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AN ABSTRACT OF THE THESIS OF Thanat Jitpraphai for the

1. VIBRATION VISUALIZATION 4 E 0 Ehotc hter Bunerworm 19 vons 20 low vons 146 VISQJALIZATI A 15 VIBRATION Signal Processor VI 05 30 97 01 29 AM Signal 05 30 97 01 29 AM ITYITTITYUUTXTITITITITTTITITITITITITICITIN 147 VIBRATION VISUALIZATION 76 148 reyes Signal Processor VI sP 05 30 97 01 29 AM IT TI DI CI ICITIT IT IIT ITITIT IT I IT X 2 y ITCERXOKJHCITIT ITIT IT p 00660 6 9 4 o e o o o oos o 6 6 6 0 0 5 6 8 6 6 8 6 9 646 6 6 056 6 816 lt 8 649 6 04610380046 9 616 8 0 616 9 4 XTITIT o o ooe 10 Signal Processor VI 05 30 97 01 29 AM wan program T yes VIBRATION TION 17 Signal Processor VI 05 30 97 01 29 AM Signal Processor VI 05 30 97 01 29 AM
2. Warning Corer Data Acquisition Controller 05 30 97 01 20 22 23 142 VISUALIZATION 11 VIPRATION Data Acquisition Controller VI 05 30 97 01 20 AM Data Acquisition Controller VI 05 30 97 01 20 AM h X Format Precision x precision J Page 24 25 143 144 Page 1 2254 Data Monitor with PS ver 2 2 s vi 05 30 97 02 26 AM SIGNAICHANNEfy GRAPH oen 4 0000000 Data Monitor with PS ver 2 2 s vi 05 30 97 02 26 AM eon cic emn Data Monttor with PS ver 2 2 s vi 05 30 97 02 26 AM VIPRATION VISUALIZATION 12 145 Page 1 i Output From Ind 1 From SR 1 From SR 2 From SR 3 From Ind 2 From Ind 3 05 30 97 01 24 AM grp or4 vi P From SR 4 From Ind 4 NObv ZI vni A IA Signal Processor VI 05 30 97 01 38 AM Processor Program X Control amp Mechatronics Lab hot S A Spiewak Oregon Sut Signal Processor VI 05 30 97 01 29 AM 010 9 910 0701 6s0lesere 0 8
3. 1 1 4 Center reference tor gen dimension2 05 30 97 01 31 Reference Gen dimemsioi ost VIBRATIDN NISUALIZATION 37 169 3 Top comer reference for gen dimension 2 vi 05 30 97 01 48 2 lt Top comer reference tor gen dimension 2 vi v 05 30 97 01 48 AM p UYITYITYITEYIIXTITIITITITITYTITITKIITIYI Plate dimension 0 L pos i 1 L neg I 2 W pos j 3 W neg j 4 H pos k 5 H neg k VISUALIZATION 2 pA TION Top comer reference for gen dimension 2 05 30 97 01 48 AM Top comer reference lor gen dimension 2 05 30 97 01 48 AM 2 2223351 tees 4 170 171 1 rr Center reference for dynamometer vi 05 30 97 01 22 AM Page 2 Center reference for dynamometer vi 05 30 97 01 22 AM Plate dimension EON Oynamometer Dimension e ie Base Dimension lt 5 lt 5 gt VIBR
4. 33 An alternative flowchart of the model based vibration visualization 35 LIST FIGURES Continued Figure Page 3 8 Major components of the dynamometer 36 3 9 Main research subjects pertaining to model based visualization 37 3 10 lt A high speed machine 38 311 simplified model of the machine tool from Fig 3 10 with the dynamometer installed 38 3 12 Simplified mechanical model of the dynamometer under consideration 40 4 1 Components of the generalized coordinate list d describing the rigid body motion of a 46 42 Flowchart of the methodology used for the visualization of machine WIDE AMONG AS SARA SESS 47 4 3 An example program virtual instrument in LabVIEW 49 44 Front panels of LabVIEW programs developed for the vibration visualization 50 4 5 A block diagram of the basic data acquisition system used in this research 52 4 6 flowchart of signal processing in vibration visualization 55 47 Diagram of the signal processing procedure 59 4 8 Coordinate systems used in describin
5. 125 Dimensions used in the corner reference drawing option 126 Birgit EeePC RAS 127 E 2 E 3 LIST OF APPENDIX TABLES Page Descriptions of sensors used in the experiment 108 Formats of data used in the visualization programs 128 Systeminfo assignment oc e o P ipu tdm 129 Example of the Systeminfo file used in the experiment 131 LIST OF ABBREVIATIONS C S Coordinate System CFR Characteristic Forced Response DAQ Data Acquisition EMA Experimental Modal Analysis FLL Front Left Lower Corner FLU Front Left Upper Corner Frequency Response Function Front Right Lower Corner FRU Front Right Upper Corner G Graphical Programming Language ICP Integrated Circuit Piezoelectric Infinite Impulse Response MBR Model Based Response MDOF Multi Degree of Freedom MIMO Multi Input Multi Output N DOF N Degrees of Freedom ODS Operational Deflection Shape RLL Rear Left Lower Corner RLU Rear Left Upper Corner Rear Right Lower Corner RRU Rear Right Upper Corner SBR Signal Based Response SISO Single Input Single Output VI Virtual Instrument NOMENCLATURE XYZ global coordinate s
6. 56 4 3 1 Conversion to Physical Units and Subtraction of the Average Value Voltage signals obtained from accelerometers require adjustment to represent the actual acceleration values A factor number needed to obtain these actual values is determined from the frequency response function FRF of the system For the piezoelectric transducer the amplitude of an acceleration acting on the sensor Ao is related to the measured output signal of the sensor Vo as follows see Eq 2 2 1 1 s Hw 4 2 A calculation of these values requires a knowledge of the sensitivity s of each employed accelerometer The sensitivity s is provided by the sensor s manufacturer units of mV g Practically the signal available directly on the output from the transducer is weak An amplifier with gain Gy is therefore included in the data acquisition system to amplify the signal Within measuring frequency range i e the working region in Fig 2 12 the can be assumed to be unity i e 1 Therefore 4 2 can be rewritten as 1 E NN 4 3 Sv where the amplitude of the actual acceleration m s2 V the voltage recorded by the data acquisition DAQ program Volts 5 the accelerometer sensitivity provided by the manufacturer m V g G the gain of the amplifier in the anti aliasing filter and the conversion factor k 9 806 m s2g The voltage recorded b
7. 40 29 3 4 Feasibility Study of the Visualization Enhancemen t 35 TABLE CONTENTS Continued Page 3 5 Modeling of the Dynamometer 37 2 531 Mechanistic eai EE 39 3 9 2 State Space Model a 41 3 5 3 Transfer Function Model odio orto trt to oan 43 3 0 MN S a 44 4 THE SIGNAL BASED VIBRATION VISUALIZATION 45 4 1 Visualization of Machine Vibrations 45 4 1 1 Overview of the Methodology 46 4 1 2 Introduction to LabVIEW Programming Environment 48 4 2 Data Acquisition System Used in Vibration Visualization 52 4 2 1 Anti aliasing Filtering tot ibus ae ipn 53 4 2 2 Attenuation of Noise in a Data Acquisition 53 4 2 3 Data Acquisition Program 2222222 55 4 3 Signal Processing in Vibration Visualization 55 4 3 Conversion to Physical Units and Subtraction of the Avelape Valles Diet veh tiit 56 4 3 2 High pass Filtering and Double Integration Procedure 57 AA Coordinate 5
8. AN ABSTRACT OF THE THESIS Thanat Jitpraphai for the degree of Master of Science in Mechanical Engineering presented on June 11 1997 Title Model Based Visualization of Vibrations in Mechanical Systems Redacted for Privacy Abstract approved Swavik A Spiewak To visualize vibrations in mechanical systems e g machine tools their movements are measured by means of suitable sensors The signals from these sensors are processed and displayed as animated pictures on a computer screen Accelerometers have been chosen as the most suitable sensors for this purpose Their main advantages include small size wide sensitivity range and frequency bandwidth In addition accelerometers measure signals with reference to the Earth so they do not require stable fixtures such as used with cameras or lasers The visualization methodology involves nine accelerometers attached to a mechanical component e g dynamometer s platform Vibration signals were acquired using a data acquisition DAQ system which is controlled by a LabVIEW based program These signals are processed to suppress errors and convert acceleration into generalized coordinate that describes motion of the visualized component as a rigid plate s movement in 3 D space The animation is accomplished by displaying a time series of pictures representing instantaneous position of the plate The animation program employs homogenous coordinate
9. File Edit Operate Controls Windows Text Help k m Eats 5 2 VIZ D THOMAS THANAT TMP DATA 3FC20 6C CTR Channel ass Comer n X direction channel Comer n Y drection channel 3 Comer n Z recton channi J Comer n X drection channel n Y drecton channel n Z drection channel Comer3 in X drection channel b in X direction rad Distance in X drection mm 46000E 0 Distance in Y direction 163 92600 0 gt Distance n Z deection mm 2 09800 0 d 45 1 2 594 64E 533 1969 91E 3 5 3 837 84E E 156 334 95E 10 25 3484 3 9 78 19 15 32 55 25169 21 676 27 01E 1 s ua 3305 3 265 File Edit Operate Controls Windows Text Help Be Be eee nimation rator Program Control amp Mechatronics Lab hathor 1 Titpnplai 2000 total time Data flen graph ane fij 100 0 0 75 0 0 50 0E 0 1 25 0 0 0 0E 0 25 0 0 50 0E 0 75 0 0 100 0 0 125 0 0 150 0E 0 175 0 0 200 0 0 225 0 0 400 0E 0 300 0E 0 200 0 0 100 0 00 0 1000 Rolshit Pitch shift Yaw shift around x around y around 3600 350 0 gt 300 4 2000 2000 0 0
10. 13 plate under investigation is shown Fig 2 8 the Cartesian coordinate system comprising three orthogonal axes and Zg defining a plate coordinate system XYZ g The origin 2 is the center of mass of the plate G Another coordinate system XYZ composed of three perpendicular axes X Y Z is introduced as a global reference coordinate system When the XYZ is fixed with reference to the Earth a motion of the plate be described as a relative movement between the XYZ G and the XYZ Variables required for describing this motion are x the translation of point G relatively to point O parallel to the axis X y the translation of point G relatively to point O parallel to the axis Y z the translation of point G relatively to point O parallel to the axis Z 0 the rotation of the plate around the X axis roll angle the rotation of the plate around the Y axis pitch angle and y the rotation of the plate around the Z axis yaw angle Linear accelerometers are employed in this research to detect linear vibration of the specific points of the plate in the X Y and Z directions The rotations are indirectly calculated from these linear accelerations using a technique proposed by Padgaonkar et al 1975 This technique is further described in Section 4 5 Piezoelectric accelerometers have been chosen because of their reasonable cost and excellent performance However
11. T The signal processing procedure used this research generates linear displacements by double integration of the signals from accelerometers as described in Section 4 3 Therefore the instantaneous translations xr y and z are obtained by applying the signal processing function SPF e Eq 4 8 to the accelerations j i x y and z j 1 2 and 3 It is symbolically written as Xi 4 14 Jr SPF ayc 4 14b Xc SPF azc 4 14c The roll 6 pitch and yaw yj angles are also calculated using displacements obtained by processing signals from the all nine accelerometers on the plate x c 2 ry xyc 2 r 4 14d XxcY 2r x c 2 rx 4 14e Wr 2 xxo 2 r 4 14f where xij SPF aij displacement obtained by double integration of the acceleration a i j the notation for axis i x y and z j 1 2 and 3 corners of the plate where the accelerometers are located 4 6 Animation of the Rigid Body Motion A knowledge of the list of generalized coordinates dy given by Eq 4 10 associated with the instantaneous position of the selected point corner C of the plate allows calculation of the remaining corners in the C S XYZ on the condition that the plate s dimensions are defined in the C S XYZ 65 animation procedure consist
12. VIBRATION VISUALIZA TION 7 Data Acquistion Controller VI 05 30 97 01 20 AM p Data Acquisition Controller VI 05 30 97 01 20 AM Warning Save to for Data saving Show graph for Display data Now displaying the acquired data Re acquinng for New acquasibon Stop program to Out the program rege w a Page 12 VKUALIZATION Z VIBRATION Data Acquisition Controller VI 05 30 97 01 20 AM Data Acquisition Controller VI 05 30 97 01 20 AM TECTITI rage io 140 Data Acquisition Controller VI 05 30 97 01 20 AM CEE III II g 2 16 04 4 True gt i 5 d limit sernir level T 5 device 1 er Data Acquisition Controller VI 05 30 97 01 20 AM The data suze s too large Plotung 4 VISUALIZATION VIDRAT ION iaa 141 VIBRATION VISUALIZATION 10 Data Acquisition Controller VI 05 30 97 01 20 AM
13. the angular velocity component of the vector around the i axis i x Fx Ty r the distances between accelerometers shown in Fig 4 9 63 Figure 4 9 Locations of nine accelerometers required for the calculation of the generalized coordinates Padgaonkar et al 1975 A computation of the linear and angular accelerations and according to equations 4 9a 4 9b and 4 9c requires at minimum six linear accelerometers These accelerometers are represented in Fig 4 9 by gray thick arrows Positive sense of each accelerometer is indicated by the arrow s heads However if only six accelerometers are used it is necessary to solve nonlinear due to the products of the angular velocities differential equations which is time consuming Worse the solution is non unique Padgaonkar 1975 Liu 1976 Padgaonkar 1975 showed that the angular accelerations can alternatively be calculated by adding three more accelerometers mounted on the plate as shown by the solid thick arrows in Fig 4 9 The addition of new equations Eq 4 12c 4 12d and 4 12e leads to the following linear relationships for the angular accelerations due to changes in angles 6 and y in Eq 4 9 az a c 2 r 7 ayc 2 r 4 13a Oy axc 2 r 7 az a c 2 rx 4 3b a 2 7 2 P 2 4 13 where d 2 d 2 d a 180
14. Data Acquisition Controller VI 5 05 30 97 01 20 132 132 Number channel Set total number of channeis that are employed in DAQ system Display Column Select data at specified channel to be displayed edge or siope edge or slope 0 Do not change the default setting default input 1 Leading edge for digital trigger positive slope for analog trigger 1 Trailing edge for digital trigger negative slope for analog trigger time limit sec time limit restricts the ammount of time LabVIEW waits for the trigger to occur The default input is 1 0 which tells LabVIEW not to change the time limit set default setting is 0 seconds no waiting trigger level analog chan and level is only used when trigger type is analog trigger abe trigger ch trigger channel is valid onty when trigger type 1 trigger channel describes the analog channel that is the source of the trigger following the syn described the Channel Addressing section of Chapter 2 Getting Started with the Data Acquisition Vis except for the following cases An empty string telis LabVIEW not to change the trigger source setting An empty string is the default input analog input trigger where x is the channel number Currently x must be 0 You can substitute the string EXTERNAL When Triggre type is 1 analog trigger 5 or 6 trigger source defaults to 0 for analog input channel 0 The following are valid trigger source v
15. Information About the System PPP Signal Processing and Generalized Coordinate Calculation Lees chasse Model Based Response Response Generalized Coordinates Estimated Model Generator Comparison and J Correction Interactive User Corrected Response Interface x Display of Vibrations Figure 3 5 Block diagram of the model based vibration visualization Signal Based Response Generalized Coordinates See Section 3 2 The complete analytical model is represented by the analytical generated structure and the estimated parameters This model is employed for generating the model based 33 response using the stimulation acting on the actual tested system Information from the estimated model is also used in the Comparison and Correction unit to generate suitable feedback commands for tuning of the Signal Processing unit and the Response Generator The comparison and correction unit is also responsible for generating messages and warning to the user A flowchart of the Comparison and Correction unit is shown in Fig 3 6 This unit comprises Data Analysis and Filter Tuning sub unit in which the actual and estimated responses of the system are compared The comparison can be based upon various indicators such as the coherence function Bendit and Piersol 1980 Results from the comparison are used for the validation of th
16. w imo noto 22 of 0006 0 49 0006 90 208 0 100 08 0 20006 0 green A wanana 2 CTR sas 10 aoto M ac 404 240 mots 00083 000083 gt edem 2 2 Veron away 1 mes FEE 2 30 0084 rene 1 1 e mn 0 1 Se cree _ Drame Pan Lang is diri mm Base Pans Lang x de vem Aet um Lor mm a O man y m Pom tan iy mem H n u Baso Mam 1 80 men s rent d m 3 moo H me Dama Lang og LO Langh ong WLO mtm Res ww AOL 2 5 Oye wa HA wan me Opname ru mm n vom bH me regna cen mm 2 4 we Page 2 3 D Animation Generator VI 05 30 97 01 14 AM nanliuuaiaolanadadaaoaad c4opadndaaaadaasnad n nooooonon 40 Make Location Vectors trom center v EN Ls ids 4 Vectors rom comer Locaton Vectors center w VIBRATION VISUALIZATION 31 3 0 Animation Generator VI 05 30 97 01 14 AM fane 3 0 Animation Gener
17. 81 Locations of nine accelerometers Kistler type 8702B25M1 mounted on the dynamomieler eon trac ictus cett secs Erbe dila 83 The data acquisition UR assia 83 Icons and wiring terminals of the major LabVIEW modules employed in the vibration visualization 84 A simplified diagram of the LabVIEW visualization programs using modules shown in ka aoreet eue 86 Definitions of characteristic time instances referred to Table 5 1 and 5 2 89 Graphs show results from the test number 5 sese 93 Graphs show results from the test number 6 sse 94 Graphs show results from the test number 7 2 95 Graphs show results from the test number 8 sess 96 Illustration of the flexible mode of platform s vibration from tle test number S i doo esr 98 LIST OF TABLES Table Page 2 1 Effect of time constant on the error in measuring various transient responses 18 4 1 Coordinates of the corners and equations used for calculations 67 42 Order of corner plotting for creating a complete 3 D rectangular plate 76 5 1 Descriptions of the test procedures used in the experiment 88 5 2 Characteristic locations of the dynamometer s platform obtained experimentally 90 5 3 Description of procedures
18. GCC and AG In response to this signal each module gets data from its predecessor rather than from a disk file Thus a new sample of data is collected processed and visualized per every run of the program When a true signal is applied to the from file line the program requests data files in a spreadsheet compatible format that were partially processed at some earlier time The default value i e no external value applied to the from file line is true i e reading data from disk files A more detailed diagram of the program is shown in Appendix F as the Simple DAC SP GCC The Systeminfo file is wired to systeminfo input terminal of the SP program The Systeminfo file contains information about transducers DAQ parameters and signal processing procedures In addition the user can design any specific signal processing algorithm and code it i e write a script for automatic execution in the Systeminfo file The data management and communication between the modules shown in Fig 5 6 are described in Appendix E Data Acquisition Controller Signal Processor YI eneralized Coordinate 3 D Animation Calculator YI Generator Vl Ez E False Signa Figure 5 6 A simplified diagram of the LabVIEW visualization programs using modules shown in Fig 5 5 87 5 3 Verification of Developed Software Modules Experiments are conducted with both simple and complex dyna
19. I would like to thank to my special person Waranush Sorasuchart for her care and patience during this work Thanks also due to all of my Thai OSU friends who help me in many things TABLE CONTENTS Page CHAPTER 1 INTRODUCTION Sulu zn aici oe bee tea ee 1 1 1 Graphical Representation of Vibration 1 1 2 Scope of W rK 2 1 3 Chapter 3 2 LITERATURE 4 2 1 Visualization in Vibration Analysis 4 2 2 System Identification in Vibration Analysis 7 2 3 Conventional Visualization Software eee 10 2 4 Measurement of Signals for Visualization 12 2 4 1 Piezoelectric Accelerometer Measurement 13 2 4 2 Conventional Low Frequency Accelerometers 20 2 4 3 Multi Directional Accelerometer 24 2 9 CLOSURE 25 3 MODEL BASED VISUALIZATION OF VIBRATIONS 26 3 1 An Overview and Definition of the Proposed Visualization Technique 26 3 2 Signal Based Vibration Visualization 28 3 3 Model Based Vibration Visualization
20. moment acting on the dynamometer s platform i x y and z rp moment caused by the force F acting away from G i x y and z lt x 3 N number of the forward and reverse coefficients respectively O origin of the XYZ n by n zero matrix P arbitrary point on a rigid plate q t vector of state variable Laplace transform of q t r position vector of point P from point C r r distances between accelerometers R reverse coefficient of the digital filter R rotational motion of the dynamometer s base S Laplace variable SP Signal Processor VI LabVIEW program SPF signal processing function 5 accelerometer sensitivity T homogeneous transformation matrix from C S 1 to C S 2 measuring time of the transient response of the piezoelectric accelerometer TC time constant of the miniature amplifier in the transducer time constant of the power supply U s Laplace transform of the input u t u t input vector U s Laplace transform of the measured input u t u t measured input signal V amplifier output voltage V voltage signal recorded by the data acquisition program x z coordinates in a different coordinate system x n element of a sequence of the discrete acceleration signal x n n element of a sequence of the discrete filtered acceleration signal x n element of a sequence o
21. the generalized coordinate of the platform and ds the generalized coordinate of the base plate option 2 D array of 34 rows and n columns n number of channels see Table E 2 Table E 1 Formats of data used in the visualization programs E 2 Systeminfo File A Systeminfo file is a data file created for describing the information of signal data obtained from an experiment signal processing procedures can also be performed automatically using information recorded in the Systeminfo A Systeminfo or a control file contains all information of data in a spreadsheet compatible format in which number of columns of the Systeminfo equals to number of channels used in the One Systeminfo file should be created for each experiment since the parameters used in an experiment setup are unique These parameters include sampling frequency DAQ limits transducer type serial number and sensitivities and sensor s locations etc user can create or edit a Systeminfo using the program Signal Processor VI An assignment of a Systeminfo that is used as a standard format in this research is shown in Table E 2 3 One column is assigned for one channel 129 0 information default 1 Loadcell 2 Proximity sensor 3 Voltage signal 4 Accelerometer Locations of sensors 2 Serial numbers of sensors Sensor s sensitivit 0 No integration default 1 Double integration is applied 2 Single integ
22. 00 00 180 0 lt 1800 LS 1800 0 200 0 0 3000E 0 4000 20 3 00 0 2 3 4 0E 3 0 00E 0 2000 3 40 00E 3 6000 3 8000E3 100 00 3 1200 Horizontal display 4X a 588 121330 1 Vertical display 71 4 22 118 period sec 350 00 6 e Platform column index option for sensor frame for dynamo comer reference J T d 3 D Animation Generator VI Figure 4 4 continued Front panels of LabVIEW programs developed for the vibration visualization 52 4 2 Data Acquisition System Used in Vibration Visualization A standard data acquisition DAQ system comprises the following basic components 1 a controller 2 a signal conditioner 3 a multiplexer and amplifier 4 analog to digital converter ADC 5 a storage unit or a memory unit and 6 a readout device Dally et al 1993 system employed in this research 15 computer based instrument The control part is implemented in LabVIEW program The memory part and the readout device are added as a desktop computer The ADC multiplexer and amplifier are provided in a plug in DAQ printed circuit board type AT MIO16E2 from National Instruments 1994 Low pass filters serve as signal conditioners to prevent signal aliasing A schematic diagram of the employed DAQ system is shown in Fig 4 5 Further details are given in Section 5
23. 149 VISUALIZATON 19 VIBRATION Signal Processor VI 05 30 97 01 29 Signal 05 30 97 01 29 AM ganasaq imi 150 14 2 VIBRATION rage Processing All Data 2 VI 05 30 97 01 40 AM Page 2 Signal Processing Filters Integration ii VI 05 30 97 01 44 151 20 VISUALIZA TION Ie 4 TIOA Signal Processing Filters amp Integration ii VI 05 30 97 01 44 AM D a4 Tros gt gt d Tros 8 m gt m 3 55 Empo Finer vi i o10 DOMINIO er Signal Processing Filters amp Integration ii VI 05 30 97 01 44 AM rage gt FX EX xit wl nnnnaataisgdandgagannnamndm VISUALIZATION 21 VIBRATION Signal Processing Fitters amp Integration ii VI 05 30 97 01 44 AM s 1 O ON Qgmmunnagnunnguna n VIE ATION VISUALIZATION 22 Datanote Changing 3 vi 05 30 97 01 32 AM Filter Controis h
24. XYZ angular velocity component of the vector i x y and z frequency variable 0 natural frequency of an example automobile natural frequency low and high limits of the measuring frequency of the accelerometer mode shape 4 damping coefficient MODEL BASED VISUALIZATION VIBRATIONS IN MECHANICAL SYSTEMS CHAPTER 1 INTRODUCTION 1 1 Graphical Representation of Vibration Graphical representation of vibration aids engineers in analysis of dynamic behavior of mechanical systems Analysts have better understanding of vibration problems by looking at actual movement of components under consideration Unlike the vibration analysis based on the finite element method and modal analysis software the visualization of the actual movement can provide information that is often beyond the estimated display generated by the analytical methods The visualization of actual vibrations in mechanical systems is accomplished by measuring the movement of these systems with suitable sensors Signals from these sensors are processed and displayed as graphical representation of the vibration To measure signals required for the visualization many different methods have been developed such as laser beam scanning schemes fiber optics based sensors vision systems or magnetic sensors The application of these systems is costly and therefore limited Accelerometers on the other hand are effective with
25. and are rearranged to the column form 1 x 4 shown below to be compatible with the homogeneous transformation Xc Ye v Ya D5 4 21 and D 4 22 1 Zc 1 ZA 1 1 Figure 4 13 Application of the homogeneous coordinate transformation for finding coordinates of point A 71 It should be noted that the coordinates of point A defined C S XYZ are converted to its coordinates in the global C S XYZ A location vector V is defined as a vector that contains local coordinates of the corner under consideration in the C S XYZ i e the coordinates with reference to the corner In this example dimensions of the plate are w and h where l the length of the plate measured parallel to the X axis w the of width the plate measured parallel to the Y axis and h the height of the plate measured parallel to the Z axis The location vector V is therefore Ve Li 0 hk 4 23 where i k are the unit vectors of the Y and Z axes respectively The column vector D 1 0 h 11 arranged from the vector V is then used in the claculation of the coordinates of the point A in the global C S XYZ as D T xo c Vc De 4 24 The same technique can be used to obtain the coordinates of the remaining corners The location vectors of the remaining corners are listed in the second column of Table 4 1 A procedure for calcula
26. graph 2 mode sampling freq gt TUI A aop 1 00 100 rage 1 25 VLJALI ZATION TION Read From Spreadsheet File 05 30 97 01 42 AM Read From Spreadsheet File w pattern vi 05 30 97 01 42 AM pattem Gee new file path Not A Path if cancelled all rows file path dialog if empty sei number of rows all 1 132 MM TY x start of read offset 132 4 3 saj 0 1 qnae Read Lines From File w pattem vi first 132 type empty max characters row limit 0 transpose no F 1321 mark after read chars instructions EOF characters If your spreadsheet string uses different separators or terminators use the Search String and Replace f examples generalstrings Ilb library or something equivalent at the output of Read Lines From File to modify the string if properly separated You can modify a copy of this VI to return the file contents into arrays of strings by changing all rows first row and tyr arrays to string arrays and by setting the format to 255 rage 1 5180 2 E BH 157 VIsUALIZATIOM 26 VIBRATION 158 1 aoc Read Lines From File w pattern vi 05 30 97 01 46 AM pea ee 40b fio path Not A Path if ca
27. identified in a straight forward mathematical manner Collins et al 1972 A block diagram representation idea of the parametric identification methods is shown in Fig 2 6 critical examination of the quality of the mode is obtained by a comparison of the system s output with the model s output where the system and the model are both excited by the same input signal The measurable system output consists of an non measurable output signal and the noise signal Iterative procedures e g least squares method Isermann 1981 are performed to find such parameter values which 10 yield the error between the model output and the system output as small as possible Unbehauen 1982 Noise Signal Non measurable Output Signal System Output Input Signal Output of the Model Parameter Values Parameter Adjustment and Adaptation Algorithm Figure 2 6 Parametric Identification Unbehauen 1982 2 3 Commercial Visualization Software The majority of commercial programs in vibration analysis visualize vibrations in mechanical systems by presenting the mode shapes of investigated system These mode shapes are calculated either from an analytical approach or an experiment In practical experimental modal analysis EMA software generates the mode shapes from a transfer function calculated using data from an experiment Once the transfer function of the system of interest is obtained based o
28. k 1 2 m is the number of data The animation is achieved by rapid display of these pictures The above programs can perform their functions individually Alternatively each of them can work as a subprogram that can be run and interfaced from other programs Figure 4 4 shows the front panels these VIs Program descriptions and usage are explained in Chapter 5 and in Appendix D Numeric Output Graph 40 60 Numeric Control Output Graph a front panel b block diagram Figure 4 3 An example program virtual instrument LabVIEW File Operate Controls Windows Text Help a T Entry number of data i _ data number urce orn Contg seve ote ae Fix paint no 6 Re scauring no time last column Edit mmm Windows Text Help Filter type High pass sensor typ EM sampling freq Nec Feubtract numb 0 STOP PROGRAM 22 0086700 00 E qm total ci sensitivity to Systeminfo un stopband high limit volts limit no attenuation dB 5o Wiite to file T EIS 001 002 0 03 0 05 0 06 0 07 010 sona 1200 _ 1400 _ 1600 1800 2000 b Signal Processing double integral ii vi Figure 4 4 Front panels of LabVIEW programs developed for the vibration visualization
29. kx2 kx3 kx4 h 0 kz2 kz3 kz4 a kz4 kz2 kz kz3 kz1 kz2 kz3 kz4 kx1 kx2 kx3 kx4 h kx4 kx3 kx2 kx1 4 kx3 kx2 kx1 b 1 ky2 ky4 a 0 ky4 kyl ky2 ha kx4 kx3 kx2 kx1 h b kyl ky2 ky3 a kx1 kx2 kx3 kx4 b Cu 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 a ay 1 0 0 az 0 0 1 0 a ax 0 0 0 1 J 111 B 2 Constant Parameters in the Spatial Matrices Mass of the platform Chen 1996 m 2 714 kg Mass moment of inertia of the platform around the center of mass G Chen 1996 Ixx 006187217392 kg m 002398407627 kg m2 Izz 008366544210 kg m Ixy 0 kg m Iyz 00007500541682 kg m Ixz 0 kg m Stiffness of the sensing elements Chen 1996 k01 262000 N m k02 1260000 N m k01 kx1 kx2 kx3 kx4 kyl 2 ky3 k02 21 kz2 kz3 kz4 Damping coefficients of the sensing elements Chen 1996 cox 140 N s m coy 125 N s m coz 275 N s m cox cx cx2 4 coy cy cy2 cy3 cy4 coz cz cz2 cz3 cz4 Distances of the sensing elements from the center of mass Chen 1996 a 0 02999 m in X direction b 0 04979 m in direction h 0 01070 m in Z direction 112 Appendix MATLAB Program Used in
30. the visualization technique presented in this thesis Variable capacitance microsensors are designed for a measurement of low frequency acceleration inclination and shock vibration Endevco 1996 They employ a capacitance sensing element that is composed of a very small seismic mass chemically etched from a single piece of silicon and positioned between two electrode plates As the seismic mass deflects under accelerations the capacitance between these plates changes With a signal conditioning circuit position of the sensing mass is detected and converted to an output voltage To prevent excessive shock motion the sensing mass is damped by a gas damper resulting in an increase in the measurement bandwidth from DC to 2000 Hz depending on the sensitivity Kistler 1995 Fig 2 14a shows example acceleration signal acquired by a capacitive accelerometer Kistler 8302B2S1 Kistler 1995 mounted on a rigid object This object is Slowly moved from one position to another 7 8 mm away The initial portion of the signal 0 0 09 sec representing an output signal of the stationary accelerometer is used to calculate the value of its DC offset This value is next subtracted from the signal and the resultant is integrated twice to obtain the velocity and displacement measured by the sensor These latter signals are shown in Fig 2 14b An inspection of the measured and actual displacement over the period shown in the figure indicates a suitabil
31. 066 2 066 750 0 00 0 01 0 02 0 03 0 04 0 05 0 05 Time sec Signal Based Generated Response pu Time Signal Based Generated Response Model Based Generated Response 0 Model Based Generated Response Figure 5 11 Graphs show results from the test number 8 From the test no 5 and 6 The response signals from both methods represented movement of the platform in the same direction Except in the Y Z and Row directions of 97 the test no 5 see Fig 5 8 no signal is generated from the model solid lines Since the model of the dynamometer is calculated under an assumption of perfect geometric structure e g locations of the sensing elements are symmetry and the mass distribution is uniform Thus the center of mass of the platform is almost the same as the center of the configuration Therefore when the input force is assigned to hit the platform in the negative X direction by arranging Cu and u matrices as shown in Table 5 3 the calculation of the model based response then provide a pure rotation around Z axis without any translation in the Y and Z directions By comparing the characteristic of the signals in the test no 5 and 6 obviously the signal based responses dashed lines contained higher frequency component than the model based responses solid lines from the analysis of periods of the signals This behavior leads to an assumption that the stiffness coefficients of the actual syste
32. 19 986 fates See Load cel Fore Table A 1 Descriptions of sensors used in the experiment 2 Power Spectrum of Signal from Accelerometer Magnitude dB 120 00 140 00 a e e 8 180 00 200 00 a frequency Figure 1 Power spectrum density of signal measured from an accelerometer that is attached to the dynamometer when no vibration applied Piezoelectric accelerometer shear type Kistler model 8702B25M1 Kistler 1996 2 Load cell piezoelectric force transducer PCB model 208B03 PCB 1996 109 A 3 Dimensions of the Dynamometer gt 169 93 Corner 3 comers lt Center of Mass G ps 11 40 units mm ZG Figure A 2 Dimensions of the dynamometer used in the experiment in units of mm the sensing elements are not shown 4 Digital High pass Filter s Coefficients filter type IIR Filter Cluster sos IR 2nd Oner Highpass filter structure IR 2nd Order onder MB Reverse Coefficients Foward Coefficients 5 i Figure A 3 Digital high pass filter s coefficients used in Eq 4 4 and Eq 4 6 110 Parameters of the Dynamometer s Model B 1 Spatial Matrices of the Dynamometer These matrices are employed in the spatial model of the dynamometer Eq 3 10 and Eq 3 11 Chen 1996
33. 200 0E 9 100 0E 9 20 5E24 100 069 200 0 9 300 0E 9 400 0E 9 500 ID 0 02 Signa Based Generated Response Time sec displacement m Model Based Generated Response 0 00 0 01 0 02 0 03 0 04 Signd Based Generated Response Time sec Model Based Generated Response jn Ya ZR Figure 5 9 Graphs show results from the test number 6 IK displacement m 95 Roll angle rad 80 0 6 60 026 400 6 20 0E 6 0 0E 0 20 0E6 Time sec Pitch angle rad 80 0E 6 40 056 20 0E 6 6 8E 21 20 0E6 40 0E6 00 066 80 0E 6 100 06 6 0 00 0 01 0 02 0 03 0 04 0 05 Signal Based Generated Response Model Based Generated Response a aw angle rad Figure 5 10 Graphs show results from the test number 7 96 displacement Roll angle rad 0 02 0 03 0 04 01 0 02 0 03 0 04 0 05 se 0 05 Signa Based Generated Response Ey Signal Based Generated 2 Model Based Generated Response Model Based Generated Response displacement Pitch angle rad 0 03 0 04 0 05 Signal Based Generated Response E Signal Based Generated Response PE Model Based Generated Response Based Generated Response displacement m 2 066 1 066 0 060 1
34. 581 Warning signal limits input limits Is an array of clusters Each array element assigns the limits for the channels specified by the corresponding element of channels If there are elements in this array than in channels the VI uses the last array element for the rest of the channels The input limits array defaults an empty array wh the input limits keep their default settings Each cluster contains the following parameters lduster input limits an array of clusters of which each array element specifies the range limits for the channel s in the corresponding elemer channels array If there are fewer elements this array than the number of channels the VI uses the last element for the rest of the channels default input is an empty array which means the input limits do not change from their default settings Each cluster contains the following parar sgi high limit specifies the maximum voltage the board measures a particular channel 501 low limit specifies the minimum voltage the board measures at particular channel 136 VIBRATION VISUALIZATION 5 rage Data Acquistion Controller VI lx 05 30 97 01 20 AM SEI high limit 10V high limit specifies the maximum voltage the board measures at a particular channel Sci low limit 10 low limit specifies the minimum voltage the board measures at a particular channel trigger level analog chan and level is only used when
35. D Animation Generator VI 11 12 MED Se As che Pewer delay s Roll shift Pitch shift Y aw shift around x around y around State Unive 360 0 360 0 S 360 0 fj total data displayed slation magnify together 2 200 0 200 0 Boas 63 109 00 00 total time display at 7180 0 1800 1800 if EA pr rere pss J pee 0 00 0 2000 3 4000E 3 6000 3 8000E 3 100 00 3 120 00 2931 6 Horizontal display ss8sE 2133E IS Vertical dieptay 7 4822 period sec 00 8 option for sensor frame s peciiced or name comer reference 1 a 6 Platform Dimension Base Dimensip _ Reference Dimension 17 Dynamo Plate Length x Ain mm Base PlatelLenath xdi mm N Plate Length di mm Dynamo Plate dir mm Base Plate Width dir mm Reff Plate Width dir mm 159 A 50000 Dynamo Plate Height 2 4 mm Base Plate Height 2 dir mm Ratt Plate Height 2 dir mm 22 10 24 N Dynamo Length from origirtdLO Bagh Length from origin bLORmm L ogth from origin 54n 2 8 9 Figure D 4 Front panel of the 3 D animation generator program AG 3 124 Before running the program Select from file switch to y
36. K kx 1 kx2 kx3 kx4 0 0 0 kx 1 kx2 kx3 kx4 h kx4 kx3 kx2 kx 1 b O ky 1 ky2 ky3 ky4 0 ky 1 ky2 ky3 ky4 h 0 ky 1 ky2 ky3 ky4 a 0 0 kz1 kz2 kz3 kz4 kz1 kz2 kz3 kz4 b kz1 kz2 kz3 kz4 a 0 0 ky 1 ky2 ky3 ky4 h kz1 kz2 kz3 kz4 b kz1 kz2 kz3 kz4 b 2 ky l ky2 ky3 ky4 h 2 kz4 kz2 kz1 kz3 b a ky4 ky3 ky 1 ky2 h a kx 1 kx2 kx3 kx4 h 0 kz1 kz2 kz3 kz4 a kz4 kz2 kz1 kz3 b a kz1 kz2 kz3 kz4 a 2 kx 1 kx2 kx3 kx4 h 2 kx4 kx3 kx2 kx 1 h b kx4 kx3 kx2 kx 1 b ky 1 ky2 ky3 ky4 a 0 ky4 ky3 ky 1 Ky2 h a kx4 kx3 kx2 kx 1 h b ky l ky2 ky3 ky4 a 2 kx 1 kx2 kx3 kx4 b 2 Identity and Zero Matrices 16 6 6 3 3 O6 zeros 6 O3 zeros 3 Cu 13 O3 0 1 az ay az 0 1 ax 1 ay ax 0 13 STATE SPACE MATRICES ssA O6 16 1 inv M K D inv M C ssB 06 inv M Cu ssC 16 06 550 06 SIGNAL CONDITIONING Remove Constant Offset u dtrend ul dtrend y1 X yY dtrend ylY yZ dtrend y1Z yTH dtrend y1TH dtrend y1PH yPS dtrend y1PS Time Scale deltaT 1 smplFreq sampling interval sec sampleTime dataSize deltaT total time of sampling sec the following vector time defines sampling instances
37. Systeminto filename Data filename Generalized coordinate output Systeminfo output q in X direction rad q in Y direction rad q in Z direction rad Systerninfo filename Data filename Data input Systeminfo input Channel assignment rx Distance in X direction ty Distance in Y direction Distance in 2 direction From file Write to file Jm Generalized Coordinate Calculator VI c Write to lt lt select signal portion Data filename Systeminfo filename Data input Systeminfo input Data filename Systeminfo filename Data output SP Systeminfo output TS graph 1 mode graph 1 graph 2 mode generated limits Signal limits emm iiie graph 2 finish program T ves From file Create new Systeminfo Attach siqnal limits to Sys Signal Processor YI b period sec Systeminfo filename Data filename Generalized coordinate input Systeminfo input magnification Platform column index Base 8 column index run 1 time angle shift pitch yaw i Platform Dimension Base Dimension Reference Dimension 3 D Animation Systeminto filename Data filename 3D graph mm adynamo 3D output return total data 1 4 Figure 5 5 Icons and wiring terminals of the major LabVIEW modules employed in vibration visualization 1 Data Acquisition Controller VI henceforth
38. based and the model based responses also occurred due to distortion of signals especially in data that were calculated from weak signals Furthermore the governed model was developed under the assumptions of perfect geometric structure and mass distribution which is not what the real system is Further development is required to improve the model to be close to the actual system 100 6 CONCLUSIONS RECOMMENDATIONS 6 1 Conclusions The algorithm of visualization of vibrations in mechanical systems using sensors with model based enhancement technique is proposed and partially developed in this thesis The first stage designated the signal based vibration visualization has been implemented as computer programs written in LabVIEW Within these programs signals from sensors attached to the mechanical systems are acquired processed and presented in 3 D graphical animation representing the actual vibrations As contribution part the second stage designated the model based vibration visualization is emphasized on the improvement of the proposed visualization programs The underlined concepts involved in this stage are outlined in the thesis that can be summarized as the use of mathematical models of the mechanical systems to detect and correct errors in the signal based vibration visualization scheme The developed visualization programs were verified in the experiments governing an experimental model o
39. component of the coordinates that is on the same axis as the viewpoint e g the component xc is neglected as the eye is assigned on the X axis in Fig 4 15 Rybaczyk 1989 73 Projected 2 D picture representing the object on the screen Viewpoint Figure 4 15 Illustration of the projection of a 3 D object a 2 D planer 4 6 4 Changing the Viewpoint Vibration visualization can be enhanced by employing viewpoint changing feature The method is accomplished using an additional coordinate transformation technique Discussed in this section are techniques used in rotating a viewpoint around the object and creating an effect of a perspective that distorts the dimensions such that the object size is reduced with increasing distance from the viewpoint Thus when a viewer looks at the picture on the screen he feels like looking at an actual object in 3 D environment Rogers and Adams 1990 Mathematically a rotation of the object in one direction relative to the eye results in the same way on the computer screen as the rotation of the eye in a negative direction relative to the object Since the viewpoint i e the eye and the projection plane are fixed together with the computer screen If a user wants to rotate the plate by certain angles 6y and yy around the X Y and 2 axes respectively the calculation of new coordinates on the screen is performed by transforming the coordinate system of
40. considered as constant matrices For zero initial conditions a new set of equations is obtained s Q s A Q s B U s 3 17a De s C Q s 3 17b where Q s is the Laplace transform of the state vector q t Dg s is the Laplace transform of the generalized coordinates vector 1 By applying rules of linear algebra the above equations can be converted into the following input output form 5 Dc s U s 5 1 2 12 A B 3 18 The above equation encapsulates the relationship between the input vector external forces and moments and the output vector generalized coordinates in the transfer function matrix Gg There are several methods to generate responses to actual input signals by using the transfer function model of the dynamometer One approach that is conceptually simple involves the following equation d L Ge s Um s 3 19 where U s Laplace transform of the actual input signal u t dm t generated model based response L inverse Laplace transform operator Commercial software packages such as Matlab or Mathematica allow quick and easy calculation of these responses 44 state space model of the dynamometer presented above 15 used Section 5 4 to generate the response vector dm The input Um is obtained using an actual force signal from an impact hammer in the experiment The calculated and recorded responses are compared and their differences are dis
41. denoted as DAC This module controls the DAQ card to acquire analog signals with desired parameters and also serves as a readout instrument The parameters that can be set are listed at the end of Section 5 1 A front panel of the module is shown in Fig 4 4a and the corresponding icon used in the Block Diagram in LabVIEW is shown in Fig 5 54 85 2 Signal Processor henceforth denoted as SP This program performs filtering and double integration of the acquired signals using the procedures described in Section 4 3 2 A front panel of the program is shown in Fig 4 4b and its corresponding icon is shown in Fig 5 5b The program also converts voltage signals to the units of acceleration and force and subtracts the average values of these signals using the calculation method described in Section 4 3 1 Digital filters are applied twice before and after the first signal integration to attenuate drift in the acceleration signals Parameters of these filters are adjustable All results shown in this thesis were obtained with the following setting e High pass 8th order elliptic filters e Cut off frequency fc 20 Hz 3 Generalized Coordinate Calculator VI henceforth denoted as GCC The program calculates generalized coordinates of the corner point C using the equations described in Section 4 5 Front panel of the program is shown in Fig 4 4c and its corresponding icon is shown in Fig 5 5c
42. detected and eliminated by comparing quesi static values of acceleration signals obtained by these two essentially different approaches One significant advantage of this method is the elimination of high pass filters see Section 4 3 2 which as an effect introduce distortion into the signals recorded from the sensors The above discussion of the algorithms facilitating enhanced visualization of vibrations in mechanical systems addressed only the main aspects of this complex problem There are numerous ways to implement these algorithms and broad spectrum of additional signal processing methods that can be applicable More research is needed to find the most reliable and efficient methods 35 Input Mechanical System Information About the System 0 Vibrations gt DAN D Measured Output x Model Structure System Identification Identification x Signal Based Response Model Based Response Accelerations Response Accelerationa Y H and Feedback orrection Command Interactive Corrected Response User Signal Processing and Interface Generalized Coordinate Calculation Display of eee Vian PE Figure 3 7 An alternative flowchart of the model based vibration visualization 3 4 Feasibility Study of the Visualization Enhancement An application of the mathematical model is investigated to generate responses of a mechanical system s
43. distortions can be clearly visible in signals having broad frequency contents such as a squarewaves or impulses It goes without saying that high pass filters are acceptable only for visualizing vibrations of objects with a frequency higher than the selected cut off frequency Acceleration m s 005 0 06 a Acceleration 0 04 0 05 0 0 b Velocity Displacement m 15 0E6 10 0 6 5 6 0 06 0 5 0E6 0 00 0 01 0 02 0 03 0 04 0 05 0 07 c Displacement 2 4 2 Conventional Low Frequency Accelerometers 0 07 20 DOD time sec time sec time sec 008 Figure 2 13 Illustration of the impact of a low frequency drift on the displacement obtained from acceleration signal Solid lines in figures b and c represent integrated and double integrated signals Dashed lines represent the same signals after using digital high pass filters see Section 4 3 2 The application of piezoelectric accelerometers can cause serious difficulties when measuring low frequency vibrations Possible errors include drift and noise as discussed above A significant improvement can be achieved by using high performance accelerometers In this section various types of accelerometers which have recently been developed with a purpose of improving the low frequency performance are 21 reviewed These new accelerometers recommended for application
44. graphite accelerometer cable and taping or gluing the cable down to a surface as close to the accelerometer as possible Br el amp Kj r 1982 4 2 3 Data Acquisition Program A controller program for data acquisition DAQ is required to read data from transducers for further use Controller VI written in the LabVIEW s G language is an interface program between the user and a DAQ board AT MIO 16E2 National Instrument 1995 employed for the digitization of signals The user is able to command the board to acquire analog voltage signals with desired parameters The user can easily inspect the acquired signals select suitable sampling parameters for these signals and acquire data again A front panel of this program is shown in Fig 4 4a 4 3 Signal Processing in Vibration Visualization Signal processing procedure transforms signals acquired from nine accelerometers for each rigid plate referred to in Section 4 1 into the generalized coordinate list dc in Eq 4 1 The signal processing steps are summerized in Fig 4 6 Section 4 3 1 Subtracting the Average or Moving average Values Filtering with Digital High pass Filter First Numerical Integration AE Signals NER Filtering with Digital High pass Filter Second Numerical Integration Section 4 3 2 Figure 4 6 A flowchart of signal processing in vibration visualization
45. high end of the frequency bandwidth is limited by the natural frequency of the transducer An estimated values of this limit are in the range of 0 1 to 0 3 Dally et al 1993 Br el amp Kj r 1982 At low frequency the drop of gain is mainly due to the amplifier characteristics which define the discharge rate of the piezoelectric sensor Concisely expressed by the discharge time constant of the amplifier which has properties of a high pass filter This time constant is used to calculate lower limit of the bandwidth For practical applications 5 error the minimal measuring frequency is oy 3 TC 7 Kail and Mahr 1984 Kistler 1995 Therefore the overall measurement bandwidth fn is 3 TC lt lt Hz 2 3 Magitade 4 50 IH 0 a i Working region 2 Pi GO 107 107 10 10 10 10 10 10 Frequency rad sec Figure 2 12 Magnitude plot of the FRF of a piezoelectric accelerometer McConnell 1995 18 Application of piezoelectric accelerometers to measure transient vibrations requires additional consideration Transient signals may contain DC signal components that are strongly attenuated as shown in Fig 2 12 The result can be a drift in the response signal The amplifier s time constant must be considered in the error analysis in such cases Table 2 1 presents the time constant necessary to limit the error in measuring various transien
46. m m 00 0 0 00 m 00 OO 0 m 0 0 0 0 0 xx Ixy Ixz 0 0 0 Ixy Iyz 0 0 0 Ixz 2 Izz c cx4 cxl 2 cx3 0 0 0 cx4 2 cx3 cx3 1 cx4 2 b 0 cy2 0 cy4 h 0 cy2 cyl cy3 a 0 0 cz4 21 cz2 cz3 cz4 cz3 cz2 cz1 b cz4 cz2 cz cz3 a 0 0 cy4 h cz4 cz3 cz2_cz1 b cz4 21 cz2 cz3 b 4 cy1 cy3 h cz2 cz3 cz4 21 cy3 cyl 2 haa cx4 1 2 cx3 h 0 cz4 cz2 21 cz3 a cz2 cz3 cz4 cz ba cz4 cz cz2 cz3 4 exl cx2 cx3 hb cx3 cx1 cx4 cx2 h b cx3 cx cx4 cx2 cy2 cy3 a 0 cy3 cyl 2 4 ha cx3 cx cx4 cx2 h b cyl 2 cy3 4 1 cx2 cx3 b kx1 kx2 kx3 kx4 0 0 0 kx1 kx2 kx3 kx4 kx4 kx3 kx2 kx1 b 0 kyl ky2 4 0 ky1 ky2 ky3 4 h 0 kyl ky2 ky3 4 a 0 0 kz1 Kz2 kz3 kz4 kz1 kz2 kz3 kz4 b kzl kz2 kz3 kz4 a 0 0 ky1 ky2 ky4 kz1 kz2 kz3 kz4 b kz1 kz2 kz3 kz4 b ky2 ky3 ky4 kz4 kz2 kz1 kz3 ky4 kyl ky2 h a 1
47. reasonable cost They also provide advantages including small size wide sensitivity range and frequency bandwidth In addition accelerometers measure the signals with reference to the Earth so they do not require stable fixtures such as used with cameras or lasers Therefore the accelerometers have been chosen as the most suitable sensors in this research Algorithms have been developed and implemented as programs for the visualization purpose In the course of this project numerous problems have emerged associated with the use of accelerometers These problems centered around errors measured signals These errors were attributed either to the environment or characteristics of the sensors themselves A suitable method for suppressing the errors has been proposed in this research Despite the precautions taken to suppress errors in the signals used for visualization there is no guarantee that the displayed motion accurately represents the actual behavior of tested system To improve the reliability of visualization a methodology have been proposed that allows the detection and correction of errors This methodology involves concepts developed in control theory as well as analytical models of systems whose vibrations are visualized 1 2 Scope of Work The research discussed in this thesis addresses two major areas and has two objectives The first objective is the development of a visualization program capable of acquiring ac
48. signal reaches the first peak after the initial contact of the impact hammer with the platform In general the characteristic force responses generated by the tested program agree with the expected behavior of the tested dynamic system see Table 5 1 Impact force N 0 50 0 0 40 0 0 30 0E 0 20 0E 0 10 0E 0 0 0 0 10 0 0 1 1 D 1 1 1 1 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 60 Data Number Vibratory Response m 2 0 6 0 0 0 1 0E 6 2 0E 6 3 0E 6 0E 6 1 1 1 1 1 1 1 1 1 1 1 i i 1 400 410 420 430 440 450 460 470 480 490 50 510 520 530 540 550 560 570 580 590 Data Number Initia Position Impacting Position Characteristic Forced Response Figure 5 7 Definitions of characteristic time instances referred to in Table 5 1 and 5 2 90 Characteristic Forced Response Magnification factors H H 10 Initial Position Characteristic Forced Response Magnification factors 10 Initial Position Characteristic Forced Response Magnification factors H H 107 Initial Position Characteristic Forced Response Magnification factors H H 10 Table 5 2 Characteristic locations of the dynamometer s platform obtained experimentally and H are magnificatio
49. signals that are acquired when the system is not excited Power spectra of these signals clearly indicate the range of low frequency errors and aid in choosing suitable cut off frequencies Recursive digital filters type infinite impulse response IIR are employed Because of sharp drop of the frequency response function FRF at the cut off frequency Elliptic filters are selected National Instruments 1994 They are described by the following formula 1 ej Np 1 x i al SF xli j R xli al i 0 1 2 1 4 4 29 ici where the n element of a sequence of the discrete input signal acceleration x n the element of a sequence of the filtered output signal acceleration R the k reverse coefficient 58 the j forward coefficient Nr the number of the reverse coefficients the number of the forward coefficients m the number of data i e number of elements in x and x The number and values of the reverse and forward coefficients depend on types and orders of filters After removing low frequency noise from acceleration signals a numerical integration National Instruments 1994 is performed to calculate velocities The integral x i of an input time series signal 1 is calculated as National Instruments 1994 x i 1 4x x i 1 Ar i 0 1 2 m 1 4 5 j 0 where x n the n element of a sequence of the discrete acceleration
50. sin Zresp i 4 cos Zresp i 4 0 0 0 0 1 T34 0 0 0 Zresp i 1 0 0 0 Zresp i 2 0 0 0 Zresp i 3 0 0 0 1 DGC ICG wCG hCG 1 DOC T01 T12 T23 T34 DGC for 1 3 ZofC ij DOC j end 116 for 4 6 ZofC i j Zresp i j end end Spread Output into Separate Array initial array genX array dataSize 1 Before Transform for i 1 dataSize genGX i 1 Zresp i 1 genGY i 1 Zresp i 2 genGZ i 1 Zresp i 3 genGTH i 1 Zresp i 4 genGPH i 1 Zresp i 5 genGPS i 1 Zresp i 6 end After Transform for i 1 dataSize genX6 1 ZofC i 1 genY 1 1 2ZofC 1 2 genZ i 1 ZofC i 3 genTH i 1 2ZofC 1 4 genPH i 1 ZofC i 5 genPS i 1 ZofC i 6 PLOT SIGNAL BASED AND MODEL BASED RESPONSES IN COMPARISON TranGain 1 RotGain 1 ForceGain 0 plot time 1 PDSZ ForceGain u 1 PDSZ y time 1 PDSZ genX 1 PDSZ m time 1 PDSZ TranGain yX 1 PDSZ g xlabel Time sec title Simulated solid and Actual dash in X direction disp Simulated solid and Actual dash Response in X direction plotted Hit any key to continue text dataSize 1 150 deltaT 0 name pause plot time 1 PDSZ ForceGain u 1 PDSZ y time 1 PDSZ gen Y 1 PDSZ m time 1 PDSZ TranGain yY 1 PDS2 g xlabel Time sec title Simulated solid and Actual dash
51. the piezoelectric sensors are charge generators that require special amplifiers Without proper amplifiers these sensors can significantly distort the motion measurement Details are discussed in the following section 2 4 1 Piezoelectric Accelerometer Measurement Piezoelectric transducers utilize piezoelectric effects occurring in certain crystals The deformations of these crystals due to an applied pressure produce electrical charges on the external crystal surfaces These effects occur in materials such as single crystal 14 quartz polycrystalline barium titanate lead zirconate Kial and Mahr 1984 piezoelectric accelerometer consists of a piezoelectric sensing element sandwiched between the transducer s body and a seismic mass as shown in Fig 2 9 By mounting the transducer on a surface of a vibrating object internal forces from the seismic mass cause deformations in the sensing element which in term produce an electrical charge Dally et al 1993 Threaded Hole for Attachment a Compression type b Shear type Figure 2 9 Configurations of piezoelectric accelerometers including a compression type and b shear type Dally et al 1993 Since the output of the piezoelectric accelerometers is a high impedance charge suitable amplifier circuits are required to obtain low impedance output Charge amplifiers are the most popular instruments used for this purpose because of their advantages i
52. z lj sin 0 kz bsin 0 a Lumped parameter system b Free body diagram Figure 2 2 2 DOF model of an automobile Thomson 1993 The automobile s dynamics is simplified such that it can be described by a lumped mass that moves only in a vertical direction and rotates around its center of mass Therefore this simplified automobile requires two coordinates z and 6 to described its motion or it has two degrees of freedom 2 DOF To define z and 0 coordinates completely reference axis of position and orientation is required As shown in Fig 2 2b the system is reduced to a free body diagram FBD in which only external forces are concerned The reference point is chosen at the center of mass Any vertical translation from the reference position is described by z and a rotation around the center of mass is defined by 0 as shown in the FBD To illustrate the mode shape constant parameters are chosen to represent the automobile s properties These parameters include m 3220 lb k 2400 Ib ft k 2600 Ib ft l 4 5 ft L 5 5 ft and J 51520 lb ft By assuming that the responses of the model have the forms z t Z e and 0 t O e the natural frequencies are calculated as 6 90 rad sec and w 9 06 rad sec The corresponding mode shapes are calculated Thomson 1993 uz 6 m 6 _ ra T The mode shapes can be symbolically represented as shown in Fig 2 3 9 06 ra
53. 1 Signal Couplers and Amplifiers Kistler 5128A PIEZOTRON coupler Impact Force and Accelerometers Kistler 1996 Kistler 8702B25MI Kistler 1996 1 Anti aliasing Filters Precision 88B Precision 1989 and Datel FLJ D6LA2 Datel 1987 ADC Multiplexers Amplifiers National Instrument 16 2 National Instruments 1995 LabVIEW DAQ program Data Acquisition version 2 6 s vi Figure 4 5 A block diagram of the basic data acquisition system used in this research 53 4 2 1 Anti aliasing Filtering The aliasing phenomenon can occur when converting an analog signal to a digital data with sampling frequency lower than the minimum required rate To prevent this phenomenon a minimum sampling rate should be set at least twice of the highest frequency component of the measured signal or the Nyquist frequency f Practically the sampling frequency fs is set to be 2 56 f Doebelin 1990 Anti aliasing filters are employed to assure the aliasing prevention regardless the actual spectral contents of recorded signals They are analog low pass filters with the high cut off frequency equal to f which reduce signal components that may produce aliasing to an insignificant level Anti aliasing filters are useful for primary signal conditioning However these devices can severely distort transient signals with high frequency components The sampling rate should be c
54. 1s calculated as dc xc zc vcl 4 15 where xc wo lo Qc zc ho H zr and Vc Wo H Wi 66 flowchart of the procedure used to calculate the list of absolute generalized coordinates is shown in Fig 4 10 Summation Figure 4 10 A flowchart of the procedure calculating the list of absolute generalized coordinates 4 6 2 Calculating Coordinates of the Plate Corners To create a wireframe image of the plate for visualization the coordinates of its corners are required Since one of the corners designated C in Fig 4 8 coincides with the origin of the C S XYZ only 7 remaining corners need to be calculated For the sake of clarity each corner of the plate is designated by a symbol shown in Fig 4 11 and further described in Table 4 1 The homogeneous coordinate transformation Wolovich 1987 is employed to calculate coordinates of these corner points The transformation and its application are described in this section Figure 4 11 Abbreviations used to designate the plate s corners Corner DE Front Left Lower FLL DP T DE Rear Right Upper RRU DIF T4 DRY Rear Right Lower RRL T Rear Left Upper RLU T T denotes the homogeneous transformation matrix Eq 4 20 Table 4 1 Coordinates of the corners and equations used for calculations H
55. 4 3 D Animation Generator VI henceforth denoted as AG The animation program creates and displays a wire frame picture of the moving plate in the 3 D space Front panel of the program is shown in Fig 4 4d and its corresponding icon is shown in Fig 5 5d Methodology used in the program is described in Section 4 6 Dimensions and positions of the dynamometer s platform base and reference plate used in the animation are listed in Appendix A The four software modules discussed above can be used as major building blocks in various specific programs needed for the visualization of vibrations As shown in Fig 5 5 each module is represented by an icon with wiring terminals The input and output variables of these modules are represented by the terminals and signals wired to the icons Different appearance of wires represents different data types according to the standard LabVIEW designations For example a single line represents a simple 86 numeric data type while pair of parallel lines represents 2D array data type Further discussions of these modules are briefly discussed in Appendix D A simplified diagram of the example visualization program comprising the modules discussed above is shown in Fig 5 6 When the program is executed the sequence of actions can be considered as moving from left to the right A false dotted line signal shown at the bottom of the diagram is wired to the from file terminals of SP
56. 4 19 Z 5607 op 000 1 a Z Z a Yaw Transformation Pitch Transformation X X4 c Roll Transformation T 0 d Translation Transformation T x 2 Figure 4 12 Definition of coordinate transformation matrices Wolovich 1987 The definitions of operations accounted for by these matrices are graphically shown in Fig 4 12 succession of coordinate transformations each represented by particular transformation matrix can be represented by a matrix product of individual transformation matrices Wolovich 1987 Therefore the overall transformation from the first coordinate system XYZ o to the final coordinate system XYZ is obtained using the following matrix Ty x y z 0 0 W T x y z T T 9 Ty v 4 20 70 As already mentioned a position of the point C in the C S XYZ is obtained directly from the measured signals This position is described by the vector xc yc zc that links the point C and the point in the global C S XYZ shown in Fig 4 13 Hence these values in V are given by the list of adjusted absolute generalized coordinates dc described by Eq 4 15 Homogeneous coordinate transformation technique is applied to find coordinates of the remaining corners For example to calculate the coordinates of corner A whose coordinate vector is V x yas ZA as shown Fig 4 13 the following steps are performed First vectors V
57. 9 4 5 Calculation of the Generalized Coordinates from Experimental Data 61 4 6 Animation of the Rigid Body Motion 64 4 6 1 Finding Absolute Position of the Reference Corner Point 65 4 6 2 Calculating Coordinates of the Plate Corners 66 4 6 3 Projection of 3 D Object to a Planer 2 D Screen 71 4 6 4 Changing the Viewpoint 73 4 6 5 Drawing a Single 3 D Picture 75 TABLE OF CONTENTS Continued Page 4 6 6 Animation of Generated 3 D Pictures 76 4 7 Transformation of Generalized Coordinates in a Rigid Plate 78 LEGE PIU Te DP qp E 79 CHAPTER 5 EXPERIMENTAL IMPLEMENTATION AND RESULTS 80 5 1 Experimental Set 80 5 2 Data Acquisition and Vibration Visualization Software 84 5 3 Verification of the Developed Software Modules 87 5 3 1 Experimental Procedure 87 5 3 2 Results and Discussion 88 5 4 Comparison of Signal Based and Model Based Responses 91 5 4 1 Experimenta
58. 9 cont1 Location of Impacting Point From Center of Mass ax 37 325 ay 68 141 11 40 Inputs are kx1 ky1 kz4 01 262 1073 02 1260 1 1073 01 2 01 3 01 4 01 01 2 01 ky3 k01 4 01 21 02 22 02 23 02 kz4 k02 Location of Sensing Elements a 2 999 0A 2 b 4 979 10A 2 hz1 07 10 2 Damping Coefficients cox 140 coy 125 coz 275 cx l cox 2 cx3 cox cx4 cox cy l coy cy2 coy cy3 coy cy4 coy 21 2 cz2 coz 23 2 cz coz M m 0 0 0 0 0 0 m 0 0 0 0 0 0 m 0 0 0 0 0 0 1 2 0 0 0 1 2 0 0 0 1 2 1 2 1221 4 1 2 3 0 0 0 cx4 cx1 cx2 cx3 h cx3 cx 1 cx4 cx2 b 0 cy4 cy 1l cy2 cy3 0 cy4 cy3 cy2 cy 1 h 0 cy4 cy2 cy 1 cy3 a 0 0 24 21 22 23 24 23 22 21 24 22 21 73 0 0 4 3 2 1 24 cz3 cz2 cz1 b cz4 cez1 cz2 cz3 b 2 cy4 cy 1 cy2 cy3 h 2 cz2 cz3 cz4 cz1 b a cy3 cy l cy2 cy4 h a cx4 cx 1 cx2 cx3 h 0 cz4 cz2 cz1 cz3 a cz2 cz3 cz4 cz1 b a cz4 cz1 cez2 cz3 a 2 cx4 cx 1 cx2 cx3 h 2 cx3 cx 1 cx4 cx2 h b cx3 cx1 cx4 cx2 b cy4 cy2 cy l cy3 a 0 cy3 cy l cy2 cy4 h a cx3 cx 1 cx4 cx2 h b cy4 cy 1 cy2 cy3 a 2 cx4 cx 1 2 3 2
59. A v 1 87 9 Comerd in X direction channel v 2 98 46000 0 a 9 Y 3 ry Dist in Y direction mm in Y direction rad 0 ection Q channel 4 0 F From file Write to wie Y 4 169 92600 0 v0 2219 YES YES Comer in X direction channel 5 2 rz Dsstance Z direction mm in Z arecton rad Comer in Z direction channel Systeminfo input 22 098006 0 0 44 816 12 40 400 2 in Y direction channel y 7 sze 1 N Comer2 in Z direction channel 8 30 a output Comers in X direction channel 9 size 2 a 0 10 Comerd in Y direction 0 channel 10 6 2048 Data input Generalized coordinate output 0 400 4000 4000 4000 4000 400 400 4000 13 27E 1 8 57 12 147 85 18769 221E9 184 81E PO 859 12 339706 39469 44269 466232 000 0 00 000 000 000 0 00 000 000 000 000 000 000 000 000 000 000 0 00 000 000 000 000 000 000 00 000 000 000 000 000 000 000 000 000 000 000 000 000 00 3000 2000 A a a a a a 000 000 000 000 00 000 000 2000 445E 12 594 64 533 19 5 91E 9 6 08E 9 837 84 14 31 1 10 35E 1 730 25 797 9 7 48 9 1 31 9 41 96 1 15 16E 1 934 95 1025E 9 8 496 9 1 88E 9 78 78 1 19 18E 1 1 15 9 12 55 9 901 9 2 51 9 121 67E 27 01 1 1 36 9 14 66 9 9 05 9 3 20E 9 Generalized Coordinate Calculator VI 05 30 97 02 10 AM
60. ATION Center reference for dynamometer 05 30 97 01 22 AM JC YIY IT EW TTYITIT TTIiRSTITWIT TITI ITI 2 52040 070 a2 Dt Center reference for dynamometer vi 05 30 97 01 22 AM 2 Reterence Dimension ___ FL Cec 6114 CEL TION VISUALIZATION 40 Page 4 VISUALIZATITI 41 VIBRATION 173 Page 1 Make Location Vectors from center vi S 05 30 97 01 41 General dimensions ost 1 Locational vectors Plate dimension 1 Make Location Vectors from Comer vi 05 30 97 01 47 Plate dimension Locational vectors os VISUALIZATION 42 TON Draw a 05 30 97 01 36 AM Transtormed point ost Draw box vi 05 30 97 01 36 AM 030 0 04070 8 6820085018 58 808 84 DONEREN 20167058 10 058 OCOC LOOTA 174 VISUALIZATION 43 VIBRATION rage Draw a box vi gt 05 30 97 01 36 Horizontal 18 Dr
61. F forces acting on the platform M t M torques applied on the platform around and 26 respectively Dr t Xr Yr Zr translational motions of the dynamometer s base and Rr t rotational motions of the dynamometer s base The output vector dg is a set of variables describing position and orientation of the rigid platform This set is called a generalized coordinate list dg that is defined as follows de xyz 0 3 4 where the x y 2 0 ware defined in Section 2 4 Under simplifying assumption that the dynamometer s base is fixed and Dr and Rr are equal to zero the input vector in Eq 3 3 reduces to a 6x1 vector u F 3 5 5 Definitions of axes are introduced in Section 2 4 40 XG Center of Mass G Figure 3 12 Simplified mechanical mode of the dynamometer under consideration Chung 1993 When the workpiece is being machined an external force F t is acting on the platform as shown in Fig 3 12 The acting point of the force is located at a ig ayjg where ig jg and kg are unit vectors of the Xg respectively The quantities and a are signed numbers measured from point along the and 2 directions The external force contains three components F and F which also generate three associated moments with respect to the center of gra
62. IBRATIONS The background and essential characteristics of the model based vibration visualization are presented in this Chapter Algorithms implemented in this thesis and an outline of future work are discussed The model based vibration visualization comprises two stages The first stage a signal based vibration visualization is developed in this thesis The second stage proposed as the future work is an improvement of the above visualization by using the system s mathematical model An experimental model of commercial dynamometer is selected as a mechanical system to be visualized Since a mathematical model of this dynamometer is employed for generating the estimated responses to applied excitation signals a technique used to obtain this latter model is reviewed at the end of this chapter The same model is also used to interpret experimental results in Chapter 5 3 1 Overview and Definition of the Proposed Visualization Technique The proposed visualization program has been developed to use real signals from sensors attached to vibrating rigid bodies for generating animated motion of these bodies Together with the acquired signals a mathematical model of the dynamic system comprising the bodies is also used to detect and suppress errors in the acquired signals The entire visualization scheme is implemented in two stages The first stage referred to as signal based vibration visualization is the focal point of this resear
63. M transducer from Kistler involves a pair of cantilever beams made of a piezoelectric ceramic material as shown in Fig 2 17 The beams are sensing elements that are constructed in a T shape By comparing the phase difference of the electrical charge signals from both beams the linear and angular accelerations can be calculated Kistler 1996 Figure 2 17 A cross section of the PiezoBEAM accelerometer Dally et al 1993 25 2 5 Closure Visualization of vibration theoretical analysis is introduced The mode shapes from the theoretical vibration analysis provide graphical representation of coordinates relative motion in the system at certain frequencies The visualization technique is proposed in which actual movement is detected experimentally using accelerometers By studying characteristics of the piezoelectric accelerometer measurement system problems in the visualization method are presented as limitation of measurement at low frequency caused by discharge of output voltage Higher performance accelerometers are reviewed and suggested for improvement direction In the next chapter concepts of the proposed visualization technique are described These concepts include the use of sensors in the signal based vibration visualization and the use of a mathematical model to improve the visualization in the model based vibration visualization 26 CHAPTER 3 MODEL BASED VISUALIZATION OF V
64. O system To transform the equation of motion into the state space form Eq 3 11 is first rearranged as follows d m k d m c d m Fe 3 12 Next by introducing an auxiliary vector dgp such that da d be written as Peters and Mergeay 1976 q A q B u 3 13 dg C q D u 3 14 where dc the vector containing 6 generalized coordinates q the vector of state variables defined as q de dg 3 15 dynamic matrix control matrix contain elements of the matrices m and k from the equation of motion model Eq 3 11 Their form is 6 3 168 m c 1 Je 3 16b m where I is a 6x6 identity matrix and is a 6x6 zero matrix The observation matrix C and the direct transmission matrix D represent relationships between d and q as well as d and u respectively If the output of the state space equation is desired to be the generalized coordinates vector dc the form of matrix C is given by Eq 3 16c Since no direct relationship exists between the generalized coordinates and the input vector u the matrix D is a zero matrix C 16 3 16 D O 3 16d 43 3 5 3 Transfer Function Model The state space model can be transformed into the transfer function model by taking Laplace transformation of Eq 3 13 and 3 14 Under the assumption of the linearity and time invariance the matrices A B C and D are
65. P 2 a 979 Ry j 0 Wxel Az 1 Az0Y 2 R1 H Av3 AvOV 27 R31 TD W Wet AxO AxOV 2 Rz3 AZ2 AzOV 2 P2 159 VISUALIZATION 23 VIBA TION 160 Generalized Coordinate Calculator VI 05 30 97 02 10 AM O S NEG TIT 61018 0 010 8 0 8 838 80040 In Seve ons Data w nou Channel assignment 5 Rat form s dimensions Data 2841089 18 Generalized Coordinate Calculator 05 30 97 02 10 AM 29 VIBRATION VIGUALIZA TION Generalized Coordinate Calculator VI 05 30 97 02 10 AM OOO Generalized Coordinate Calculator VI 05 30 97 02 10 AM Generalized coordinate output E you to save data please save n COO 161 VIZUALIZA TON 30 VIBRATION Page 1 3 D Animation Generator VI 05 30 97 01 14 AM aa
66. able in this research It divides dynamic systems based on their features relevant to parameter identification as shown in Fig 3 3 Shadowed boxes indicate the path applicable in this research SISO Time invariant MIMO Stochastic Dynamic SISO systems Time varying Time invariant SISO Single Input Single Output MIMO Multi Input Multi Output Time varying MIMO SISO SISO LEEK MIMO Figure 3 3 Classification of dynamic systems Chung 1991 30 system that has several input signals that stimulate responses several outputs as shown diagrammatically in Fig 3 4 is termed The multi input multi output MIMO system Input Output System u t G 5 Figure 3 4 system Dynamic mechanical systems under consideration MDOF systems and also MIMO systems which can be described by the following vector metric equation Chung 1993 Y s G s U s 3 1 where Y s Laplace transform of the output vector y t i e a vector whose elements are all output signals U s Laplace transform of the input vector u t and G s the transfer function matrix of the system The matrix G s has nxm elements 5 where i 1 2 n andj 1 2 m Each of these elements is a rational polynomial representing a transfer function between input j and output for a system with m inputs and n outputs It is usually difficult to
67. acterized followed by an introductory discussion of the proposed visualization technique Since piezoelectric accelerometers have been used to measure vibrations of system in this research the design and performance of these accelerometers 15 described Finally the latest advances in relevant sensor technology i e low frequency accelerometers and multi directional sensors are reviewed 2 1 Visualization in Vibration Analysis Vibration analysis is a study of dynamic behavior of a system As a rule the analysis involves an attempt to define mechanical systems by means of mathematical models The systems are usually described by one of the following model forms a spatial model a modal model and a response model Ewins 1984 These three forms are related to each other and one form can be transformed into another as shown in Fig 2 1 The spatial model describes the system based on its physical parameters including masses stiffness k and damping coefficients c These properties are usually formed in the equation of motion The modal model is obtained by applying free vibration analysis to the spatial model which results in the information given by a set of natural frequencies damping ratios and corresponding mode shapes y Natural frequencies 0 Mode shapes V Damping ratio 6 Figure 2 1 Equivalent forms of a mathematical model used in vibration analysis By applying forced vibration an
68. alu n where n is an analog input channel number PFIO E series set trigger channel to PFIO trig1 in this program Sui trigger level level is the voltage value the analog source must cross for a trigger to occur You must also specify whether level must be crossed on a leading c with the edge or slope input The default input for level is 0 0 V Time to acquire sec Enter time to be acquire Program calculates sampling frequency by Sampling treq Number of data includes pre trig time to acquire Page 6 Data Acquisition Controller Vl DAC 05 30 97 01 20 AM B H BB B m 4 s 5 save option Users can select to save data with time information attached to last column using time first column If this option is selected program creates a column with equal size as acquired data in which this additional column contains time values gradually increa according to real period in measurement This column is attached after las signa column pre trig number is set time will begin with negative values before time 0 sec Fix point no Number of digits to the right of the decimal point of saved data remote T read only one time y precision Number of digits to the right of the decimal point of saved data y tormat x precision Number of digits to the right of the decimal point of saved data x format limit setting current limit acquired data
69. alysis to either the modal or the spatial model response model is obtained This model describes the response of the system when exposed to an external excitation The response is usually expressed in a standard form as the system s response to a unit amplitude sinusoidal force applied to each point on the system individually and at every frequency within a specified range Therefore the response model consists of a set of frequency response functions FRFs Ewins 1984 In the visualization aspect the vibration analysis is applied obtained and interpreted the movements of an excited system The motion of a system usually does not occur in one direction but involves various translations rotations as well as deflection Such motion is referred to as having more than one degree of freedom A multi degree of freedom system MDOF requires more than one coordinate to describe its dynamic motion If a system requires N coordinates to characterize its motion such the system is termed N degrees of freedom system or briefly an N DOF system Mode shape W is a set of relative amplitudes of the coordinates at certain frequency of vibration It describes how one coordinate behaves relatively to the others An N DOF system has N natural frequencies with N corresponding mode shapes Thomson 1993 An example of the mode shape representation applied to an automobile is shown in Fig 2 2 Equilibrium position k
70. arefully selected for a specific high frequency measurement to prevent distortion on one hand and aliasing of the signal on the other Doebelin 1990 4 2 2 Attenuation of Noise a Data Acquisition The noise introduced into measured signals by imperfect sensors causes distortion on the data used for visualization of motion A suitable hardware configuration of DAQ can the decrease level of the noise This section provides information about sources of signal disturbances and some techniques to eliminate them Drift is defined as gradual changes in the sensor s output or in the amplifier offset voltage and bias current resulting from temperature time and line voltage that can be minimized by keeping temperature and or line voltage nearly constant The effect of drift can also be reduced using an ac coupling acting as a high pass filter or compensating the drift with automatic zero balancing techniques or subtracting average in digital systems 54 Doebelin 1990 Cable shielding grounding design within the instruments should also be taken into account since inductive pickup electrostatic pickup and ground loops can cause large error voltage Any conductor carrying alternative current generates magnetic field that can produce interfering voltages to signal wires by the inductive pickup These effects can be reduced by removing equipment such as power lines motors and transformers from the neighborhood of sensitive signa
71. atic errors Angular components of the generalized coordinate list dg are then calculated from the processed displacement signals using equations proposed by Padgaonkar et al 1975 as further explained in Section 4 5 Finally a sequence of generalized coordinate lists representing consecutive snap shots is generated for the animation of motion The animation procedure uses the homogeneous coordinate transformation Wolovich 1987 to draw 3 D wireframe pictures that present the consecutive positions of visualized solid Rapid displaying of pictures generated from each data point 6 coordinates creates the effect of animated motion Detailed discussion of animation is presented in Section 4 6 Data Acquisition LabVIEW DAQ Controller Section 4 2 Signals Transducers Signal Processing subtracting moving average conversion to physical units e digital high pass filtering Section 4 3 Selection of Transducers Idealization and Decomposition Simplified Generalized Coordinate Calculation Rigid Body application of formulae given by Model Structure of the Eq 4 10 Idealized System Section 4 5 3 D Animation Section 4 6 Figure 4 2 Flowchart of the methodology used for the visualization of machine vibrations 48 4 1 2 Introduction to LabVIEW Programming Environment LabVIEW is a program development application that
72. ator Vl 05 30 97 01 14 AM 163 3 0 Animation Generator VI Paqe 14 3 D Animation Generator VI 05 30 97 01 14 VIBRATION VISUALIZATION 32 35 VISUALIZA TION 165 rage 1 Changing coordinate using transtormation x 05 30 97 01 33 AM I 2009 um UE 41 pat P i H Page 2 Changing coordinate using transformation 05 30 97 01 33 AM n degree vet jew Cornate Array weathed iocaton agoncrEannnnna nnumnunnumnnr s anmzrnumramunnmnuunamaaocun VIBRATION VISUALIZATION 34 166 Page 3 Changing coordinate using transformation 05 30 97 01 33 AM Page 4 Changing coordinate using transformation 05 30 97 01 33 VISUALIZA TION VIBRA TION 167 Page 1 Center reference for gen dimension2 vi 05 30 97 01 31 139 37 2 Center reference for gen dimension2 05 30 97 01 31 AM Dynamometer Dimension unit factor 1000 Plate dimension wu ns W pos j ost W neg j H pos H neg 56 VISUALIZA IB ATION 168 Page 3 Center reference for gen dimension2 vi 05 30 97 01 31
73. aw box vi 05 30 97 01 36 AM 2 VIQJALIATION 44 VIB A TION Draw axes vi 05 30 97 01 39 AM Draw axes vi 05 30 97 01 39 AM LW Changing coordinate using transformation shift angle degree Shift p ft posit ost i COE idi kd 176 ERATION VISUALIZATION AG 177 rage 3 Draw axes vi 05 30 97 01 39 T ITU YT IT YT TT EL 2858 Page 5 Draw axes vi 05 30 97 01 39
74. by a signal from the impact hammer 83 Piezoelectric Accelerometers 7d KISTLER 8702B25MI X Y 7 Platform of the Dynamometer ai the acceleration at the corner j the i direction where i C 1 2 and 3 j x y and z see Section 4 4 Minus sign indicates an inverted signal as compared with the values in equation 4 10 Figure 5 3 Locations of nine accelerometers Kistler type 8702B25M1 mounted on the dynamometer The symbols indicate accelerations assigned according to Eq 4 10 Data Acquisition DAQ Card Signal Conditioner for Load Cell id National Instrument PCB 408D06 rane AT MIO 16E2 LabVIEW DAQ Controller Program Anti aliasing Filters Precision 88B and Datel FLJ D6LA2 Kistler type 5128A Coupler Used with Accelerometers Figure 5 4 The data acquisition system 84 5 2 Data Acquisition Vibration Visualization Software Four major program modules written in LabVIEW have been developed as stated in Section 4 1 2 A brief discussion of these modules follows time limit sec Type of DAG Pretrigger data number Number of channel Entry number of data scan rate scans sec limit setting trigger level pre setting Power Save to file remote T read only one time Real scans sec acquired data signal limits a trigger level Data Acquisition Controller VI a
75. celeration signals that represent vibrations These signals are processed to obtain three dimensional movements of mechanical systems and displayed as graphical animation representing the vibrations This objective is accomplished and presented in this thesis The second objective is an enhancement of the above visualization program by utilizing information encapsulated in analytical models of investigated systems to detect and eliminate errors of visualization The work in this area is only outlined here and recommended as the future development However a preliminary experiment was conducted in this thesis to study the feasibility of using analytical models for the purpose of visualization enhancement The visualization technique and software developed the thesis apply to a rigid body motion In particular the motion of a plate suspended on four three dimensional springs is investigated This plate is one component of a multi component force sensor dynamometer A model of this dynamometer is known from previous research Chung 1993 Chen 1996 and is ready to be used in a comparative experiment for the feasibility study of the model based enhancement of visualization 1 3 Chapter Overview Visualization by means of vibration analysis is introduced in Chapter 2 System identification technique is also described since it will be used as part of the model development necessary for enhancing the visualization program Basic
76. ch A suitable methodology discussed in detail in chapter 4 facilitates visualization of vibrations of rigid bodies by using signals from acceleration sensors attached at precisely defined locations of these bodies Set of data collected from these sensors are referred to as 27 signal based responses SBR Typically conducting one experiment generates SBR In the second stage of the visualization each specific SBR is analyzed to detect possible errors in recorded signals by using a mathematical model of the system under consideration This is accomplished by comparing the SBR with a hypothetical response which is generated by stimulating the model with the actual input signals that acted on the physical system These generated hypothetical output signals are referred to as model based responses MBR In addition to detecting errors e g drift in signals from the actual sensors the knowledge of mathematical model facilitates significant suppression of these Thus the obtained enhanced visualization is referred to as the model based vibration visualization Fig 3 1 shows in the form of a flowchart the essential units of the developed visualization and software Stage 1 Signal Based Response Comparison Corrected and Display of Correction Vibrations Measured Model Based Vibrations in Sienal Mechanical AS Signal Based System Vibration Visualization Math
77. component list do whose first three components are coordinates of the origin of the Reference C S given in the Global Reference C S the three remaining components are angles that define the spatial rotation 61 of the Reference C S with respect to the Global Reference C S entire list do 15 defined as do lo wo ho Wo 4 9 4 5 Calculation of the Generalized Coordinates from Experimental Data The shape of objects whose motion is to be visualized imposes constrains upon the mounting locations of the accelerometers In the case under consideration i e plate accelerometers can not be easily mounted at the center of mass Since they can be easily mounted at the corners of the plate see Fig 5 3 it is advantageous to redefine the generalized coordinates describing the instantaneous spatial location of the plate The generalized coordinates introduced in Chapter 3 i e dg that are defined at the center of mass of the plate are now transformed to new coordinates that are defined at one specific comer point C of the plate The first three components of the new list of generalized coordinates dy define the position of point in the reference C S XYZ g Three positions are designated y and The remaining three components define orientation of the plate with respect to XYZ g These later components are designated and Thus the entire new list of generalized co
78. coordinates of point C dc Eq 4 15 calculated from the measured signals represent location of the plate s corner To obtain the generalized coordinates of the center of mass G in Fig 4 1 the translation variables of C are transformed using the following homogeneous transformation matrix D Ty tc zo Vc De 1 4 26 where D acolumn vector see Eq 4 27 defining a vector from the point C to the point G where this vector is wg j D he 11 4 27 a case of a rigid plate with uniform distribution the center of mass coincides with the plate s geometric center Therefore can be obtained from the plate dimensions as lg 1 2 w 2 and h 2 The rotation variables of the generalized coordinate list dg at point G are equal to those of d the plate is a rigid body Therefore dg becomes dc xc We 4 28 the other hand generalized coordinates of point be calculated from the generalized coordinates of the point G as follows T xc zo v c DC 4 29 where le he 1 4 30 Compatible with homogeneous transformation 79 4 8 Closure To generate an animation of 3 pictures from actual signals measured from sensors four procedures are employed The first procedure controls both hardware and software during data acquisition A
79. ctual dash Response in PSI rotation plotted Hit any key to continue text dataSize 1 150 deltaT O name pause Bode Plot Zo Bode ssA ssB ssC ssD title Bode Plot of Dynamometer model Xz Fz text dataSize 1 150 deltaT O name tosave yX y Y yZ genX genY genZ u save MDNERZA COD tosave ascii 118 Appendix D Descriptions of the LabVIEW Visualization Programs Type of DAQ E Le Ea P VEU PRAE dE Da i 2 FIS Acquire again no error code bource Group Contig saveto fie trol amp Mechatronics Lab save cption nthor 1 Titpraphai No time last Stop program Figure D 1 Front panel of Data Acquisition Controller program DAC 119 Pre acquisition Mode user can set all parameters e g sampling frequency and number of data before applying these parameters to the program DAC Press Setting button 1 if wanting to activate a presetting mode Program does not acquire data but display acquisition time 13 and sampling frequency 14 calculated from specified DAQ parameters Determine suitable DAQ parameters number of data sampling frequency DAQ type and trigger conditions and release APPLY button to begin the acquisition ram DAC reads data When the p
80. cussed Matlab program used to generate response of the dynamometer s model to the actual excitation is listed in Appendix C 3 6 Closure The model based vibration visualization proposed in this research is performed in two stages The first stage is the signal based visualization in which vibrations of a mechanical system are measured and recorded by means of accelerometers and presented in 3 D graphical animation In the second stage the mathematical model of the visualized system is used to detect errors that can occur in the first stage A MIMO model of the actual system is applied to generate estimated responses to the actual excitation signals The system identification technique is used to assure high fidelity of the model A dynamometer is used as an example dynamic system Its model developed in previous researches Chung 1993 Chen 1996 is employed This model obtained by means of Lagrange s method treats the actual dynamometer as a linear time invariant MIMO system Details of a methodology developed for the signal based vibration visualization applied to a rigid plate are explained in the next chapter The application of this methodology and its experimental evaluation are presented in Chapter 5 45 CHAPTER 4 THE SIGNAL BASED VIBRATION VISUALIZATION The underlying theory and algorithms employed in the signal based vibration visualization are presented in this chapter Convenient coordinate systems and se
81. d s Node a Vibration at and b Vibration at and Figure 2 3 Vibrating lumped system two different mode shapes Thomson 1993 2 2 System Identification in Vibration Analysis Theoretically estimated parameters m c k in the model are usually not adequate to describe an actual system Some parameters are difficult to identify such as the damping coefficients and stiffness of the system An efficient method is required to estimate the model parameters which are as close to the actual system as possible System identification provides means to obtain such parameters for the purpose of developing mathematical models which describe the static and dynamic behavior of systems in a sufficiently accurate manner Unbehauen 1982 In system identification a model equivalent of the system of interest is determined based upon an input and an output of the system as shown in Fig 2 4 Natke 1982 Excitation Response Physical System Identification Procedure The System s Model Figure 2 4 Obtaining the system s model by means of an identification procedure Natke 1982 Output utput Algorithms used in the system identification can be classified as shown in Fig 2 5 Mainly determination of the model can be performed either in the frequency domain or in the time domain In the frequency domain the estimation of model parameters is based on fitting the model frequency response
82. e acquired data and diagnosis of likely errors Signal Based Response SBR i Response Display of Vibrations Control and Results Data Analysis and Filter Tuning Interactive User Interface Model Based Response MBR Estimated Model Structure and Parameters 2 yasaya TES Information about the System Feedback Command To Signal Processing unit and Response Generator Figure 3 6 Block diagram of the Comparison and Correction unit 3 For example a force impulse from an impact hammer 34 For example if the results show discrepancy between both responses above a certain pre set level the control unit informs the user about the incompatibility of the data via the Interactive User Interface The user may insist to display the response by sending a command to the Data Analysis and Filter Tuning unit Relevant correction methods may be employed for example subtracting mean values of the estimated errors from the signal based responses or re processing the signals using new parameters set in the Feedback Command An alternative method for the visualization improvement is to compare directly the acceleration signals acquired from the sensors with acceleration responses calculated from the model A block diagram corresponding to this better method is shown in Fig 3 7 The drift embedded in the signals can be
83. e coordinate vector of the same point A in C S pl Perspective Rotation D Matrix Matrix 4770 Figure 4 17 Diagram of coordinate transformation procedure for changing the viewpoint 4 6 5 Drawing a Single 3 D Picture To create a complete wireframe picture of the plate its edges are plotted on X Y graph display object provided by LabVIEW These edges are the lines connecting the corners defined in the rotated C S XYZ y To accomplish this the set of experimental data is arranged as 2 D array that consists of three columns representing xc yc and with 8 rows representing 8 corners for each column After projecting these corner s coordinates on the 2 D screen see Section 4 6 3 the projected coordinates are arranged in the order such that consecutive plot of these corners with lines connecting between two successive points creating a structure of a rectangular plate without any line passing through the plate s volume Table 4 2 shows the order of the plotted corners designed in 76 the visualization program that results as wireframe diagram of the plate shown Fig 4 11 Plotting order FRE Table 4 2 Order of corner plotting for creating a complete 3 D rectangular plate 4 6 6 Animation of Generated 3 D Pictures To animate vibrations of the plate one generalized coordinate list are selected from a series of the generalized coordinates format
84. e electrodes When subjected to an acceleration the cantilever tip deflects thus changes tunnel current between the electrodes The feedback circuit then controls the electrostatic voltage such that it returns to its initial position The variations in voltage required to maintain constant distance between electrodes indicates the sensor s acceleration This voltage is free from hysteresis and drift and insensitive to temperature The FRF cover the range from DC up to 5000 Hz Rockstad et al 1992 Another approach of the low frequency measurement is the use of a piezoresistive element to detect the motion of the seismic mass These elements are mounted on a beam that suspends the micromachined silicon seismic mass as shown in Fig 2 16 Bending of the beam caused by the accelerated mass yields a strain that is detected by the piezoresistor With a bridge circuit the changes in resistance are converted to an output voltage 24 Piezoresistive Elements Silicon Cap Seismic Mass Silicon Figure 2 16 cross section of the piezoresistive accelerometer 1997 2 4 3 Multi Directional Accelerometer To describe a rigid body motion in 6 DOF at least six linear accelerometers are required see Section 2 4 The number of transducers can be minimized by using multi axis accelerometers resulting in a reduction of equipment cost PiezoBEA
85. e number identifying an error A value of 0 means no error a negative value means an error and a positive value is Appendix A Error Codes for a code description Page 8 fm Data Acquisition Controller VI DAC 05 30 97 01 20 AM aes Signal source source shows where an error occurred The source string is usually the name of the VI that produced the error Graph Color Code 1 Light green color program is reading data 2 Dark blue color acquired data is plotted on graph Users can zoom and re scale graph using palette below Total time used in acquisition Realtime sec Ga Real freq scans sec Sampling frequency used in acquisition 137 VIBRATION VISUALIZATION 6 Data Acquisition Controller VI 05 30 97 01 20 AM Data Acquisition Controller VI 05 30 97 01 20 AM CYTITITITITITITITITECITITITI B S IT IT IT TT TD PS to begn acqushon Waw gt gt number of data r 7 fi emote read only one ime Visible remote T reed one me Visible Display monitor data apture agen Vebi L atis Power 2030101070 4 ITITITITITITITIITTITITITITITITITITITITITIT
86. ed to the experimental model of a Kistler dynamometer type 9257A Kistler 1996 using an impact hammer Several acceleration signals the system s responses that represented vibrations were recorded and processed to obtain animated motion of the model Instruments were set up according to the schematic diagram shown in Fig 5 1 The Impact Hammer shown in Fig 5 2 was employed as an exciter to stimulate the dynamometer The hammer comprised of a Piezoelectric Load Cell PCB model 208B03 PCB 1996 attached at the hammer s head and a Signal Conditioner type 408006 PCB 1996 The signal from the load cell was also used as a trigger to initiate the data acquisition 10 Vibration oscillatory motion is intuitively understood as a repeating change of position However it can also be described in terms of the first and second time derivatives of this position i e velocity and acceleration respectively 81 Impact Hammer with Piezoelectric Load Cell Signal Conditioner Tri Star Desktop Computer Controller Program Accelerometers Dynamometer Accelerometer Anti aliasing Coupler Filters Interface Panel Figure 5 1 Schematic diagram of the experimental setup Load Cell piezoelectric force transducer type 208B03 Figure 5 2 Impact hammer PCB type 208B03 used for exciting the dynamometer 82 Nine shear type low impedance piezoelectric Accele
87. educed Suitable software should be selected as the final application program for implementing the visualization with consideration of efficient time memory management and cost are considered The software programs under the consideration include OS 2 GPF based on C program and Visual Basic 103 BIBLIOGRAPHY Bendat J S and Piersol G Random Data Analysis and Measurement Procedures New York John Wiley amp Sons Inc 1985 Br el amp Kj r Measuring Vibration Available from Br el amp Kjeer Instruments Inc 1982 Chen Ben Computer aided Derivation of State Space Models for Linear Dynamic Systems Diss Oregon State U 1996 Chung Y L Adaptive Compensation of Dynamic Characteristics of In process Sensors Diss U of Wisconsin Madison 1991 Model Based Adaptive Compensation of Dynamic Characteristics of In process Sensors Diss U of Wisconsin Madison 1993 Collins R et al Methods and Applications of System Identification in Shock and Vibration System Identification of Vibration Structures Mathematical Models from Test Data Ed Plikey W D and Cohen R New York ASME 1972 Dally J W Riley W F and McConnell K G Instrumentation for Engineering Measurement 2nd ed New York John Wiley amp Sons Inc 1993 Datel FLJ D5 D6 Digital programmable High order Low pass Filter Product Data Sheet Available from Datel 1987 Doebelin E O Measuremen
88. ematical Model Response of the System Model Based Vibration Visualization Figure 3 1 Flowchart of model based visualization of vibrations 28 3 2 Signal Based Vibration Visualization A flowchart of the methodology used in the signal based vibration visualization is illustrated in Fig 3 2 Vibrations of a mechanical system are detected by means of accelerometers attached to the system The acceleration signals acquired using a data acquisition DAQ system are processed in two steps to obtain generalized coordinates2 These steps are designated in the figure as the Signal Processing and Generalized Coordinate Calculation Finally the processed data are transformed into three dimensional 3 D animated pictures representing the vibrations of the mechanical system Vibrations Data Generalized of Acquisition Signal Coordinate 3 D Mechanical System Processing Calculation Animation System Figure 3 2 A flowchart representation of the methodology developed for the signal based vibration visualization The major purpose of the signal processing stage is the calculation of variables necessary for describing spatial motion of the system from measured data This is accomplished by double integration of the acceleration signals to obtain translational and rotational displacement variables required for the visualization An implementation of the integration is difficult since various errors distort the measu
89. er 2 in the direction Column 6 corner 2 in the Z direction Column 7 x 3 corner 3 in the X direction Column 0 corner 3 in the direction Table E 3 shows an example of the Systeminfo file that is designed for the 131 TTE 9 986 a 5 as a o 5 N 2 N 20609 4 kiss 0000 20000 0000 10000 20 01 60 Ko 100 10 0 5 Er n E EXE Eel ed EE T S Ds EHE Table E 3 Example of the Systeminfo file used in the experiment Appendix F Block Diagrams of the LabVIEW Visualization Programs 132 VIJALIZARN VIBRATION Simple DAC SP GCC AG VI 05 30 97 02 19 Simple DAC SP GCC AG 05 30 97 02 19 AM detect ponor T remote enn Premog 050 Number ol chanig Entry number of de UD raye 1 UAL Las Come in Y channel Comer n 2 recton channel ER Comerd m X dwecbon channel eman anan o genera cooranaies osu automate anmavor program ws rage Data Acquisition Controller VI 05 30 97 02 22 AM 1 Scan datapoinvsec 4 Contin
90. ertical bar 15 Data in the generalized coordinate are displayed on the graph 16 The displayed column can be selected using the column selection 17 The moving bar 16a shows the currently displayed data 18 Color of the plates and the axes on the 3 D graph can be selected 20 Control length of three reference axes X Y and Z Press Power button to off to finish the program 11 16 125 Figures D 5 and D 6 show definitions of dimensions used for generating wireframe plates that are displayed on the program s 3 D graph The pictures shown in Fig D 5 are generated under drawing option of center reference option whereas the corner reference drawing option is defined in Fig D 6 Y X C C Center of Configuration Center of Top Surface Figure D 5 Dimensions used in the center reference drawing option 126 Gi RHo Base Plate C C Center of Configuration 2 Center of Top Surface Figure D 6 Dimensions used in the corner reference drawing option 127 Data Management the LabVIEW Visualization Programs E 1 Types of Data Since many processes have been performed to transform the acceleration data into the generalized coordinates explanations of the data types and their meaning are provided in this Section Figure E 1 shows transformation
91. es to read data from a spreadsheet file Specify drawing option Two types of drawing are available 1 Center reference option Program animates plate s vibrations using generalized coordinate data of the center of mass G that refers vibrations of C S XYZ g to the global C S XYZ Origin of reference axes XYZ is at a center of a top surface of the reference plate with axis of the XYZ are drawn Fig D 5 2 Specify for dynamometer corner reference Program animates plate s vibrations using the generalized coordinate data of the corner point C that refers vibrations C S XYZ to the global reference C S XYZ Axes of the XYZ are drawn with its origin at the corner Specify horizontal axis and vertical axis of the viewing screen whose the 3 D object is projected on when no rotation of a viewpoint is applied reference coordinate system XYZ B reference coordinate system XYZ referring to reference coordinate system XYZ Run the program AG 10 Set magnification factors of translations H and rotations H to be displayed on the 3 D graph Specify animation speed Delay control is used to set a delay time between two consequent plots in the animation Step control is used to specify number of pictures to be skipped in the animation 15 Total data number and current data number are displayed Percentage of the displayed data is also shown graphically in the v
92. estimate the exact transfer function G s of an actual physical system Therefore an Equivalent System is often introduced which retains the major properties of the original system but at the same time has a transfer function model that can be experimentally estimated If this latter transfer function Gg s is obtained by 31 means of a modeling technique the estimated response yg t can be calculated from the input u t acting on the original physical system and the transfer function of the equivalent system Gg using the following relationship ye t 171 Ge s U s 3 2 where L denotes the inverse Laplace transform operator Since the model based vibration visualization employs the estimated response yr to compare with the signal based response see Section 3 2 the accuracy of yg is of utmost importance Thus high accuracy of the system s modeling and estimation are necessary This can be achieved by applying accurate system identification techniques introduced in Section 2 2 A structure of the model i e form of equations is derived such that it captures the essential physical laws governing the system s behavior Parameters in this model structure are estimated from the input and the output signals The estimation procedure can be performed in real time which means that the identification takes place at the same time as the system s vibrations are measured The model based vibration visualization is an e
93. f a dynamometer As a result the animated pictures generated from the programs represented actual vibrations of the dynamometer in logical manners In addition the vibrations obtained experimentally from the programs responded accordingly to the vibrations generated analytically from a dynamometer s model thus confirmed reliability of the programs However some errors were presented in the display of excessive vibrations that can be a result of the sensor s characteristic the signal processing procedure used in the visualization programs or noises in the data acquisition system Even though this misrepresentation is difficult to be distinguished from the 101 actual vibrations the errors able to be detected using comparison with model based generated vibrations as studied in the experiments These errors may also be reduced by applying the second stage to the visualization procedure 6 2 Recommendations The model based vibration visualization technique may be accomplished based upon the following options of developments 1 Application of the system identification technique will be implemented by developing programs that perform the identification of system s parameters on real time data 2 A methodology of the comparison and correction unit see Fig 3 6 will be investigated A study for the development will be emphasized on the signal analysis for example random data analysis digital signal proces
94. f the terminology used henceforth is warranted The explanation begins with the Coordinate Systems C S shown in Fig 4 8 Spiewak 1994 1 Reference Coordinate System of the Plate XYZ p This C S is associated with the initial position of the plate The origin of is at the corner of the plate and the three orthogonal axes Yr and Zp are parallel to the plate s edges 60 Reference Coordinate System of the Plate XYZ p XR Instantaneous Coordinate System XYZ 2 Global Reference Coordinate System XYZ Figure 4 8 Coordinate systems used in describing the plate motion 2 Instantaneous Coordinate System XYZ Vibrating plate assumes varying translations and rotations The C S XYZ moves together with vibrating plate Therefore the coordinates of characteristic points of the plate its corners in the XYZ system remain unchanged Consequently converting these coordinates into the reference coordinate XYZ system is easy 3 Global Reference Coordinate System XYZ The above coordinate systems suffice to visualize the motion of one only plate However if multiple plates are involved it is advantageous to introduce one global coordinate system With this better system each plate has its unique Reference and Instantaneous coordinate systems Each individual Reference C S describes the initial location of the plate associated with it This location is defined by a six
95. f the discrete velocity signal x n element of a sequence of the discrete filtered velocity signal x n element of a sequence of the discrete displacement signal x y z translations of point relatively to point O parallel to X Y and Z axes Yo Zc translations of point C relatively to point O parallel to X Y Z axes Yo Z translations of point G relatively to point O parallel to X Y and Z axes X z translations of point C relatively to point C parallel to X Y and Z axes Xp Z translations of the dynamometer s base X Y and Z directions Y s Laplace transform of the output y t y t output vector Y t estimated output vector from G s and U s 2 amplitude of the automobile s translation angular acceleration of point P in XYZ angular accelerometer component of the vector i x and z At sampling period amplitude of the automobile s rotation y rotations of C S XYZ around X Y and Z axes respectively 6 0 W rotations of C S XYZ around X Y and Z axes respectively w rotations of C S XYZ around X Y and 2 axes respectively 0 V rotations of C S XYZ around X Y and Z axes respectively 6 9 Y rotations of dynamometer s base around X Y and Z axes respectively 0 Py V rotations of C S XYZ around X and Z axes respectively 0 angular velocity of point P in C S
96. g for the rest of the cr The input for coupling input config is an empty which means the parameters keep their uster contains 1 t settings Each following parameters N 5 a 016 coupling no change 0 5 m coupling Refer to Chapter 2 Hardware Overview for information about coupling on your specific type ot board 0 Do not change the coupling setting lt 1 DC MS 2 AC 25 3 Ground 4 Internal reterence 5 lt e Input config 0 input contig 0 Do not change the mput config setting 1 Differential defaun 2 Referenced single ended 3 Nonreterenced singie ended VIBRATION VWIAJALIZA 2 gt Data Acquisition Controller VI 05 30 97 01 20 Us Ga 5 Data Acquisition Controller V 05 30 97 01 20 1 The device number is the ID number of the plug in data acquisition card to be used tor the acquisition You can check the device numbers using the configuration utilty The configuration utility displays a board number or siot number next to each National Instruments board you use that number as the device number for the board zero array DBL Acquire again Press this button to begin acquisition again Users can select pre setting mode to adjust parameters again Display monitor Sho
97. g the plate motion 60 4 9 Locations of nine accelerometers required for the calculation of the generalized coordinates e VEA epo ea eR 63 4 10 flowchart of the procedure calculating the list of absolute generalized duse ioi aet ASS 66 411 Abbrevations used to designate the plate s corner 67 4 12 Definition of coordinate transformation matrices 69 4 13 Application of the homogeneous coordinate transformation for finding coordinates of point eet 70 4 14 Flowchart shows procedure of calculating coordinates of all corners 72 4 15 Illustration of the projection of a 3 D object on a 2 D planer 73 4 16 Illustration of the effect from changing the viewpoint on the 2 D picture 74 4 17 Diagram of coordinate transformation procedure for changing the viewpoint 75 4 18 flowchart of the entire animation procedures T Figure 5 1 5 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 5 10 5 11 5 12 LIST OF FIGURES Continued Page Schematic diagram of the experimental setup 81 Impact hammer PCB type 208B03 used for exciting the dynamometer
98. generalized coordinate s column direction Eq 4 12 29 Column 0 Platform generalized coordinate s XXX index column index r Distance between accelerometers in the X XXX mm Column 0 Platform s width in Y direction PW Column 1 Base s width in Y direction BW Column 2 Reference s width in Y direction RW Column 0 Platform s height in Z direction PH Column 1 Base s height in Z direction BH Column 2 Reference s height in Z direction RH Column 0 Platform s length from origin PLo Column 1 Base s length from origin BLo Column 2 Reference s length from origin RLo Column 0 Platform s width from origin PWo Column 1 Base s width from origin BWo Column 2 Reference s width from origin RWo Column 0 Platform s height from origin Column 1 Base s height from origin BHo Distance between accelerometers in the direction Eq 4 12 r Distance between accelerometers in the 2 direction Eq 4 12 Assignment of data column for calculation in Eq 4 13 or Eq 4 14 Table E 2 continued Systeminfo assignment experiment in Chapter 5 Column 0 corner C in the X direction Column 1 corner C in the Y direction Column 2 x c corner in the 7 direction Column 3 x corner 1 in the X direction Column 4 x corner 1 in the Z direction Column 5 corn
99. gned to each decomposed component 3 synthesis level in which generalized coordinates are assigned to the components and the mathematical model is built and 4 modification level where the generated model is checked the analytical model specifications or experiment results Chen 1996 38 This modeling methodology is applied for modeling the dynamometer under consideration installed on a machine tool Fig 3 10 The machine tool is treated as a linear rigid body system and it is decomposed into four major units shown in Fig 3 11 Spring dashpot elements represent bolts joints and guideways the link all components Chen 1996 Figure 3 10 A high speed machine tool Kalpakjian 1992 ul Spindle Assembly and Tool a Column of the I Machine attached to the dynamometer pd Dynamometer s Platform Four Sensing Elements Part of the System dealt with in Sections 3 5 1 3 5 3 Figure 3 11 simplified model of the machine tool from Fig 3 10 with the dynamometer installed Chen 1996 39 3 5 1 Mechanistic Model The dynamometer described in the preceding section whose simplified mechanical diagram is shown in Fig 3 12 is considered as a MIMO system with an input vector u t and output vector dg The input vector has in a general case twelve components Chung 1993 ut 33 where F t F
100. he specified Systeminfo and displays data on the graph 4 and 5 5 Use the z 121 Display of filter parameters and subtracting average parameters that used signal processing of data in the selected and displayed channel Display of sensor information and DAQ parameters that are associated with data in the selected and displayed channel 9 Display signal processing procedure that is applied to data in the selected and displayed channel Press Change filter control button to change any parameters in signal processing parameters in the Systeminfo A Systeminfo changing window will pop up for the user to set suitable processing parameters After the APPLY button is pressed the program SP will process the data in the selected channel according to new parameters Before stopping the program Select portion of data to be saved or used in next programs by moving the beginning bar 12a and the ending bar 12b to specify the range of data 13 Select save data switch to yes to save data and Systeminfo Program will do a save procedure after the program is stopped 14 Press Stop program to finish the program The user can also wire true signal to the finish program terminal for controlling the program SP to run only one time The program SP will process all data according to the information in the Systeminfo a
101. igh cutoff freq fh s lection Eliptic hter onginal seiecbon titer E Filter type m stopband ai pressed attenuation dB changeabiet 27071 Signal Processing Controis 2 Parameters cwm Eo votta pressed2 changeabie2 45 T E EE rage i DATANOTE Descnpoon 0 sensor 0 data 1 loed 2 sensor 3 voltage signal 4 accelerometer 1 location 2 serial number 3 transducer sensarety mV uni 4 double integrate 7 13 represent data n micrount 0 no convert 1 conver to mecrount 14 inverse signal directon 0 no inverse 1 inverse 15 offset index 16 offset length 17 gain 18 Filter setection 0 fitter 1 Butterworth filter 19 hugh limes 20 low limits volts Datanote Changing 3 vi 05 30 97 01 32 AM 154 VIBRATION VISUALIZATION 23 155 Page 3 hn Datanote Changing 3 vi 3 05 30 97 01 32 AM Page 4 Datanote Changing 3 vi 3 05 30 97 01 32 9 21 FA LIZATION VIBRATION Graph Plotting Preparation VI 05 30 97 01 34 Graph Plotting Preparation VI 05 30 97 01 34 Processed data Original data set
102. in Y direction disp Simulated solid and Actual dash Response in Y direction plotted Hit any key to continue text dataSize 1 150 deltaT 0 name 117 pause plot time 1 PDSZ ForceGain u 1 PDSZ y time 1 PDSZ genZ 1 PDSZ m time 1 PDSZ TranGain yZ 1 PDS2 g xlabel Time sec title Simulated solid and Actual dash in Z direction disp Simulated solid and Actual dash Response in Z direction plotted Hit any key to continue text dataSize 1 150 deltaT O name pause plot time 1 PDSZ ForceGain u 1 PDSZ y time 1 PDSZ genTH 1 PDSZ m time 1 PDSZ RotGain y TH 1 PDSZ g xlabel Time sec title Simulated solid and Actual dash in THETA rotation disp Simulated solid and Actual dash Response in THETA rotation plotted Hit any key to continue text dataSize 1 150 deltaT O name pause plot time 1 PDSZ ForceGain u 1 PDSZ y time 1 PDSZ genPH 1 PDSZ m time 1 PDSZ RotGain yPH 1 PDSZ g xlabel Time sec title Simulated solid and Actual dash in PHI rotation disp Simulated solid and Actual dash Response in PHI rotation plotted Hit any key to continue text dataSize 1 150 deltaT O name pause plot time 1 PDSZ ForceGain u 1 PDSZ y time 1 PDSZ genPS 1 PDSZ m time 1 PDSZ RotGain yPS 1 PDS2 g xlabel Time sec title Simulated solid and Actual dash in PSI rotation disp Simulated solid and A
103. in the experiment 4 484 4 92 Appendix Appendix Appendix Appendix D Appendix E Appendix F LIST OF APPENDICES Page Experiment Specifications a 108 Parameters of the Dynamometer s 110 112 MATLAB Program Used in the Experiment Descriptions of the LabVIEW Visualization Programs end 118 Data Management in the LabVIEW Visualization Programs 127 Block Diagrams of the LabVIEW Visualization Programs 132 Figure A 1 A 2 A 3 D 1 D 2 D 3 D 4 D 5 D 6 E 1 LIST OF APPENDIX FIGURES Page Power spectrum density of signal measured from an accelerometer 108 Dimensions of the dynamometer used in the experiment in units of mm the sensing elements are not shown a a a 109 Digital high pass filter s coefficients used Eq 4 4 and 4 6 109 Front panel of Data Acquisition Controller program DAC 118 Front panel of the signal processor program SP 120 Front panel of the generalized coordinate calculator program GCC 122 Front panel of the 3 D animation generator program AG 123 Dimensions used in the center reference drawing option
104. ity of the sensors under consideration for visualization of vibrations with frequencies down to as low as 2 5 cycles per second 22 Acceleration m s2 time sec a Acceleration signals Velocity m s 150 0 3 100 0 3 50 0 3 0 0E 0 50 0 3 time sec 0 00 0 10 b Velocity signals Displacement m time sec c Displacement signals Figure 2 14 Signals obtained by means a variable capacitance accelerometer and its results from double integration Kistler 1995 Still better sensors can be developed by improving the measuring technique for detecting position of the seismic mass since the acceleration can be calculated from the relative displacement between the seismic mass and the sensor s body For example the electron tunneling effect can be used in the measurement of this displacement Rockstad et al 1992 The transducer consists of a miniature silicon cantilever that acts as a sensing mass as shown in Fig 2 15 23 Suspended proof mass Sensor housing Measured tunnel current Apply deflection voltage Figure 2 15 Schematic illustration of a single element electron tunneling accelerometer Rockstad et al 1992 The cantilever is a gold coated electrode and supported over the other electrode by a small spacing that is maintained constant by a feedback circuit applying voltage bias to th
105. ization is shown in Fig 4 2 First a simplified mechanical model is developed by dividing the machine into a suitable number of rigid solids e g plates For example a simplified machine shown in Fig 3 11 comprises four prismatic solids representing the column spindle assembly table and the dynamometer s platform with workpiece Multiple sensors are installed on the machine to measure vibrations of each prismatic solid The solution employed in this research requires nine accelerometers per each solid to measure its list of generalized coordinates Padgaonkar 1975 Acceleration signals from these transducers are recorded by a data acquisition DAQ system which is controlled by a Lab VIEW based Data Acquisition Controller program see Fig 4 4 In the second step the voltage signals from accelerometers are 1 amplified 2 low pass filtered anti aliasing and 3 converted to acceleration using respective transducer s sensitivities Average quasi static offset voltage having its source in thermal drift in each sensor is eliminated by subtracting the average or moving average value calculated for each signal Adjusted signals are numerically integrated twice to obtain displacement signals Since signals from piezoelectric accelerometers are affected by 47 drift and noise direct integration of these signals yields strongly distorted results Therefore digital high pass filters are employed to remove quasi st
106. l Procedure 91 5 4 2 Results and Discussion 92 5 5 GIOSUFE 99 6 CONCLUSIONS AND RECOMMENDATIONS 100 6 1 Conclusions 100 6 2 Rec mmen datI BS 101 BIBLIOGRAPHY L a 103 APPENDICES esses aaa uh INN 107 24 22 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 11 2 12 2 13 2 14 2 15 2 16 2 17 3 1 3 2 3 3 3 4 3 5 3 6 3 7 LIST OF FIGURES Page Equivalent forms of a mechanical model used in vibration analysis 5 A 2 DOF model of an automobile 2 4447 40440400 6 Vibrating lumped system in two different mode shapes 7 Obtaining the system s model by means of an identification procedure 8 Classification of the system identification techniques 9 Parametric identification asesores MES CE 10 An animation of vibration on the display of ME ScopeTM 11 Six coordinates describing motion of a rigid plate 12 Configurations of piezoelectric accelerometers incl
107. l circuits Doebelin 1990 The use of an enclosing conductive shield can further decrease electrostatic pickup The shield captures unintentional charges and drains them off to a satisfactory ground However a shield required for a magnetically induced noise is quite thick and is not practically used The inductive noise can be minimized by twisting the two signal conductors so that the loop area available for inducing error voltages is balanced providing a canceling effect Commercially available cables include twisted conductors wrapped foil shields and grounding drain wire Deobelin 1990 A ground loop is created by connecting a signal circuit to more than one ground Since a conductor that serves as ground generally carries currents and also has some resistance potentials between two points grounds are not identical Any potential difference produces current that flows through the shield or signal circuit yielding large noise voltage The ground loop is simply broken by grounding a shield at only one point and using a floating input isolated amplifier to break the other loop which results in great reduction of noise pickup Doebelin 1990 Potter 1992 Tribo electric noise is often induced into the accelerometer cable by mechanical motion of the cable itself It originates from local capacity and charge changes due to dynamic bending compression and tension of the layers making up the cable This problem is avoided by using a proper
108. lay of ME Scope Vibrant Technology 1996 Another approach to visualize vibrations in commercial software is to display the operational deflection shape ODS in which the forced dynamic deflection is determined at the operating frequency Powell 1992 This technique is utilized to find the mode shape of the system directly from an experiment By mounting sensors usually accelerometers at specified nodes on the system s surfaces and then exciting the system with a suitable excitation the relative deflection of each node is obtained by capturing all signals from all nodes simultaneously when the system is subjected to excitation Powell 1992 2 4 Measurement of Signals for Visualization An approach of visualization technique in this research is similar to the Operational deflection Shape technique discussed above Vibration of the system under consideration are measured in actual operating conditions using suitable sensors and then displayed directly Therefore this visualization method presents the actual movement of the system The technique proposed in this research is explained by way of example A rigid plate fixed with springs and dampers represents a mechanical system To visualize the complete motion of the plate six coordinates are required which include three for rotations and three for translations Discussion of these coordinates follows Figure 2 8 Six coordinates describing motion of a rigid plate
109. lso attach time information by pressing button 11 to ON in which the program DAC creates an additional column containing time data and attaches this column to the last column of the data set Press Stop program button to finish the program 120 D 2 Signal Processor VI Edit zm Controls Windows Text r rr _ a Lab Filter t type posue yes 1 2 ais sampling freq 20000 00 Anus Soke high cutoff freq fh 10000 00 eaten Systeminfo fl ODE serial number DAQ gain n 3 offset i 10086700 000 n limits sensitivity to ee mV unit 21 41 stopband high limit volts low limit attenuation dB po oA 8 fiter 1 Integrate filter 2 Intergate filter 3 TM ME In microunit 0 06 0 07 008 X 003 0 10 0 1 400 50 800 1000 120 140 1600 1900 Before running the program SP Select from file switch to yes to read data and Systeminfo from spreadsheet files Select create new Systeminfo switch to yes to build an empty Systeminfo file eram will ignore the Systeminfo file that is read from step 1 Select yes to replace information of signal limits in the Systeminfo with limit values that are wired from the program DAC to the DAQ limit terminal Run the program SP Program processes data in all channels using t
110. m are higher than estimated stiffness coefficients of the model In addition the decay of the model based response is faster than the signal based response thus the damping coefficients of the model are also assumed to be higher than those of the actual system Therefore the experimental result showed incompatibility of the parameters in the governed model In the test no 7 and 8 the signal based responses dashed lines showed fluctuation around zero level even at the portion of signals before the impact especially in X and direction of the test no 8 see Fig 5 11 and in Yaw of the test no 7 see Fig 5 10 This oscillation of the based line of the signal based response is suspected to be caused by the digital filters used in the signal processing procedure see Section 4 3 2 The fluctuation errors in the displacement signals also have an influence on the rotations from the calculation described in Section 4 5 The results are also the fluctuation of rotation signals as shown in the Roll Pitch and Yaw signals in the test no 8 However the same type of error does not have much influence on such strong signals found in the test no 5 an 6 The recorded signals from the sensors are stronger in these tests because the impact locations are closer to the accelerometer s mounting locations than those in the 98 test 7 and 8 Therefore signal processing technique the visualization program the program SP
111. mic systems to verify the calculations in the programs SP and GCC as well as the graphical animation generated by the program AG One representative experiment is discussed in this thesis This experiment involved a mechanical system whose structure was similar to a high performance dynamometer Kistler type 9257A Kistler 1996 This system is henceforce referred to as the dynamometer 5 3 1 Experimental Procedure Several tests were conducted by exciting the dynamometer with an impact hammer Signals from nine accelerometers and the impact force were collected and saved in spreadsheet compatible files Table 5 1 shows the locations where the impact occurred and the directions of the force white arrows Expected response to each impact is briefly characterized in the same table The Characteristic Forced Response CFR of the dynamometer is the response which is dominant during the short period of time when the force from the impact hammer is applied were presented in 3 D graphical display on the front panel of program AG The orientation of the picture showed immediate response direction which was then compared with the expected response ASCII tab delimited file 88 Location and direction of the Comment The impact hammer excites the dynamometer in the positive direction at a center of the dynamometer s platform Translation parallel to the Zp axis should dominate the CFR The impac
112. n factors applied to translatory and rotational motions respectively 91 5 4 Comparison of Signal Based and Model Based Responses 5 4 1 Experimental Procedure A linear time invariant model of the dynamometer developed by Chung 1993 has been adopted to investigate the feasibility and potential advantages of the model based vibration visualization A computer program written in Matlab generates responses of this model to the actual force recorded during the experiment A list of this program is provided in Appendix C Four representative data sets discussed in this Section are characterized in Table 5 3 Each data set contains the following variables 1 Force signal from the impact hammer 2 Nine signals from accelerometers mounted on the plate These signals are converted to the measured generalized coordinates of the plate signal based responses according to Sections 4 4 and 4 5 3 The predicted generalized coordinates calculated from the model model based responses subjected to the actual measured force impact Force signals from the impact hammer are read from the data files These signals are used to stimulate the mathematical model of the dynamometer in the previously described state space form given in Chapter 3 The computer generated responses of the model are plotted in together with the recorded and processed acceleration signals generated by the developed signal based vibration visualization software Alth
113. n in Fig 5 8 to Fig 5 11 The solid lines represent the model based responses calculated using the mathematical model of the dynamometer Chung 1993 The dashed 93 lines represent signal based responses calculated using the signal based vibration visualization technique IX displacement m Roll angle rad 0 00 0 01 0 02 0 03 0 04 0 02 0 03 0 04 0 05 0 05 Signal Based Generated Response Signal Based Generated Model Based Generated Response Model Based Generated Response s displacement m Pitch angle rad 0 03 0 04 0 02 0 03 0 04 0 05 0 05 Signal Based Generated Response E Signal Based Generated Response dd Time sec Model Based Generated Response Time sec Model Based Generated Response displacement m aw angle rad 75 066 50 0E 6 25 0E 6 6 8E 21 25 0656 50 06 6 75 066 100 066 125 0E6 0 00 0 01 0 02 0 03 0 04 0 05 0 01 0 02 0 03 0 04 0 05 Signal Based Generated Response Signal Based Generated Response d Time sec Model Based Generated Response Time sec Model Based Generated Response Figure 5 8 Graphs show results from the test number 5 94 Roll angle rad rad 40 066 20 066 68 21 20 0E6 30 0E6 80 0E6 0 00 0 01 0 02 0 03 Time sec Pitch angle rad 400 0E 9 300 0E8
114. n the input and output signals the natural frequencies and corresponding mode shapes are determined Powell 1992 The vibration is presented as a relative movement of one part of the system with respect to the others For example if a user decomposes the visualized system into a sufficient number of elements each point or node is assigned to represent the vibration of that particular element In the animation of vibration one reference node is displayed as oscillating sinusoidal with corresponds to a forced harmonic FRF responses at the selected resonance frequency Positions of the other nodes are then calculated by using the relative displacement information provided the identified mode shapes Therefore graphical display shows vibrating shape or deflected shape of the system at this selected resonance frequency Vibrant Technology 1996 An example of visualization display presenting mode shape of a plate at a certain frequency is illustrated in Fig 2 8 Available software in the EMA area are LMS CADA X STAR system EMODAL M I DEAS PC MODAL and ME Scope Lang 1990 Spectral Dynamics 1995 Structural Dynamics Research Corporation 1996 Vibration Engineering Consultants 1997 Animate PLTMODES STR 4 01 340080 n FM a HE 2E C752 1 eer jose iu i 123 43 52 88 Heels Hrrla Figure 2 7 An animation of vibration on the disp
115. nceled read entire file number ot nee ak 1 1 1 AGIS E b A fave esa OR T CE read zero lines 2 Read Lines From File pattern vi 42 05 30 97 01 46 start of read offset chars 0 pos mode rel to begin new file path Not A Path if cancelled 1332 0 convert no F oF pattem 42 prompt Abe read one or more lines line string type of dialog continue or stop message file path dialog empty i Read File string vi 2 General Error Handler vi prompt fi Close File vil Choose file to read max characters per line no limit 0 number of lines 1 The read subV inside the While Loop searches for the last end of line but does retum any data Once that location is four the read subVI outside the loop reads the entire string This method is faster for a large number of lines because it does not requi the of a stnng concatentation function the loop and its associated memory suftling but this method is slower for a few lines VIAJALIZATION 27 VIBRATION GCC Generalized Coordinate Calculator VI 05 30 97 02 10 AM Data filename 2 VIZ Systerunto filename 0 THOMAS THANAT TMP DATAGFC20 8C CTR 8 m X direction rad Channel assignment rx Distance in X direction mm
116. ncluding high linearity good frequency response low influence of the cable capacitance and high stability Kistler 1995 A typical measurement system that involves a high impedance piezoelectric accelerometer and a charge amplifier is shown in Fig 2 10a Kistler 1995 Integrated circuit piezoelectric sensors been developed with advanced 5 technology transducers consist of high impedance piezoelectric accelerometers combined with miniature built in charge to voltage converter circuits as shown in Fig 2 11 PCB 1996 As a result the charges generated by piezoelectric sensing elements are converted into voltage inside of the transducers providing low impedance that can be measured by any typical readout instrument McConnell 1995 Low noise coaxial cable High impedance accelerometer Charge amplifier Readout instrument a High impedance piezoelectric measurement system Coaxial cable Low impedance accelerometer with miniature amplifier circuit Constant current Power Readout instrument supply Coupler b Low impedance piezoelectric measurement system Fig 2 10 Comparison diagrams of instrument setup between the high and low impedance system Kistler 1995 Therefore the low impedance piezoelectric systems requires only constant current power supply generated from an external low impedance power supply or a coupler to supply the tra
117. nd then stop 122 D 3 Generalized Coordinate Calculator VI 4 i GCC File Edit Operate Controls Windows Text Help Bh EE NNI 8 TMP DATA CENT R D ATHOMASVTHANAT TMPYDATAVFC20 8C CTR 2 direction rad Channel as n x Distance in X dition mm ri canet channel 5 4600 0 in Y direction channel 0 in Y direction tad 1 Come in Z direction channels 4 FG xXx Comer in X direction chann inZ dwechon channel 6 7 2 in Z direction channels 8 z Distance 2 direction mm inZ direction 22 098006 0 Wrisa 12 0 8 57E 12 L147 85 7 9 221 3 p 53E 12 339 70 MES SERES SEES JEEZ FETE SHEN FEE 121 67 27 36 3 9 9 3 Select from file switch to yes to read data from a spreadsheet file A user can select data sources weather to read data from a spreadsheet compatible file or to use data wired from another put terminal Select write to file switch to yes to save data Program will do a save procedure after the ram is stopped Assign channel numbers for specified variables used in Padgaonkar s equations Eq 4 13 or Eq 4 13 or Eq 123 D 4 3
118. nsducer as shown in Fig 2 10b Kistler 1995 MOSFET is abbreviation of Metal Oxide Semiconductor Field Effect Transistor Figure 2 11 Cross section diagram of the accelerometer 1996 The low impedance systems offer potentially lower cost and more convenient implementation than the sensors described earlier Kial and Mahr 1984 Kistler 1995 However the high impedance sensors used together with charge amplifiers offer significantly wider range of operation since the time constant and gains can be controlled at the charge amplifier In addition the high impedance transducers contain no built in electronics therefore they work in wider temperature range than the low impedance system Kistler 1995 The overall FRF of a measurement system of the low impedance ICP accelerometer H is Dally et al 1993 Vo 1 1 0 j TC O lat a q j4TCD o 1 70 0 where Sy the system s sensitivity volts g Ao amplitude of the acceleration applied to the transducer g Vo the amplifier output voltage volts 0 the natural frequency of accelerometer C the damping ratio of the accelerometer TCl the time constant of the power supply 17 the time constant of the miniature amplifier the transducer and J the imaginary unit j Wale A magnitude plot of the above FRF is given in Fig 2 12 It is clearly seen that the
119. nt signal from the accelerometer after subtracting the DC offset is shown Fig 2 13a It is known that given enough time after the force impulse vibrations die down and the investigated point of the plate returns to its initial position Although no drift is noticeable in the acceleration signal its presence is evident after its integration i e in the velocity signals which rise gradually as shown in Fig 2 13b The displacement signal shown in Fig 2 13c which is obtained by applying another integration has clearly visible drift strongly exceeding the level of vibrations caused by the force impulse The primary solution to the drift problem in this research is the use of high pass filters to suppress the low frequency components drift in the acceleration signals Suitable high cut off frequency has been selected by determining experimentally characteristics of the transducer For this purpose a power spectrum of the acceleration measured from the stationary i e at rest sensor was examined The cut off frequency was selected to be above the frequency of the dominating components appearing in the spectrum The results obtained by applying high pass filters between each integration to the signals measured from the excited vibrating system are shown as dashed lines in Fig 2 13b and 2 13c A side effect of using high pass filters are distortion of the signals manifesting themselves as a phase shift at lower frequencies These
120. nti aliasing filters are employed to prepare signals for digital acquisition In the second step acceleration signals are processed to obtain displacements Digital high pass filters are applied to the signals to suppress the drift The third stage is the calculation of the generalized coordinates of the plate Linear equations for determining angular accelerations Padgaonkar et al 1975 are employed in these calculations The final step is drawing 3 D pictures using the homogeneous coordinate transformation Wovolich 1987 and presenting the animation in which a series of the calculated pictures is rapidly displayed 80 5 EXPERIMENTAL IMPLEMENTATION AND RESULTS The signal based visualization programs introduced in previous chapters are complicated and prone to conceptual as well as programming errors It is therefore necessary to verify these programs experimentally before further development and application Several experiments discussed in this chapter have been designed and conducted with the following objectives 1 Verification of the visualization programs applied to a test object namely a rigid plate suspended by four three dimensional springs 2 A comparison of signals obtained from the test system with the signals generated by a mathematical model of the system to study the feasibility of the model usage for the visualization enhancement 5 1 Experimental Set Up Transient excitation was appli
121. omogeneous transformation matrices Wovolich 1987 are utilized in coordinate transformation Each matrix represents a relationship between arbritary x y z coordinates of a point one coordinate system and corresponding x and z coordinates in another coordinate system For a 6 DOF system four transformation matrices are required to change one coordinate system to another The matrices are defined as follow 8 The form is compatible with the homogeneous transformation 68 1 Yaw transformation matrix T v This matrix accounts for a rotation around the Zo axis by an angle y shown in Fig 4 12a This matrix is defined as follows cos y sin y 0 0 _ siny cosy 0 0 D Pon 4 16 0 0 0 1 2 Pitch transformation matrix T 9 This matrix accounts for a rotation around the Y axis by an angle shown in Fig 4 12b Its form is 0 sin d 0 0 quos 4 17 sin 0 0 0 0 0 1 3 Roll transformation matrix T This matrix accounts for a rotation around the axis by an angle 0 shown in Fig 4 12c This matrix is defined as follows 1 0 0 0 cos 0 sin 0 0 O sin 0 cos 0 O 0 0 0 1 T 0 4 18 4 Translational transformation matrix T x y z This matrix accounts for x and z translations along the and Z axes respectively as shown in Fig 4 12d 69 000 x 5 5 0
122. on acting on the platform s center of mass components of the force vector x and z Fr vector of the measured force signal Nyquist frequency gt measurement frequency bandwidth of the piezoelectric accelerometer sampling frequency G center of mass Generalized Coordinate Calculator VI LabVIEW program G s transfer function matrix of the equivalent system G gain of the amplifier the anti aliasing filter element of the transfer function matrix G s G s transfer function matrix of the system of the low impedance piezoelectric accelerometer system rotational magnification factor translational magnification factor L n by n identity matrix index denoting the direction io unit vectors in X Y Z axes respectively i j k unit vectors in X Y and Z axes respectively J mass moment of inertia J index denoting the corners of the plate k stiffness coefficient k stiffness matrix of the spatial model k conversion factor l w h dimensions of the plate in X Y and Z directions l l length of the automobile model lg Wo h distances from point C to point G in X Y and Z directions 1 Wo ho distances from point O to point C in X Y and Z directions m mass mass matrix of the spatial model vector of torques acting on the dynamometer s platform
123. ordinates is di x z 0 wil 4 10 The values of x y and z can be obtained by integrating twice the filtered acceleration signals as discussed in Section 4 3 2 The rotations and calculated according to a method proposed by Padgaonkar et al 1957 briefly summerized below 7 This corner point coincides with the origin of the coordinate system XYZ in Fig 4 8 62 For rigid body the relative acceleration of point P is given by the formula Hibbeler 1995 4 11 where P the arbritary point on a rigid plate in this case one of the corners P 1 2 and 3 shown Fig 4 9 ap the acceleration of point P in ac the acceleration of point C in 2 the angular velocity of point P in XYZ a the angular acceleration of point P in 2 and the position vector of point P from the origin C of XYZ By applying Eq 4 7 to coordinates of the plate corresponding to points 1 2 and 3 in Fig 4 9 the following equations are obtained Padgaonkar et al 1975 Ox dz a c r Dy 0 4 122 2 0 0 4 12b 4 120 Ay3 By D 4 12d axc r 4 12e Oz axi axc r 4 12f where Q the angular acceleration component of the vector around the i axis i x y lt
124. ough the model s responses are calculated using the actual input force data it is necessary to provide program information regarding locations and directions of the forces acting on the platform Therefore in each set of experimental data matrices Cu and ug in Eq 3 10 and Eq 3 19 are formed differently from the others if the acting points 92 of the force ay and and the force directions not identical The values of and a in the ux and Cu are shown in Table 5 3 The Fr is an mx1 matrix obtained from the measured force signals containing m data point Locations and Parameters Directions of the Comment of Cu mm Impact The impact hammer hit the dynamometer in a negative direction 00000 the side plane hitting location was close to corner 2 in Fig 4 9 The impact hammer hit the dynamometer in a Zp direction on the 00Fr000 top plane The hitting location was e near the origin of XYZ p The impact hammer hit the dynamometer several times in a Za 00Fr000 direction on the top plane The hitting location was near the farthest corner from the origin of 2 The impact hammer hit the dynamometer several times a 00Fr000 direction at a center of the top plane Table 5 3 Description of procedures in the experiment 5 4 2 Results and Discussion The comparison of graphs representing signal based and model based responses are show
125. r the Computer Aided Monitoring and Diagnosis of Machine Tools Diss U of Wisconsin Madison 1989 Spectral Dynamics STAR System Available from Spectral Dynamics Inc 1995 Spectral Dynamics Research Corporation I DEAS Model Available from Structural Dynamics Research Corp 1996 Spiewak S A Modeling of Cutting Point Trajectories in Milling ASME Journal of Engineering for Industry Vol 116 No 4 pp 440 448 1994 Thomson W T Theory of Vibration with Applications New Jersey Prentice Hall Inc 1993 106 Unbehauen H Some Modern Developments System Identification Using Parameter Estimation Methods Identification of Vibrating Structures NY Springer Verlag Wien 1982 Vibrant Technology ME scope Demo Guide Available from Vibrant Technology Inc 1996 Vibration Engineering Consultants PCMODAL Modal Analysis Software for the PC Available from Vibration Engineering Consultants Inc 1997 Wolovich W A Robotics Basic Analysis and Design New York CBS Collage Publishing 1987 APPENDICES 107 108 Appendix Experiment Specifications A 1 Information of Sensors Used in the Experiment Comment Location Channel Amplifier Accelerometer ac 60088 20292 Accelerometer C1083 201 2045 6 203 3069 394 5 BE C101812 20 394 C100862
126. ration is applied 0 Low pass filter default 1 High pass filter 2 Band pass filter 3 Band stop filter Eilter ON OEE selection Set number based on binary number basis for controling 3 filters e g 1 ON filter2 OFF filter3 OFF 001 4 filter1 OFF filter2 ON filter3 OFF 010 16 All filter is ON 111 Converse data to microunit If YES is selected program will multiply data the associated column by 1 million Inverse direction of signal If YES is selected program will multiply data in the associated column by 1 Beginning index number of data to calculate averaged value used in the subtracting average procedure 0 16 0 is the default 0 NO default 1 YES 0 NO default 1 YES 0 NO default 1 YES use Systeminfo Column 0 selection to use information from Systeminfo file in the program GCC Column 1 selection to use information from Systeminfo file in the program AG Column 0 Platform s length in X direction PL Column 1 Base s length in X direction BL Column 2 Reference s length in X direction RL Table E 2 Systeminfo assignment 130 27 mm Column 2 Reference s height from origin RHo 28 Option for plotting picture 0 Reference frame at plate s origin O 1 Reference frame at plate corner Cpr Column 1 Base
127. red signals as discussed in Section 2 4 1 As a result the animated vibration of the system may be quite different from the actual ones software package developed in this thesis comprises all functions represented Fig 3 2 It also suppress a large portion of errors in the measured signals by using rudimentary techniques namely subtraction of the average value of the signal and high pass filtering These techniques are well documented in the professional gt Definitions of the generalized coordinates is introduced in Section 3 5 29 literature applied commercial systems see Section 2 3 achieve further attenuation of errors it is recommended to use a constitutive mathematical model of the system whose vibrations are visualized A full implementation of this new approach is a complex task beyond the scope of this thesis The remaining part of this chapter contains a discussion of the ideas involved in implementing constitutive mathematical models for the purpose of error suppression 3 3 Model Based Vibration Visualization The model based vibration visualization is based on an assumption that vibrations of a system can be estimated if its mathematical model is known To model the system a priori knowledge of the system is required There are various classifications of dynamic systems depending upon the features of these systems that are interest A classification proposed by Chung 1991 is applic
128. rogram is reading data the acquired data are displayed on the graph 2 with a light green color on a background However in a continuous DAQ mode these data are not recorded The user can record data at any time by pressing capture button 6 If the acquisition is complete captured data are then displayed on the graph 2 and the background color is changed into dark blue Scan through all channels using the channel selection control 3 and set suitable gain for each channel using limit setting control 4 Precision of the displayed data can be set using the precision selection 15 If data size is bigger than 60 000 points program will suppress the data plotting on the graph to prevent a memory failure However the user can force the program to plot these data on the graph by switching mandatory plot 5 switch to ON Press Re acquiring button 7 to capture a new set of data Program will immediately begin new data acquisition using previously set DAQ parameters unless the Presetting button 1 is pressed 8 Press Show graph button 8 to activate data monitor program that plots acquired data on a big graph 9 Press Save data button 9 to save data in a spreadsheet compatible file that contains m rows and n column m number of data n number of channels A user can set a precision of data by selecting number of digits after point 10 The user can a
129. rometers Kistler type 8702B25M1 Kistler 1996 were used to measure vibrations of the dynamometer s platform They were attached by beeswax Doebelin 1990 Br el amp Kj r 1982 at the specific locations discussed in Section 4 5 as shown in Fig 5 3 Technical specifications of the accelerometers are provided in Appendix A Signals from the accelerometers were conditioned by an Accelerometer Coupler Kistler model 5128A Kistler 1996 and passed through a low pass Anti aliasing Filters Precision model 88 B and Datel model FLJ D6LA2 programmable filters Precision 1989 Datel 1987 The cut off frequency was set to 1000 Hz and the filters provided signal gains of 10 Filtered and amplified signals were passed through an Interface Panel to the DAQ Card Data Acquisition Card National Instrument AT MIO 16E2 installed inside of a Desktop Computer Tri Star The DAQ card used a 12 bit Analog To Digital Converter ADC a multiplexer and additional amplifiers The gains of these amplifiers were set by a program written in LabVIEW Figure 5 4 shows the entire data acquisition system The system allowed selection of the following parameters during visualization e Sampling frequency 0 40 000 Hz e Number of recorded signals 10 impact force and nine accelerations Further details are provided in Appendix A e Number of data 512 16 384 per signal with 0 1 024 pre trigger data points e Signal acquisition mode triggered
130. s of data in the signal processing and the generalized coordinate calculation programs Data types are displayed in round boxes with shadow whereas the processing programs are shown in rectangular boxes Formats of these data are shown in Table E 1 Acquired Data from Experiment from the Program DAC Systeminfo File CTR Information about Sampling Frequency Filter Parameters Sensors Sensitivities Gains Acquired Data in Voltages DAT eConversion to Physical Units Filtering and law Double Integration Using the Program SP eChange and Save New Parameters of the Signal Processing in the Systeminfo file Displacement Data in Meters VIZ Information about Assignment of Channel to Implement Eq 4 14 Generalized Coordinate Calculation using the Program GCC Generalized Coordinate Data COD Information about Plate s Dimensions and Positions Transferring into Array of X Y Data and Plot the 3 D Graph using the Program AG Figure 1 Data system 128 2 D array of m rows and n columns m number of data n number of channels used in the DAQ system 2 D array of m rows and n columns m number of data n number of channels used in the DAQ system 2 D array of m rows and 6 columns for one plate or 12 columns for two plates m number of data Each row contains data in a format of dp 45 where d
131. s of five steps 1 finding the absolute position of the reference corner point 2 calculating coordinates of the plate s corners in the C S XYZ using homogeneous coordinate transformation 3 modification of the corner coordinates according to the chosen viewing point 4 drawing a single 3 D picture representing instantaneous position of the plate and 5 animation of the 3 D pictures These steps are discussed in the following subsections 4 6 1 Finding Absolute Position of the Reference Corner Point Data obtained from the Generalized Coordinate Calculator see Section 4 1 2 defines only the relative position and orientation of the C S XYZ with respect to C S 7 in Fig 4 8 Furthermore the position and orientation of the reference C S XYZ r in the global C S XYZ is defined by additional six variables contained in the list do Eq 4 9 Absolute generalized coordinates of the point C in Fig 4 8 in the global C S XYZ can be calculated by combining the list of instantaneous position dr Eq 4 10 and the list of reference position do Eq 4 9 As the plate s movements can be very small compared to its dimensions a translational magnification coefficient H and a rotational magnification coefficient H are introduced to multiply the vibration signals instantaneous position of the plate before adding them to the reference position Therefore the list of adjusted absolute generalized coordinates dc
132. s of piezoelectric accelerometers dealt within this research are reviewed with respect to their applications and possible errors resulted from their characteristics Alternative accelerometers are discussed as a suggestion for the research improvement In Chapter 3 a concept of the model based enhancement of visualization is introduced A terminology used in the research is defined at the beginning of the chapter followed by a brief explanation of a methodology of the visualization Possible extension of work is also delineated Finally the mathematical model of a dynamometer employed from previous researches is discussed as this model underlines the proposed extension of Work In Chapter 4 theory and algorithms used in the visualization program developed in this thesis is explained Experiments conducted for the program verification are described with results and discussion in Chapter 5 The experiments also involve a comparison between results from the visualization program and those from the mathematical model Finally conclusions and a future direction of the research are stated in Chapter 6 CHAPTER 2 LITERATURE REVIEW The objectives and application of vibration analysis are discussed and a motivation for the visualization of vibration is presented System identification techniques as an enhancement of the vibration analysis are also reviewed Representative visualization techniques and commercial programs are briefly char
133. signal input the n element of a sequence of the velocity signal At the sampling period calculated from inverse of the sampling frequency Before performing the second integration of the velocity signal the Elliptic filter is applied to this signal again A position signal is then obtained by integrating the filtered velocity signal using the above integration method sun v x i jl YR ti a 4 6 R j 0 k 1 4n Le x Lj 1 At i 0 1 2 m l 4 7 where the n element of a sequence of the filtered velocity signal xs n the n element of a sequence of the displacement signal 59 Figure 4 7 graphically illustrates the signal processing described above A signal processing function SPF is henceforth introduced to represent all operations performed on the signals available as outputs of the accelerometers xs SPF V 4 8 where SPF e the processing steps that transform the readout voltage V into the displacement values xs Conversion to Subtraction of the Units of the Average Acceleration Value 1 High pass 1 Discrete 274 High pass 2 Discrete Elliptic Filter Integrator Elliptic Filter Integrator Figure 4 7 Diagram of the signal processing procedure 4 4 Coordinate Systems Several variables vectors and coordinate systems relevant to the vibration visualization are defined in this Section To avoid confusion a brief explanation o
134. sing and fuzzy knowledge 3 The graphical display of the animation program AG should be enhanced The perspective effect as well as the presentation of the plate as a solid block should be applied to the 3 D display In addition to the LabVIEW a number of software packages with excellent graphical display e g Mathematica and Matlab can be used for this purpose 4 The model of the system can be improved by considering flexible mode of vibrations Since the model employed in the first stage is designed under the assumption of a rigid body the generated responses using this model may not be precisely close to the actual vibration 5 In similar to the model development the flexible mode of vibrations should be considered in the visualization program The animation program can be modified 102 such that it presents plate comprising of many elements movement of each element is calculated from individual data The signal processing and the generalized coordinate calculation procedures can also be extended to be able to calculate more data without significant time delay More sensors can be employed and a modal analysis technique can also be applied to detect the flexible mode of vibrations Low frequency accelerometers and the multidirection accelerometers may be used to improve the visualization Inclinometers can also be employed where the angular movement is directly measured thus a number of sensors is r
135. still have problems when this technique is applied to weak signals As analyzing the result from the test no 8 in which the impact hammer hit the platform at the plate s center Even though the major movement occurs in the Z direction as expected the signal based responses also show unexpected rotations of the platform in the Pitch and Yaw directions see Fig 5 11 The signal based response in the Roll direction also contains an additional rotation besides the oscillatory similar to those of the model based response These errors are assumed to be caused by the flexible vibrations of the plate A diagram of the dynamometer shown in Fig 5 12 illustrates a possible mode shape of the response when a flexible mode is taken into account Rigid Platform Bending Platform Impact Force Platform Sensing Element Zr Reaction Forces a Impact force acting on the platform s center b Response of the platform Figure 5 12 Illustration of the flexible mode of platform s vibration from the test number 5 When the impact hammer hits the platform gray arrow in Fig 5 12a the platform should move purely downward shown as the dashed plate in Fig 5 12b since it is assumed to be rigid However the movement of the platform against the sensing elements yielding reaction forces acting on the platform shown as solid arrows in Fig 5 12b These reaction forces cause the platform to bend Thus besides
136. sualization algorithm shown in Fig 3 2 Detailed discussion of the experiments is presented in Chapter 5 Measured input forces and moments Model based response Modeling of generalized coordinate Force transducer Dynamometer Information about Section 3 5 process and sensor Comparison Section 5 4 3 D Animation of Vibrations Section 4 6 Signal Processing Section 4 3 and Section 4 2 Coordinate Measured Calculation Excitation Signal based response output 4 Input P Section 4 5 generalized coordinate force Mechanical System under Consideration Figure 3 9 Main research subjects pertaining to model based visualization 3 5 Modeling of the Dynamometer Practical application of the model based visualization hinges upon the capability to derive analytical models of the systems under consideration efficiently and rapidly modeling technique of the dynamometer proposed by Chung 1993 has been used in this thesis This technique using Lagrange s equation has been combined with computer aided symbolic computation proposed by Chen 1996 The modeling involves the following four stages 1 decomposition level in which the physical system is converted Into components where motion attributes are assigned to each part 2 mapping and identification level where the energy attributes are assi
137. t Systems Application and Design 4th ed New York McGraw Hill Inc 1990 Endevco Endevco Piezoelectric and Variable Capacitance Accelerometers Available from Endevco Corp 1996 Ewins D J Modal Testing Theory and Practice New York Research Studies Press Ltd 1984 104 Franklin F Powell J D and Abbas E Feedback Control of Dynamic Systems 3rd ed New York Addison Wesley Publishing Company Inc 1994 Hibbeler R C Engineering mechanics Statics and Dynamics 7th ed New Jersey Prentice Hall Inc 1995 IC Sensors Understanding Accelerometer Technology Available from IC Sensors Inc 1991 Kalpakjian S Manufacturing Engineering and Technology 2nd ed New York Addison Wesley Publishing Company Inc 1992 Kial R and Mahr W Piezoelectric measuring Instruments and their Applications Trans Kial R and Mahr W Kistler Instrument Corp 1984 Kistler Advanced Instrumentation for a World of Applications Available from Kistler Instrument Corp 1995 Kistler K BEAM Accelerometer Available from Kistler Instrument Corp 1996 Lang G F PC Based Modal analysis Comes of Age Sound and Vibration Jan 1990 pp 20 30 McConnell K G Vibration Testing Theory and Practice New York John Wiley amp Sons Inc 1995 National Instruments AT MIO E Series User Manual Available from National Instruments Corp 1995 National Instruments LabVIEW for Windo
138. t hammer excites the dynamometer at a mid width of the platform near the edge e Rotation around axis and translation in the Zp axis should dominate the CFR The impact hammer excites the dynamometer at mid length of the platform near the edge coinciding with the Y axis e Rotation around axis and translation in the axis should dominate the CFR The impact hammer excites the dynamometer at a mid height of the side of the platform near to the edge e Rotation around Zp axis and translation in the axis should dominate the CFR Table 5 1 Descriptions of the test procedures used in the experiment 5 3 2 Results and Discussion The results of experiments are summerized in Table 5 2 Representative positions of the dynamometer s platform shown in the table are obtained as snap shots of the screen generated by the AG program module The dash dotted wireframe box represents a location of the platform before the impact initial position The wireframe drawn with the solid line represents the Characteristic Forced Response CFR i e the location of the platform shortly after the force impact The criteria for selecting presented pictures from the displacement response signals are illustrated in Fig 5 7 The initial position of the platform corresponds typically to the time instance 25 ms 50 data samples before the 89 impact corresponds typically to the time when the response
139. t signals Dally et al 1993 To visualize motion of the 6 DOF plate considered in Section 2 4 acceleration signals have to be converted to displacements Although performing double integration on acceleration signals for this purpose seems possible in theory its implementation is difficult Combined distortion of signals measured by piezoelectric accelerometers usually results in a gradual change of the output voltage drift This change is as a relearn unknown function of time since drift may result in addition to the distortions caused by the dynamics of the amplifier from changes of temperature line voltage or amplifier characteristics Pulse Shape Rectangular Pulse Error P ad LN Triangular Pulse Time Constant Required for Limiting Error i i 16 61 31 127 Table 2 1 Effect of time constant the error measuring various transient responses Dally et al 1993 Half sine Pulse 19 Low frequency measurement errors discussed above strongly amplified when integration is applied in the process of converting accelerations into displacements These errors are illustrated by the following example involving monitoring displacements of one point of the plate see Section 2 4 in response to an impulse force acting on it Kistler model 8702B25M1 piezoelectric accelerometer Kistler 1995 is employed to measure the acceleration of this poi
140. ted in 2 D array see Section 4 5 This list is transformed and rearranged such that a single 3 D picture is created and plotted on the X Y Graph display object using procedures described in Sections 4 6 1 4 6 5 After the complete picture is displayed next generalized coordinate list is selected from the initial 2 D array and processed to create another 3 D picture The new picture is then plotted replacing the previous one Consequent plotting yields an animation of the plate movement according to the actual vibration All steps of the animation process described in this section are summarized in Fig 4 18 77 Setting pointer to the first plate location first frame Magnifying vibration signals in generalized coordinates see Eq 4 15 Summing with reference position of XYZ Section 4 6 1 Calculating positions of the plate s corners in the global coordinate system XYZ see Eq 4 24 Changing the viewpoint see Eq 4 25 gt SEGUON O Planer representation of the plate s corners projection on the computer screen Section 4 6 3 Drawing edges of the plate generating the wireframe Plotting on the X Y Graph Display Object no Advancing the pointer to the next location of the plate Figure 4 18 A flowchart of the entire animation procedures 78 4 7 Transformation of Generalized Coordinates Rigid Plate The generalized
141. the Experiment This program is applied to the experiment number 10 described in Section 5 3 1 clear format short e Read Data from File InputColm 1 Assign column number of force signal Xcolm 1 Assign column number of Response signal Ycolm 2 Zcolm 3 THETAcolm 4 PHicoim 5 PSIcolm 6 smplFreq 20000 sampling frequency Hz load NEARZA VIZ enter specific name of your force data file forcel NEARZA ul forcel InputColm select the first column as input dataSize size ul force1 0 load NEARZA COD enter specific name of your Response data file response y1X responsel Xcolm select the second column as ylY responsel Ycolm ylZ responsel Zcolm responsel THETAcolm responsel PHIcolm 1 responsel PSIcolm response 1 0 PDSZ 1000 Plot Data Size text dataSize 1 150 deltaT 0 name Plate Dimension Lall 3 939 0254 Wall 6 69 0254 Hall 0 87 0254 Center of Gravity Location Vector ICG 1 9695 0254 wCG 3 1827 0254 hCG 0 4488 0254 Plate Moment of Inertia cont1 025445 7 841 1043 2 714 74 6372 Iyy 28 9323 cont1 Izz 100 9267 cont1 0 lyz 9048 cont1 Ixz 0 Ixy 0 05 cont1 Iyzz 9048 cont1 Ixz 0
142. the eye XYZ y to be rotated around the C S XYZ with the same angles shown in Fig 4 16 74 2 D picture the screen 3 D object Rotation of the vlewpoint New viewpoint Change of the Perspective Figure 4 16 Illustration of the effect from changing the viewpoint on the 2 D picture An effect of a perspective on the picture is accomplished by changing mathematically the viewpoint s location on the axis whose it is located on Rybaczyk 1989 This procedure also governs a homogeneous coordinate transformation matrix that its first three components of the forth column are not zero Rogers and Adams 1990 The calculation of new coordinates from the rotation and translation of the viewpoint therefore governs two transformation matrices shown in Fig 4 17 Rybaczyk 1989 The visualization program written in this research performs the rotation of the viewpoint where the rotation matrix in Fig 4 17 is modified from the homogeneous transformation matrix see Eq 4 20 with elements as the rotation angles Qv and yy Therefore new coordinates of each characteristic point of the plate in the viewpoint C S XYZ y can be calculated using the homogeneous transformation matrix modified from Eq 4 24 as follows 75 D 0 0 0 6 4 25 where the coordinate vector of point A in C S XYZ D th
143. the movement in 99 the Z direction the corner also moves toward the platform s center the X Y directions Since the visualization program assumes the component to be a rigid plate the vibrations in the X and Y directions from the actual bending platform result as the plate s rotations shown in the Roll Pitch and Yaw directions in Fig 5 8 However these errors can be indicated by observing the predicted responses from the model From the test no 8 the model based responses solid lines show the indication of no significant rotation in the response according to a calculation from the location and direction of the impact force Hence the discrepancy between the signal based dashed lines and the model based responses solid lines in the X and Y directions in the test no 8 indicates the possible bending of the platform Thus the model based responses can be used to detect the error in the signal based visualization 5 5 Closure Programs designed for visualizing vibrations in mechanical systems have been tested in the experiment The program provided sufficient results in presentation of vibration The results from a comparison between the signal based responses and the model based responses showed similarity in movement directions however some characteristics of the responses frequencies and decay rates were different due to the primary estimation of the parameters The discrepancy between the signal
144. time O deltaT sampleTime deltaT 115 plot time 1 PDSZ 10 1 u 1 PDS2 y time 1 PDSZ 10 6 yX 1 PDSZ r time 1 PDSZ 10 6 yY 1 PDS2Z c time 1 PDSZ 10 6 yZ 1 PDS2 g xlabel Time sec title Input y and Output Y c Z g micro m Signals disp Sample input and output in X Y Z direction data plotted Hit any key to plot angles text dataSize 1 150 deltaT O name pause disp plot time 1 PDSZ 10 1 u I PDSZ y time 1 PDSZ 10 6 yTH 1 PDSZ r time 1 PDSZ 10 6 yPH 1 PDSZ c time 1 PDSZ 10 6 yPS 1 PDSZ g xlabel Time sec title Input y and Output Theta r Phi c Psi g micro rad Signals text dataSize 1 150 deltaT O name disp Sample input and output in Theta Phi Psi rotation data plotted Hit any key to plot model based responses pause disp Simulate Impulse Signal t 0 dataSize for i 1 dataSize 0 1 0 T 50 1 end z5 zeros size u RealUz z5 z5 u z5 z5 z5 time2 Zresp lsim ssA ssB ssC ssD RealUz time FIND GENERALIZED COORDINATE OF CORNER for i 1 dataSize T01 cos Zresp i 6 sin Zresp i 6 0 0 sin Zresp i 6 cos Zresp i 6 0 0 0 0 1 0 0 0 0 1 T12 cos Zresp i 5 0 sin Zresp i 5 0 0 1 0 0 sin Zresp i 5 0 cos Zresp i 5 0 0 0 0 1 T23 1 0 0 0 0 cos Zresp i 4 sin Zresp i 4 0 0
145. timulated by known input signals This system is a physical model of a high performance dynamometer Kistler type 9257A Kistler 1995 The dynamometer 18 a multi component force sensor comprising the major parts shown Fig 3 8 A platform is supported by four piezoelectric based sensing elements When a force is applied to the workpiece attached to the platform the platform is moved from its equilibrium position thus causing the piezoelectric sensing elements to deform Electric 36 charges generated from the deformations measured combined to calculate the applied force If the platform is considered as a rigid body its motion is described by six degrees of freedom three translations and three rotations as previously described in Section 2 4 Figure 3 8 Major components of the dynamometer Kistler 1995 The signal based vibration visualization technique proposed in this Chapter is applied to visualize vibrations of dynamometer s platform A linear model of the dynamometer Chung 1993 and Chen 1996 is used for calculating theoretical vibration responses Calculated and measured generalized coordinates are then compared Results and errors of the visualization are analyzed The main research subjects dealt with in this thesis and pertain to model based visualization are schematically shown in Fig 3 9 These subjects are laid out in the figure such that they resemble the flow chart of signal based vi
146. tion of the coordinates of all corner is illustrated in Fig 4 14 4 6 3 Projection of 3 D Object to a Planer 2 D Screen The calculated coordinates of each corner of the plate comprises of three variables i e xc yc zc representing the plate as a 3 D object To display this 3 D picture of the plate on a planer 2 D computer screen a projection technique is performed 72 to project the 3 D object 2 D plane This technique is accomplished by assigning a viewpoint to be located on one of the axes of the global C S XYZ shown as an eye symbol in Fig 4 15 The eye e g on the X axis looks toward the origin O of the C S XYZ yielding a screen or a plane defined by the Y and Z axes shown as a gray plane in Fig 4 15 START Creating T T Q0 4 transformation matrices Creating location vectors see Table 4 1 Begin calculations for the first corner Obtaining the resultant transformation matrix T T ost E 0 i TO T yc Transforming to global C S XYZ 22 4 D T x yes no Figure 4 14 Flowchart shows procedure of calculating coordinates of all corners The line of sight from the eye projects the 3 D object on this 2 D screen resulting in an image of 2 D picture representing this object This projection procedure is simply accomplished in the visualization program by discarding the
147. to the measured frequency response of the system In the time domain approach the estimate of system parameters are based directly on the measured system s excitations and responses Collins et al 1972 The methods are also classified based on the spatial model or the modal model Both approaches govern either one or both techniques direct or iterative schemes The direct or non parametric identification approach is based on developing a single step solution procedure that estimates all required parameters The iterative parameter optimization or parametric identification is based on the use of algorithms that relate change in parameter values to change in model responses in which a priori knowledge of the system is required to derive a mathematical model Collins et al 1972 Frequency Domain Test Data Time Domain Iterative Parameter Modal Model Spatial Model Optimization Iterative Parameter Optimization Spatial Model Iterative Parameter Optimization Figure 2 5 Classification of the system identification techniques Collins et al 1972 A distinction between the complete and incomplete methods are determined by comparing a number of the model s degrees of freedom to the number of normal modes possessed by the mathematical model If the number of the model degrees of freedom and measured modes are equal a unique equivalent structure mode can be
148. transformation to draw 3 D wireframe pictures Since various errors distort the measured signals the animated movement may be inaccurate The knowledge of a mathematical model of the system whose vibrations are animated allows detection suppression of distortions For this purpose the signals measured from the actual dynamic system are compared with the signals simulated by the system s model subjected to the same excitation as the actual system Discrepancies between the actual and simulated signals are detected They are analyzed to identify possible sources and forms of distorting signals As the next step the measured actual signals are corrected by removing estimated distortions A methodology and software package capable of performing all functions necessary to implement the visualization of vibration have been developed in this research using LabVIEW programming environment As compared with commercial software for experimental modal analysis the most distinctive feature of the developed package is improved accuracy achieved by applying concepts utilized in control theory such as modeling of multi input multi output MIMO systems and on line system identification for the model development and correction of signals Copyright by Thanat Jitpraphai June 11 1997 All Rights Reserved MODEL BASED VISUALIZATION VIBRATIONS MECHANICAL SYSTEMS by Thanat Jitpraphai A THESIS s
149. trigger type is analog trigger Ges trigger channel trigger channel is valid onty when trigger type is 1 trigger channel describes the analog channel that is the source of the trigger following the syn described in the Channel Addressing section of Chapter 2 Getting Started with the Data Acquisition Vis except for the following cases An empty string telis LabVIEW not to change the trigger source setting An empty string is the default input EXTx denotes an extemal analog input trigger where x is the channel number Currently x must be 0 You can substitute the string EXTERNAL sten More detail tt When Triggre type is 1 analog trigger 5 or 6 trigger source defaults to 0 for analog input channel 0 The following are valid trigger source n where n is an analog input channel number E series only set trigger channel to PFIO trig1 in this program trigger level 0 V level is the vollage value the analog source must cross for a trigger to occur You must also specify whether level must be crossed on a leading slope with the edge or input The default input for level is 0 0 V DAQ error error out contains error information If the error in cluster indicated an error the error out cluster contains the same information Otherwise error out descr status of this VI status status is TRUE if an error occurred status is TRUE this VI does not perform any operations code code is the error cod
150. ubmitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science Presented June 11 1997 Commencement June 1998 Master of Science thesis of Thanat Jitpraphai presented on June 11 1997 APPROVED Redacted for Privacy Major Professor representing Mechanical Engineering Redacted for Privacy Head of Department of Mechanical Engineering Redacted for Privacy Dean of School I understand that my thesis will become part of the permanent collection of Oregon State University libraries My signature below authorizes release of my thesis to any reader upon request Redacted for Privacy Thanat Jitpraphai Author ACKNOWLEDGMENT I would like to express my appreciation to my advisor Dr Swavik A Spiewak I first met Professor Spiewak in his Smart Product and Design class where he showed me the power of computer programs that have leaded to the LabVIEW programs in this thesis Without the inspiration as well as his advice encouragement and great effort this thesis would not have been accomplished I would also like to thank the students in my research group Thomas Nickel Brian Brisbine Ben Chen and all of my friends in helping me out of many difficulties I would like to thank sincerely to my parents Dr Phaibul and Dr Chattaya Jitpraphai as well as my brothers Peera and Siros for their encouragement and support for my study
151. uding a compression type and b shear type 14 Comparison diagrams of instrument setup between high and low impedance asses 15 Cross section diagram of the ICP accelerometer 16 Magnitude plot of the of a piezoelectric accelerometer 17 Illustration of the impact of a low frequency drift on the displacement obtained from acceleration 20 Signals obtained by means of a variable capacitance accelerometer and its results from double integration 22 Schematic illustration of a single element electron tunneling accelerometer 23 A cross section of a piezoresistive accelerometer 24 A cross section of the PiezoBEAM accelerometer 24 Flowchart of model based visualization of vibrations 27 A flowchart representation of the methodology developed for the signal based vibration obl carmen sanders aac suis 28 Classification of dynamic systems 29 The MINITORSVSIE RI con 30 Block diagram of the model based vibration visualization 32 Block diagram of the Comparison and Correction
152. ulates DAQ parameters User can justify feasible parameters before applying to real measurement After Apity button is pressed data acquisition then begins Mandatory graph plot In case of large data numbers more than 60 000 scans program deny 10 plot data on graph to prevent memory failure Users can torce program to plot graph by press Mandatory graph plot button Number of data Set number of Scans In Manual user specity number by setting number in Entry number of data contro below Type of DAQ 1 Trigger mode acquisition begins after getting a trigger signa at channel PFIO or Trig 0 2 Continuous mode program just read signals unti Capture data button is pressed Data is then acquired 3 Read only one time program acquires data immediately atte Apply button is pressed Show X axis display option 1 Show time sec on X axis 2 Show point data number on X axis Ground display option 3 Show signal and ground 0 vott in time axis 4 Show signal and ground 0 volt in point axis Power spectrum option 5 Show auto power spectrum of signal in dB set freq options for setting DAQ parameters 1 Set Scan rate datapoint sec is to specify a sampling frequency With data number this program calculate acquired time 2 Set Time to acquire sec with data number this program calculate sampling frequency VIBRATION VISUALIZATION 4 rage gt
153. uous mode put 5 n Manual 10000 0 DE EUR 9 1 trigger 0 000 l Apply ET X axis in dT _ press Apply to begin acquisition ZRROF code 10240 source Al Group Config x z DAQ Control amp Mechatronics Lab Paprogram Author T Jitpraphai Oregon State University Page 2 Data Acquisition Controller 05 30 97 01 20 AM coupling amp Input config no change 0 cluster coupling amp input config an array of clusters of which each array element specifies the coupling and input configuration for the channel s in the corresponding element of the channels array there are fewer elements in this array than the number of channels the VI uses the las elemen for the channels The default input is an empty array which means the parameters do not change from their default settings Each cluster contains the followir parameters u16 coupling 0 Do not change the coupling setting DC AC Ground relerence 016 input config 0 Do not change the input config setting Differential Referenced single ended Nonreferenced single ended coupling amp input config ts an array clusters Each array element assigns the configuration for the channels specified by the corresponding ele channels If there are fewer elements in this array than in channels the VI uses the last element of coupling amp input conti
154. user constructs a graphical source code of the VI by wiring together objects represented by icons that send or receive data perform specific functions and control the flow of execution Fig 4 3 shows an example of the VI variable amplitude sinewave generator which displays a knob and graph in the front panel a while the calculations take place in the block diagram b 49 visualization programs this research were written LabVIEW The developed set of programs consists of the following major VIs 1 Data Acquisition Controller VI This program facilitates acquisition of up to 16 vibration signals from transducers attached to a machine It is further discussed in Section 4 2 2 Signal Processor VI This program performs the signal conversion subtraction of the moving average filtering and numerical integration on the acquired data Calculations used in this VI are further discussed in Section 4 3 3 Generalized Coordinate Calculator VI This program calculates a sequence of the generalized coordinate lists dg t introduced in Section 4 1 Each list is calculated from nine linear acceleration signals recorded at any given time instant for each prismatic solid representing an element of the simplified machine Equations employed in the calculations are further explained in Section 4 5 4 3 D Animation Generator VI This program generates 3 D pictures using a time series of coordinate lists dg k
155. uses graphical programming language G to create programs in a flow chart like form called a block diagram National Instrument 1994 Unlike other programming languages such as C FORTRAN or BASIC LabVIEW relies on graphic symbols rather than textual statements to describe programming actions National Instruments 1994 These symbols icons such as switches knobs and displays represent and resemble elements found in physical instruments Because of its appearance the LabVIEW programs are called virtual instruments VI LabVIEW offers extensive libraries of functions and subroutines for example application specific libraries for data acquisition data analysis data presentation and data storage In data acquisition aspect LabVIEW can command DAQ boards to read analog input signals with A D conversion generate analog output signals D A conversion read voltage data for processing or other manipulation and write digital signals into storage devices National Instruments 1994 A typical virtual instrument VI consists of two parts a front panel and a block diagram displayed in two separate windows In the panel window a user interfaces with the VI via Controls and Indicators The Controls simulate instrument input devices and supply data to the block diagram of the VI The Indicators simulate instrument output devices that display data acquired or generated by the block diagram of the VI In the block diagram window the
156. veral crucial procedures are discussed in detail The entire visualization process is partitioned into four major steps namely 1 data acquisition procedure 2 signal processing 3 generalized coordinate calculation and 4 3 D animation procedure 4 1 Visualization of Machine Vibrations The visualization of machine vibrations presented below assumes that each specific part of the machine is a rigid plate with six degrees of freedom 6 DOF motion This means that six variables three for translation and three for rotation are required to describe three dimensional 3 D motion of the plate These variables are formed into the generalized coordinate list dg Eq 4 1 Graphical representation of this list is shown in Fig 4 1 dg x yz 0 4 1 One generalized coordinate list represents one specific location of the plate in space and is used to generate one picture of the plate in the 3 D space Hence a time series of coordinate lists is needed to create a sequence of pictures that represent motion of the vibrating plate A technique to calculate the generalized coordinate lists from experimental data is discussed below A method employed to present the results as 3 D graphical animation is also described 46 Figure 4 1 Components of the generalized coordinate list dg describing the rigid body motion of a plate 4 1 1 Overview of the Methodology A flowchart of the algorithm used in the visual
157. vity Myr az 3 6 a F 3 7 a ay Fx 3 8 The three components of the measured force and three resultant moments acting on the platform represent a complete set of independent variables in the equations of motion This set is henceforth referred to as a forcing function vector Fe It is obtained 41 combining six elements of the simplified input vector u given Eq 3 5 with the moments Myr M r Fe F F F Myr M Mic 3 9 F F and F are signed numbers that are positive if they in the positive directions of the respective and Zg axes The above equation can be rewritten in terms of the input vector u and a matrix C a whose exact form is given in Appendix B 3 10 By applying Lagrange s method the equation of motion can be derived matrix vector form Chung 1993 m de c de k de Fe 3 11 where matrices m and are listed in Appendix 3 5 2 State Space Model Since the dynamometer is a MIMO system its analysis can be greatly simplified by introducing the state space model Franklin et al 1994 One of the main reason of using the state space model in this research is the availability of commercial software packages such as Matlab or Mathematica that allow easy implementation of complicated algorithms developed in control theory for the investigation of MIM
158. w acquired data on larger graph Data monitor ver 2 4s Power Valid after program has acquired data sucesstully graph color is dark blue and this button blinks Entry number of data scan is defined as a reading taken from each channel in the channel list taken in the order specified The number of scans to acquire is the number of times to scan through all channels in the list which also is the number of samples to acquire from each the hist Capture data Valid only in Continuous mode Program just read data and plots on graph Until Capture data button is pressed the acquisition then begins scan rate scans sec The scan rate determines how many scans per second to acquire Because all channels in the channel list are sampled during each scan each channel sampled at the chosen scan rate Pretrigger data number This number is valid in Trigger mode DAQ Pretrigger data number specify number of data to be acquired betore a trigger signal occurs at time 0 Save to file Program writes acquired data to a spreadsheet file ASCII format save data press save to button Pricision of saved data is specified by setting fix point number of digits to the right of the decimal point Users can select to save data with time information a data set shows gradual time number to last column using time first column pre setting In pre setting is shown program will not read data but calc
159. ws User Manual Available from National Instruments Corp 1994 Natke H G Identification of Vibrating Structures an Introduction Identification of Vibrating Structures Italy Springer Verlag Wien 1982 Padgaonkar A J Krieger K W and King A I Measurement of Angular Acceleration of a Rigid Body Using Linear Accelerometers Journal of Applied Mechanics Transactions of the ASME Vol 42 pp 552 556 1975 105 PCB Piezoelectric Accelerometers Available from Piezotronics 1996 Peters J and Mergeay M Dynamic Analysis of Machine Tools Using Complex Modal Method Annals of the CIRP Vol 25 1 1976 pp 257 261 Potter D Keys to Success in Data Acquisition System Design In Tech Vol 39 Aug 1992 pp 24 28 Powell C P Machinery Troubleshooting Using Vibration Analysis Techniques Sound and vibration Jan 1992 pp 42 54 Precision 88 Series 8 Pole 8 Zero Programmable Filter Modules Available from Precision Filters Inc 1989 Rehsteiner F et al Modeling and Assessment of Machine Tool Dynamics and Accuracy To be published Rockstad H K et al A Miniature High Resolution Accelerometer Utilizing Electron Tunneling Mocromachinical Systems ASME DSC Vol 40 1992 pp 41 Rogers D F and Adams J A Mathematical Elements for Computer Graphics 2nd ed New York McGraw Hill Publishing Company 1990 Rybarczyk P R An Integrated System fo
160. xtension of the signal based vibration visualization presented in Section 3 2 in which the system identification and model response analysis technique are included A flow diagram of the enhanced visualization is shown in Fig 3 5 The visualization begins with the measurement of vibrations of a mechanical system by means of sensors accelerometers which is performed by the Data Acquisition DAQ system The measured signals are processed in the Signal Processing unit and the Generalized Coordinate Calculation unit yielding the signal based response In a parallel thread a suitable mathematical model of the system is derived analytically in the Model Derivation unit The modeling technique that involves symbolic computations can be applied to generate a structure of the model while the essential unknown parameters are represented as symbolic coefficients in model s equations Rehsteiner et al 1997 To complete the model a system identification is applied to estimate the unknown parameters As already explained in 32 Section 2 2 this technique employs iterative procedure that adjusts values of the model s parameters in order to minimize a difference between the signals recorded from the sensors and calculated responses of the model Input Mechanical System Vibrations Model Derivation eee DAQ System Measured Input Measured Output Model Structure System Identification Feedback Command
161. y the data acquisition DAQ system is not necessarily equal to zero when no acceleration is applied to the transducer This is a result of noise in the DAQ system To eliminate DC component of this error which is strongly amplified The gain of anti aliasing filters is adjustable see Chapter 5 57 by signal integration an average value V of each signal V is subtracted from it The obtained zero centered signal is henceforth denoted x see Eq 4 4 4 3 2 High pass Filtering and Double Integration Procedure The measurement of position required to visualize vibrations by means of accelerometers has one serious drawback namely high sensitivity to low frequency noise superimposed on the measured accelerations To overcome this drawback a suitable digital signal processing is proposed in this research in addition to the zero centering procedure described in the preceding Section A priori knowledge of the system s model allows to predict likely characteristics of the measured signals at low frequency for any specific excitation By selecting excitations which do not have low frequency components the system of interest should not have low frequency components in responses to these excitations Such components in the measured signals are treated as errors and eliminated The correction of signal is performed using digital high pass filters The cutoff frequency of these filters is determined by analyzing
162. ystem comprises of X Y and Z axes XYZ coordinate system at the plate s center of mass comprises of X Y and Z axes XYZ instantaneous coordinate system of the plate comprises of X and Z axes XYZ reference coordinate system of the plate comprises of X Y and Z axes XYZ viewpoint coordinate system of the plate comprises of X Y and Z axes A B C D coefficient matrices of state space equations AG 3 D Animation Generator VI LabVIEW program amplitude of acceleration applied to accelerometer aoa accelerations of point C and P respectively in XYZ acceleration in i direction at corner j i x y and z 1 2 and 3 a a a distances between the force application point and G in the X Y and 2 directions respectively damping coefficient c damping matrix of the spatial model C origin of XYZ origin of XYZ Cu constant matrix converting actual input u into Fe D 1x4 matrix describes a vector from point 1 to point 2 DAC Data Acquisition Controller VI LabVIEW program d generalized coordinates of point C and P respectively with respect to XYZ d generalized coordinate of point C with respect to XYZ d generalized coordinate of point C with respect to XYZ D vector of translational motion of the dynamometer s base F vector of the forces acting on the dynamometer s platform Fe vector of the forcing functi

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