MODELING AND IDENTIFICATION OF AN AXIALLY-MOVING CANTILEVER BEAM Liyan Deng © November 15, 2002 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE MScENG DEGREE IN CONTROL ENGINEERING FACULTY OF ENGINEERING LAKEHEAD UNIVERSITY THUNDER BAY, ONTARIO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. National Library Bibliotheque nationale of Canada du Canada Acquisitions and Acquisisitons et Bibliographic Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 0-612-83404-2 Our file Notre reference ISBN: 0-612-83404-2 The author has granted a non­ L'auteur a accorde une licence non exclusive licence allowing the exclusive permettant a la National Library of Canada to Bibliotheque nationale du Canada de reproduce, loan, distribute or sell reproduire, preter, distribuer ou copies of this thesis in microform, vendre des copies de cette these sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format electronique. The author retains ownership of the L'auteur conserve la propriete du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts from it Ni la these ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent etre imprimes reproduced without the author's ou aturement reproduits sans son permission. autorisation. Canada Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract An axially-moving cantilever beam is used to study identification o f time-varying systems. A circuitry for DC motor current control and sensor conditioning is built. The circuitry meets the design requirement o f controlling the axial motion o f the beam and amplifying the sensor signals. A linear time-varying model governing lateral vibration o f the beam is developed. Computer simulation is conducted to study the dynamic properties o f the system, such as transient responses, varying state transition matrices, “frozen” modal parameters, “pseudo” modal parameters, etc. A previously developed algorithm is applied to identify the system. Two identification tasks are carried out. The system identification determines the discrete-time state space model o f the system. The modal parameter identification determines the “pseudo” modal parameters o f the system. In both cases, an ensemble o f freely vibrating responses are used. The study addresses several critical issues encountered in the experiment such as excitation, data preprocessing, the beam motion control, etc. The study also investigates several important factors that affect the accuracy o f identification, such as the number o f necessary experiments, model order, the block row number, etc. An algorithm based on the moving-average method is developed to select the “pseudo” natural frequencies o f vibratory modes. The study shows that the algorithm is capable o f estimating the “pseudo” natural frequencies o f the vibratory modes, present in responses, while it fails to give good estimates for the “pseudo” damping ratios. I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknow ledgem ents I gratefully acknowledge my supervisor Dr. K. Liu for his support and responsible supervision. The thesis would not have been possible without his knowledge and guidance. I would like to thank my co-supervisor Dr A. Sedov for his comments on the thesis. I would like to thank Dr. K. Natarajan and Dr. A. Tayebi for their valuable suggestions in designing and testing the circuitry, Mr. M. Klein and Mr. W. Paju for their assistance in building the circuitry, and Mrs. N. Behman and Mrs. D. Lehtinen for their proofreading o f my thesis. I dedicate this work to my wife and my parents for their love, patience and support. II Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table o f C ontents Abstract.................................................................................................................................................. I Acknowledgements............................................................................................................................ II Table of Contents............................................................................................................................. Ill List o f Figures...................................................... V List o f T a b les ...................................................................................................................................XII Chapter 1. Introduction................................................................................................................... 1 1.1 Overview o f the Previous Studies on Axially-Moving Cantilever Beams............ 1 1.2 Overview o f the Previous Studies on Time-Varying Systems.................................. 2 1.3 Identification o f Linear Time-Varying Systems.......................................................... 3 1.4 Objectives o f the Thesis Research.................................................................................. 4 1.5 Outline o f the Thesis.......................................................................................................... 4 Chapter 2. The Experimental S ystem .......................................................................................6 2.1 Axially-Moving Cantilever Beam Apparatus..............................................................7 2.2 DAQ Board and Computer............................................................................................... 7 2.3 Motor Current Control and Sensor Conditioning Circuitry.......................................8 2.3.1 Description o f Individual Circuits....................................................................9 (1) Motor Power Supply Module.....................................................................9 (2) Bridge Module.............................................................................................10 (3) Current Control Module.............................................................................10 (4) Opto-Isolation Module............................................................................... 11 (5) Signal Separation Module..........................................................................12 (6) Power Supply Module................................................................................ 12 (7) Sensor Module............................................................................................. 12 A. Potentiometer Circuit...................................................................12 III Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B . Strain Gauge Circuit.....................................................................13 2.3.2 Testing Results o f the Circuitry Board.......................................................... 13 (1) Testing o f the Bridge Module...............................................................13 (2) Testing o f the Motor Power Supply Module......................................14 (3) Testing o f the Current Control Module...............................................14 (4) Testing o f the Opto-Isolation Module................................................. 15 (5) Testing o f the Signal Separation Module............................................16 (6) Testing o f the Power Supply Module...................................................17 (7) Testing o f the Sensor Module................................................................17 A. Potentiometer Circuit.................................................................... 17 B. Strain Gauge Circuit...................................................................... 18 (8) Testing o f the Overall Circuitry...............................................................18 2.4 Summary........................................................................................................................... 20 Chapter 3. Dynamics o f an Axially-M oving Cantilever Beam ........................................ 21 3.1 Review o f Dynamics o f a Fixed-Length Cantilever Beam......................................22 3.1.1 Mathematical Modeling.................................................................................... 22 3.1.2 Response to a Concentrated Force Input.......................................................24 3.1.3 Observability and Controllability o f the System......................................... 26 3.2 Modeling o f an Axially-Moving Cantilever Beam................................................... 29 3.2.1 Modeling with Considering the Contribution o f the Axial Force............29 3.2.2 Modeling without Considering the Contribution o f the Axial Force.. ..34 3.2.3 Continuous-Time State Space Representation o f the System.................. 35 3.2.4 “Frozen” Modal Parameters.............................................................................37 3.2.5 Controllability and Observability o f the System......................................... 37 3.3 Simulation o f the Axially-Moving Cantilever Beam................................................41 3.3.1 Influence o f the Axial Force.............................................................................41 3.3.2 Generalized Coordinates and Their First-Order Derivatives....................43 3.3.3 Transient Responses.......................................................................................... 45 1) Strain Gauge Outputs........................................................ 47 2) Deflection Outputs...................................................................................... 49 3) Velocity Outputs.......................................................................................... 52 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4) Acceleration Outputs..................................................................................55 3.4 Evaluation o f Varying Discrete-Time State Transition Matrix..............................59 3.4.1 Discrete-Time State Transition Matrix and “Pseudo” Modal Parameters............................................................................................................ 59 3.4.2 Numerical Evaluation o f the Discrete-TimeS tate Transition Matrix....................................................................................................................60 3.4.3 Comparison between “Frozen” and “Pseudo” Modal Parameters.........61 3.5 Conclusions.........................................................................................................................63 Chapter 4. System Identification o f an Axially-M oving Cantilever Beam ..................64 4.1 A Subspace-Based Identification Algorithm..............................................................64 4.1.1 State Space Representation o f LTI system.................................................. 65 4.1.2 Hankel Matrix and Observability Matrix...................................................... 66 (1) Hankel Matrix............................................................................................. 66 (2) Extraction o f the Range Space o f the Observability Matrix............. 67 4.1.3 The Computational Procedure........................................................................69 4.1.4 Experimental Results........................................................................................70 A. Identified Modal Parameters.....................................................................70 B. Simulated Transient Responses................................................................ 72 4.2 Identification Algorithm................................................................................................. 75 4.2.1 Transition Matrix...............................................................................................75 4.2.2 Identification o f a LTV System Using an Ensembleo f Freely Vibrating R esponses............................................................................................................76 4.2.3 Computational Procedure for the Identification Algorithm.....................80 4.2.4 Identified “Pseudo” Modal Parameters........................................................ 80 4.3 Comparison o f the True and Identified “Pseudo” Modal Parameters...................81 4.4 Experimental Identification o f the System................................................................. 84 4.4.1 Identification Results Using the Original Ensemble Data.......................89 4.4.2 Identification Results Using the Selected Ensemble Data....................... 95 4.4.3 Relationship between the Model Order and Model Accuracy................97 4.4.4 Relationship between the Block Row Number and Modal Accuracy...98 4.4.5 Comparison o f Transient Responses............................................................99 V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Experimental Identification o f the “Pseudo” Natural Frequencies..................100 4.5.1 Identified “Pseudo” Natural Frequencies.....................................................101 4.5.2 Selection o f the Identified “Pseudo”N atural Frequencies o f the Vibratory Modes...............................................................................................104 4.5.3 Results o f Selection o f Natural Frequencies o f the Vibratory Modes.................................................................................................................. 105 4.6 Conclusions .............................................................................................................. 109 Chapter 5. Summary and Future Work........................................................................111 Bibliography........................................................................................................................113 Appendix..............................................................................................................................117 VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Figures 2.1 The experimental system........................................................................................................6 2.2 Relationship between setpoint voltage and motor current.............................................. 8 2.3 Motor current control and sensor conditioning circuitry.................................................9 2.4 The P effort and I effort for IV square waveform input................................................15 2.5 The 1V square waveform input and PI effort waveform...............................................15 2.6 Output A and 1V square waveform input......................................................................... 16 2.7 Output B and 1V square waveform input......................................................................... 16 2.8 Rectified magnitude waveform and square waveform for direction..........................17 2.9 Potentiometer output waveform.......................................................................................... 18 2.10 The strain gauge signals at two positions.........................................................................18 2.11 Waveform comparison between setpoint and feedback.................................................19 2.12 Waveform comparison when setpoint amplitude changes...........................................19 2.13 Waveform comparison when setpoint frequency changes...........................................19 2.14 The waveforms o f the variable resistor and the setpoint.............................................20 2.15 High-amplitude setpoint waveform and saturated feedback waveform................... 20 3.1.1 Lateral vibration o f the cantilever beam and a free-body diagram o f the beam element....................................................................................................................................22 3.1.2 The first three mode shape functions...............................................................................28 3.1.3 The second derivatives o f the first three mode shape functions................................. 28 3.2.1 The apparatus o f the axially-moving cantilever beam.................................................. 29 3.2.2 (a) deformation; (b) free-body diagram o f A s ; (c) remaining part o f the beam....30 3.2.3 Axial motion profiles............................................................................................................38 3.2.4 Varying magnitudes o f 0 { a jn) ......................................................................................... 40 3.2.5 Varying magnitudes o f O, ( a oun) / Is and O, (ccout2 ) / L 2 .......................................... 40 3.3.1 “Frozen” natural frequencies / and damping ratios C, for scenario A ...................42 3.3.2 “Frozen” natural frequencies / and damping ratios £ for scenario B ................... 42 VII Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.3 The generalized coordinates and velocities for the axial extension........................ 43 3.3.4 The generalized coordinates and velocities for the axial retraction..........................44 3.3.5 Strain outputs for motion scenarios A and B .................................................................47 3.3.6 Strain output components y u (i = 1,2,3) at the base location...................................48 3.3.7 Strain output components y 2i (z = 1,2,3) at the middle location...............................49 3.3.8 Deflection outputs for motion scenarios A and B ........................................................ 49 3.3.9 Deflection output components y 4j(i = 1,2,3) at the base location.............................50 3.3.10 Deflection output components y 5i(i = 1,2,3) at the middle location ........................ 51 3.3.11 Deflection output components y 6j(i = 1,2,3) at the tip location................................ 51 3.3.12 Velocity outputs for scenarios A and B ...........................................................................52 3.3.13 Velocity output components y 7i (/ = 1,2,3) at the base location............................52 3.3.14 Velocity output components y 7i (z = 4,5,6) at the base location...........................53 3.3.15 Velocity output components y g(. (z = 1,2,3) at the middle location.......................53 3.3.16 Velocity output components y Sj (z = 4,5,6) at the middle location......................54 3.3.17 Velocity output components v9, (z = 4,5,6) at the tip location.............................. 55 3.3.18 Acceleration outputs for scenarios A and B..................................................................55 3.3.19 Acceleration output components y 10( (z = 1,2,3) at the base location................... 56 3.3.20 Acceleration output components y 10, (z = 4,5,6) at the base location................... 56 3.3.21 Acceleration output components y 11( (z = 1,2,3) at the middle location.................57 3.3.22 Acceleration output components y Uj (z = 4,5,6) at the middle location................57 3.3.23 Acceleration output components y Uj (i = 1,2,3) at the tip location.........................58 3.3.24 Acceleration output components y ni (z = 4,5,6) at the tip location........................58 3.4.1 The “pseudo” and “frozen” modal parameters for axial extension..........................61 3.4.2 The “pseudo” and “frozen” modal parameters for axial retraction..........................62 4.1.1 The experimental setup for a fixed-length cantilever beam.......................................70 4.1.2 Comparison o f analytical and identified natural frequencies.................................. 72 4.1.3 FFT plots o f the transient responses at the three observed positions.......................73 4.1.4 Comparison between the measured and simulated responses................................... 73 VIII Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.5 Responses near the tail o f the data records..................................................................... 74 4.3.1 Comparison o f the true and identified “pseudo” modal parameters under scenario A ...............................................................................................................................................82 4.3.2 Comparison the true and identified “pseudo” modal parameters under scenario B .............................................................................................................................................. 82 4.3.3 Relationship between RMS error indices o f modal parameters and varying M ...84 4.4.1 The experimental setup....................................................................................................... 84 4.4.2 Command voltage profiles..................................................................................................86 4.4.3 Pot signal profiles.................................................................................................................87 4.4.4 The singular values for each combination o f the motion scenarios and speeds...................................................................................................................................... 90 4.4.5 Comparison o f the first twenty singular values at different time instants and average.................................................................................................................................... 91 4.4.6 Comparison o f the first twenty average singular values between scenarios A and B ............................................................................................................................................... 92 4.4.7 Comparison o f the first twenty average singular values between two motion speeds...................................................................................................................................... 92 4.4.8 (5 - n x) curves for the fast and slow motions o f the two scenarios......................... 97 4.4.9 (5 - M ) curves for the two motion speeds and the two scenarios........................... 98 4.4.10 Comparison between the simulated and measured responses for scenario A ........99 4.4.11 Comparison between the simulated and measured responses for scenario B ......100 4.5.1 Identified “pseudo” natural frequencies f i (k ) using the thresholds nx .............. 102 4.5.2 Comparisons o f IPNFs using nx = 4 and nx = 12 under scenario A ............................................................................................................................................. 103 4.5.3 IPNFs o f vibratory modes for fast m otion .................................................................. 106 4.5.4 IPNFs o f vibratory modes for slow m otion ................................................................. 106 4.5.5 Selected IPNFs using M, = 10 and M, = 50 for fast motion o f scenario A ............................................................................................................................................. 107 4.5.6 Selected IPNFs using M, = 10 and M x = 50 for slow motion o f scenario A ............................................................................................................................................. 108 IX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5.7 Selected IPNFs using M ] =100 and M x = 2 0 0 for fast motion o f scenario B ..............................................................................................................................................108 4.5.8 Selected IPNFs using =100 and M, = 2 0 0 for slow motion o f scenario B ..............................................................................................................................................109 X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 2.1 Voltage between gates and sources o f the MOSFETs................................................... 13 3.1 p jL and a y o f the first four modes..................................................................................23 4.1 The identified natural frequencies and damping ratios..................................................71 4.2 Relationship between the command voltage and motor current.................................85 4.3 RMS error 8 for the models identified using the original ensemble data................. 94 4.4 RMS error 8, for the models identified using the original ensemble data...............94 4.5 RMS error 8 for the models identified using the selected ensemble data.................96 4.6 RMS error 8, for the models identified using the selected ensemble data..............96 XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Acronym s • BSG Base Strain Gauge • LTI Linear Time-Invariant • LTV Linear Time-Varying • M SG Middle Strain Gauge • M OSFET Metal-Oxide Field-Effect Transistor • M AC Middle Accelerometer • SNR Signal to Noise Ratio • SVD Singular Value Decomposition • TAC Tip Accelerometer • IPNFI ith Identified “pseudo” Natural Frequency XII Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction The dynamics of axially-moving cantilever beams and strings have received a good deal of a tten tion in connection with vibration problem s of tim e-varying mechanical system s, such as band saw blades, paper and m agnetic tapes, threadline in textile industry, high-rise elevators, spacecraft antennae and tethered satellite in space exploration, robotic arm s w ith prism atic jo int [1-7]. This study is m otivated to develop an axially-moving cantilever beam system for study of identification and control of time-varying mechanical systems. The m ain task of the research is to develop an experim ental system, to develop an analytical model of the system, and to identify the system. The rest of this chapter is organized as follows: Section 1.1 overviews the previous studies on axially-moving cantilever beams, Section 1.2 overviews the previous studies on tim e-varying systems, Section 1.3 reviews identification m ethods of linear time-varying systems, Section 1.4 lists the objectives of the thesis research, and Section 1.5 outlines the thesis. 1.1 Overview of the Previous Studies on Axially-Moving Can­ tilever Beams The studies on axially-moving cantilever beams can be classified into two areas: m odeling and control. Efforts in modeling have been made in two different aspects. One is analytical modeling; The other is identification as described in section 1.3. Studies dealing w ith the m athem atical modeling of axially-moving cantilever beam s are re- 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ported in [1-9]. The models are derived by applying N ew ton’s second law, extended H am ilton’s principle or Lagrangian form ulation based on the assum ption th a t the deflection gradients of the beam are small and the beam is axially rigid. The axial m otion influences the dynam ics of the axially-moving cantilever beams. A positive dam ping effect is induced by axial extension and a negative dam ping effect is induced by axial retraction respectively [2]. The extending and retracting m otions of a flexible robot arm have destabilizing and stabilizing effects on the arm vibration based on the fact the deflection a t the tip of the beam becomes large during axial extension and small during axial retraction respectively [4]. The m otion-induced vibration is considerable and the vibration of the tail section of the robotic arm can cause appreciable posi­ tion errors [5]. The deflection and velocity of an axially-moving cantilever beam are sim ulated numerically [7]. It is found th a t the axial extension increases the am plitude of the deflection due to a reduced stiffness while the axial retraction reduces the deflection of the beam due to an increased stiffness. Also, the axial extension decreases the am plitudes of the vibration velocity because of dissipation of v ibration energy while the axial retraction increases the am plitudes of the vibration velocity because of absorption of vibration energy [7]. To date, little effort has been m ade in modeling the axially-moving cantilever beam systems using sta te space representation. Also, little work has been done in analyzing the contributions of generalized coordinates, generalized velocities, an d /o r v ibratory modes to the transient re­ sponses including strain , deflection, velocity and acceleration in the axially-moving cantilever beam system. This is the first m otivation of the present study. 1.2 Overview of the Previous Studies on Time-Varying Systems The studies on tim e-varying system s may be classified into two groups. One is the theoretical analysis of controllability and observability of time-varying systems [10-13]. The o ther is the analysis of modal param eters of time-varying systems. It is noted th a t a LTV ( linear time- varying) system violates one of the assum ptions of the conventional m odal analysis, th a t is, stationarity. The concept of the “pseudo” modal param eters was introduced in [14,16]. They are obtained by conducting eigendecomposition of the varying discrete-tim e sta te transition m atrices [16]. The identification of the “pseudo” modal param eters for LTV system s was 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. extended to the cases of forced responses and forcing inputs in [14]. The applicability of the “pseudo” m odal param eters based on the “pseudo” transfer function are also reported in tim e- varying structures in [15]. However, the algorithm s developed have only been verified using com puter simulation. To implement the algorithm experim entally is the second m otivation of this study. 1.3 Identification of Linear Time-Varying Systems In order to design a high performance active control system, an accurate system model is a prerequisite. It is difficult to obtain an accurate m athem atical model due to the fact th a t irregularities, including physical dam ping effect and nonlinear factors, exist in the system . On the other hand, identification obtains a system model based on experim ental da ta . Such a model can represent the actual system better. A daptive m ethod is a popular m ethod in tim e-varying system identification. I t uses recursive algorithm to estim ate or identify the tim e-dependent param eters of the model which is assumed to be a polynomial. The feature of this m ethod is the use of d a ta from a single experim ent [17]. Ensemble m ethod is an alternative approach to tim e-varying system identification. The feature of the m ethod is the use of m ultiple input and outpu t d a ta from m ultiple experim ents, and each of the experim ents m ust experience the same time-varying change. The key of the ensemble m ethod is concerned w ith three steps. F irst, a series of Hankel m atrices are formed by an ensemble of the freely v ibrating responses from multiple experiments; Second, the varying s ta te transition m atrix at each moment is estim ated through the SVD (singular value decomposition) of two successive Hankel matrices; T hird, the “pseudo” modal param eters are obtained by conducting the eigendecomposition of the varying sta te transition m atrix [16]. The varying sta te transition m atrices can also be estim ated using forced responses and forcing inputs. The “pseudo” m odal param eters are evaluated by conducting eigendecomposition of the varying sta te transition m atrices [14], The identification algorithm s m entioned in [14] and [16] have been successfully verified using com puter simulation. A wavelet-based approach for the identification of a linear tim e-varying lumped-mass system is reported in [19]. However, it is noted th a t experim ental identification of LTV system s rem ains a relatively inactive area th a t deserves 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. more attention. Meanwhile, little effort has been made to identify the axially-moving cantilever beam system with time-varying sta te transition m atrix and the tim e-varying o u tp u t m atrix . To capture the dynamics of the system, the model to be identified should be overparam eter­ ized [20-25]. The overparameterized model contains both the system m odes and com putational modes caused by the noise or irregularity of the system. It is im portant to distinguish between these modes. To qualify the contribution of individual modes to pulse responses, m ode singular value is defined in [23]. Modal response m agnitude is also proposed to qualify the m aximum contribution of the individual modes to the responses [21-22]. However, th is research is lim ited to time-invariant systems. It can be inferred th a t the overparam eterized model contains system modes and com pu­ tational modes at each tim e moment if a system is time-varying. I t is expected th a t for the case of time-varying systems, selection of the system modes among the identified modes is more challenging than the case of tim e-invariant system s as identification m ust be conducted at each moment. An effective m ethod to select v ibratory modes rem ains to be found. This is the th ird m otivation for the present study. 1.4 Objectives of the Research 1. The first objective of the research is to build a m otor current control and sensor signal conditioning circuitry in order to control axial m otion of the cantilever beam and amplify the sensor signals. 2. The second objective of the research is to develop an analytical model and conduct a com puter sim ulation in order to understand the dynamics of the system. 3. The th ird objective of the research is to apply an identification algorithm to the axially- moving cantilever beam system and to identify the model of the system. 1.5 Outline of the Thesis The following chapters of the thesis are organized as follows: C hapter 2 describes the develop­ m ent of the experim ental system, C hapter 3 develops an analytical model for the axially-moving 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cantilever beam and presents some com puter sim ulation results, C hap ter 4 focuses on identifi­ cation of the system, and C hapter 5 draws the conclusions of the s tudy and recom m ends fu ture work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 The Experimental System This chapter presents the development of the experim ental system. The entire system consists of three subsystems: ( 1 ) an axially-moving cantilever beam apparatus, (2 ) a m otor current control and sensor signal conditioning circuitry, (3) the system of a d a ta acquisition (DAQ) and PC computer; all shown in Figure 2-1. X X 7! m o to r c u r r e n t1 \---------- I ' c o n tro l c i rc u i t / i fal-DA '■ (e le c tr ic a l in te r f a c e b o a rd ] C o m p u t e r s e n s o r s ig n a l £ f 1 / a m p lif ie r |* : B e lt & P u lle y s R o lle r S u p p o r ts r a c k DC Motor C la m p e r B a s e S tr a in G a u g e M id d le S tr a in G a u g e ................ B eam Figure 3.1 The experim ental system The chapter is organized as follows: Section 2.1 describes the axially-moving cantilever 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. beam apparatus, section 2.2 describes the DAQ board and PC com puter, section 2.3 presents the development of the m otor current control and sensor conditioning circuitry, and section 2.4 is a brief summary. 2.1 Axially-Moving Cantilever Beam Apparatus The apparatus was designed and built by Mr Ahmed Hage as his Bechelor Degree project [29]. It consists of a 12V DC perm anent m agnet reversible m otor, a belt and pulley set, rack and pinion, and the beam. The m otor has a built-in gearbox w ith a transm ission ratio of 13:1. The m otor idle speed is 180 rpm. The speed under a load of 24 in-lb on the shaft is 160 rpm @ 6.2 to 7.2 amps. Under a 45 in-lb load, the outpu t speed is 145 rpm @ 10.5 to 11.6 amps. The transmission ratio of the belt and pulley set can be 1.9, 2.9, or 5.2 by changing the large pulley. The center distance of two pulley shafts is 3.96 inches. In order to m easure the angular position of the large pulley shaft, or the axial displacem ent of the cantilever beam , a po t is a ttached at one end of the large pulley shaft. The cantilever beam is m ade from 6061-T6 aluminum- magnesium-silicon alloy and its dimension are 1850 mm (length) x50.6 mm (width) x3.175 mm (thickness). A clam p is used to ensure the boundary conditions a t the clam ped end. In order to m easure the lateral vibration signals of the beam, the stra in gauge sensors are bonded on the beam surface at two positions. One refers to as base strain gauge (BSG), the o ther refers to as middle strain gauge (MSG). 2.2 DAQ Board and Computer The com puter used to control the system is a Pentium III with a speed of 1000MHz and 128MB RAM. The DAQ board is a National Instrum ents PC I Series Model PCI-M IO-16E-4, which has a resolution of 1 2 bits. 16 single-ended or 8 differential analog input channels w ith maximum sampling ra te of 500KS/s, 2 D /A outpu t channels w ith a maximum update rate of IM S /s, eight digital IOs, and two counters. Labview is used for programing. The voltage comm and from one of the analog output channels (DACO and D A C l) of the DAQ board in the com puter is referred to as setpoint. It serves as an input th a t can be separated into two signals: one is the m agnitude signal and the o ther is the direction signal in 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. signal separation circuit (see in section 2.3.1). The m agnitude and direction signals are used to control the m agnitude and direction of the m otor current, respectively. The relationship between m otor current and the setpoint voltage is shown in Figure 2.2 and the testing m ethod is described in section 2.3.2. A proportional gain between the m agnitude of the setpoint voltage in Channel DACO and m otor current is 0.5 A /V within the range of 0V to 3V. As long as the setpoint voltage exceeds 3V, the m otor current saturates. o 0.6 O 2 10 setpoint voltage (V) Figure 2.2 Relationship between setpoint voltage and m otor current 2.3 Motor Current Control and Sensor Signal Conditioning Circuitry One of the m ain tasks of this project is to build an electrical circuitry. The decision of building the circuitry in house was due to the lim ited budget and a training opportunity for the au thor to gain some practical experience in electrical engineering. The requirem ents for the circuitry are 1 . to control the m agnitude and direction of m otor currents; 2 . to amplify the signals of potentiom eter and stra in gauges. The circuitry design was based on the one used in previous projects [26-27]. Some m istakes in the original schematic drawings have been found and corrected accordingly. The overall circuitry, shown in Figure 2.3, consists of m otor power supply module, bridge module, power supply module, current control module, signal separation module, sensor m odules including 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. potentiom eter circuit, strain gauge circuits. P otentiom eter Term inal Board Motor Pow er Supply ♦12 V TO AcHi --------- End Gnd(1) Output TO ACH1 ACO 0 ln_AC2 12 V -12 V TO ACHO ACO Gnd Bridge M odule ACI Gnd Current control m odule TRV-------------------- 12 V (m otor) InputM agnitude 51 & m otor 1 ate?* 3 * * Gnd(1) 52 A m otor 2 FeedEack -12 V S ignal Separation Circuit O pto-lsolation Circuit (3) P ow er Supply Circuit (3) -7WV ♦ b V (4 ) ♦5 V(1) ♦12 V(4) In AC1 Input Gnd(1) Output B InputD lreetion ♦5V(4) In AC2 O utputM ag ♦5V (4) Strain G au ge (MSG) Gnd (1) Gnd (4) O utputD ir Output A -12 V(4) Output -12V Gnd (4) From MSG 1 < O pto-lsolation Circuit (2) P ow er Supply Q rcuit (2) ♦12 i/(3) ♦5 V(1) ♦12 V(3) In ACI Output B Inp u tD irectior ♦5V<3) In AC2 ♦ 5V P ) Gnd (1) Gnd (3) Strain G auge (BSG) Output A -12 V(3) P ow er Supply Circuit (4) Gnd (3) output ♦12 V _ IN AC1 ♦12 V Opto-lsolation Circuit (1) From BSG 1 A 2Pow er Supply Circuit (1) Gnd (1) Inputl _ ♦5 V ♦12 V<2) -12 VIN AC2 ♦5 V(1) Input2♦12 V(2) In AC1 Gnd (1) Output B InputD ireclion ♦5V(2) In AC2 - 12 V ♦5 V(2) Gnd (1) Gnd (2) Output A IT - -12 V(2) Gnd (2) Figure 2.3 M otor current control and sensor conditioning circuitry 2.3.1 D escrip tion o f Ind iv idual C ircuits ( 1 ) M o to r P o w e r S u p p ly M o d u le The schematic of m otor power supply circuit is shown in Figure A2.1 of A ppendix A. The module provides an DC power source to the current amplification circuit of transistor 2N6059 in bridge m odule in order to change the m agnitude and direction of the m otor current. It is capable of providing m aximum 5A DC currents at 12V. The module consists of a transform er, a fuse, a diode bridge, a RC filter, and a current boost circuit. A transform ed AC waveform is rectified by the diode bridge. Then, the rectified DC waveform is passed through the RC filter to obtain a sm ooth waveform. The transistor 2N6052 is used to amplify or to boost the current. LM 317 is an adjustable voltage regulator and R3 pot resistor is used to obtain and 9 A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. adjust a desired ou tpu t voltage. The voltage regulators 7805 and 7912 are used to ob tain +5V and -12V voltage respectively. (2) Bridge M odule As shown in Figure A2.2 of Appendix, there are four M OSFETs ( M etal-O xide Field-Effect Transistors ) Q \, Q 2 ,Qs, and Q 4 in the circuit. They are used to switch the current directions. If Q iand Qy, are on, the m otor current flows in one direction; If Q 2 and Q 4 are on, the m otor current flows in the opposite direction. Thus, the m otor ro tation can be reversed. The ou tpu t of the current control module is applied at the base of the transistor of 2N6059 to obtain a desired current. The actual m otor current is detected by m easuring voltage of resistor Rg. Since there is an extreme buildup of voltage during the M OSFET switching which can reduce M O SFE T ’s life, the capacitors, resistors and diodes between drain and source of each M O SFET m ust be used in order to dam p high voltage buildup. The resistors a t the gates of the M O SFETs are used to dam p the noise from the ou tputs of opto-isolation circuits. (3) Current Control M odule The circuit is used to control the m agnitude of the m otor current. It is a proportional (P) and integral (I) controller. The values of corresponding resistors and capacitors are given in the schematic showrn in Figure A2.3 of Appendix. The voltage across l f l resistor Rg in the bridge m odule shown in Figure A2.2 is fed to the non-inverting term inal of the operational amplifier circuit consisting of an O pam p LM1458 and the two resistors, i? i3 and R 1 4 . in the bridge module. The feedback gain is designed to be 2. The output of the amplifier circuit is compared with the reference input from m agnitude ou tpu t of the signal separation module. The ou tpu t voltage of the current control m odule is given by (2-1) where e is the error between the reference input and the feedback voltage. R io /R g = R io /R g = 1. 2 / 1 . 2 = 1, R u / R n = 5.6/82 = 0.0683, R 7 / R 5 = 180/1200 = 0.15, and 1 /R e C i = 1/(18000 x 1.2 x 10~9) = 46269. Because ratio i? i2 /.R iiis very small, the ou tpu t voltage or 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. control effort can be approxim ated by u = K pe + K i [ edt (2-2) Jo where K v = 0.15 is the proportional gain and K i = 46269 is the integral gain. (4) Opto-Isolation M odule Figure A2.4 in Appendix A shows the schem atic of the opto-isolation circuit. The function of the m odule is to obtain two ou tpu ts th a t are inverted w ith each other, which can guarantee th a t M O SFET's pair ( Q \ k Qz) and (Q2 k Q 4 ) in the bridge m odule do not tu rn on and off a t the same time. Thus, the direction control of the m otor current can be realized in th is way. There are three opto-isolator circuits. As shown in Figure 2.3 of the overall circuitry, the ou tpu t B and its ground in opto-isolation circuit (2) are connected to G ate 1 and Source 1 of M O SFET Q\ respectively. The ou tpu t A and its ground in opto-isolation circuit (1) are connected to G ate 2 and Source 2 of M OSFET Q 2 respectively. Since M OSFETs Q3 and Q \ shares the same point at the source, only one opto-isolation circuit (3) is needed. The ou tpu t A, ou tpu t B and their shared ground in opto-isolation circuit (3) are connected to G ate 4 of M O SFET Q 4 , the G ate 3 of M OSFET Qz, and the shared Source S 3 k S 4 of M OSFETs Q 4 and Q 3 respectively. Careful grounding should be considered because the output grounds in three opto-isolation circuits are independent with each other and they do not share the same ground as the common ground in the circuitry. On the other hand, the m agnitude of ou tpu t voltage with respect to its ground must be greater th an 10V, because 10V voltage is needed to tu rn on M O SFETs fully. D S0026C N is a clock chip which can drive capacitive load at the M O SFET w ith a peak of one ampere. The opto-isolation chip is HP261A. The opto-isolation circuit m ust be able to operate at high frequency, the same as the sampling frequency in d a ta acquisition. Since each of these circuits m ust be isolated with each other, they require different power supplies. The +5V supply is used for HP261A opto-isolation and +12V for DS0026C1V. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5) S ig n a l S e p a ra t io n M o d u le The schematic is shown in Figure A2.5 of Appendix A. The function of the signal separation circuit is to separate the comm and voltage from the DAQ board into a m agnitude signal and a direction signal. The gain .R2 /-R1 = 1 and diode D \ are used to rectify and to obtain the absolute value of the command voltage. The direction ou tpu t is a b inary value. If the input signal is positive, the direction ou tpu t is zero because of the diode D 2 action; If input signal is negative, the direction outpu t value is v ~ - r a K + ( 2 - 2 ) where V+ is the ou tpu t voltage of O pam p LM1458. ( 6 ) P o w er S u p p ly M o d u le x Figure A2.6 in Appendix A shows the schematic of the power supply module. To provide ± 1 2 P and + 5V DC voltages to various chips including opto-isolation chip HP261A, clock chip D S0026C N , LM1458 opamps etc., four power supply modules are needed. The circuit consists of a transform er, a diode bridge, voltage regulators, and capacitors. (7) S e n so r M o d u le The sensor m odule includes one potentiom eter circuit, two stra in gauge circuits. A . P o te n t io m e te r C ir c u i t Figure A2.7 in Appendix A shows the schem atic of the poten­ tiom eter circuit. The circuit is a simple voltage follower. The ou tpu t voltage V( t ) a t tim e instant t can be expressed by V « ) = + V' L ' ~ Vl L ' (2-3) L 1 — JU2 — L i 2 where V\ is the voltage reading when the beam is L \ , V2 is the voltage reading when the beam is L 2 , and x(t ) is the axial position of the cantilever beam at tim e instant t. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B . S tra in G a u g e C ir c u i t Figure A2.8 in appendix shows the schem atic of s tra in gauge circuit. The function of the circuit is to amplify the stra in gauge signals. The ou tp u t V( t ) of the circuit is determ ined by v { t ) = " w 2V2) = 100 x 10{Vl ~ V2) (2_4) where Vi is the input from one term inal of the stra in gauge sensor and V2 is the input from the o ther term inal of the stra in gauge sensor. 2.3 .2 T esting R esu lts o f th e C ircu itry B oard After the circuits have been set up, they are tested individually. Then, the assembly circuitry is tested. The testing results are presented below. ( 1 ) T e s tin g o f t h e B r id g e M o d u le Figure A2.9 in Appendix A shows the hookup for testing bridge module. The testing was done by m easuring the voltage between the gate and source of each M O SFET to make sure th a t M O SFET’s pair (Qi& Qs) and pair {Q2i i Q$) do not tu rn on and off a t the same tim e. Note th a t Case 1 is the testing condition for shorting common ground and input direction of one of the opto-isolation modules, and Case 2 is the testing condition for shorting input direction and +5V term inal of the m odule on the left. The results are shown in Table 2-1. Table 2-1. Voltage between gate and source of M OSFET M O S F E T s No Voltage in Case 1(V) Voltage in Case 2 (V) Qi 11.53 0 Q2 0 1 0 . 8 6 < P M T B 5 . 0 C i m s Figure 2.6 O utpu t A and IV square input waveform p m .? .:5. 7 0 1 - f .7 h i : c.» v f* h l : pkp Nr- 4 - <• . r;., ’vr ; I Lt..... _. J ~ 1 | M—-------- p*,— — j --- .J r !.CH 1 1 . 0 0 V -- : fcCHX’ 1 . 0 0 }/ ■■ B5 . OC m.s Figure 2.7 O utput B and IV square input waveform (5) T e s tin g o f th e S ig n a l S e p a ra t io n M o d u le Figure A2.14 of Appendix A shows the hookup for testing diagram of the circuit. Figure 2-8 shows the m agnitude output and direction ou tpu t waveforms when a IV sine waveform with 60Hz frequency from function generator is input to the circuit. It seems th a t the circuit can behave as expected, i.e., producing a rectified waveform for the m agnitude and a square waveform for the direction. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chjmtil 4.01X1 ---------- Databloclc- Name * C ham td Chamal2 Daw *4/13101 4119101 3.0DC Tima • 21251PM 2:1252 PM YScala > 1 VIDn 1 VID» 2.01H Y1V5QX > 0.OH0 V 2525011 XScalr ■ 5 mslDw 5 mlDiv 1.0110 XAtOy. * 0.0 IM 0.0 ms XSn. * 50015121 500(5121 0.01DV Mawmum • 156S0V 11275 V Mjrwuti • -0.6545 V 0.0400 V -0.3300 - Cursor V a b n - X1 125 ms -13300 X2 375 ms dX 250 ms -23300 Y1 -05756V Y2 34775V -3 3300 dY 40531 V O.Orr 5ms/Dw Figure 2.8 Rectified m agnitude waveform and square waveform for direction ( 6 ) T e s tin g o f th e P o w e r S u p p ly M o d u le The testing m ethod of power supply m odule is the same as th a t of testing m otor power supply module. It was done by measuring the voltage ou tpu ts m arked as 12V, -12V and 5V. The waveforms of three ou tputs were checked using a Fluke scope. (7) T e s tin g o f th e S e n so r M o d u le s The experim ental results are given as follows: A . P o te n t io m e te r C ir c u i t Figure A2.15 shows testing diagram of potentiom eter circuit. By turning the pot, ou tpu t waveform is shown in Figure 2.9. The result shows th a t the circuit can change angular position signal into voltage signal properly. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P M 3 3 7 O B chi STOP c m M1D5.PC c h l Figure 2.9 Potentiom eter ou tpu t waveform B . S tra in G a u g e C ir c u i t The testing diagram of the circuit are shown in Figure A2.16. Figure 2.10 shows strain gauge signals. W hen the beam was im pacted, the amplified stra in gauge signals follow the oscillation of the beam, which indicates th a t the circuit works properly. PM33 7 0 B \ , \ :CH l :cH2 ms c h l +■ Figure 2.10 The stra in gauge signals a t two positions ( 8 ) T e s tin g o f t h e O v e ra ll C i r c u i tr y The testing diagram of the overall circuit assembly is shown in Figure A2.17. Figure A .2.18 of Appendix A. In the testing, the variable resistor was used as the m otor. Figure 2-11 shows the comparison between the setpoint waveform and feedback one. Figure 2-12 shows the com pari­ son between the setpoint waveform and feedback one when m agnitude of the setpoint changes. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2-13 shows the comparison between the setpoint waveform and feedback one when fre­ quency of the setpoint changes. Figure 2-14 shows voltage waveforms of the setpoint and the variable resistor. Figure 2-15 shows th a t sa tu ration occurs when resistance of the variable resistor an d /o r am plitude of the setpoint waveforms from function generator increases. 3.6750 Nam# • ChanrwH Cham#42 2 6750 Dai# -4n9TO1 4/W 1Tim# ■4S0 5SPM 4:5856 PM YScal# • 1 Vfttw 5 WOlw 1.8750 Y At 5054 ■ -0.1250 V -0 7250V XSc4# ■ 5 ms/Ow 5 ms/O* 0.8750 XAtO* • 0.0 ms 0 0 ms XSit# - 500(512) 500(512) -0.1250 V Mwamum • 10025 V 5.2031 V Mnimun > -00806 V -0.0344 V -11250 -21250 X1 125 msX2 : 37 5 ms dX : 250 ms -312S0 Y1 : 0 6750 V Y2 0 6750 V -4.1250 dY 0 0000V 0.0 ms Figure 2.11 Waveform comparison between setpoint and feedback — D a tB b io 0 * N a m # • C h a n n e l 1 C h e n n a i 2 O a t # - 4 /1 B /0 1 4 /1 9 /0 1 9 700 T im # - 1 0 3 02 P M 1:03 0 5 P M V S c o l # - 1 V /D n / 5 0 0 m V /O iv Y A t 5094 • 1 9 7 0 0 V 0 9 9 2 5 0 V 9700 X S e a l # - 5 m # /D r v 5 m s / O b ' X A J0 V . - 0 0 m s 0 0 m s X S i f # - 5 0 0 (512) 5 0 0 (51 2) 9 7 0 0 M a x im u m • 3 1 9 6 3 V 1 5 9 7 5 0 V M in im u m • -0 0 6 3 9 V -0 0 1 5 9 4 V C u r s o r V a iu # > — ■ — 9 7 0 0 V X I 1 2 5 m s X 2 37 5 m s d X 2 5 0 m s 970 0 V I 2 6 3 6 3 V Y 2 2 5 9 5 0 V d V -0 0 4 1 2 V 030 0 03 0 0 Figure 2.12 Waveform comparison when setpoint am plitude changes 3.8750 CHann#i2 2 6750 4(18/01 4:5740 PM 1.0750 -01250 V -0 7250 V 0.0750. 0 0 0.0 mi 500(512) 500(512] -0 1250 V 10.3584 V -0.2313 V -11250 -2.1250 12.5 -3.1250 Y2 . 16450 V 0 0413 V 0.0 ms Figure 2.13 Waveform comparison when setpoint frequency changes 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Nsrrw 4Crht9#nn#l 1 Chwvwi2 Dat» «1 4/19/01 Tim# 7:5128 PM 75129 PM YSo*SiO# 2 VOv 50 mV/Ow yai x 16400V 37.000 mV X XX A SctOdX* 5 ma/Dw 5 m*/DwSu# 0.0 ms 0.0 ms500(5121 500(512) M amun 2.0941 V 47.625 mV -0.D63 V -SSTtt mV -Cmsoi Vr«iu#s- XI 12.5 ms X 2 37 5 ms dX 250 ms Y1 13320/ Y2 13320/ -6 3630 dY 0.0000/ Figure 2.14 The waveforms of the variable resistor and the setpoint 9700 N a m * - C > a n n * l 1 O a n n * 1 2 D a l e * 4 / 1 8 / 0 1 4 /1 0 /0 1 9700 T im * - 1 : 0 4 3 0 P M 1 0 4 4 0 P M V S e a l * - 1 V /D rv 5 0 0 m V /O w V A t 5 0 % • 1 9 7 0 0 V 0 9 9 2 5 0 V 9700 X S c a l # - S m i / D v 5 m s /D iwX A 1 0 V . 0 0 m s 0 0 m s X S i i * - 5 0 0 (512 ) 5 0 0 (512 ) M a x im u m - 3 5281 V 1 4 3 9 1 3 V M in im u m ■ * 0 .0225 V - 0 0 3 6 2 5 V ■ ■ C u r s o r V a l u * s - .................... i 9 7 0 0 V X I 12 .5 m s X 2 37 5 m s O X 2 5 0 m s 9 7 0 0 V I 2 8 0 1 3 V Y 2 2 8 0 1 3 V d Y 0 0 0 0 0 V 0 3 0 0 03 0 0 S m » /D » Figure 2.15 High-am plitude setpoint waveform and sa tu ra ted feedback waveform It is concluded th a t the output follows the setpoint no m atter how frequency of the setpoint changes and under the condition of small variations of the resistance of the m otor and small variations of setpoint amplitudes. However, the output does not follow the setpoint or satu ration occurs when the am plitudes of both the resistance of the m otor and the setpoint become too high. 2.4 Summary The circuitry functions properly. It can meet the requirem ents of controlling the m angitude and direction of the m otor current and amplifying the sensor signals. The testing m ethod is useful in m aintenance and troubleshooting for the circuitry. A valuable experience has been obtained from the building and testing of the circuitry. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Dynamics of an Axially-M oving Cantilever Beam The second objective of the present work is to develop an analytical model to study the dynamics of an axially-moving cantilever beam. F irst, the dynamics of a fixed-length cantilever beam is briefly reviewed. Then, a state-space model is derived for a cantilever beam engaged in axial motion. The model is time-varying as the system m atrix, input m atrix , ou tpu t m atrix and direct transm ission m atrix are all functions of tim e when the beam is axially moving. As an analytical solution of the model is not possible, a com puter sim ulation is conducted. The sim ulation results are given acccordingly to show how the axial m otion of the cantilever beam influences the dynamics of the system, including transient responses, “frozen” m odal param eters, and “pseudo” m odal param eters. The chapter is organized as follows: Section 3.1 reviews the dynamics of the fixed-length cantilever beam. Section 3.2 describes the modeling of the axially-moving cantilever beam, Section 3.3 shows the com puter sim ulation results, Section 3.4 discusses the evaluation of varying discrete-tim e transition m atrix and “pseudo” m odal param eters, and Section 3.5 is a brief summary. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 Dynamics of the Fixed-Length Cantilever Beam 3.1 .1 M ath em atica l M od elin g A schematic diagram of a fixed-length cantilever beam is shown in Figure 3.1.1. w ( j , 0 ♦ ds f M h L .0* 4 os V(s,t) \ V ( s , t ) + ^ ^ d sds k M s } t ) Undeform ed ds axisd s+ S Figure 3.3.1 Lateral vibration of the cantilever beam and a free-body diagram of the beam element To simplify the problem, the following assum ptions are made: (1) The beam is uniform along its longitudinal direction, both in mass d istribu tion and elastic properties. (2) R otary inertia and shear deform ation can be neglected. (3) The beam is composed of a linear, homogeneous, isotropic, elastic m aterial w ithout axial load such th a t plane sections rem ain plane and the plane of sym m etry of the beam is also the plane of vibration so th a t ro tation and translation are decoupled. The beam th a t satisfies the above assum ptions is referred to as the Euler-Bernoulli beam. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If a physical dam ping q (kg/m .s) exists in the system, a partial differential equation gov­ erning the m otion of the beam element is given as [28] + + = (3-1-1) where m (kg/m ) is the length density of the beam, or m = pAb , Ab (m2) is the cross-section area of the beam, p (kg /m 3) is the density of the beam m aterial, E (N /m 2) is Young’s m odulus of elasticity for the beam, I (m4) is the moment of inertia for the cross-sectional area, and f { s , t ) (N /m ) is the applied external force per unit length of the beam. To transform the partial differential equation (3-1-1) into a set of ordinary differential equa­ tions, it is assumed th a t the deflection w ( s , t ) can be expressed as n w ( s , t ) = (s)q(t) = ^ 2 $ j ( s )Qj(t) (3-1-2) j = 1 where q(t) = [qi(t) qiit) ... qn (t)]T is a column vector of the generalized coordinates, $ (s ) = [$ i(s) $ 2 (s ) ••• $n(s)] is a row vector of mode shape functions, and n is the num ber of the vibratory modes considered. The mode shape function is expressed as $ j(s ) = cosh(/3js) - c o s (^ s ) - (Tj(sinh(/3js) - sin (fijs)) (3 - 1 -3 ) where Oj = [sinh(/3jL) - sin(,3^1)] / [cosh(/3JL) +cos(/3JL)] [28]. The values of the constants d j L and aj for the first four modes are listed in Table 3.1. Table 3.1 (3jL and <7 j of the first four modes mode M o 1 2 3 4 33L 1.8751 4.69409 7.8547 10.9955 0.7341 1.0185 0.9992 1 . 0 0 0 0 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The natural frequency of the j t h mode is given as 1 ( { 0 dL ) \ 2 I E l ( 3 - M ) 3.1 .2 R esp on se to a C oncentrated Force If a concentrated force F(t) is applied a t s tn, f ( s , t ) can be expressed by f ( s , t ) = F ( t ) S ( s - s in) (3-1-5) where 6 (s — Sjn ) is a Dirac D elta function. Substituting equation (3-1-2) and (3-1-5) into equation (3-1-1) results in m $ (s) 9 (f) + 7 (s) 9 (t ) + E I $ " " ( s ) q ( t ) = F ( t ) 6 (s - s in) (3-1-6) where dots denote derivatives with respect to tim e t and primes derivatives w ith respect to s, or •• . d 2 q(t) ■ dq(t ) »« d*$(s) V (*) = ~d ta22 r~ <* (*) = ~d^t r $ («) = ds 4 Prem ultiplying the above equation by $ T (s) and integrating it w ith respect to s from 0 to L results in I $ T&ds q (t) + — f $ T $ d s 9 (t ) + — [ d s q(t) = f $ T S(s — Sin )ds. Jo m J o m Jo m Jo Using the orthogonality of the mode shape functions, the following relations exist p L r L / $ r $ d s = I and A = / $ T$""ds = diag [(/3jL)4, ( 0 2 L ) 4 , {0nL ) 4} (3-1-7) J o Jo where I is an n x n identity m atrix and A a n n x n diagonal m atrix. Equation (3-1-6) becomes q (t) 4 - —I 9 (t ) + — A q(t) = * T{S'n)F(t). (3-1-8) m m m 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The above equation can be rew ritten into a state-space representation x (t ) = Ax( t) + B u ( t ) (3-1-9) where the sta te vector is (3-1-10) 9 (*) the svstem m atrix is Onxn I A = (3-1-11) and the input m atrix is (3-1-12) and the input “ (*) = F ( t) . (3-1-13) Here the convention 0 ,XJ is adopted to denote an i x j m atrix of zeros. It is noted th a t the num ber of sta te variables is n x = 2n. Several different responses or ou tputs can be obtained, depending upon the means of m easurem ent. If a stra in gauge sensor located at sout is used, the ou tpu t will be a voltage signal proportional to the m agnitude of the strain at Sout. In this case, the output is given as y = K ag$ " (soutMt) (3-1-14) where K sg is the gain of the strain gauge m easurem ent system. If the deflection a t can be measured, the ou tpu t is y = K d$ { s out)q(t) (3-1-15) where K d is the gain of the deflection m easurem ent system. If the velocity at sout can be measured, the ou tpu t is y K v W ( S o u t t ) Q ( t ) (3-1-16) 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where K v is the gain of the velocity m easurem ent system. If the acceleration a t s ^ t can be measured, the output is y = K a w ( s ^ t ) = K a$ ( s 0ut) q (t) (3-1-17) where K a is the gain of the acceleration m easurem ent system. Substituting Q (t) solved from equation (3-1-8) into equation (3-1-17) results in V - K a - — $ ( s out)Aq(t) - l $ ( Sout)I q (t ) + ± $ ( Sout) $ T (sin)F(t) (3-1-18)m m m In general, the output vector is expressed as y = C x + Du (3-1-19) where C is the ou tpu t m atrix and D is the direct transm ission m atrix. For example, the use of a strain sensor at s ^ t i , a displacement sensor a t sout2 , a velocity sensor a t s<>ut3 and an accelerometer at sout4 results in (Scuti ) Ol xn Kd®(Sout2 ) O lx n C (3-1-20) O lx n KyQ^Soat 3 ) - K a^ ( s out4)A - K a2-$ (Sout4) I and 0 0 D = (3-1-19) 0 K a m 3* ( Sout4 ) ( Sin ) 3 .1 .3 O bservability and C ontrollab ility o f th e S ystem The controllability m atrix of the system w ith the system order n x is given by [36, 37] Q = B A B A 2B . . A Hx~ l B (3-1-20) 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The system is controllable if the m atrix Q has full rank, i.e., rank[ Q ] = n x . The observability m atrix of the system with the system order n x is given by [36, 37] C C A C A 2 (3-1-21) C A 11* - 1 The system is observable if the m atrix P has full rank, i.e., rank[ P } = nx . To understand these two im portant m atrices, it is assumed th a t the system is excited by a concentrated force a t sin and two stra in gauge ou tputs are observed a t s ^ t i and Sout2 positions, respectively. The Q and P m atrices are given by l Onxl $ T {sin) $ T(Sm) Q = (3-1-22)m ^ {Sin § T {Sin) and K Sg$ (Soutl) Ol xn K Bg* " { Sout2) O lx n O lx n K s g ^ ' i s ^ n ) O l X T l K s g $ " ( S o u t 2 ) P = (3-1-23) O l x n Ksg*& (^out l) O l x n K Sg<& (Scut2) The controllability depends on the mode shape function values at sm , and the observability depends on the values of the second derivatives of the mode shape functions a t Soutl and Sout2 - Figure 3.1.2 shows the first three mode shape functions. It can be seen th a t if the force is applied at one of the node points, such as s / L = 0.5,0.755 or 0.864 where the m ode shape 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. functions become zero, the system is uncontrollable. 0 . 5 - 0 . 5 1 st m ode — 2 n d m o d e — 3 r d m o d e - 1 . 5 0.2 0 .4 0.6 0.8 ratio s/L Figure 3.1.2 The first three mode shape functions Figure 3.1.3 shows the second derivatives of the mode shape functions. I t is noted th a t to ensure the system to be observable, the stra in gauge sensors cannot be placed at the location where $ ” (s ^ t ) is zero. 150 1st mode 1 0 0 2nd mode 3rd mode 50 O -50 -100O 0.2 0.4 0.6 0.8 1 ra t io s /L Figure 3.1.3 The second derivatives of the first three mode shape functions 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 Modeling of an Axially Moving Cantilever Beam The apparatus of the axially-moving cantilever beam is shown in Figure 3.2.1. M I er SLfcjport s P i n i o n w impact s Figure 3.2.1 The apparatus of the axially-moving cantilever beam It is noted th a t the lateral m otion takes place on the horizontal plane such th a t gravity has little effect on the motion. The beam is assumed to be an Euler-Bernoulli beam. In addition, it is assumed th a t the axial m otion of the beam is a function of tim e only. Two modeling cases are considered: case one includes the contribution of the axial force to the lateral vibration of the beam and case two does not. Each case is further divided into two subcases. Subcase one considers the physical dam ping and subcase two does not. 3.2 .1 M odeling: th e C ontrib ution o f th e A xia l Force C onsidered The following development is based on the work presented in [4]. Figure 3.2.2 (a) shows the beam th a t is laterally deflected by w(s , t ) . Figure 3.2.2 (b) shows the free-body diagram of a beam element A s and Figure 3.2.2 (c) shows the rem aining part of the beam, where V and M are the shear force and bending moment respectively, T is the axial force acting on the beam element A s at s perpendicular to the face of the cross section, T\ is the axial force acting at the beam section at (s + A s), 6 = d w / d x is the slope of the beam w ith respect to the undeform ed 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. elastic axis. clamper s { t ) L i t ) d M A s M + A s se A s elastic axi A M L i t ) Figure 3.2.2 (a) deformaion; (b) free-body diagram of As; (c) rem aining p a rt of the beam Applying New ton's second law to the element A s in the lateral direction gives d2w (s t) d V 8 6 m A s — — = V — (V + - ^ A s ) + T\ sin (0 + ^ A s) - T s i n O + / ( s , t ) A s. (3-2-1-1) It is assumed th a t 6 , 8 6 / d s , and A s are very small, dO dO sin(# + t t - A s ) « { 6 + - r r A s ) and s in 0 « 0 . ds ds Equation (3-2-1-1) can be changed into d2 w(s , t ) d V 8 6 m A s - * * = - ^ A s + f f i - r ) S + r i ^ A S + / ( M ) A S. (3-2-1-2) Applying N ew ton’s second law to the element A s in the axial direction gives m A s L = T\ — T. (3-2-1-3) 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If the ro tato ry inertia can be neglected, the sum m ation of the m om ents induced by all external forces and moment about any point equals zero, i.e., ( A / + ^ A s ) - M - V A s = 0 or V = (3-2-1-4) os os Applying Newton’s second law to the rem aining beam (L — s) in the axial direction gives pA (L — s) L = —T\. (3-2-1-5) It is noted tha t d w dw M = and 6 = ^ (3-2-1-6) o s 2 os where E l is the flexural rigidity of the beam. Substituting equations (3-2-1-3), (3-2-1-4), (3-2-1-5), and (3-2-1-6) into equation (3-2-1-2) gives . d2 w(s . t ) ^ Td 4 w(s , t ) A A dw ■ d 2w m A s — ^ — = - E l — ^ — A s + m A s L - m ( L - s) L + / ( M ) A s . Thus, the dynamic model, when dam ping is neglected, is expressed as d2 w (s , t ) E l d*w(s, t ) -_dw ■_ d 2w f ( s , t ) dt 2 m d s 4 ds ( > d s 2 m ' (3-2-1-7) If the system dam ping 7 (k g / m . s ) is considered, the vibration equation in lateral direction is given as d2w (s , t ) 7 dw(s . t ) E l &lw (s , t ) ■ dw ■ d 2w f ( s , t ) -dht - - + -m d1t + --m--- -----a--s--- ------------d-s-- -+ { l - s ) l ^2 4 d s1 = j- ^m- l . 3 -2 - 1 - 82 To transform the above partial differential equation into a set of ordinary differential equa­ tions. let s = a L 0 < a < 1. (3-2-1-9) 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus, the mode shape functions in term s of equation (3-1-3) becomes $,-(a) = cosh(0 jLa) - cos(03 La) — a j ( s inh(0 jLa) — s in (0 jLa ) ) and the value (3jL and rrj are given in Table 3.1. Now the beam deflection becomes n w(a, t) = = 9(a)q{t) (3-2-1-10) j =i where $ (a ) = [$ i(a ) $ 2 (0 ) ••• $«(<>)] is the row vector of the mode shape functions in term s of a . The following relations exist dw(s , t ) $ (q ) q (t) + $ (a)q ( t ) a dt d2 w ( s , t ) d / . . . . . < , , N . n ~ d F ~ = s ( * W ? W + * ( < » ¥ t ) a = $ ( a ) q (t) + 2 $ (a) q (t ) a + $ (a)q(t) d 2 -(-$ (a)q ( t ) a a = | ( 1 - a ) (3-2-1-11) • 2 a = ( l - a ) ( | - 2 ^ ) where dots denote derivatives with respect to tim e t and primes represent derivatives with respect to variable a. Assume th a t the force acting laterally F( t) is concentrated a t sin = a inL f{ s , t) = F ( t ) 6 (a - a in) (3-2-1-12) where 6 (a— a*n ) is a Dirac D elta function. It is noted th a t Sjn is the axial distance between the concentrated force and the clamp (see Figures 3.2.1 and 3.2.2) and it is a function of time. The following relation exists Sin + Vin = L (3-2-1-13) 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where rm is a constant axial distance between the concentrated force and the tip of the beam. Thus, the variable a in can be expressed as - 7*tn &in — 1 • (3-2-1-14) Substituting equations (3-2-1-11), (3-2-1-12), (3-2-1-13) into equation (3-2-1-8) , and rear­ ranging results in • 2 4>(a) q (t ) + 2 ^ ( 1 - a ) $ ' ( a ) + “ *(«) «(<) + j ^ ( l - a ) 2 + (1 - a ) j ^ " ( a ) q{t)-fL m • 2 E l //// / v / v 4> {a)q(t) + = (3-2-1.15) m i 4 m Prem ultiplying both sides of the above equation by term $ T (a) and integrating it from 0 to 1 with respect to the variable a results in Ai q (t) + ( ^ A 2 + — A i ) q ( i ) + L m . . . 2 • • - 2 /L *L *7 L \ . L . L . L . E l q(t) = ^ -): A 7 ~ 2 Z 2 + m Z " L A + + L M + m & M (3-2-1-16)z m where A\ f 4>r (a)(a)da A 2 = [ ( 1 — a ) $ T ( a ) $ ' ( a )d a Jo Jo As f § T ( a ) $ (a)da A 4 = / ( l - a ) 2 $ T (a) (a)di Jo Jo r l ^5 / ( 1 — cr)4? (a)4> (a)da A § = [ $ T( a ) $ / (a ) d a Jo Jo a 7 = I $ T (a)S(a - a in)da = $ T(a ,n) . (3-2-1-17) Jo Using the orthogonality of the mode shape functions 4?(a), it can be proven th a t A , = I A 6 = d ia g [((31 L ) 4 ■ ■ ■ {(3nL ) 4] . 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A compact form of equation (3-2-1-16) is given Q (t ) + D d(t) q (t ) + K(t )q( t ) = B \ (t)u(t) (3-2-1-18) where the time-varying param eter m atrices D S ) , B \{ t ) and K ( t ) are defined as D d[t) = ^L~ r A 2 + ~mA \ B S ) = -m$ T (ain) (3-2-1-19) K ( t ) = ( l - 2 j 2 + ^ l ) A 2 - l ^ + j 2 ^ + ^ A b + ^ A 6 . If the system is free of damping, the Dd{t) and K ( t ) m atrices become . 2 - 2 D S ) = ^ A 2 K ( t ) = ( j r - 2 j ^ ) A 2 - jrAs + j r z A 4 + | a 5 + — I As . (3-2-1-20) 3 .2 .2 M odeling: th e C ontrib ution o f th e A x ia l Force N eg lec ted If the effect of the axial force is not considered, the lateral m otion of the system is expressed as follows (see equation 3-2-1-7) d2 w (s , t ) 7 dw(s , t ) E l 8 *w(s, t) _ f ( s , t ) dt 2 m dt m ds 4 m The same equation as equation (3-2-1-18) is derived using the same procedure m entioned above. However the D S ) , B\{ t ) and K ( t ) param eter m atrices are different, as shown in the following D S ) = ^ A 2 + ^ A 1 H 1 (t) = - $ r ( a m ) (3-2 -2 -2 ) L m m ■ 2 - 2 , , / J_j _ L \ a L a E l . A( t ) = L ~~ £ 2 + ^ L ^ + mL* ' If the system is free of damping, the D S ) and K( t ) m atrices become • 2 - 2 D S ) = 2~j - A 2 K ( t ) = ( | - 2 j ^ ) A 2 + A 4 + ^ A e . (3-2-2-3) 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 .3 C ontinuou s-T im e S ta te Space R ep resen tation o f th e S y stem Let F(t) = u(t). The sta te space representation can be obtained from equation (3-2-1-18) as follows x (t) = A ( t ) x ( t ) + B(t)u{t) (3-2-3-1) where Onxn f A(t) = B( t) = _ - K ( t ) —D d(t) _ and x(t) = [qT (t) qT (t)}T is the sta te variable vector. The ou tpu t signals depend on the means of m easurem ents. If a stra in gauge sensor located at s ^ t = a outL is used, the output is a voltage signal proportional to the m agnitude of the strain: y(t) — K sg (otout)q{t)/ L — [ K sg (aout) /T 2 0i> (3-2-3-2) g ( t ) If a deflection sensor located at sout = aoutL is used, the ou tpu t is a voltage signal proportional to the m agnitude of the deflection: y ( t ) = K d $ ( a out)q(t) Kd${aout) Olxn (3-2-3-3) _ 9 (t) _ If a velocity sensor located at sout = a outL is used, the ou tpu t is a voltage signal proportional to the m agnitude of the velocity: y(t) = k v $ ( c w ) - ( l - a o u t ) q ( t ) + $(<*out) 9 ( t ) Q(t) = K v ^ ( a 0ut)4‘( 1 ®out) ^(aout) (3-2-3-4) q(t) If an acceleration sensor located at Sout = aoutL is used, the output is voltage signal proportional to the m agnitude of the acceleration. Using the equation (3-2-1-11), the aceleration ou tpu t is 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 o3 ■ X1-0 t . . 1 expressed as y(t) = d Wj ^ — = $ ( Q) ^ (f) + 2$ ( a ) 9 (t) a +$"(a)q ( t ) a + $ ' (a)q(t) a (3-2-3-5) Substitu ting a and a in equation (3-2-1-11) and q (t ) solved from equantion (3-2-1-18) into (3-2-3-5) results in q{t) Ck(cXout) Cdî out) + — $ ( a out) $ T (ain)u(t) (3-2-3-5)771 _ « ( * ) _ where C fc(cioui) — (ci(w () (® out) dou( $ { O l o u t ) K ( f ) (3-2-3-6) Cd(ctout) = 2 *̂ (ttout) dout It is noted th a t a ou( can also be expressed as (3-2-3-7) where rout is the axial distance between the sensor and the tip of the beam and a constant. If there are one stra in gauge sensor a t a ^ t i , one deflection sensor a t a ( m t 2 , a velocity sensor a t Ck0 ut3 ,and a acceleration sensor a t Q0 ut4 , the outpu t m atrix and the direct transm ission m atrix are given as K 8g$ " ( a m t l ) / L 2 O lx n 0 Kd^{ocmlt2 ) O lx n 0 C(t) = and D(t) = A ,,$ ( Q o u i 3 ) ^ ( l — Ô outs) A”lj$(Q:<}jit3) 0 KaCkiQoutA) K aCd (c*out4) A T a ^ ( a cmt 4 ) ^ T ( Q i n ) / m (3-2-3-8) Generally, the output can be given as y(t) = C( t)x( t ) + D(t)u( t) (3-2-3-9) 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where C(t) is a time-varying ou tpu t m atrix and D(t) a tim e-varying direct transm ission m atrix. 3 .2 .4 “Frozen” M odal Param eters The concept of the modal param eters, including natu ra l frequencies and dam ping ratios, is no longer valid for the time-varying system. Instead, the concept of the “frozen” m odal param e­ ters can be used [14, 16]. If the system is considered to be frozen a t a tim e instan t t, the eigendecompositon of A(t) becomes A{t) = V { t ) A ( t ) V ~ \ t ) (3-2-4-1) where V(t) is a “frozen” eigenvector m atrix at the tim e instant t, A (t) = diag [Ai(t), ■ ■ •, \ Ux is a “frozen” diagonal eigenvalues m atrix a t the tim e instant t. A pair of complex “frozen” eigen­ values can be expressed as A (f) = - Q ( t ) u i ( t ) i w W / l - d W (3-2-4-2) where u n .{t) is defined to be the ith “frozen” natu ra l frequency at the tim e instan t t, Q(t) is defined to be the i th “frozen” dam ping ratio a t the tim e instant t, and j = y/—l. 3.2 .5 C ontro llab ility and O bservability o f th e S ystem The controllability and observability of a time-varying system is no longer defined by the ranks of the Q and P m atrices in (3-1-20) and (3-1-21) respectively. However, for an approxim ate analysis, the concept of the “frozen” system can be used again. The controllability m atrix of the time-varying system th a t is “frozen” at a tim e instant t is defined as Q{t) = [ B { t ) B ( t )A ( t ) B ( t ) A 2 (t) . . B { t )A n*~x{t) ] (3-2-5-1) The “frozen” system is controllable if the m atrix Q(t) has a full rank at the tim e instan t t , i.e., r a n k [ Q(t) ] = n x . Similarly, the observability m atrix of the time-varying system th a t is “frozen” at tim e instan t 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t is defined as C ( t ) C(t)A(t ) C ( t ) A 2 (t) P(t) = (3-2-5-2) The “frozen” system is observable if the m atrix P(t) has a full rank a t the tim e instan t t, i.e., r a n k [ P(t) ] = n x . To understand these two im portan t m atrices, an example is used. In th is example, the first three vibratory modes are considered, or n = 3. Thus, the system order is 6 , or n x = 6 . The trapezoid velocity profiles are used as axial m otion profiles, as shown in Figure 3.2.3. Two kinds of scenarios are considered. Scenario A: axial extension in which the beam length varies from L min to L max: Scenario B: axial retraction in which the beam length varies from L max to L min. In either case. L m\n= 0.66 m and L max = 1.09 m. S c e n a r i o A S c e n a r i o B E E ■»e «> E•> 0 9 «Eu *jo a. □ e um m Mo. 0 6 TJ ■o Jg M IB 0 6 IB KIB 0 2 I M £ 0 15 / I -0 05 0 1 // \ \ Uo« 0.05 /' \ >// ■ K 0I \ m -0.2 0 2 0.2 1 ' 0 1 0.1 €s 0 0 - 0 1 •0 1 -0 2 -0 2 tim e ( s e c ) tim e ( s e c ) Figure 3.2.3 Axial motion profiles The input is applied at rin = 0.528 m in scenario A and rm = 0.875 m in scenario B respectively. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The locations of two stra in gauge sensors roun and are assumed to be 0.60 m and 0.30 m respectively w ith respect to the tip of the beam, or rout\ = 0.60 m and rout2 = 0.30 m. The C(t), Q(t) and Pi t ) m atrices are of the forms ipoutl)/ L 0xx3 C(t) = (3-2-5-3) K . g & ' i a ^ / L 2 0 l x 3 03x1 $ T (ctin) ^ ( O in ) Q ( t ) = m _ $ T(a ,n ) - 2 ( f ) $ T(a in) ( - 2 ( f ) ) $ T(a in) (-2(£))V(tt(n) _ (3-2-5-4) and K s g & i a ^ / L 2 0 l x 3 K s g & ' i a o u ^ / L 2 0 l x 3 0 l x 3 K s g & ' i a ^ / L 2 P{t) = (3-2-5-5) 0 l x 3 K s g & ' i a ^ / L 2 0 l x 3 K s g & ' i a ^ / L 2 0 i x 3 K s g & ' i a ^ / L 2 Obviously, the “frozen" controllability depends on the mode shape function values $ T {ocin). The “frozen” observability depends on the values of {ocouti) / L 2 {aout2 ) / L 2. Figure 3.2.4 shows § i ( a i n ) versus tim e. Figure 3.2.5 shows $ / ' ( c w i ) / L 2 and § ”{ot(mt2 ) /L 2 versus time. In these figures, a solid line represents the first mode, a dashed line the second mode, and a 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dashdot line the th ird mode. Scenario A Scenario B 14 12 10 8 ? E 4 2 0□ i 2 3 4 time (sec) time (sec) Figure 3.2.4 Varying m agnitudes of 4>("(acmti)/L2 and (a out2 )/ L 2 It can be seen from Figure 3.2.5 th a t for the two scenarios, the “frozen” system becomes tem porarily unobservable at two tim e instants . It is interesting th a t such tem porary unobserv­ ability occurs at the first sensor location, indicating th a t the axial position of the stra in sensor influences the “frozen” observability of the system. The closer to the clam p the s tra in sensor, 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the more unobservable the “frozen” system. 3.3 Simulation of the Axially-Moving Cantilever Beam To understand how the axial m otion influences the dynamics of the axially-moving cantilever beam system, it is necessary to sim ulate the dynamic responses of the system . In the sim ulation, the density p and the modulus E of elasticity of the beam m aterial are 2800 k g /m 3 and 70.9 G Pa respectively. The thickness b and height h of the cross-section of the beam are 3.175 mm and 50.6 mm respectively. The beam length varies from Lm,n = 0.66 m to L max = 1.09 m. The axial m otion profiles used in the sim ulation are the same as those in subsection 3.2.4. The first three flexible modes are considered in the sim ulation, i.e. the system order nx is 6 . Since the highest frequency / max is 110.66 Hz when L = L min, the tim e step A t is chosen to be 0.0005 second in order to capture the dynamics of the system. The beam responses are observed at three locations. Location one is a t r olltl = 0.6435 m and referred to as base location. Location two is a t r t im e ( s e c ) t im e ( s e c ) Figure 3.3.1 “Frozen” natu ral frequencies / and dam ping ratios £ for scenario A 2X 10"* 0 -2 -4 -6 -eo 1 2 3 4 x 10“* t im e ( s e c ) t im e ( s e c ) Figure 3.3.2 “Frozen” natu ral frequencies / and dam ping ratios £ for scenario B From the figures, the following observations can be drawn: (1) The axial force has little influence on the “frozen” modal param eters, as the two curves in each figure overlap each other. Thus, the model not considering the contribution of the axial force can be chosen in the following simulation, including the generalized coordinates 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and derivatives, transient responses, true “pseudo” modal param eters and identified “pesudo” modal param eters. (2) The axial extension makes the “frozen” natu ra l frequencies decrease and the axial re­ traction makes the “frozen” natu ral frequencies increase. Such behaviors are expected as the variations of stiffness and mass of the beam. The stiffness of the beam decreases and the mass of the beam increases as the beam extends. The stiffness of the beam increases and the mass of the beam decreases as the beam retracts. (3) The axial extension induces positive “frozen” dam ping ratios, and the axial retraction induces negative “frozen” dam ping ratios. The reason for th is is the sign of the dam ping m atrix Dd{t) = 2 L A 2 / L th a t depends on the sign of the axial velocity. Therefore, the axial extension makes the system stable because of the positive dam ping effects, and the axial retraction make the system unstable because of the negative dam ping effects. 3 .3 .2 G eneralized C oord inates and V elocities Figure 3.3.3 and Figure 3.3.4 show the generalized coordinates and velocities for the axial extension and the axial retraction, respectively. 0.2 □.□05 0.005 ■0.1 ■0.2 •0 01 0 1 2 3 A t im e ( s e c ) t im e ( s e c ) Figure 3.3.3 The generalized coordinates and velocities for the axial extension 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 05 •05 t im e ( s e c ) t im e ( s e c ) Figure 3.3.4 The generalized coordinates and velocities for the axial retraction From the figures, the following observations can be drawn. (1) The m agnitudes of the generalized coordinates and velocities in scenario B are greater than those in scenario A. The reason for this is th a t the initial stiffness of the beam in scenario B is smaller than th a t in scenario A, because the initial length of the beam in scenario B is longer than th a t in scenario A. (2) An axial extension results in an increase of the m agnitude of the generalized coordinates and a decrease of the m agnitude of the generalized velocities. An axial retraction results in a decrease of the generalized coordinates and an increase of the m agnitude of the generalized velocities. This can be explained by the variation of the beam stiffness K ( t ) . The axial extension reduces the stiffness of the beam, which makes the beam ’s flexing easier. The axial retraction increases the stiffness of the beam, which makes the beam ’s flexing more difficult. (3) The m agnitude of the first mode is much greater th an the m agnitudes of the second and the th ird modes in the generalized coordinates. The m agnitude of the th ird mode is smaller than the m agnitudes of the first and second modes in the generalized velocities. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 .3 Transient R esp on ses In the following, the sim ulation results of transient responses, including stra in , deflection, veloc­ ity and acceleration signals at the three ou tpu t positions namely the base location, the middle location and the tip location are presented. In this case, the ou tpu t m atrix can be given as y(t) = C{t)x( t ) + D ( t )u ( t ) (3-4-1) where u(t) is the d istributed impact force, the outpu t m atrix C(t) w ith the dimension 12 x 6 is shown as K sg$ " ( a outl) / L 2 K s g & ' i a ^ / L 2 0 l x 3 0 l x 3 KtfQiccovxi) 0 l x 3 Kd^idout 2 ) 0 i x 3 0 i x 3 K v*& ( a o u t l ) ^ ( l t t o u t l ) K v*&(&outl) K v*& ( o out2 )^ j ( l (%out2 ) Kv*&{Ckout2 ^ (& cmi3 )^ ( l — &out3) K vQ{.Ckout 3 ) ^ a C k i p L o u t l ] KaCdiOtoufi) KaCkfaout?) KaCd{ckout2 ) KaCk(ckoUt3) Ka Cd ( Oout3 ) 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O wX 0 0 0 0 0 0 D(t) = = [£>,ijj (3-4-3) 0 0 0 # a $ ( < W i) $ T (a in )/m K a^ { a out2 ) ^ T {otin) / m Ka${aout3)$T {a in) /m and the output vector is given as y( t ) = l y i ( t ) 2/2( 0 2/3( 0 2/4(0 2/5(0 2/e( 0 1/7(0 i/s ( 0 2/9(0 y io ( 0 2/ n ( 0 1/12(0 ]T (3-4-4) where j/i(t), 2/2 O) and ?/3 ( 0 are the strain gauge ou tputs a t the base location, the middle location and the tip location, respectively; 2/4(0, 2 / 5 (0 and 2/6(0 are the deflection ou tpu ts a t the base location, the middle location and the tip location, respectively; 2/7 (0 , 2/8 ( 0 and 2/9 ( 0 are the velocity ou tputs at the base location, the middle location and the tip location, respectively; and 2/io(0, 2 /n (0 and 2/12(0 are acceleration ou tputs a t the base location, the middle location and the tip location, respectively. In order to understand how the vibratory modes, the generalized coordinates, an d /o r gener­ alized velocities make the contribution to the transient responses, the ou tpu ts can be expressed as Ti-x 2/7 ( 0 Cji(t )xi (t ) = ] T yji{t) 1=1 i= l In the following sim ulation, the gains of output m easurem ents are K sg = 1000 (V), Kd = 1 (V /m ), K v = 1 (V /m .s- 1 ), and K a = 0.01 (V /m .s-2 ). 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1) Strain Gauge Outputs Figure 3.3.5 shows the stra in ou tputs for motion scenario A and B respectively. S c e n a r i o A S c e n a r i o B A / W V V V V * 0 4 0.2 0 5 0 0 -05 -0.2 -0 4 t im e ( s e c ) t im e ( s e c ) Figure 3.3.5 Strain ou tputs for m otion scenarios A and B It can be seen tha t for the base and middle locations, the axial extension results in a decrease of the m agnitudes of the strain ou tpu ts while the axial retraction results in an increase of the mag­ nitudes of the strain outputs. The behavior can be explained by the variations of $ " ( a o u t ) /L 2 and generalized coordinates q(t). As the beam length increases, the term 3>" (Qou<)/ L 2 and q(t) becomes smaller. And as the beam length decreases, the term (ctout)/ L 2 and q(t) becomes larger. For the tip location, the stra in signals rem ain at zero as outs) = 0 (?. = 1 ,2 ,3 ) be­ cause of Qout3 = 1- It is also noted th a t the m agnitude of the strain signal a t the base location is greater than th a t at the middle location. To see how the v ibratory modes contribute to the stra in output signals a t the base and the middle locations, the stra in ou tpu t components, or y n and y u ( i = 1 ,2 ,3 ), are p lo tted in the following figures. Figure 3 .3 .6 show s three com ponents o f th e strain ou tp u t at th e base loca tion for th e tw o 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m otion scenarios. S cen a rio A S cen a rio B 0.2 06 04 02 -04 •02 0 1 2 3 4 0.2 £CM •0.1 -02 «■»» t im e ( s e c ) t im e ( s e c ) Figure 3.3.6 S train ou tpu t components y u ( i = 1 ,2 ,3) a t the base location It can be seen th a t the higher the mode index, the less it contributes to the to ta l response. The response com ponents from the second and th ird modes become zero when the sensor location coincides with the points where ^ ( c w ) = 0. The axial extension decreases the m agnitude of the first mode component y n while the axial retraction increases the m agnitude of the first mode component y \ \ . Figure 3.3.7 shows the three components of the strain ou tputs at the middle location for the two m otion scenarios. It can be seen th a t the to ta l response is dom inated by the contribution of the second mode. Only the th ird mode component experiences the instantaneous zero point of $ 3 (ciout) = 0. The response components of the first mode and the second mode, or t/ 2 1 and 1)2 2 , decrease during the axial extension and increase during the axial retraction. However, it seems th a t the component of the th ird mode, or 1 / 2 3 , increases during the axial extension and decreases during the axial retraction. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S cen a rio A S c e n a rio B 'vwWVWi -0 05 □ 05 05 •□05 t im e ( s e c ) t im e ( s e c ) Figure 3.3.7 S train ou tpu t components y 2i(i = 1 ,2 ,3) a t the middle location 2) D e fle c tio n O u tp u ts Figure 3.3.8 shows the deflection ou tputs a t the three locations for the axial extension and the axial retraction. S c e n a r i o A S c e n a r i o B 0 5 £ S' 8 •0 05 •0 1 -05 2 3 t im e ( s e c ) t im e ( s e c ) Figure 3.3.8 Deflection ou tputs for motion scenarios A and B 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The figure shows th a t the axial extension results in an increase of the deflection responses and axial retraction results in a decrease of the deflection responses. The further the observation location away from the clam per, the larger the response m agnitude. I t should be pointed out tha t the deflection ou tputs behave differently from the stra in outputs. T he former diverges as the beam extends while the la tte r diverges as the beam retracts. The behavior of the deflection can be explained by the variation of the stiffness of the beam. The stiffness of the beam decreases as the beam extends and increases as the beam retracts.. To evaluate how the vibratory modes contribute to the transient responses of deflections, the m agnitudes of deflection ou tpu t components, or y ^ , and y& (i = 1 ,2 ,3 ) are p lo tted in Figures 3.3.9, 3.3.10, and 3.3.11, showing the deflection ou tpu t components a t the base location, the middle location and the tip location respectively. S c e n a r i o A S c e n a r i o B 4 0 2 ■ 4G 1 2 3 4 x 10'5 •001 □ 02 001 «*» o *7 -0 01 -0 02 t im e ( s e c ) t im e ( s e c ) Figure 3.3.9 Deflection outpu t components y ^ ( i = 1 ,2 ,3) a t the base location 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 005 •0 01 •0 05 005 0.005 <*> t im e ( s e c ) t im e ( s e c ) Figure 3.3.10 Deflection ou tpu t com ponents y ^ ( i = 1 ,2 ,3) a t the middle location □ 02 S c e n a r i o A S c e n a r i o B £ «N •0 02 0 02 o 1 0 01 005 £ £ c as* •0 01 -0 05 •0 0 2 -0 1 2 3 2 3 t im e ( s e c ) t im e ( s e c ) Figure 3.3.11 Deflection ou tpu t components yei(i = 1 ,2 ,3) a t the tip location As expected, it can be seen th a t the higher the mode index, the less it contributes to the to ta l response. The contribution of the first mode is dom inant in the three deflection ou tpu t components. The further the observation location from the clamp, the greater the m agnitudes of the deflection ou tpu t components. It can be explained th a t the m agnitude of $ i (£*0 ^ 3 ) 9 1 (f) at the tip location is much larger than those of $i(a:out2 )9 i(£) and $ i ( a out i) 9 i(t) . 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3) Velocity Outputs Figure 3.3.12 shows velocity ou tpu ts for the two motion scenarios. Scenario A Scenario B at- % ° wVV\A time (sec) time (eec) Figure 3.3.12 Velocity ou tpu ts for m otion scenarios A and B It can be seen th a t the further the observation from the clamp, the larger the m agnitudes of velocity responses. To see how the vibratory modes contribute to the velocity signals a t the base location, the middle loction and the tip location, the velocity ou tpu t com ponents, or yn , ysi and yg, (i = 1 ,2 ,3 ,4 ,5 ,6 ) , are p lotted in the following figures. Figure 3.3.13 and Figure 3.3.14 show the velocity outpu t components y-^ (i = 1 ,2 ,3 ) and the velocity ou tpu t components y n (i = 4 ,5 ,6 ) at the base location, respectively. Scenario A Scenario B •0.005 WWWWVW'— ■— "̂ âaaAAAAA/̂ 0 1 2 3 3 1 2 3 4 * to'3 ^ 0 005 —^ ■ 1 -o oos 0 1 2 3 i c 1 2 3 4 x 10'3 _ 0 006 £» O ° -0 006 — time (sec) time (sec) Figure 3.3.13 Velocity output components yn( i = 1 ,2 ,3) at the base location 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S cen a rio A S c e n a rio B VVAAAa/w 0 05 •0 05 -0 05 t im e ( s e c ) t im e ( s e c ) Figure 3.3.14 Velocity ou tpu t com ponents yji(i = 4 ,5 ,6 ) a t the base location It is noted th a t the components yji (i = 1 ,2 ,3) are induced by th e axial m otion and such term s m ay be referred to as m otion-induced term s. Com pared w ith the com ponents yn (? = 4 ,5 ,6 ) th a t associated w ith the mode shape functions and generalized velocities, the motion-induced term s have much smaller m agnitudes. Figure 3.3.15 and Figure 3.3.16 show the velocity components y$i (i — 1 ,2 ,3 ) and the velocity components ygj (i = 4 ,5 ,6 ) a t the middle location respectively. S c a n a n o A S c a n a r i o B □.01 002 0 005 □ 01 S 0 - w v W W t / W W ---------- S 0~ ^ a / \ a a A A A A A A / V v ~ - 0 005 ** -0 01 •0 01 I3 1 2 3 - c3 1 2 3 4 0.01 _ 0.005 _ 001 * . -□.□os 1 | -0.01 ■ -0 01I3 1 2 3 - c 1 2 3 4 D 01 □ 005 0 01 0 - •0 005 ^ -0 01 -0 01 •0 02 t im e ( s a c ) t im a ( s a c ) Figure 3.3.15 Velocity ou tpu t components ysi(i = 1 ,2 ,3) a t the middle location 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.3.16 Velocity ou tpu t components y$i(i = 4 ,5 ,6 ) a t the m iddle location Similarly, the m agnitudes of m otion-induced response components y%i (i = 1 ,2 ,3 ) are much smaller than those of the components y& (i = 4 ,5 , 6 ) associated w ith the generalized velocities and mode shape functions. For the m otion-induced term s, the th ird mode makes larger con­ tribution to the responses th an the first mode and the second mode do. For the com ponents ySi (i — 4 ,5 , 6 ) associated w ith generalized velocities, the contributions of the first mode and second mode are dom inant in the two scenarios. The th ird mode com ponents experience a tem porary vanishing. W hen Ciaufi — 1 -the motion-induced velocity response components is zero due to fact th a t i(&out3 ) i (1 — CW 3 ) = 0, and the components 1/9 , (i = 4 ,5 ,6 ) associated w ith com bination between the generalized velocities and the mode shape function is constant, or (a out3 ) = [ 2 —2 2]. Figure 3.3.17 shows such components. It is noted th a t none of them vanishes 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. temporarily. S cen a rio A S c e n a rio B £ £ * 2 3 t im e ( s e c ) t im » ( s a c ) Figure 3.3.17 Velocity ou tpu t components y§i{i = 4 ,5 ,6 ) a t the tip location 4) A c c e le ra tio n O u tp u ts Figure 3.3.18 shows the acceleration ou tputs for the two scenarios. Scenario A Scenario B t i m e t s e c ) t u n e ( s e c ) >avvwv\\WAVW t i m e ( s e c .) t u n e ( s e c ) t u n c ( s e c ) t i m e ( s e c ) Figure 3.3.18 Acceleration ou tputs for motion scenarios A and B 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It can be seen th a t the further the observation location from the clam p, the larger the m ag­ nitudes of the acceleration responses. To evaluate how the v ibratory m odes contribu te to the acceleration signals at the base location, the middle loction and the tip location, the accelera­ tion output components, or yio,, y m and y \ 2 i (i = 1 ,2 ,3 ,4 ,5 ,6 ) , are p lo tted in the following figures. Figure 3.3.19 and Figure 3.3.20 show the acceleration ou tpu t com ponents yyoi (i = 1 ,2 ,3 ) and ywi (i = 4 ,5 ,6 ) a t the base location, respectively. Scenario A Scenario B o .o i 0.04 0 005 s 002 r ° ~ o -0 005 -0 02 \A/WVVWv̂“; - 0.01 0 1 2 3 ' ) 1 2 3 4 time (sec) t ime (see) 0.01 0.005 g- 002 O ~ O -0.005 ^ -0.02 - 0.01 0 1 2 3 ' ■"•“ c> 1 2 3 4 t ime (sec) t i me (sec) 0.01 0 .0 0 5 y . 0 02 o -0.005 -0.02 - 0.01 -O 04 lime (sec) time (sec) Figure 3.3.19 Acceleration outpu t components yioi(i = 1 ,2 ,3) a t the base location Scenario A Scenario B 0.005 0.01 0.005 0.01 0.01 T i m tim 0.01 0.02 0 005 0.01 o .o o s 0.01 - 0:01 0.02 t i m e ( s e c ) time (sec) 0.01 0.02 0.005 0.01 0.005 0.01 0.02 time (see) time (sec) Figure 3.3.20 Acceleration ou tpu t components yioi(i = 4 ,5 ,6 ) a t the base location Obviously, under scenarios A and B, for the components j/ioi (i = 1 ,2 ,3) associated w ith the 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. generalized coordinates, the first mode makes a larger contribution to the response th an the second and th ird modes do; for the components ywi (i = 4 ,5 ,6 ) associated w ith the generalized velocities, the contribution of the th ird mode to the response is the m ost dom inant, and the contribution of the first mode is the least dom inant. Figure 3.3.21 and Figure 3.3.22 show the acceleration components y m (i = 1 ,2 ,3 ) and y m (? = 4, 5,6) a t the middle locations. Sccnurio A Scenario B 1 2 3 t u n e ( s e c ) t u n e ( s e c ) t i m e ( s e c ) t i m e ( s e c ) 0.05 - 2 -0.05 tim e (sec) t i m e ( s e c ) Figure 3.3.21 Acceleration ou tpu t components y m ( i = 1 ,2 ,3) at the m iddle location Scenario A Scenario 3 O 04 0.02 0.02 0.04 t u n e ( s e c ) t i m e ( s e c ) 0.04 0 02 0.02 0.04 tjiTie (sec) t u n e ( s e c ) 0.04 O 02 0.02 0.04 t i m e ( s e c ) t i m e ( s e c ) Figure 3.3.22 Acceleration ou tpu t components y m ( i — 4 ,5 ,6 ) at the middle location For the components y m (i = 1 ,2 ,3 ) associated with the generalized coordinates, the m agnitude 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the second one is the larger than those of the first and th ird ones. For th e com ponents y Ui (i = 4 ,5 ,6 ) associated with the generalized velocities, the m agnitude of the th ird one is larger than those of the first and second ones. It should be pointed out the m agnitude of the second component associated w ith the generalized coordinates is much larger th an the any other components associated w ith the generalized coordinates and velocities. Thus, the second mode associated with the generalized coordinates is dom inant in the response. Figure 3.3.23 and Figure 3.3.24 shows acceleration com ponents y i 2 i (i = 1 ,2 ,3 ) and y i 2i (i = 4 ,5 ,6 ) a t the tip location, respectively. Scenario A Scenario 13 4 .S 3 *« .S 2 3 tune (»ec) tame ( te c ) -1 .5 -3 tim e (Hec) tune (wee ) tim e (rcc) t u n e d e c ) Figure 3.3.23 Acceleration output components y \ 2 i(i = 1 ,2 ,3) a t the tip location Scenario A Scenario B v A /W W ' t i m e ( n e e ) Wa*'— time < te c ) tunc (sec) 0.01 tu n e ( te c ) Figure 3.3.24 Acceleration output components y \ 2 i{i = 4 ,5 ,6 ) a t the tip location 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Similarly, for the components y ^ i (i = 1 ,2 ,3) associated w ith the generalized coordinates, the m agnitude of the th ird one is much larger than those of the first and second ones; for the components y ^ i {i = 4 ,5 ,6 ) associated w ith the generalized coordinates, the m agnitude of the second one is larger than those of the first and the th ird ones. A pparently, the m agnitude of the th ird component associated w ith the generalized coordinates is also much larger th an th a t of any other components associated w ith the generalized coordinates and velocities. Thus, the th ird mode associated w ith the generalized coordinates is dom inant in the response. 3.4 Evaluation of Varying Discrete-Time State Transition Ma­ trix The concept of the “pseudo” m odal param eters was proposed to characterize the global prop­ erties of a time-varying system in [14,16]. The “pseudo” modal param eters are based on the eigenvalues of the discrete-tim e sta te transition m atrix. This section defines the “pseudo” m odal param eters and evaluates them numerically for the model under study. A com parison between the “pseudo” m odal param eters and the “frozen” m odal param eters is presented. 3.4.1 D iscrete-T im e S ta te T ransition M atrix and “ P seu d o ” M od al P aram e­ ters A discrete-time state-space representation of a time-varying system under an initial condition is given as [14] x (k + 1) = G(k + 1, k)x(k) , x(0); y(k) = C(k )x (k ) (3-4-1-1) where G ( k + 1 , k) is an n x x nT discrete-time s ta te transition m atrix. x(0) is the initial condition. C(k) is an n y x n x ou tpu t m atrix. Note th a t the two im portant properties of G (k -I-1, k) are G(k + l , h ) = G(k + l , k ) G ( k ,h ) , k > h; G{h ,h) = I (3-4-1-2) If the varying discrete-tim e sta te transition m atrix G(k -t- 1, k) is non-singular, the corre- 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sponding eigendecomposition exists, i.e., G(k + 1 , k) = V(k)A (fc)V- 1 (ib) (3-4-1-3) where. V(k) is the eigenvector m atrix and A(k) = d ia g [Ai(fc), A2 (fc),. . . , \ n (k)} is the eigenvalue m atrix. Since the elements of the G(k + 1, k) are real, the complex eigenvalues occur in pairs. If the ith eigenvalue is complex, then the following relation can be used A i(k) = \*+nx/2 {k) = exp -Q(k)u>i(k) ± jLOt(k)\Jl — Ci(k)] A t (3-4-1-4) where ( l (k),uJl (k) are defined as the i th “pseudo” dam ping ratio and “pseudo” n a tu ra l frequency respectively. 3 .4 .2 N um erical E valuation o f th e D iscrete-T im e S ta te T ransition M atrix The varying discrete-tim e s ta te transition m atrix G(k + 1, k) can be found using an ensemble of n x sets of states since it satisfies a m atrix equation given by X ( k + l ) = G(k + l , k ) X ( k ) (3-4-2-1) where X ( k ) = [x1 (k) x 2 ( k ) , . . . . , xnx(k)} is the sta te m atrix a t tim e instant k and X ( k + 1) = [x1(k + 1) x 2(k + 1), , . . . ,xnx(k + 1)] is the sta te m atrix a t tim e instant k + 1. Note th a t x J(k) is generated by an initial condition x l (0). The transition m atrix can be found by G{k + l , k ) = X { k + l ) X ~ 1 {k). (3-4-2-2) 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To ensure the existence of X 1 (k), the n x sta tes m ust be independent of one another. T he n x sta te sequences are numerically found by the R unge-K utta algorithm . 3.4 .3 C om parison b etw een th e “Frozen” and “P seu d o ” M od al P aram eters To obtain “pseudo” modal param eters, including natu ral frequencies and dam ping ratios, the true varying discrete-tim e s ta te transition m atrices are evaluated using an ensemble of nx sets of states. The two scenarios m entioned in subsection 3.2.4 are used. The n x sets of initial states are generalized random ly in order to ensure independence of the sets of initial states. The param eters used in the sim ulation, including the duration of the axial m otion, tim e step, and system order nx etc., are the same as those used previously for the comparison. The n x sets of states were numerically obtained using the R unge-K utta m ethod. The comparisons between the “pseudo” and “frozen” m odal param eters are shown in Figure 3.4.1 and Figure 3.4.2 for axial extension and retraction. In these figures, a solid line represents the “pseudo” modal param eters and a dashdot line represents the “frozen” m odal param eters. . * 10** 2 3 time (sec) time (sec) time (sec) time (sec) 0 - I' 0 1 2 3 4 time (sec) time (sec) Figure 3.4.1 The “pseudo” and “frozen” m odal param eters for axial extension 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 •4 -60. 1 2 3 4 time (sec) 2 time (sec) 0 -2 -4 £ -6 0 1 2 3 4 time (sec) 2 time (sec)120 100 0 •2 0 1 2 3 4 time (sec) time (sec) Figure 3.4.2 The “pseudo” and “frozen” modal param eters for axial retraction The figures show th a t the curves representing the “pseudo” values and the curves repre­ senting the “frozen” values virtually overlap each other except for a small discrepancy in the dam ping ratios of the th ird mode. This indicates th a t the two types of the m odal param eters are almost identical for this particular system. The reason th a t the two types of the m odal param eters are identical can be explained by the slow variability of the system m atrix A(t) [14]. An approxim ate form of a general discrete-tim e sta te transition m atrix can be obtained using a series expansion G { k + l , k ) = I + A { k ) A t + ^ [ A 2 {k)+ A (k)}A t 2 + ^ [ A 3 ( k ) + A ( k ) A (fc)+2 A (*:)+ A (k )]At3+ ... (3-4-3-1) If the derivatives A {k), A {k) , ..., are sm a ll, the discrete-time state transition m atrix G ( k + 1 , k) can be approxim ated by G(k + l , k ) = I + A ( k ) A t + \ A 2 {k )A t 2 + 1 A z {k )A t3 + .. = e xp(A(k )A t ) . (3-4-3-2) z. o. Thus, the evaluation of the “pseudo” modal param eters using this approxim ate transition ma- 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. trix will result in the “frozen” modal param eters. 3.5 Conclusions Based on the results presented in this chapter, it is concluded th a t 1. The model is a linear time-varying system. The contribution by the axial force to the dynamics of the system is negligible. 2. The axial m otion influences dynamics of the system, including “frozen” m odal param eters, generalized coordinates and velocities, as well as the transient responses. The contribu­ tions of individual v ibratory modes to the transient responses are different for the different types of sensors and their locations along the beam. 3. To evaluate the varying discrete-tim e transition s ta te m atrix numerically, the system m ust undergo the same variation and an ensemble of n x sets of initial s ta tes m ust be generated randomly. The “pseudo” m odal param eters are very close to the “frozen” modal param eters for the system under study. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 4 Identification of the Axially-Moving Cantilever Beam The th ird objective of the thesis is to verify an identification algorithm using the axially-moving cantilever beam system. The algorithm was presented in [16]. It identifies the discrete-tim e s ta te space model of a LTV system using an ensemble of freely vibrating responses. The ensemble of the responses are obtained through m ultiple experiments on the system while the system undergoes the same change. The rest of the chapter is organized as follows: Section 4.1 reviews a subspace-based iden­ tification algorithm for linear tim e-invariant (LTI) systems; then, the algorithm was applied to identify the fixed-length cantilever beam system. Section 4.2 briefly reviews the identification algorithm using an ensemble of the responses, Section 4.3 presents the identification results based on the freely vibrating responses from the analytical model, Section 4.4 presents the identification results based on the experim ental responses, Section 4.5 presents the identifica­ tion results of identified “pseudo” natural frequencies based on the experim ental responses, and Section 4.6 contains the conclusions. 4.1 A Subspace-Based Identification Algorithm 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 .1 S ta te Space R ep resen tation o f LTI S ystem s The sta te space representation of an nx/2-degree-of-freedom freely vibrating LTI system is given by x (t) = Ax( t) , a?(0); y ( t ) = Cx(t) (4-1-1) where x(t ) is a vector of n x s ta te variables, x(0) is the initial condition, A is an n x x n x constant system m atrix, C is an n y x n x ou tpu t m atrix, and y(t) is a vector of n y responses. The corresponding discrete-tim e sta te space is expressed as x (k + 1) = G x ( k ) x(0); y(k) = C x(k ) (4-1-2) where G is an n x x nx s ta te transition m atrix given by G = exp(.4Af) (4-1-3) where A t is the sampling interval. The solution of equation (4-1-2) is y(k) = C x(k ) = C G kx{0). (4-1-4) It is noted th a t the description of equation (4-1-2) is not unique. Let T be an n x x n x non-singular m atrix and define a new sta te vector 2 = T x . Replacing x by T ~ l z in equation (4-1-2) results in z ( k + 1) = Gz(k) , y(k) = C(k)z{k) (4-1-5) where G = T G T ~ l and C = C T ~ \ (4-1-6) Equation (4-1-5) is another representation of the system defined by equation( 4-1-2). Such operation is referred to as sim ilarity transform ation. One of the im portan t properties of the sim ilarity transform ation is th a t G and G share the same eigenvalue, i.e., G = VA'S> - 1 or G = ( T V ) A ( m ) - 1 (4-1-7) 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where A = d ia g (Ai, A2 ,...., An i ) is the diagonal eigenvalue m atrix and $ is the eigenvector m atrix. Note th a t for an underdam ped vibratory system , the n x eigenvalues and eigenvectors occur in complex conjugate pairs. Thus, the eigenvalues can be arranged as (4-1-8) where j = y / ^ l , C, is the i th dam ping ratio, and uil is the ith na tu ra l frequency. 4 .1 .2 H an k el M a tr ix an d O b serv a b ility M a tr ix (1) Hankel M atrix Using a series of freely vibrating responses y(k) , k = 0 ,1 ,2 , . . .Ks — 1, and K s is d a ta length, a block Hankel m atrix is formed as y{ 0) 2/(1) y ( N - 1) 2/ ( 1 ) 2/ (2 ) y (N ) H = (4-1-9) y ( M — 1) y ( M ) . . . y ( M + N - 2 ) where M is the block row number, and N is the column number. Using the relation of equation (4-1-4), the Hankel m atrix can be factored as H = T X C C G r = c g 2 (4-1-10) C G M~ 1 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is the observability m atrix of the system, and X = x ( 0 ) x ( l) x ( N - 1) (4-1-11) is the sta te m atrix. Define two m atrices T i and T2 which are obtained by retaining the first ( M - 1) blocks and the last ( M — 1) blocks of the observability m atrix T, i.e., c CG C G C G 2 r i = C G 2 r 2 = C G 3 (4-1-12) C G M ~ 2 C G M~ l Therefore an estim ate of the G m atrix can be obtained by solving g = [Ti]+ r 2 (4-1-13) where (,)+ denotes the M oore-Penrose pseudo inverse. (2) E xtraction o f th e R ange Space o f th e O bservability M atrix A m atrix T is said to have the same range space as T, or i.e., C C G T = I T -1 = C G 2 (4-1-14) C g m ~ 1 if T is an n x x n x non-singular transform ation m atrix. Thus the first block row of T is C and G is given as g = [ r 1] + r 2 (4-1-15) 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where C C G C G C G 2 Ti = C G 2 and r 2 = C G 3 (4-1-16) c g m ~ 2 c g m~ 1 In practice, the m easured responses are contam inated by noise such th a t y(k) — y(k) + w(k) k = 1,2, ... , N (4-1-17) where y(k) is the m easured response vector and w(k) is the noise vector. The Hankel m atrix formed by the noisy responses becomes m y( i ) y ( N — 1) y( i ) m y( N) H = H + W = (4-1-18) y ( M — 1) y{M ) . . . y ( M + N - 2) The singular value decomposition (SVD) is used to ex tract the observability range space because of its numerical stability. For the case where ou tpu t m easurem ents are free of noise, it can be proved th a t [21, 23, 24] Range [H H T] = Range (T) (4-1-19) where H H T is a covariance m atrix. The above equality shows th a t the system observability range space can be obtained from the Hankel m atrix constructed from the experim ental data . In practice, Range [H H T] does not exactly equal to Range(T) due to the fact th a t the measured ou tpu t is contam inated by the noise. To estim ate Range(T), applying the SVD to [H H t ], i.e., r Jnx 0 H H = U n x U nu u nx u n 200 I ;■m 0^- V \ 0 0 5 0 10 0 150 I 5 0 10 0 15 0 freq u e n c y (H z) fre q u e n c y (H z) o 3 0 0 r 6 0 0 § - ex p erim en ta l o CO — e x p e r im e n ta l — iden tified id e n tified CD 2 0 0 4 0 0 * £ h 100 200 £ m oL 0 0 5 0 10 0 15 0 ( 5 0 10 0 1 50 fre q u e n c y (H z) f r e q u e n c y (H z) < > 1 5 0 r 4 0 0 < i— ex p erim en ta l - e x p er im e n ta l iden tified 3 0 0 — id en tified 100 S Z3 •8 200 & 50 3 g fe 100I r 0 ^ m 0 0 5 0 10 0 150 5 0 100 1 5 0 fre q u e n c y (H z) fr e q u e n c y (H z) Figure 4.1.3 F F T plots of the transient responses a t the three observed positions L = 0 .6 6 m L = 1 .0 9 i - identified . I — iden tified experim ental 0 .5 |H) lit |i I e x p er im e n ta l /VVmWVvw^vww I to -0 .5 CD -1 I W ^ V V V v v v v ^ 2 4 2 4 tim e ( s e c ) tim e ( s e c ) iden tified id en tified 0 .5 ex p erim en ta l ex p er im e n ta l 2 4 tim e ( s e c ) tim e ( s e c ) iden tified iden tified ex p erim en ta l e x p er im e n ta l M i/W ^̂WMAA/W/vwvww« < -0 .5 2 4 tim e ( s e c ) tim e ( s e c ) Figure 4.1.4 Com parison between the measured and sim ulated responses 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From these two sets of figures, it can be seen th a t (1) At the shortest length, from the BSG and MSG outputs, the first two m odes can be observed, bu t the th ird mode is very weak. From the TAC ou tput, all the three m odes can be observed. 2) At the longest length, from the BSG and MSG outputs, the first three m odes can be observed, but the fourth mode is hardly observable. From the TAC ou tpu t, the first four modes can be observed. 3) The sim ulated responses agree well w ith the m easured responses. However, there are some phase shifts near the tail of the responses, shown in Figure 4.1.5. It is noted th a t the solid lines represent the m easured responses and the dashdot line represents the sim ulated responses. The main reason is th a t the responses near the tails of the d a ta records have a low SNR (signal- to-noise ratio) as the response m agnitudes decay toward the end of the d a ta record. The poor SNR responses used in identification result in a large error between the sim ulated responses and the m easured responses. L=0.66m L= 1.09m f t 5.4 5.6 5.2 5.4 5.6 5.6 time (sec) time (sec) % f 5.4 5.6 time (sec) time (sec) ^ -0.05 5.4 5.6 time (sec) time (sec) Figure 4.1.5 Responses near the tail of the d a ta records 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Identification Algorithms 4.2 .1 T ransition M atrix The discrete-time sta te space representation of a LTV system subject to an initial condition is given in equation (3-4-1) of subsection 3.4.1. The solution of equation (3-4-1) is given by y(k) = C(k)G(k ,0)x(0) , (4-2-1) where G(k, 0) is the initial s ta te transition m atrix. The sim ilarity transform ation m atrix is no longer constant for the LTV systems. If the non-singular m atrices T ( k + 1) and T ( k ) exist, the sim ilarity transform ation is defined as G(k + l , k ) = T ( k + l )G{k + l , k ) T ~ 1 {k) C(k) = C ( k ) T ~ l (k) (4-2-2) where G(k + 1, k ) and C(k) represent another realization of the system , G( k + 1, k ) is an n x x n x s ta te transition m atrix of another realization of the system, C(k) is an n y x nx ou tpu t m atrix of another realization of the system, T ( k - f 1) is an nx x n x transform ation m atrix a t tim e instant k + 1, and T(k) is an n x x n x transform ation m atrix at tim e instant k. A series of G( k + 1, k ) and C(k) can be identified using the algorithm presented in [16] if an ensemble of responses are available. It is noted th a t G(k + 1, k) and G( k + 1, k) do not share the same eigenvalues. As ‘‘pseudo” m odal param eters are based on the eigenvalues of the m atrix G ( k + l , k ) , identification of the “pseudo” modal param eter presents a challenge. To find the eigenvalues of G(k + 1, k), a m atrix is defined as G(k + 1) = T ( k ) G( k + 1, /c)T_1 (k) (4-2-3) Thus. G(k + 1, k) can guarantee the invariability of the eigenvalues. Generally, the C(k) , G( k + 1, k), and G(k + 1, k) cannot be obtained through the d a ta from a single experim ent, the identifications of these are conducted by the ensemble m ethod. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 .2 Identification o f a LTV S ystem U sin g an E nsem ble o f Freely V ib ratin g R esp on ses It is assumed th a t N experiments have been conducted on the system whose param eters undergo the same variation. A freely vibrating response vector is denoted by y^(k), where j = 1,2, ....JV denotes the index of the experiments, k = 0 ,1 ,2 ..., K s — 1 denotes tim e instan t fc, and K s is the to ta l length of data. A general block Hankel m atrix is formed using k to k + M — 1 successive responses of N experiments, shown as follow y ' ( k ) y 2 ( k ) y N (k) y 1(k + l) y 2 (k) y N ( k + 1) H( k ) = (4-2-4) y 1(k + M — 1) y 2(k + M - 1) . . . y N (k + M - 1) The m atrix H( k) can be factored as H( k ) = T(k)[x1 (k) x 2 (k) x 3 (k) ... ^(jfc)] (4-2-5) where the observability m atrix T(fc) is given as C(k) C ( k + l ) G ( k + l , k ) T ( k ) = C ( k + 2)G{k + 2 , k) (4-2-6) C( k + M - l ) G ( k + M - l , k ) 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Its corresponding range space becomes C{k) C(k + l )G(k + l , k ) C(k + 2)G(k + 2,k) (4-2-7) C(k + M - l ) G ( k + M - l , k ) To extract G( k + 1, k), a successive Hankel m atrix H ( k + 1) is formed using the k + 1 to k + M successive responses of N experiments. The m atrix H ( k + 1) can be factored as H ( k + l ) = T ( k + l ) [ x 1 ( k Jr l ) x 2 ( k - 1-1) rr3(fc -b 1) ... x N (k + 1 )] (4-2-8) where T(k + 1) has a sim ilar form as equation (4-2-6) and its range space T(/c + 1) is given by C( k + 1 ) C( k + 2)G(k + 2 , k + l) r (k + 1) = r ( k + i ) T - \ k + 1) = C( k + 3)G(k + 3 , k + l ) . (4-2-9) C( k + M ) G ( k + M , k + 1) Let Ti ( k + 1) be the first (M - 1) block rows of T{k + 1), and IVfc) be the last (M - 1) block rows of T(fc), i.e., C(k + 1) C( k + 2)G(k + 2 , k + 1) ri(* + i) = C( k + 3)G(k + 3 , k + l) = r l ( k + l ) T ~ l (k + l) (4-2-10) C ( k + M - 1 )G(k + M - l , k + l) 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and C{k + l )G{k + l , k ) C( k + 2 )G(k + 2, k) r2(fc) = C ( k + 3 )G(k + 3, k) = r2(fc)r-1(fc) (4-2-11) C ( k + M - 1 )G(k + M - 1, k) _ where C ( k + 1) C ( k + 2)G(k + 2 , k + l ) r i(fc + i ) — C ( k + 3)G{k + 3 , k + l ) (4-2-12) C( k + M — 1 )G(k + M - l , k + l) and C{k + l )G{k + l , k ) C ( k + 2)G(k + 2 , k ) r a(AO = C(fc + 3)G(k + 3, fc) (4-2-13) C(fc + M - l ) G( k + M - 1, Jfc) Thus, the m atrix G(A; + 1, k) can be found by G(k + 1, k) = [Ti(Jb + 1)]+ T 2 {k). (4-2-14) In com putation, the range space m atrices T(k) and T(k + 1) are ex tracted by the SVD of H( k ) and H ( k + 1), i.e., H{k) = U( k ) i : ( k ) V(k ) T and H ( k + 1) = U(k + l)E(fc + l ) V ( k + l ) r . (4-2-15) Let Unx(k) be the first n x columns of U(k) and Unx(k + 1) the first n x columns of U(k + 1). Use the first (M - 1) block rows of UUx (k + 1) as Unxi{k + 1) and the last ( M - 1) block rows 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of Unx{k) as Unx2{k). Invoking equation (4-2-14) results in G(k + 1 ,k) = {Unxl(k + 1 )}+ Unx2(k) = T(k + 1 )G(k + 1 ,k ) T - ' ( k ) . (4-2-16) However, since T(k-f 1) and T(k) are unavailable, the exact solution of G(k + l , k) cannot be found. An approxim ate solution for G(k + l , k ) is proposed in [16]. In th is study, an alternative solution is used. Instead of forming Unx\ (k + 1 ) using the first (M — 1) block rows of UUx (k + 1), let Unxi(k) be the first ( M — 1) block rows of Unx(k). Note th a t Unxi(k) = T i ( k ) T~1(k) and Unx2(k) = r 2(k)T~l (k) (4-2-17) where C(k) C(k + l )G(k + l , k ) Ti (k) = (4-2-18) C(k + M - 2 ) G ( k + M - 2 , k ) and C( k + 1) C(k + 2)G(k + 2,k + 1). T 2 (k) = G(k + l , k ) . (4-2-18) C(k + M - 1 )G(k + M - 1, k + 1) The following relation exists G(k + l , k) = [C7^i(Ar)]+t/^2(fc) = = [ r - T(fc )rf(fc )r1(A r)r-1(fc)]-1r - r ( fc ) rf (fc )r^ (fe )r -1(A:) = T W W i ' i Q W i W G i k + ^ Q T - ' i k ) (4-2-19) where A / - 1 w1(fe) = rf(fc)r1(fc) J 2 G T{k + j - 1 , k ) CT(k + j - 1 )C(k + j - 1 )G(k + j - l , k ) j =i (4-2-20) 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and A / - 1 W2(k) = J 2 G T(k + j - 1 ,k) CT (k + j - 1 )C(k + j )G ( k + j - i , k + l ) (4-2-21) j =i If the system variation from moment k to moment k + 1 is small, i.e., C(k) is close to C ( k + 1) and G ( k + l , k ) close to G ( k + 2 , k + l ) , Wx(fc) is close to W 2 ( k ) or W ^ 1 ( k ) W 2 (k) a I. Therefore, an approxim ate solution of G(k -I-1, k) is given by G ( k + l , k ) a G( k + l,fc). (4-2-22) 4 .2 .3 C om pu tation al P roced u re for th e Identification a lgorith m To identify C( k ) , G ( k + 1 , k) and G(k + 1, k), k = 1 ,2 ,3 ..., K s — M — 1; 1. Form H(k) . Conduct the SVD on H( k ) to obtain U(k). Form Unx{k) by using the first n x columns of U(k): 2. Form Unx2 {k) ,Unxi(k) by retaining the last ( M — 1) block rows of UUx(k) and the first ( M - 1) block rows of Unx(k). Form C(k) by retaining the first block row of Unx(k); 3. O btain G{k + 1 , k) by using equation (4-2-19); 4. Form H ( k + 1). Conduct the SVD on H ( k + 1) to obtain U(k + 1). Form UUx{k + 1) by extracting the first n x columns of the U(k + 1). Form Unx\ (k + 1) by retaining the first ( M — 1) block row of U(k + 1); 5. Substitu te Unx2 (k) and Unxi (k + 1) into equation (4-2-16) to obtain G { k + l,fc); 6. Unx(k + 1) in Unx(k). If k < K s - M + 1, increase k by 1 and go to step 2. 4 .2 .4 Identified “P seu d o ” M odal Param eters The m atrix W ^ 1 ( k ) W 2 ( k)G(k + 1, k) can be used to evaluate a set of identified“ pseudo”m odal param eters. The eigendecomposition of the m atrix is given as G(k + 1, k) = $ (fc)A (fc)$-1(fc) (4-2-23) 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where &(k) is defined as an identified “pseudo” eigenvector m atrix and A(k) an identified “pseudo” eigenvalue m atrix formed by A(fc) = d ia g [Ai(Ar), A2( f c ) , A„ , ( f c ) ] - (4-2-24) Since the elements of G{k + 1, k) are real, the complex eigenvalues occur in pairs. If the zth eigenvalue is complex, then the following relation can be used A i(k) = A *+Bs/2(fc) = exp (4-3-25) where C,-(fc) and u5j(fc) are defined as the zth identified “pseudo” dam ping ratio and the zth identified “pseudo” natural frequency, respectively. 4.3 Comparison of the True and Identified “Pseudo” Modal Parameters To compare the modal param eters based on eigendecomposition of W ^ 1 ( k ) W 2 (k)G{k + 1 , k) or G( k + 1, k) and those based on eigendecomposition of G( k + 1 ,k ) or G( k + l , k ) , the analytical model developed in the previous chapter is used to com pute these m odal param eters numerically. Again, two motion scenarios are considered, namely scenario A: axial extension in which the beam length varies from L mm to Lmax and scenario B: axial retraction in which the beam length varies from L max to L mtn. The values of L max to L min are the same as those in chapter 3. The trapezoid velocity profiles are also used as the axial m otion profiles. Two stra in outpu ts, namely BSG and MSG whose axial locations w ith respect to the tip of the beam are 0.643 m and 0.335 m respectively are used. The system order n x is 6. A series of the m atrices G ( k + 1, k) are com puted using an ensemble of n x sets of states th a t are generated randomly. A series of the m atrices C(k) can be determ ined in equation (3-2-3-3) according to the prespecified axial m otion profiles and stra in sensor locations with respect to the tip of the beam. Thus, the m atrices Wi(fc) and W2(fc) can be found. Figures 4.3.1 and 4.3.2 show a comparison of the true “pseudo” m odal param eters and identified “pseudo” ones for scenario A and scenario B, respectively. In com putation of G{k + 1, k), the block row num ber M = 35nx = 210 is used. It 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is noted th a t solid lines represent identified “pseudo” m odal param eters and dashed lines the true “pseudo” modal param eters. 006 004 002 *0 02 •j 6 6 4 2 00 1 2 3 4 6 4 & 2 0 •2 ■4 2 3 0 1 2 3 4 time(sec) tim*(see) Figure 4.3.1 Com pariosn of the true and identified “pseudo” modal param eters under scenario A time(sec) time(sec) Figure 4.3.2 Com parison of the true and identified “pseudo” modal param eters under scenario B 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It can be seen th a t the identified “pseudo” natural frequencies agree w ith the tru e values well and the identified “pseudo” dam ping ratios deviate from the true values significantly. A natural question is how the choice of M affects the “pseudo” m odal param eters. To evaluate the closeness between the identified “pseudo” modal param eters and true ones, two dimensionless root-m ean-square (RMS) error indices are defined as e \ M k ) - m k= 1 ____________; e i m \ k= 1 E [CiW-CiWl2 /c=l _____________ = K s i = 1,2, . . . ,n x / 2 (4-3-1) E Ci(k)2 \ k= 1 where Sft and <5̂ . are the RMS error indices of the zth natural frequency and dam ping ratio respectively, f i (k) and f i (k) are the i th true “pseudo” natu ral frequency and identified “pseudo” one, ( i (k) and Ci(k) are the ith true “pseudo” dam ping ratio and identified “pseudo” one. Then, two overall indices are defined as (4-3-2) 2 = 1 1=1 where 6 f and 6 ̂ are the RMS error indices of the natural frequencies and dam ping ratios respectively. Figure 4.3.3 shows the relationship between the defined error indices and M . I t can be seen th a t, for the “pseudo” natu ral frequencies, the errors between approxim ate values and the true ones increase with an increase of M ; for the “pseudo” dam ping ratios, the errors between the identified values and the true one decrease w ith an increase of M. I t should be pointed out th a t the errors between the identified “pseudo” dam ping ratios and true values are much greater th a t the errors between the identified “pseudo” natural frequencies and true values. This indicates th a t the m atrix G( k + l , k ) can give a good estim ate of the “pseudo” na tu ra l frequencies, bu t it cannot give a meaningful estim ate of the “pseudo” dam ping ratios. It seems th a t the “pseudo” 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. damping ratios are more sensitive to the modification on the m atrix G( k + 1, k ) caused by the term W ^ 1 {k)W2 {k) than the “pseudo” natural frequencies. Scenario A Scenario B $ 0 .0 2 5 0 .0 2 5 u B. 0 02 0.02 • 0 .0 1 5 0 0 1 5 B 001 X 0.01 S 5 0 .0 0 5 0 .0 0 5 2CO n t r 50 100 1 5 0 2 0 0 2 5 0 3 0 0 3 0 6 0 9 0 1 2 0 1 5 0 1 6 0 2 1 0 2 4 0 2 7 0 3 0 0 M M 16 1 6 8 e 14 14 12 ■s b 10 10tr 4 ) C t2 O r t r 3 0 6 0 9 0 120 1 5 0 1 8 0 2 1 0 2 4 0 2 7 0 3 0 0 3 0 6 0 9 0 1 2 0 1 5 0 18 0 2 1 0 2 4 0 2 7 0 3 0 0 M M Figure 4.3.3 Relationship between RMS error indices of modal param eters and varying M 4.4 Experimental Identification of the System This section presents the experim ental identification results conducted on the axially-moving cantilever beam apparatus. The experim ental setup is shown in Figure 4.4.1. Impact Excitation 1 I Motor Current Axially-M oving Command to the Jt A /D 1 " j Control Circuit Cantilever Beam DC Motor N I-D A Q Board Data Kile D /A Sensor Conditioning Circuit Data Acquisition Computer Pentium III Figure 4.4.1 The experim ental setup 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Once again, two m otion scenarios are considered, namely, scenario A: axial extension and scenario B: axial retraction. To command the DC m otor to accomplish the desired axial m otion scenarios, a DC m otor control strategy needs to be chosen. To th is end, the relationship between the command voltage generated by the DAQ board and the m otor current was exam ined by connecting an ampere m eter in series to the DC m otor and applying a constant voltage input to the m otor current control circuit. Thus, the relationship between the com m and voltage and m otor current can be obtained and listed in Table 4.2. Table 4.2 Relationship between command voltage and motor current Scenario A Scenario B Command Peak/Steady motor current Command Peak/Steady motor current voltage(V) (A) voltage (V) (A) 0.5 - 0.245 / 0.245 -0.5 0.248 / 0.248 1.0 -0.486 / 0.486 -1.0 0.495 / 0.495 1.5 -0 .712 /0 .712 -1.5 0.743 / 0.743 2.0 -0.992 / 0.992 -2.0 1 .014 /1 .014 2.5 -1.211 /1.211 -2.5 1.235/ 1.235 3.0 -1 .463 / 1.463 -3.0 1.485/ 1.485 3.5 -1 .657 / 1.525 -3.5 1.701 / 1.520 4.0 -1.943 / 1.520 -4.0 1 .975/ 1.525 4.5 -2 .113 /1 .522 -4.5 2 .210 /1 .523 5.0 -2 .313 /1 .518 -5.0 2 .3 4 2 / 1.519 5.5 -2 .454 / 1.526 -5.5 2.473 / 1.516 6.0 -2 .792 / 1.525 -6.0 2 .6 8 2 / 1.525 6.5 -2 .962 / 1.523 -6.5 3.023 / 1.523 7.0 -3 .145 / 1.522 -7.0 3.203 / 1.521 It is noted th a t a proportional relationship exists between the voltage and the steady current up to 3V. A further increase in the applied voltage does not result in an increase in th e steady m otor current. However, it was observed th a t when a voltage greater th an 3V was applied, the 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m otor current had a short-tim e surge and then quickly settled down at the steady value. Such a surge in the m otor current did result in a fast acceleration m otion of the m otor. I t was desired to use two m otion speeds in the test. For the fast m otion, it is desired th a t the beam completes its motions in about 4.3 seconds in both scenarios. For the slow m otion, the beam completes its motions in about 9 seconds in bo th scenarios. After trial-and-error tests, a two-step control strategy was employed. The command voltages used are shown in Figure 4.4.2. Scenario A Scenario B > 4> If fa s t m o tio n ">O >4 f a s t m o tio n •o O 2 3 4 .3 4 .3 time (sec) tune (sec) s lo w m o tio n s to w m o tio n time (sec) tune (sec) Figure 4.4.2 Com m and voltage profiles For scenario A of the fast motion, a voltage of 7V is applied first until the pot voltage reaches 4.3V and then a voltage of 1.5V is applied until the maximum beam length is obtained. For scenario B of the fast motion, a voltage command of -TV is applied first until the pot voltage reaches -4.3V, and then a voltage of-1.5V is applied until the minimum beam length is obtained. For scenario A of the slow motion, a voltage of 2.8V is applied first until the pot voltage reaches 4.3V and then a voltage of 1.75V is applied until the maximum is obtained. For scenario B of the slow motion, a voltage of -2.8V is applied first until the pot voltage reaches -4.3V and then a voltage of -1.75V is applied until the minimum beam length is obtained. A Labview program was used to execute such comm ands and collect data . Figure 4.4.3 shows the pot signal profiles 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. obtained by such commands. Sceanrio A Sccanrio B 5 ■Sr•aB P 0 £ s. S. fast motionf a s t m o tio n 50 1 2 3 4 time (sec) tim e (sec) 5 5 ©C0 00 *c3s 0 0 '• S CQ e s tow m o tio n I , slow motion •50 2 6 8 •54 0 2 4 6 8 time (sec) tim e (sec) Figure 4.4.3 Pot signal profiles Three sensors were used. They are the base stra in gauge (BSG) sensor located at 0.643 m, the middle accelerometer (MAC) located at 0.335 m, and the tip accelerometer (TAC) located at 0.004 m. All the sensor locations are m easured w ith respect to the tip of the beam. The accuracy of an identified model is m easured by several dimensionless RMS error in­ dices. The indices are defined by comparing the m easured responses with sim ulated responses generated by an identified model G(k + l , k ) and C(k) . The closeness between the ith m easured response yj (k ) from the j t h experiment and its sim ulated counterpart yf (k ) is m easured by E \y{(k ) - y i ( k )' k= 1 L_ _ _ _ _ _ _ _ _ _ _ _ _ _ ; (4-4-1) K . r . - ,2 \ e mfc=1 L The closeness between all the m easured responses from the j t h experim ent y-7 and their simu- 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. lated counterpart ip is m easured by (4-4-2) The closeness between the ith m easured response from all the experim ents and their sim ulated counterparts is m easured by (4-4-3) where <5i denotes the overall m easure using the BSG response, 6 2 overall m easure using the MAC response, and <*>3 overall measure using TAC response. An overall m easure using the to ta l m easurem ent is defined as (4-4-4) To generate an ensemble of freely v ibrating responses, a proper initial excitation to the beam is very critical. A suitable excitation m ust excite the system dynamics fully and ensure the responses of current experiment to be independent of the previous ones. Each excitation was produced by tapping the beam at a different location with a different m agnitude of force in the two different directions. After the beam was tapped, the Labview program was executed to s tart the beam m otion and the responses of each experiment were displayed and visually examined to determ ine the suitability of the data. Numerous experim ents were conducted. In what follows, forty experim ents are considered for each com bination of the m otion speeds and the m otion scenarios. The d a ta acquired during the experiment are referred to as original ensemble data. The rest of the section is organized as follows. F irst, the identification results using original ensemble d a ta are presented in order to estim ate the system order n x , to understand some fac­ tors influencing the model accuracy. An ensemble of the responses d a ta of twenty experim ents is selected from the original d a ta of the forty experiments in order to reduce the com puta­ tional tim e and to improve model accuracies. Second, using the selected ensemble data , the relationship between the system order n x and the RMS error index are presented in order to understand how the system order n x affects the model accuracy, and the relationship between the block row num ber M and model accuracy are also given in order to understand how M 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. affects the model accuracy. Meanwhile, the comparison between the sim ulated responses and the measured ones are presented. 4.4 .1 Identification R esu lts U sin g th e O riginal E nsem ble D a ta The identified model accuracies are influenced by the quality and independency of the m easured responses, as well as the used model order n x and block row num ber M [16]. Due to the fact th a t the m easured ou tputs are contam inated by noise or may be affected by the system nonlinearity, the identified model should be overparam eterized in order to catch the dynam ics of the system [24, 25]. In what follows, using the original ensemble data , first, the variation of singular values obtained by the eigendecomposition of the Hankel m atrix in (4-2-4) are discussed for each combination of the m otion scenarios and m otion speeds, and a proper model order n x can be estim ated; Second, the model accuracies are calculated using the estim ated model order for various combinations of ou tpu ts used. The singular values a t tim e instant k are defined based on the SVD (singular value decom­ position) of Hankel m atrix H( k ) in (4-2-15), i.e., H( k ) = U ( k ) Z { k ) V T (k) k = 0 , l , - - - K s - M - 1 (4-4-1) where U(k) is an n xM x nxM left singular vector m atrix , V( k ) is an N x N right singular vector m atrix. 2(/c) is an n xM x N m atrix and can be partitioned as [^N{k)]pjxN E(*) = (4-4-2) Q( n x M - N ) x N where T,N {k) = diag ((Ti(k),a2(k), ■ ■ ■ , a N (k)) w ith eri(fc) ^ a 2{k) ^ > <7at(/c) ^ 0. The num ber ai(k) is called the ith singular value at tim e instant k. Since the singular values are varying, the average of the i th singular values (rl (k) ranging from (1 ^ k ^ K s — M — 1) is used to describe its variation for the time-varying process of the system and defined as K s- M - 1 K 5s - M - 1 GiW (4-4-3) fc=i 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure <1.4.4 shows the variations of all singular values for each com bination of the two motion speeds and scenarios. Figure 4.4.5 shows the comparisons of the first twenty singular values a t tim e instant / = 0 (sec), / = 2 (sec), / = 4 (sec) and their averages for fast motion, a t tim e instant / = 0 (sec), / = 4.5 (sec), / = 8.9 (sec) and their averages for slow motion, for axial extension and retraction. Scenario A fas t motion fast motion number of singular value number of singular value slow motion ■6 100 motion ~ 100 0 8 9 5 0 number of singular value number of singular value Figure 4.4.4 The singular values for each combination of the motion scenarios and speeds 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Scenario A Scenario B 200 average average 150 + t=2.0 •VO 2 261000 O lSl C 1 4 8 12 16 20 number of singular value number of singular value Figure 4.4.5 Comparison of the first twenty singular values a t different tim e instan ts and average Obviously, the m otion speeds and the m otion scenarios affect the variations of singular values. The m agnitudes of singular values become small as the beam becomes longer. The singular values for axial extension decrease much faster than those for axial retraction. This indicates th a t the m agnitudes of the modes present in the responses of axial extension reduces faster than those in the responses of axial retractoin toward the end of the axial motion. They reveal th a t the fewer modes are present in the responses of axial extension th an in the responses of axial retraction. Figure 4.4.6 shows the comparison of the first twenty average singular values between scenarios A and B for the two m otion speeds. Figure 4.4.7 shows the comparison of m agnitudes of the first twenty average singular values between the fast m otion and slow m otion 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. under scenarios A and B. 100 6 0 o s c e n a r io A o s c e n a r io A — s c e n a r io B * s c e n a r io B 90 5 0 8 0 ( 7 0 40< 0) 60 50 > 60 jp S c e n a r i o A 3w i S c e n a r i o B g» 40c et 5 0 - 0V) uO) 4 0 1 30 E - « • ------------------- ©=3 % -5 3 -4 □me (sec) Figure 4.4.11 Comparison between the sim ulated and m easured responses for scenario B It can be seen th a t the sim ulated ou tpu ts agree well with the m easured ones. Obviously, the outputs in scenario A decay faster than those in scenario B for the two m otion speeds. The main reason is th a t the positive dam ping effect under scenario A is greater th an th a t under scenario B since the axial extension causes the combination between the positive physical dam ping effects and the positive m otion-induced dam ping one and the axial retraction causes the combination effect between the positive physical dam ping and the negative m otion-induced damping, respectively. 4.5 Experimental Identification of “Pseudo” Natural Frequen­ cies In subsection 4.2.2. it was shown th a t the identified “pseudo” natu ra l frequencies can be de­ term ined by eigendecomposition of the identified m atrices G(k + 1, k). Here, the experim ental identification of the “pseudo” natu ral frequencies is addressed. F irst, the identified “pseudo” natural frequencies are presented using the selected ensemble data . Next, the moving-average m ethod is developed in order to select the “pseudo” natu ral frequencies of the v ibratory modes. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, the results of selection of identified “pseudo” natu ra l frequencies of the v ibratory modes are presented. 4.5 .1 Identified “P seu d o ” N atural Frequencies Using the experimental d a ta and the com putational procedure in section 4.2.3, the m atrix G(A-+1, k) in equation (4-2-19) can be obtained. Since the m easured responses are contam inated by the noise and other unknown irregularities, the identified “pseudo” eigenvalue m atrix A(k) contains the complex pairs of eigenvalues and the real eigenvalues. Thus, the eigenvalue m atrix A(k) in equation (4-4-24) can be represented in another form A (k) — Ai (k) X2(k) ■■■ Ani_ \ (k) Xn i (k) Ani+i(fc) ••• A nx (k) (4-5-1) where k = 0 ,1 , • • • K s — AI — 1 is the tim e instant. It is assumed th a t there are n \ / 2 pairs of complex eigenvalues and eigenvectors a t each tim e instant k. Thus, Xl+i(k) = Xi (k) for ? = 1,3, • • • n i — 1 are the pair of complex eigenvalues and eigenvectors, A,-(fc) for i = n\ 4-1, • • • n are real eigenvalues. As shown in (4-5-1), the eigenvalues are arranged in such a way th a t the complex pairs of eigenvalues and real eigenvalues can be separated and grouped individually. Thus, the identified “pseudo” natural frequencies corresponding to the complex pairs of eigen­ values can be arranged as f i ( k ) h ( k ) ••• f ni - i ( k ) f n i (k) (4-5-2) where f i{k) =Qi ( k ) / 2n . (4-5-3) To understand how the identified “pseudo” natural frequencies are influenced by model order n x, the identified “pseudo” natural frequencies are shown in Figure 4.5.1 using the threshold values of model order nx and the values of block row num ber M m entioned in section 4.4.5. It is noted th a t the acronym I P N F i ( ith identified “pseudo” natu ral frequencies) is used, or I P N F i ( k ) — f i (k) , in the following figures. It should be noted th a t the identified “pseudo” natu ra l frequencies were grouped by sorting them in an ascending order according to their 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m agnitudes at each moment. The blue color represents the na tu ra l frequencies of th e first group, green color of th e second group, red color of the th ird group, cyan color of th e fourth group, m agenta color of the fifth group, yellow color of the sixth group, and black color of the seventh group. Scenario A Scenario B 70 120 60 J. ; nx = 12 for fSst motionn = 4 for fast motion 100 \ . isle i 50 g 80 g 60 Z 40 20 0’ 2.8 4.25 1 . 4 2 . 8 4.25 time (sec) time (sec) n = 14 for slow motion nx = 4 for slow motion m 3 4 5 6 7 8 8.95 2 3 4 5 6 7 8 8.95 time (sec) time (sec) Figure 4.5.1 Identified “pseudo” natu ra l frequencies f i (k) using the threholds n x Based on Figure 4.5.1 and the tru e “pseudo” m odal param eters in section 3.3, the following observations can be drawn: 1. For scenario A, the identified “pseudo” natu ra l frequency of th e first v ibratory m ode can be observed using the the model order n x = 4 while the “pseudo” na tu ra l frequency of the second vibratory mode can be observed for a short tim e period in begining. 2. For th e case o f th e fast m otion in scenario B , th e variation trends o f th e first th ree “pseudo” natu ra l frequencies are visible from clustering of th e values of different groups. For example, the natu ral frequency of the th ird v ibratory mode mainly consists of the values of the “cyan” group in begining, then th e values of the “purple” group, finally the 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. values of th e “yellow” group. In the case of slow motion, the identified values appear even more disorganized. This indicates th a t the sorting by m agnitude ranking fails to group the identified “pseudo” natu ra l frequencies properly. 3. If th e model order is low, such as the case of scenario A, only the first m ode was identified for th e entire period of motion. In the case of scenario B, the model order of 12 was used and the first three v ibratory modes were captured. However, w ith an increase of the model order, the com putational modes were introduced. fast motion slow motion N SC N U SC,h f t £ fcc"g 1.4 2.8 4.25 8.95 time (sec) time (sec) 120 n=12 n = 1 2100 N 80 SC 5 60 CL) r. a ‘ ' . i £ i *'''*■40 20 0 1.4 2.8 4.25 8.95 time (sec) time (sec) Figure 4.5.2 Com parison of I P N F s for n x = 4 and n x = 12 under scenarios A Figure 4.5.2 shows the comparison of the “pseudo” natural frequencies identified using n x = 4 and nx = 12 for the fast and slow motions. I t is noted th a t the color codes are the same as those m entioned in Figure 4.5.1 except for the seventh group of na tu ra l frequencies. This figure shows th a t th e frequencies of the first th ree vibratory modes can be observed if the overparam eterized model order n x = 12 is used. From the observations, it is concluded th a t the overparam eterized model order should be used in order to observe the variations of identified 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ‘•pseudo” natural frequencies of system vibratory modes. 4 .5 .2 S election o f th e Identified “P seu d o ” N atu ra l Frequencies o f th e V ibra­ to ry M odes From the descriptions given previously, the identified “pseudo” natu ra l frequencies correspond­ ing to the complex pairs of eigenvalues contain the natu ral frequencies of the v ibratory modes and the com putational modes a t each tim e instant k if an overparam eterized model order n x is used. A natural question to ask is how the identified “pseudo” na tu ra l frequencies of the vibratory modes given by f f ( k ) i = 1,2, ..n, and n ^ « i , k = 0, (4-5-5) are selected from the identified “pseudo” na tu ra l frequencies corresponding to the complex pairs of eigenvalues in equation (4-5-2), where n is the v ibratory modes considered. The answer to the question is th a t f f ( k ) can be determ ined based on the principle th a t the values of f f ( k ) should be as close as possible to the reference values of the natural frequencies of the v ibratory modes given by f l { k ) i = 1 ,2 ,..., n. (4-5-6) The reference values f [ ( k + 1) at the tim e instant k + 1 can be determ ined using the moving- average m ethod. The m ethod is used to calculate the reference values of the natu ra l frequencies f [ ( k + 1), based on average of the identified “pseudo” natural frequencies of the vibratory modes th a t are selected from the identified “pseudo” natural frequenices in the tim e period block (k ^ k\ ^ k + M\ — 1). The selected natural frequencies expressed in f i ( k i ) (fc O t < * + Aii - 1) (4-5-7) should be as close to the reference values f [ ( k ) at tim e instant ki in the varying tim e period block, where the M i is the tim e period block number. To select the “pseudo” natu ral frequencies of the vibratory modes given in (4-5-5) for k = 0 ,1 ,2 , ...K s — M — 1, the following procedure is developed. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. Specify the initial reference values of the natu ral frequencies of v ibratory modes /,r (0); 2. Set up the tim e period block from k to k + M i - 1; 3. Select all f s(ki) from f ( k i ) in (4-5-3) in the tim e period block (k ^ ki < k + M\ - 1) based on the principles th a t all f s (k\) should be as close to the reference values f r (k) as possible, calculate the average value of all f s(ki) in the tim e period block given as fc+Mi-l ~k[k) = w ^ i £ / '(* > )• i4-5-8* k\=k \ { k < K s — M — M i — 1, increase k by 1, let f i ( k + 1) = f [ ( k ) , and go to step 2; If the k — K s — M — M i — 1, f i ( k ) are selected from f ( k ) based on the principle th a t f j ( k ) should be as close to the reference values f [ ( K s — M — M i — 1) as possible in the tim e period block (K s — AI — M\ — 1) ^ k ^ ( K s — M — 1). 4 .5 .3 R esu lts o f S election o f th e N atu ra l Frequencies o f th e V ib ratory M odes The natural frequencies of the v ibratory modes at the two extrem e fixed lengths listed in Table 4.1.1 are used as the initial reference values of the natu ral frequencies of the v ibratory modes. For three v ibratory modes considered, or n = 3, the natu ral frequencies of the first three vibratory modes at the shortest fixed length given by f r (0) = [4.9325 31.5350 87.66] are used as the initial reference values for m otion scenario A, and the natu ral frequencies of the first three vibratory modes a t the longest fixed length given by f r (0) = [1.9635 12.389 35.016] are used as the initial reference values for m otion scenario B. Figure 4.5.3 and Figure 4.5.4 show the identified “pseudo” natu ral frequencies of the first three vibratory modes considered using the param eters n z = 12 and M = 50 for the fast and slow motions of scenario A, n z = 12 and M = 60 for the fast m otion of scenaro B, n z = 14 and M = 70 for the slow m otion of scenario B. The tim e period block num ber M \ = 200 are also 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. used. Scenario A Scenario B time (sec) lime (sec) 40 40 30 30 20 20 10 1.4 2.6 4.25 1.4 2.8 4.25 100 time (sec) 100 time (sec) 80 60 60 60 40 20 20 1.4 2.8 4.25 1.4 2.6 4.25 time (sec) Figure 4.5.3 I P N F s of vibratory modes for fast motions under scenarios A and B Scenario A Scenario B time (sec) 40 30 20 10 6.95 time (sec) 100 time (sec)150 80 60 50 40 20 8.95 8.95 time (sec) time (sec) Figure 4.5.4 I P N F s of vibratory modes for slow motions under scenarios A and B It can be seen th a t the variations of the identified “pseudo” natu ral frequencies of the first three vibratory modes are similar to those of the true values m entioned in C hapter 3, i.e., the identified “pseudo” natu ral frequencies of the first three vibratory modes vary from high values to low values for scenario A and from low values to high values for scenario B. It is noted th a t 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the identified ‘‘pseudo’' natural frequencies of v ibratory modes fluctuate around their overall trends due to the fact th a t the m easured ou tpu ts used in identification are contam inated by the noise. Meanwhile, M i is an im portant param eter in selecting the identified “pseudo” na tu ra l fre­ quencies of v ibratory modes. Figures 4.5.5 and 4.5.6 show the comparison of the identified “pseudo” natu ral frequencies of the v ibratory modes for M\ = 10 and M \ = 50 for fast and slow motions under scenario A, respectively. Figures 4.5.7 and 4.5.8 show the comparison of the identified “pseudo” natu ra l frequencies of v ibratory modes for M\ = 100 and M \ = 200 for fast and slow m otions in scenario B, respectively. M =10 M = 5 0 1.4 2.8 4.25 1.4 2.8 4.25 time (sec) time (sec) 40 ~M 30I ~ 20 i. ■fi. .•:• 14 2.8 4.25 2 8 4.25 time (sec) 100 100 _ 80 XN 50 — 60 V* 40 * t 1.4 2.8 4.25 20 1.4 2.8 4.25 time (sec) time (sec) Figure 4.5.5 Selected frquencies using M i=10 and those using M i =50 for fast m otion of scenario A 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. time (sec) time (sec) 3 6 time (sec) time (sec) 150 150 £ 100 3 : 100 50 - « 50 8.95 time (sec) time (sec) Figure 4.5.6 Selected frquencies using M i= 10 and those using M i =50 for slow m otion of scenario A M ^IOO M 1= 2 0 0 40 15 |--------------------------------------------------------- TJ 30 - r* X x 10 L 5 20 z l*• 10 z r a f | ..................... 0I 1 2.8 4.25 1.4 2.8 4.25 time (sec) time (sec) 40 40 _ 30 3: £ 30 ' Z 20 ■£r •*w 201 0 L 10‘ 1.4 2.8 1.4 2.8 4.25 time (sec) time (sec) 100 x 80 40 20 1.4 2 .8 4.25 1.4 2.8 4.25 time (sec) time (sec) Figure 4.5.7 Selected frquencies using A /i=100 and those using M i =200 for fast m otion of scenario B 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 ^ = 1 0 0 M 1= 20 0 40 15 30 i.20 10 895 8.95 time (sec) time (see) 100 40 30 £. 50 -~20 10 8.95 100 time (sec) time (sec)100 80 40 20 8.95 8.95 time (sec) Figure 4.5.8 Selected frquencies using M i=100 and those using M i= 200 for slow m otion of scenario B From the above figures, it is concluded th a t the identified “pseudo” na tu ra l frequencies of the vibratory modes selected are not reasonable if M \ values are too small. Similarly, the variations of identified “pseudo” natu ral frequencies of the v ibratory modes are also not reasonable if M i is too large. How to determ ine a proper value of M i is a topic for the fu ture work. 4.6 Conclusions Based on the results presented in this chapter, the following conclusions can be drawn: 1. The identification algorithm can only obtain a good approxim ation for the natu ra l fre­ quencies of the v ibratory modes. However, it cannot give a meaningful estim ate of the dam ping ratios of the vibratory modes. 2. To obtain the best identification results, it is necessary (a) to ensure th a t the system m ust undergo the same time-varying variation; (b) to generate the m easured responses as independently as possibly; 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (c) to use the multiple m easured ou tputs instead of one m easured output; (d) to force the m easured responses to be zeros mean; (e) to select a small set of ensemble d a ta from the original ensemble data . 3. The m otion speed and m otion scenario have a direct effect on the model accuracies. 4. To observe as many “pseudo” natu ral frequencies of the v ibratory modes as possible, it is necessary to use the overparameterized model order. 5. The moving-average m ethod obtains the sensible results for the variations of the identified “pseudo” natural frequencies of the v ibratory modes for each com bination of the m otion speeds and the scenarios. 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Summary and Future Work The sum m ary of the thesis study is described as follows 1. A circuitry for DC m otor current control and sensor signal conditioning has been built. The circuitry meets the requirem ents of controlling the axial m otion of the cantilever beam and amplifying the sensor signals. 2. An analytical modeling procedure has been conducted to model the lateral v ibration of an axially-moving cantilever beam. The sim ulation results confirm th a t the axial mo­ tion influences the dynamics of the system, including transient responses and “ frozen” modal param eters. To evaluate the discrete-time s ta te transition m atrix reasonably, the system m ust undergo the same variation and an ensemble of nx sets of initial sta tes m ust be generated randomly. The “pseudo” modal param eters are evaluated by conducting eigendecomposition of the discrete-time sta te transition m atrix. The sim ulation results indicate th a t the “pseudo” modal param eters are almost identical to the “frozen” ones for the system under study. 3. A previously developed algorithm has been applied to identify the system. The study has addressed several im portant issues, such as the m ethods of exciting system , eval­ uation of model accuracies, factors influencing model accuracies, relationships between model accuracies and param eters used in identification, selection of na tu ra l frequencies of the v ibratory modes from identified “pseudo” natural freqeuncies. It has also been shown th a t the identification algorithm can provide a sensible approxim ation for na tu ra l 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. frequencies of the vibratory modes but cannot provide a reasonable estim ation for dam p­ ing ratios. The moving-average algorithm can be used to select the identified “pseudo” natural frequenciees of vibratory modes among the identified modes. Recommendations for future work are: 1. to determ ine a proper value of tim e block num ber M \ in order to obtain a b e tte r result of identified “pseudo” natural frequenices of the vibratory modes. 2. to implement identification algorithm s based on forced responses of the axially-moving cantilever beam to ensure th a t the v ibratory modes can be fully excited. 3. to develop a model reduction and updating m ethod. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] B. Tabarrok, C. M. Leech and Y. L. Kim, “ On the Dynamics of an Axially Moving Beam ” , Journal o f the Franklin Institute (1974) , 297(8), pp.201-220. [2] A. K. M isra and V. J. Modi, “ Deployment and Retrival of Shuttle Supported Tethered Satellites ” , Journal o f Guidance (1982), 5(3), pp.278-285. [3] N. G. Chalhoub and A. G. Ulsoy, “ Dynamic Simulation of a Leadscrew Driven Flexible Arm and Controller ” , Journal o f Dynam ic Systems, Measurement, and Control (1974) , 108, pp.201-220. [4] P. K. C. W ang and Jin-Duo Wei, “ V ibration in a Moving Flexible Robot Arm ” , Journal o f Sound and Vibration (1987) , 116(1), p p .149-160. [5] K. Krishnam urthy, “ Dynamic Modeling of a Flexible Cylinderical M anipulator ” , Journal o f Sound and Vibration (1989) , 132(1), p p .143-154. [6] S. H. Hyun and H. H. Yoo, “ Dynamic Modeling and Stability Analysis of Axially Oscil­ lating Cantilever Beam ” , Journal o f Sound and Vibration (1999) , 228(3), pp.543-558. [7] W. D. Zhu and J. Ni, “ Energetics and Stability of Translating M edia w ith an A rbitrary Varying Length ” , Journal o f Vibration and Acoustics (2000) , 122, pp.295-304. [8] A. Kumaniecka and J. Niziok, “ Dynamic Stability of A Rope w ith Slow Variability of the Param eters ” , Journal o f Sound and Vibration (1994) , 178(2), pp .211-226. [9] A. A. Renshaw, C. D. Rahn, J. A. W ichert, and C. D. Mote, J r, “ Energy and Conserved Functionals for Axially Moving M aterials ” , Transaction of the ASME (1998), 120, pp.634- 636. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [10] B. Jacob, “ A Formula for the Stability Radius of Time-Varying System s ” , Journal o f Differential Equations (1998), 142, pp.167-187. [11] H. Xiao and Y. Liu, “ The Stability of Linear Time-Varying Discrete System s w ith Time- Delay ” , Journal o f M athematical Analysis and Applications (1994), 188, pp.66-77. [12] S. 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McConnell, “Instrum entation for Engineering M easurement” . 1984, John W iley & Sons, Inc. [31] J . D. Irwin, C. Wu, “ Basic Engineering Circuit Analysis” , 6 th Edition, 1999, John W iley &; Sons, Inc, New York, U. S. A. [32] J. R. Hanly, E B. Koffman, and J. C. Horvath, “C Progam Design for Engineer” , 1995, Addison-Wesley Publishing Company, Inc, Menlo Park, California, U. S. A. [33] R. H. Bishop, “Learing w ith Lab V IEW ” , 1999, Addison-Wesley Publishing Company, Inc, Menlo Park. California, U. S. A. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [34] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Num erical Recipes in C” , 2nd Editon, 1992, Cambridge University Press. [35] J. E. Slotine, and W . Li, “Applied Nonlinear Control” , 1991, Prentice Hall, Englewood Cliffs, New Jersey, U. S. A [36] G. F. Franklin, J. D. Powell, and M. W orkman, “Digital Control of D ynam ic System ” , 3rd Edition, 1998, Addison wesley Longman, Inc, Menlo Park, California, U. S. A. [37] K. O gata, “M odern Control Engineering” , 2nd Edition, 1990, Prentice Gall, Englewood Cliffs, New Jersey, U. S. A. [38] M. W .Spong, and M. Vidyasagar, “Robot Dynamics and Control” , 1989, John W iley & Sons, Inc, New York, U. S. A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A Schem atic Drawings of the Circuits and The T esting Setups 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Motor Power Supply Module 2N6052 —v — 7805 Diode Bridge +12 V output LM317 R3 +5 V output AC R2; C2 input Ground 7912 +12 V output R1 =51 ohm C1 = 470 uF R2 = 4.7 kohm C2 = 11000 uF R3 = 270 ohm C3= 2700 uF R4 = 5 k ohm (variable) C4= 2.2 uF Figure A2.1 Motor Power Supply Module Bridge Module n +12 V (3> 5A from motor power supply module Gate 1 R2 R1 Q1 IRF540 Gate 2 IRF540 T c R5 Source 1 Q4 Source 2 R3 R4 G ate 3 R7 Q3, R8 D 3iL 2tD4 IRF540 Source 3 & 4 IRF540 Gate 4 nFrom output of current control circuil 2N6059 i To emitter of current control circuit R9 Ground R1 = R2 = R3 = R4 = R9 = 1 ohm R5 = R6 = R7 = R8 = 12 ohm C1=C2=C3=C4=2.2 uF Figure A2.2 Bridge Module 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Current Control Module +12 V power supply F5 R8 Input Magn itude LM 1458 LM 1458 LM 1458 LM 1458 R2 R3 R4 F6 R11 R12R9 LM 1451 Ground •12 V power su jply Fn>m 2N6059 emitter of bridge module R14 R1 = R2 = R3 = R4 = 2.2 khom R5 = R8 = R9 = R10 = 1.2 khom R6 =18 Kohm R7=180 ohm R11=82 kohm R12 = 5.6 kohm R13 = R14 = 1 kohm C1=1.2nF Figure A2.3 Current Control Module Opto-Isolation Module "A & B" ♦6V<1| ♦» V(2) R 2 output B - r C 1 R 3 C 3 inputdiroctioi R 1 M P S 3 6 4 6 Gnd D J 0 0 2 6 C N R7 CS 0 0 2 6 C N Gnd (1) Hp 2 6 1 A Gnd (2) C 1 = C 2 = C 3 = 0 .1 u F R 1 = 4 .7 k o h m R 2 = 5 8 0 o h m R 3 = 1 6 0 o h m R 4 = 1 .0 k o h m R 5 = 2 7 0 o h m R 6 = 1 0 .0 k o h m R 7 = 5 .6 k o h m Figure A2.4 Opto-Isolation Module " A & B " 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Signal Separation Module +12 V power supply input from , \CO_0 LM 1458 LM 1458 magnitude LM 1458 output R3 LM 1458 R4 D2 directionoutput Ground -12 V power supply R1 = R2 = 10 kohm R3= 2.2 kohm R4 = 1 kohm Figure A2.5 Signal Separation Module Power Supply Module +12 V output 7812 Diode Bridge +5 V output 7805 AC T C1 T C2 Input Ground T C 3 7912 -12 V output C1 = 470 uF C2 = 2700 uF C3= 470 uF Figure A2.6 Power Supply Module 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Potentiometer +12 V power supply eo output LM 1458 12 V power supply Figure A2.7 Potentiometer Strain Gauge Module +12 V power supply +5 V power su p ply R7 R5 R8 R9 LM 1458 output V1 LM 1458 LM 1458 R6 V2 R3R10 R11 POT 10 K LM 1458 Ground -12 V power supply R1 = R2 = R5 = R6 = 1 kohm R8 = R9= R10= R11 = 120 ohm R3 = R4= 100 kohm R7 = 10 kohm Figure A2.8 Strain Gauge Circuit 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Motor Power Supply ln_AC1 Gnd ln_AC2 12 V Bridge Module Current control module ♦12 V 12 V (motor) lnput_magnitude . 3 4 4 8 1 4 motor 1 Gnd(1) S2 4 motor 2 •12 V n ■ £ © ? DC voltage Opto-ieolation Circuit (3) Powor Supply Circuit (3) TTTWr 12 V DC Motor ♦« V<1) ♦12 V<4) In AC1 Voltage input outputs ^Amp lnput_direction ♦«V(4) ln_AC2 +6V(4) . motor Gnd (1) Gnd (4) output A •12 V(4) Gnd (4) Opto-iaolation Circuit (2) Power Supply Circuit (2) ♦12V(J) ♦12 V(3) ln_AC1 output B direction +*V(3) ln_AC2 ♦6 V(3) Gnd (3) output A -12 V(3) Power Supply Circuit (4) Gnd (3) ____ IN AC1 +12 V OptOHSoiation Circuit (1) Powor Supply Circuit (1) ♦6 V ♦6V1 +12V(2) IN AC2 +12 V(2) ln_AC1 Gnd (1) outputs Input direction ♦6V(2) ln_AC2 •12 V +6 V(2) Gnd (1) Gnd (2) output A •12 V(2) Gnd (2) Figure A2.9 Testing Bridge Circuit and Measuring Motor Current Voltage meter V+ Motor Power Supply ln_AC1 Gnd Gnd In AC2 12V Fluke Scope Bridge Module CH1 Gnd (1) CH2 Gnd (2) Figure A2.10 Testing Diagram of Motor Power Supply 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Function Generator - w a v e f o r m C H 1 G n d ( 1 ) G n d Current Control Module C H 2 G n d ( 2 ) +12 V pow er su pply R7 F10 F 5 R8 LM 1458 LM 1458 LM14& LM 1458 R2 R3 R4 F 6 R11 R12 Ground LM 14511 •12 V pow er su pply s h o r t i n g t h e f e e d b a c k w i t h g r o u n d P o w e r S u p p l y m o d u l e ( I) + 1 2 V R14 + 5 V G n d ( 1 ) - 1 2 V Figure A2.ll Testing of P and I Control Effort F u n c t i o n G e n e r a t o r SSiZBfc r a v e f o r m -G H 1 G n d ( 1 ) - G n d -G H 2 G n d ( 2 f Current Control Module +12 V pow er supply F5 R8 Input M agnitude R 1 r - 2- LM 1458 LI l/l 1458 LM 14J8 LI MR 1458 R2 R3 R 4 F6 R11 R12 Ground -12 V pow er su pp ly shorting the feedback with ground P o w e r S u p p l y m o d u l e / 1 ) — J-+ -12V R14 + 5 V G n d ( 1 ) 1 2 V Figure A2.12 Testing of PI Control Effort 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fluke Scope CHI Function generator Gnd (1) CH2 waveform Gnd (2) Gnd Opto-isolation module Power supply (1) Power supply (4) ------------- TiTV +12 V(2) +5 V(1) +12 V(2) ln_AC1 IN-AC1 v output B +5 lnput_direction +5V(2) ln_AC2 +5 V(2) ,N- AC2 Gnd (1) Gnd (1) Gnd (2) output A -12 V •12 V(2) Gnd (2) Figure A2.13 Testing Diagram of Opto-isolation Module Function generator Fluke Scope waveform Slgnal-separation circuit T B T " Gnd +12 V Output_mag Gnd (1) input Output_dir CH2 •12 V Gnd Gnd (2) +12 V IN_AC1 +5 V IN AC2 Gnd (1) -12 V Figure A2.14 Testing of Signal-Separation Module 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Potentiom eter F luke S c o p e +12 V +12 V output CH1 Slider Slider Gnd Gnd(1) -12 V -12 V CH2 p ow er su p p ly Gnd(2) +12 V IN_AC1 +6 V IN AC2 Gnd(1) -12 V Figure A2.15 Testing Diagram of Potentiometer Circuit Fluke S c o p e CH1 Gnd(1) Strain G au ge (B SG ) CH2 +5 V Gnd(2) +12 V ^Clamper Gnd (1) input2 P ow er S upply (4) -12 V in p u tl ♦12 V : rom BSG 1 & 2 IN AC1 +5 V IN AC2 Gnd (1) Strain Gauge(M SG ) -12 V +5 V output- +12 V Gnd (1) input2 -12 V in p u tl From MSG 1 & 2 Figure A2.16 Testing Diagram of Strain Gauge Circuit 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH1 (aetpoint meaaurement) Motor power aupply Potantiometar Gnd(1) Gnd(1) IN.AC1 -CH2 (feedback maaauramant) ♦12 V IN.AC2 -Ond(2) ( fluke S co p e) Bridge Module Function Generator Currant control module ♦12 V 12 V (m otor) Input magnitude 81 8 motor 1 Rf Gnd(1) Output 82 & motor 2 — ■12 V Feedback 0.5 • 5.5 ohm Signal-eepa ration Circuit Opto-ieelation Circuit (3) Power Supply Circuit (3) ♦12V ♦12 V(4] ♦12 V(4) ln_AC1 Input Gnd(1) output Biut direction ♦6V(4) ln_AC2 outputm ag Strain Gauge (MSG) Gnd (4) outputdir output A ♦«V output -12 V(4) 12V Gnd (4) ♦12 V Gnd (1) input2 Opto-iaolation Circuit (2) Power Supply Circuit (2) -12 V inputl ♦12 V(3) ♦12 V(3) ln_A From MSG 1 8 2 output B direction ♦6V(3) ln_AC2- - ♦6 V(3) Gnd (3) Strain Gauge (BSG) output A -12 V<3) +£ V output Power Supply Circuit (4) Gnd (3) ♦12 V IN AC1 ♦12 V Opto-isolation Circuit (1) Power Supply Circuit (1) Gnd (1) input2 +6 V ♦12 V{2] -12 V inputl IN AC2 +6V<1) -e32V (2) In.Ai Gnd (1) output B From BSG 1 8 2 Input.diroction -*SV(2) In. -1 2 V Gnd (1) -Gnd (2)output A -12 V(2) Gnd (2) Figure A.2.17 Testing Diagram for Setpoint and Feedback Comparison 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH1 (setpoint measurement] Motor power supply Potentiometer GikMD •rm Gnd(1) IN_AC1------------ CH2 (motor waveform output ♦12 V IN AC2 Gnd(1| -12 V (R u k eS cop e) Function Generator Current control module ♦TF7 12 V (motor) lnput_magnttude § 3 4 4 31 & motor 1 Gnd(1) Output 3 2 4 motor 2 12 V Feedback 0.5 - 6.5 Ohm gnal Separation Module Opto-isolation Cirourt (3) Power Supply Circuit (3) TT7 +12 V(4) ♦6 V(1) ♦12V(4) IruACI Gnd(1) output B Input.direetion output.mag ♦6 V(4) Strain Gauge MSG)Gnd (1) Gnd (4) output_dir output A output •12 V(4) 12V Gnd (4) ♦12 V Gnd(1) biput2 Opto-isolation Cirouit (2) Power Supply Cirouit (2) 12 V Inputl ♦6V(1) ♦12 V(3) ♦12 V(3) tn_AC11 From MSG 1 4 2 output B lnput_direetior +6V(3) In A Gnd (1) Gnd (3) Strain Gauge (BSG) output A 12 V(3) x v Power Supply Circuit (4) Gnd (3) IN AC1 +12 V Opto-isolation Cirouit (1) Power Supply Cirouit (1) Gnd (1) inputs ♦6 V liv(2) IN AC2 ♦6V(1) ♦12 V(2) ln_AC4- - Gnd (1) outputs From BSG 1 4 2lnput_dlreetion -12 V ♦6 V(2) Gnd (1) output A -12 V(2) Gnd (2) Figure A2.18 Testing Diagram for Setpoint and Motor Current Waveform 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.