace structure: experimental identification and modal interactions. Journal of Sound and Vibration,333(12), pp.2588-2607. [43] Zavodney, L.D., 1991. Identification of nonlinearity in structural systems: Theory, simulation, and experiment. Applied Mechanics Reviews, 44(11S), pp.S295-S303. [44] Shaw, S.W. and Holmes, P.J., 1992. A periodically forced piecewise linear oscillatort. Chaotic Oscillators: Theory and Applications, 90(1), p.304. [45] Sarkar, S., Venkatraman, K. and Dattaguru, B., 2004. Dynamics of flexible structures with nonlinear joints. Journal of vibration and acoustics, 126(1), pp.92-100. [46] Chen, G.X. and Zhou, Z.R., 2005. Experimental observation of the initiation process of friction-induced vibration under reciprocating sliding conditions. Wear, 259(1-6), pp.277-281.[47] Lee, Y.L., Pan, J., Hathaway, R. and Barkey, M., 2005. Fatigue testing and analysis: theory and practice (Vol. 13). Butterworth-Heinemann. [48] Weibull, W., 2013. Fatigue testing and analysis of results. Elsevier. 5AppendicesAppendix I Coherence function in frequency domain is defined as ߛሺ߱ሻଶ ൌ ܵ௫௬ሺ߱ሻଶܵ௫௫ሺ߱ሻܵ௬௬ሺ߱ሻܵ௫௫ሺ߱ሻ: autocorrelation of the input, ܵ௬௬ሺ߱ሻ : autocorrelation of the output, ܵ௬௬ሺ߱ሻ : cross correlation of the input and the output . The coherence magnitude at each frequency is between 0 and 1. A coherence of 1.0 indicates that the output is purely and linearly explained by the input. Deviation from unity may be the indication of noise in input or output, a non-linear behavior and/or any combination of them. Figure below shows the lateral accelerance of the mock-wheel due to lateral impulse on the tip mass. Good coherence exists at the resonance peak around 12.5 Hz. Lateral accelerance (g/N)Figure 1 : Impact hammer test with coherence 5Appendix II Stiffness of the shaker can be found based on the experiment depicted below. ShakerShort stingerTip massUp and down motion Proposed modela) b)(Vertical motion)Figure 2 : (a) Schematic of experimental setup to measure the stiffness of the shaker, (b) proposed model0 10 20 30 40 50 6000.20.40.60.81x 10-3Frequency (Hz)Vertical response (g)Figure 3: Frequency spectrum of the tip mass in vertical directionThe vertical response of the tip mass was measured and it is illustrated in Figure 3. A vertical mode exists at 22.04 Hz. Short stinger connected to the tip mass has flexible modes well above 100 Hz. Therefore, the mode at 22.04 is a rigid body mode in the system. The mass of the moving element in the shaker is negligible compared with the total mass of tip mass and stinger which is 165 grams. Therefore, stiffness of the shaker can be calculated as follows. 53161 N/m is very close to a number announced by shaker’s manual as suspension axial stiffness for the shaker which is 3150 N/m. Therefore, 3150 was used in FEM of the mock-wheel. It is worth mentioning that no significant change in the linear modes of the mock-wheel modelled in ABAQUS occurs by changing 3150 to 3160 as shaker’s stiffness. 2 2ker0.165 (22.04 * 2 * 3.14) 3161 /shatotalN mkkmZ  u   o

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