transient response of mechanical structures using modal analysis techniques. angela burgess

301 Pages·1998·5.28 MB·English

imperial College of Science and Technology
(University of London)
TRANSIENT RESPONSE OF
MECHANICAL STRUCTURES
USING MODAL ANALYSIS
TECHNIQUES.
BY
ANGELA BURGESS
A thesis submitted in partial fulfilment for the
degree of Doctor of Philosophy, and for the
Diploma of Imperial College.
Department of Mechanical Engineering, Research at a public
imperial College, London SW7. research establishment.
J a n u a r y 1 9 8 8. ARE(Portland)
c
ABSTRACT
The use of experimental modal analysis techniques together with Fourier
transform methods is considered for their application to the transient response
analysis of structures. The limitations and validity of this approach are
examined for linear structures, and a relationship derived that describes the
errors involved due to time aliasing within the inverse discrete Fourier
transform. The method is demonstrated with a simple beam using several
experimental modal analysis procedures.
The applicability of the same analysis and prediction techniques to non-linear
structures is explored. The use of constant-force stepped-sine excitation and
subsequent circle-fit modal analysis procedures for identifying non-linearities
is reviewed. Also, the reciprocal-of-receptance analysis is extended for
classifying and quantifying the non-linearity present.
Results from non-linear systems subjected to impulse excitation are examined
using various analyses. It is found that clear trends are evident.for different
non-linearities, but they do not correspond to those from a constant-force
stepped-sine test.
Techniques for predicting the transient response for non-linear systems using
data from experimental tests are examined, with the result that several
different approaches are recommended depending on the non-linearity, the
initial conditions and the type and accuracy required of the results. These
prediction methods include using a specific linear model and developing
non-linear models that accurately describe the transient response.
Multi-degree-of-freedom systems with one non-linear element are also
examined, as in many practical structures the non-linearity tends to be
concentrated in a single component. The trends exhibited in the individual
non-linear elements subjected to experimental modal analysis are found to
correspond to those trends found in the resonances of the
multi-degree-of-freedom systems. Similarly, the recommended techniques for
transient response prediction of a non-linear structure correspond to those for
non-linear elements under the same conditions.
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CONTENTS
Page
Abstract. 2
List of tables. 8
List of figures. 10
Acknowledgements. 16
1 INTRODUCTION.
17
1.1 Badground and overview.
18
1.2 Objectives.
18
1.3 Contents of the thesis.
2 A REVIEW OF TRANSIENT RESPONSE ANALYSIS
METHODS.
21
2.0 Introduction.
2.1 Time domain methods. 22
2.1 .l Numerical methods. 22
2.1.2 Duhamet’s integral. 23
2.1.3 Graphical methods. 25
2.2 Frequency domain methods. 26
2.2.1 Shock response spectra. 26
2.2.2 Fourier transform methods. 29
2.3 Discussion. 31
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Page
3 A REVIEW OF FREQUENCY RESPONSE MElHODS
FOR STRUCTURAL MODELLING.
3.0 Introduction. 32
3.1 Summary of essential modal theory. 33
3.2 Experimental testing for frequency response of structures. 37
3.3 Extracting modal properties from experimental frequency
response data. 41
3.3.1 Single-degree-of-freedom (SDOF) analysis. 41
3.3.2 Multi-degree-of-freedom (MDOF) analysis. 43
3.3.3 Time domain analysis. 45
3.4 Application of frequency response data and modal parameters. 46
3.4.1 From modal model to spatial model. 46
3.4.2 Response prediction in the frequency domain. 47
3.4.3 Frequency response coupling for structural assemblies. 47
3.5 Discussion. 49
Figures for chapter 3. 50
4 REQUIREMENTS FOR TRANSIENT RESPONSE
ANALYSIS USING FREQUENCY RESPONSE.
4.0 Introduction. 56
4.1 Fourier transform - from theory to practice. 58
4.2 Limitations of using the ‘Fourier transform method
within the OFT. 61
4.2.1 Frequency aliasing and windowing. 61
4.2.2 Errors due to time aliasing. 62
4.3 Limitations of using the ‘Fourier transform method.
resulting from errors in experimental data. 65
4.3.1 Consequences of linear damping model. 65
4.3.2 Accuracy of experimental data. 66
4.3.3 Effect of the out-of-range modes. 68
4.4 Case study - Transient response analysis of a beam. 70
4.5 Discussion. 73
Tables for chapter 4. 75
Figures for chapter 4. 78
Page
5 RESPONSE CHARACTERISTICS OF NON-LINEAR
ELEMENTS USING MODAL TESTING TECHNIQUES.
5.0 Introduction. 101
5.1 Some examples of non-linear elements found in practice. 103
5.1 .l Stiffness type non-linearities. 103
5.1.2 Damping type non-linearities. 105
5.2 Frequency response measurement of non-linear elements. 107
5.3 Frequency response characteristics from sine tests. 109
5.3.1 Characteristics in the measured Bode plot. 110
5.3.2 Modal parameters obtained from SDOF analysis. 112
5.3.3 Hilbert transform. 115
5.3.4 Characteristics using ‘standard’ sine excitation. 115
5.4 Response characteristics from impulse tests. 116
5.4.1 Characteristics of the time-histories. 116
5.4.2 Data trends in the frequency domain from impulse
excitation. 118
5.5 Comparison of response characteristics in the time domain. 121
5.6 Discussion. 123
Tables for chapter 5. 125
Figures for chapter 5. 136
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Page
6 TRANSIENT RESPONSE PREDICTION OF NON-LINEAR
ELEMENTS USING DATA FROM MODAL TESTS.
6.0 Introduction. 181
6.1 Summary of routes for transient response prediction. 182
6.2 Transient response prediction for a cubic stiffness element. 184
6.3 Transient response prediction for a system with backlash. 187
6.4 Transient response prediction for a system with bi-linear
stiffness. 190
6.5 Transient response prediction for a system with friction. 192
6.6 Transient response prediction for an element with quadratic
viscous damping. 194
6.7 Discussion. 196
Tables for chapter 6. 197
Figures for chapter 6. 198
7 RESPONSE ANALYSIS OF MDOF SYSTEMS WITH A
NON-LINEAR ELEMENT.
7.0 Introduction. 211
7.1 System definition. 212
7.2 Application of prediction techniques for transient response of
MDOF non-linear systems. 213
7.3 Modal analysis of non-linear systems. 215
7.4 Transient response of non-linear systems via the frequency
domain. 218
7.4 Discussion. 220
Tables for chapter 7. 222
Figures for chapter 7. 232
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8 CONCLUDING DISCUSSION.
8.0 Introduction. 252
8.1 Discussion. 252
8.2 Recommendations for transient response predictions. 256
8.3 Suggestions for further research. 260
9 REFERENCES. 261
10 APPENDICES.
Appendix 1 Summary of essential modal theory. 279
Appendix 2 Summary of SDOF modal analysis methods. 284
Appendix 3 MDOF curve fitting for lightly damped structures. 289
Appendix 4 Basic impedance coupling theory. 291
Appendix 5 Free decay for systems with viscous or hysteretic
damping. 292
Appendix 6 Determination of friction damping from
reciprocal-of-receptance analysis. 294
Appendix 7 Development of a non-linear model for the
transient response prediction of a friction
element. 296
7
List of tables
Page
4.1 Modal parameters for theoretical 12 mode structure. 75
4.2 Modal paramers from FRF data obtained using stepped-sine
excitation. 76
4.3 Modal paramers from FRF data obtained using random and
impulse excitations. 77
5.1 Trends in frequency domain displays from non-linear systems
subjected to sine testing. 125
5.2 Various parameters evaluated from a single constant-force
stepped-sine test of a system with cubic stiffness. 126
5.3 Various parameters evaluated from a single constant-force
stepped-sine test of a system with backlash. 127
5.4 Various parameters evaluated from a single constant-force
stepped-sine test of a system with friction. 128
5.5 Various parameters evaluated from a single constant-force
stepped-sine test of a system with quadratic viscous damping. 129
5.6 Trends in frequency domain displays from non-linear systems
subjected to impulse excitation. 130
5.7 Various parameters evaluated from a single impulse test of a
system with cubic stiffness. 131
5.8 Various parameters evaluated from a system with backlash
subjected to an impulse. 132
5.9 Parameters from the ‘modes’ of a bi-linear system subjected to
,
a single impulse. 133
5.10 Various parameters evaluated from a system with friction
subjected to an impulse. 134
5.11 Various parameters evaluated from a system with quadratic
viscous damping subjected to an impulse. 135
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6.1 Data from analysis of systems with backlash type non-linearity
subjected to constant-force stepped-sine tests and reciprocal-
of-receptance analysis using data either above or below
resonance and averaging the two results. 197
7.1 Physical description of each 2 degree-of-freedom system. 222
7.2 Parameters from mode 1 point 1 ,l. 223
7.3 Parameters from mode 2 point 1,l. 224
7.4 Parameters from mode 1 point 1,2. 225
7.5 Parameters from mode 2 point 1,2. 226
7.6 Parameters from mode 1 point 2,l. 227
7.7 Parameters from mode 2 point 2,i. 228
7.8 Parameters from mode 1 point 2,2. 229
7.9 Parameters from mode 2 point 2,2. 230
7.10 Results from using ‘Ident ’type analysis of point (2,2) of the 2
degree-of-freedom system with bi-linear stiffness. 231
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List of figures
Page
2.1 Shock response spectra for a half-sine pulse. 28
3.1 Bode plot using linear frequency axis and log response. 50
3.2 Nyquist display with frequency information included. 50
3.3 Data from sine test - Coarse log frequency sweep followed by
fine linear frequency sweeps around resonance. 51
3.4 Measured FRF from random excitation on a structure, linear
frequency spacing over the full frequency range. 51
3.5 Nyquist analysis. 52
3.6 reciprocal-of-receptance analysis. 53
3.7 Effect of MDOF extension to SDOF analysis. 54
3.8 Effect of residuals on regenerated FRF data. 55
4.1 Common windows and their frequency components. 78
4.2 Comparison of time-histories transformed from SDOF modal
models with equivalent linear damping. 79
4.3 Comparison of time-histories transformed from SDOF modal
models with equivalent linear damping. 80
4.4 Effect of errors in natural frequency on the predicted
time-history of a theoretical 12 mode system. 81
4.5 Effect of errors in the magnitude of the modal constant on the
predicted time-history of a theoretical 12 mode system. a2
4.6 Effect of errors in the phase of the modal constant on the
predicted time-history of a theoretical 12 mode system. 83
4.7 Effect of errors in the damping estimate on the predicted
time-history of a theoretical 12 mode system. 84
4.8 Displacement time-histories from a theoretical 12 mode
system. 85
4.9 Velocity time-histories from a theoretical 12 mode system. 86
4.10 Acceleration time-histories from a theoretical 12 mode
system. 87
4.11 Displacement time-histories from a theoretical 12 mode
system: effect of high frequency modes. 88
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