Estimation of Static Stiffnesses from Free Boundary Dynamic (FRF) Measurements

232 Pages·2014·12.34 MB·English

Estimation of Static Stiffnesses from
Free Boundary Dynamic (FRF) Measurements
A dissertation submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
DOCTORATE OF PHILOSOPHY (Ph.D.)
in the Department of Mechanical and Materials Engineering
of the College of Engineering and Applied Sciences
2014
by
Hasan G. Pasha
Bachelor of Engineering, Thiagarajar College of Engineering, 2001
Master of Technology, Indian Institute of Technology Madras, 2008
Committee Chair: Randall J. Allemang, Ph.D.
November 10, 2014
Abstract
Static structural stiffness is an important criterion in automobile structure design
as it impacts vehicle handling, ride comfort, safety and durability. Traditionally,
automobile manufacturers used special test rigs, which require precise setup and
expensive instrumentation, to estimate stiffness. In the past decade, dynamic fre-
quency response function (FRF) measurements along with modeling techniques
were used to estimate static stiffness. Though test setup time and expense associ-
ated with the instrumentation are considerably reduced, heroic technical measures
in terms of user experience and data processing are required to get reasonable
stiffness estimates.
The premise of this research is that static stiffness information contained in the
measured free-free FRFs can be extracted by utilizing spatial/geometric filtering
and averaging techniques. Two simple and efficient methods to estimate static
stiffness from free-free FRFs are developed in this research. The enhanced Rota-
tional and Bending Compliance Function (eRCF and eBCF) methods presented
here utilize novel tools to enhance response functions and estimate static stiffness;
suchasmassperturbationduringstructuraltesting,spatial/geometricfilteringand
averaging techniques. The stiffness estimates obtained from these methods were
comparable with the results from existing methods, while requiring significantly
less resources.
This dissertation presents the theoretical development of these methods and ex-
perimental validations on a rectangular plate (eRCF and eBCF) and an auto body
(eRCF). Sensitivity studies on the effects of varying boundary conditions, dimen-
sions and parameter estimation bandwidth on torsional and bending stiffness are
presented. The effects of off-centered loading and overhang on stiffness estimates
are also discussed in detail. Finally, the conclusions and recommendations for
future research are presented.
○c Hasan G. Pasha, 2014
All rights reserved
ii
Dedicated to Sara
iii
Acknowledgments
I would like to thank everyone who have helped me immeasurably throughout my
PhD. First and foremost, I would like to acknowledge the contributions and thank
my committee members Dr. Allemang, Dr. Brown, Dr. Kim and Dr. Phillips, for
actively guiding me in every step of my research and for supporting me with their
technical expertise. It has been a great pleasure working with them and learning
from them.
I would like to thank the BMW Manufacturing Company for funding this research
and for allowing me to use their testing equipment. I thank Jeff Poland and
Alex Young for their help during data acquisition on the auto body. I sincerely
thank UC Simulation Center and P&G for assistantships and for providing me
training/accesstovariousengineeringsoftwarethathelpedmeduringmyresearch.
I would also like to thank UC Aeromechanics Research Lab and GE Aviation for
assistantships and support. I am thankful to the UC Graduate School for the
graduate scholarship.
I would like to thank my colleagues at the UC-SDRL for their support and interest
in my research. I would like to thank my Structural Dynamic Analysis course
project team-mates from the classes of 2010 − 2013. The group projects helped
me immensely with the work I performed for my dissertation.
I want to thank my master’s advisor, Dr. Chandramouli Padmanabhan, for his
guidance and for cultivating my desire to work in the field of dynamics. The time
I spent at IITM has been influential and laid a good foundation for my PhD.
Finally, I would like to thank my friends and family: Manish, for the help I
received from him on finite element analysis and the interesting conversations I
hadwithhim; Sara, mybestfriend, forherextraordinarysupport, encouragement,
patience and for reviewing my dissertation tirelessly; mom, dad, brother and sister
for encouraging and supporting me.
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Contents
Abstract i
Acknowledgments iv
List of Figures ix
List of Tables xii
Abbreviations xiv
Symbols xv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Premise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Background – Automotive Analysis Issues 9
2.1 Purpose of the Structure . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Types of Automobile Structures . . . . . . . . . . . . . . . . . . . . 10
2.3 Structural Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Significance of Stiffness for an Automobile Structure . . . . . 13
2.3.2 Static Torsional Stiffness . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Static Bending Stiffness . . . . . . . . . . . . . . . . . . . . 16
2.4 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Basic Global Load Cases . . . . . . . . . . . . . . . . . . . . 18
2.5 Methods to Estimate Static Stiffness . . . . . . . . . . . . . . . . . 19
2.5.1 Finite Element Analysis Method . . . . . . . . . . . . . . . . 20
2.5.1.1 Force Method . . . . . . . . . . . . . . . . . . . . . 20
2.5.1.2 Displacement Method . . . . . . . . . . . . . . . . 20
2.5.2 Static Testing Method . . . . . . . . . . . . . . . . . . . . . 21
v
Contents vi
2.5.2.1 Torsion Test . . . . . . . . . . . . . . . . . . . . . . 22
2.5.2.2 Bending Test . . . . . . . . . . . . . . . . . . . . . 25
2.5.3 Dynamic Testing Method . . . . . . . . . . . . . . . . . . . 27
2.5.3.1 Modal Modeling Approach . . . . . . . . . . . . . . 27
2.5.3.2 Impedance Modeling Approach . . . . . . . . . . . 28
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Background – Structural Test Methods 31
3.1 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Classification: Analytical and Experimental . . . . . . . . . 32
3.1.2 Stages in Experimental Modal Analysis . . . . . . . . . . . . 32
3.2 SDOF Partial Fraction Model . . . . . . . . . . . . . . . . . . . . . 34
3.3 MDOF Partial Fraction Model . . . . . . . . . . . . . . . . . . . . . 35
3.4 Constrained Boundary Test Configurations . . . . . . . . . . . . . . 37
3.5 Mass Perturbation of Structural Tests . . . . . . . . . . . . . . . . . 39
3.6 Spatial/Geometric Filtering of Measured FRFs . . . . . . . . . . . . 41
3.7 Modal Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8 Impedance Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Static Stiffness from Dynamic Measurements 43
4.1 Enhanced Compliance Function Method . . . . . . . . . . . . . . . 44
4.1.1 Enhanced Rotational Compliance Function (eRCF) . . . . . 48
4.1.1.1 Formulation of the eRCF for Torsion Load Case . . 50
4.1.2 Enhanced Bending Compliance Function (eBCF) . . . . . . 53
4.1.2.1 Analysis of a Structure with Two Point-Loads at
Mid-Span Location . . . . . . . . . . . . . . . . . . 54
4.1.2.2 Formulation of the eBCF for Two Point-Loads Case 56
4.1.2.3 General Bending Load Case Producing Symmetric
Deformation . . . . . . . . . . . . . . . . . . . . . 59
5 Experimental Validation 61
5.1 Rectangular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.1 Estimation of Static Torsional Stiffness . . . . . . . . . . . . 63
5.1.1.1 Analytical Method . . . . . . . . . . . . . . . . . . 63
5.1.1.2 EnhancedRotationalComplianceFunction(eRCF)
Method . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.2 Estimation of Static Bending Stiffness . . . . . . . . . . . . 71
5.1.2.1 Analytical Method . . . . . . . . . . . . . . . . . . 71
5.1.2.2 Enhanced Bending Compliance Function (eBCF)
Method . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 78
5.1.3.1 Effect of Varying Boundary Conditions on Tor-
sional Stiffness . . . . . . . . . . . . . . . . . . . . 78
5.1.3.2 Effect of Dimension Variation on Torsional Stiffness 80
5.1.3.3 Static Torsional Stiffness Estimation Sensitivity . . 83
Contents vii
5.1.3.4 Static Bending Stiffness Estimation Sensitivity . . 88
5.1.3.5 Effect of Varying Width on Bending Stiffness . . . 95
5.1.3.6 Effect of Edge Moment(s) on Bending Stiffness . . 99
5.2 Auto Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 Estimation of Static Torsional Stiffness . . . . . . . . . . . . 103
5.2.1.1 Static Testing Method . . . . . . . . . . . . . . . . 103
5.2.1.2 EnhancedRotationalComplianceFunction(eRCF)
Method . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.2.1 Parameter Estimation Sensitivity . . . . . . . . . . 111
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Conclusions and Recommendations for Future Work 118
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . 120
6.2.1 SVD based Spatial/Geometric Filtering Approach . . . . . . 120
6.2.2 Adapting the eBCF Method to Estimate Bending Stiffness
for General Load Cases . . . . . . . . . . . . . . . . . . . . . 121
6.2.3 Bending Stiffness for an Auto Body . . . . . . . . . . . . . . 122
6.2.4 Effect of Signal Processing Parameters on Stiffness Estimates123
6.2.5 Autonomous Parameter Estimation Process . . . . . . . . . 124
A Model Calibration and Validation of a Rectangular Plate 125
A.1 Verification and Validation (V&V) . . . . . . . . . . . . . . . . . . 126
A.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2.1 Analytical Modeling and Formulation . . . . . . . . . . . . . 128
A.2.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.3.1 Analytical Modal Analysis . . . . . . . . . . . . . . . . . . . 131
A.3.2 Experimental Modal Analysis . . . . . . . . . . . . . . . . . 132
A.3.3 Modal Correlation . . . . . . . . . . . . . . . . . . . . . . . 133
A.3.4 Model Calibration . . . . . . . . . . . . . . . . . . . . . . . 134
A.3.4.1 Model Updating . . . . . . . . . . . . . . . . . . . 135
A.3.4.2 Testing Related Changes . . . . . . . . . . . . . . . 135
A.3.4.3 Modal Correlation after Calibrating the FE Model 137
A.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.4.1 Perturbed Mass Modal Analysis . . . . . . . . . . . . . . . . 140
A.4.2 Constrained Boundary Condition Modal Analysis . . . . . . 142
A.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
B Static Stiffness Estimation using Impedance Modeling Method 145
B.1 Impedance Modeling Method . . . . . . . . . . . . . . . . . . . . . 145
B.1.1 Formulation of Impedance Modeling Methods . . . . . . . . 146
Contents viii
B.1.1.1 Dynamic Stiffness Method . . . . . . . . . . . . . . 147
B.1.1.2 Compliance Method . . . . . . . . . . . . . . . . . 149
B.1.1.3 Hybrid Impedance Modeling . . . . . . . . . . . . . 152
B.1.2 Stiffness Estimation from Impedance Modeling . . . . . . . . 155
B.1.2.1 Torsional Stiffness . . . . . . . . . . . . . . . . . . 156
B.1.2.2 Bending Stiffness . . . . . . . . . . . . . . . . . . . 158
B.2 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2.1 Rectangular Plate . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2.2 Auto Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
C Overview on General Bending Load Cases 170
C.1 Effect of Applying Off-Centered Loading . . . . . . . . . . . . . . . 172
C.2 Effect of Introducing Overhang on Both Sides of the Supports . . . 176
C.3 Effect of Removing Overhung Material on Both Sides of the Supports178
C.4 Effect of Sliding the Supports and Loading Locations (a double side
unequal overhang) . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
C.5 Effect of Introducing Overhang on One Side of the Supports . . . . 185
C.6 Effect of Removing Overhung Material on One Side of the Supports 188
C.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
D Techniques for Synthesizing FRFs from Analytical Models 192
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
D.2 Synthesizing FRFs from Analytical Models . . . . . . . . . . . . . . 194
D.2.1 Full Space Method . . . . . . . . . . . . . . . . . . . . . . . 194
D.2.2 Synthesizing FRFs from Reduced Order Models . . . . . . . 195
D.2.3 Modal Superposition Method . . . . . . . . . . . . . . . . . 196
D.2.4 Implementation Details . . . . . . . . . . . . . . . . . . . . . 196
D.2.4.1 Retrieving System Matrices with ANSYS(cid:114) Work-
bench . . . . . . . . . . . . . . . . . . . . . . . . . 197
D.2.4.2 Computing the FRF . . . . . . . . . . . . . . . . . 198
D.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
D.3.1 Modeling Damping with ANSYS(cid:114) Workbench . . . . . . . . 199
D.4 Frequency Response Assurance Criteria (FRAC) . . . . . . . . . . . 200
D.5 Experimental Example . . . . . . . . . . . . . . . . . . . . . . . . . 201
D.5.1 FRF Comparison . . . . . . . . . . . . . . . . . . . . . . . . 201
D.5.2 Effect of Structural Damping . . . . . . . . . . . . . . . . . 202
D.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
References 205