STRATEGIES FOR NON-LINEAR SYSTEM IDENTIFICATION

192 Pages·2010·8.81 MB·English

STRATEGIES FOR NON‐LINEAR SYSTEM 
IDENTIFICATION 

A thesis submitted for the degree of
Doctor of Philosophy

By
Aditya Chandrashekhar Gondhalekar

Department of Mechanical Engineering
Imperial College London/ University of London
October, 2009
Abstract
The thesis presents different strategies to detect, characterize, and identify localized
and distributed non-linearities in practical engineering structures. The formulations
presented in the thesis work in the frequency domain and based on first-order
describing functions to express the non-linearities.
The novel idea of ‘non-linear force footprints’ is proposed to characterize the
type of non-linearity. A library containing footprints of different non-linearities like
cubic stiffness, clearance, and friction is compiled. This library can be used as a
look-up chart to subjectively identify the type of non-linearity in a structure. A shape-
matching algorithm is proposed to numerically compare the extracted non-linear
restoring force with the footprints from the library. This provides an automatic
identification of the non-linearity type.
Three different methods are proposed for the parametric identification of
non-linear systems. All these methods extract the non-linear restoring force, identify
the location of non-linearity, and finally estimate the non-linear parameters via a
genetic algorithm optimization. The first method uses an FE model of the underlying
linear system; while the second and the third methods use a modal model and a
response model of the underlying linear system respectively. The performance of
the three methods for a range of criteria is evaluated on common ground by using
simulated data for a representative engineering structure with localized non-
linearities.
An experimental study is undertaken on the so-called MACE structure sub-
assembly with the aim to identify non-linearities in an actual industrial structure.
Different methods proposed in this thesis, and also from the literature, are used to
detect the presence of non-linearity, and identify its type. The response of the
structure at different excitation amplitudes is measured using step-sine excitation
with constant force. An attempt is made to estimate the non-linear parameters using
the methods proposed in the thesis. It has been found that the methods accurately
detect, locate and characterize the non-linearities.
A three-stage strategy is presented for the identification of non-linear
parameters, for base-excited structures. The method is illustrated on a pyramid-like
structure by using simulated measurement data. It has been found that if the non-
linearities in the system are independent of the mass matrix; the non-linear
parameters can be accurately identified using the proposed method.

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Acknowledgements
I would like to express my sincere gratitude to my supervisors, Professor Mehmet
Imregun and Dr Evgeny Petrov, for their guidance, encouragement and support
throughout the research. It was a great honour and privilege to work with you both.
I am indebted to the Department of Mechanical Engineering for providing me the
opportunity to work in a world class research environment, and for funding my
studies. I would also like to thank the Higher Education Funding Council for England
(HEFCE) for partially contributing towards my tuition fees during the study.
Special thanks to Ms Vernette Rice for helping me in administrative matters during
my stay. Her politeness and understanding nature is worthy of an ovation.
I would like to show appreciation to Mr D. Robb and Dr Christoph Schwingshackl for
their constructive comments and suggestions during the experimental work.
I would like to acknowledge the Atomic Weapons Establishment (AWE) UK, for
providing a realistic non-linear structure for the research. I would like to thank Dr
Philip Ind, Mr Tony Moulder, Mr Alex Humber, and others who were involved in the
collaboration.
I wish to say a word of thanks to my colleagues and friends of Dynamics Section
and VUTC for making my stay at Imperial College a memorable experience. I would
particularly like to thank Dr Luca de Mare, Dr Mehdi Vahdati, Mr Armel De
Montgros, and Mr Yum Ji Chan. Big thanks to Mr Sanjay Mata and Mr Davendu
Kulkarni, who were always ready for a cup of coffee. The long discussions we had
around the coffee table acted as a stress-buster in many cases. Thanks to the
system administrators, Dr Sridhar Dhandapani and Dr Zachariadis Zacharias-
Ioannisthe, for promptly solving any computer related issues. I appreciate the help
of Ms Nina Hancock and Mr Peter Higgs in administrative issues within the centre.
A special word of thanks to the ‘94 Ellington road’ bunch, Mr Aditya Karnik, Mr
Sandeep Saha, and Mr Srikrishna Sahu. The time spent with you all, is surely
amongst the best in my life.
Finally, words would never be enough to thank my beloved wife, Kanchan, for her
un-conditional love, encouragement, and support through thick and thin of my
research.
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To my Parents,
with love and admiration

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Nomenclature
Latin symbols
a Binary multiplier corresponding to the ith non-linearity
i
c Artificial noise percentage in data
[C] Viscous damping matrix
[D] Proportional damping matrix
{F} Excitation force vector
{Fd} Excitation force vector for base-excited systems
{g} Non-linear restoring force vector in time-domain
gcub Non-linear force related to cubic stiffness non-linearity
gcle Non-linear force related to clearance non-linearity
gfri Non-linear force corresponding to friction non-linearity
{G} Non-linear restoring force vector in frequency-domain
[K] Stiffness matrix
K Tangential stiffness for stick friction
d
K Clearance stiffness
z
m Number of measured degrees of freedom
[M] Mass matrix
n Number of non-linear parameters
p
n Number of frequency lines in measurement
f
N Total number of degrees of freedom
N Reduced number of degrees of freedom
r
{Pd} Pseudo excitation force vector
r Index representing mode number
R Residual quantity to be minimized in optimization
{~ }
Res Vector containing non-linear residual
t Time
u Number of un-measured degrees of freedom
{U} Vector of relative displacement amplitude
YB Y-coordinate of benchmark curve
YE Y-coordinate of extracted non-linear force curve
{y} Displacement vector in time-domain
y Clearance gap distance
c
{Y} Displacement amplitude vector
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Greek letters
[ ]
α Receptance matrix
[ ]
Z Dynamic stiffness matrix
β Coefficient for cubic stiffness non-linearity
ε Error in estimation
η Modal damping ratio
[ ]
λ Complex eigenvalues matrix
µ Coefficient of friction
φ Vector containing ith column from mode shape matrix
i
[ ]
Φ Mode shape matrix
ω Excitation Frequency
ω Resonance frequency of the rth mode
r
{ }
χ Non-linear modal vector
Subscripts
c Index representing the measurement location close to
non-linear degree of freedom.
e Index representing excitation degree of freedom
m Index representing measured degrees of freedom
max Index representing the maximum value of a variable
M Index representing the identified modes
r
nl Index representing non-linear degree of freedom
u Index representing un-measured degrees of freedom
U Index representing the un-identified modes
r

Superscripts
T Transpose of a matrix
-1 Inverse of a square matrix
+ Pseudo-inverse of a rectangular matrix

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Abbreviations
1D, 2D,.. One dimensional, two dimensional,...
CFD Computational fluid dynamics
DOF Degree of freedom
DFM Describing function method
FEA Finite element analysis
FEM Finite element method
FFT Fast Fourier transform
FRF Frequency response function
GA Genetic algorithm
HBM Harmonic balance method
HMT Hybrid modal technique
I-HMT Improved hybrid modal technique
LMA Linear modal analysis
MDOFs Multi-degrees of freedom
NMG Non-linear modal grade
NMV Non-linear modal vector
POD Proper orthogonal decomposition
SDOF Single-degree of freedom
SSD Sum of squared distance

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Contents
Abstract ................................................................................................................................. 2 
Acknowledgements ............................................................................................................ 3 
Nomenclature ....................................................................................................................... 5 
Abbreviations ....................................................................................................................... 7 
List of Figures .................................................................................................................... 12 
List of Tables ...................................................................................................................... 16 
Chapter 1 ............................................................................................................................. 18 
Introduction ........................................................................................................................ 18 
1.1 Background of the problem ..................................................................................... 18 
1.2 Non-linear system identification ............................................................................. 19 
1.3 The describing function method ............................................................................. 20 
1.4 Objectives of the thesis ............................................................................................ 21 
1.5 Organization of the thesis ........................................................................................ 22 
Chapter 2 ............................................................................................................................. 24 
Literature Survey ............................................................................................................... 24 
2.1 Introduction ................................................................................................................ 24 
2.2 Non-linear system identification ............................................................................. 25 
2.2.1 Non-linearity detection ...................................................................................... 27 
2.2.2 Non-linearity characterization .......................................................................... 28 
2.2.3 Non-linear parameter extraction: Spatial methods ....................................... 29 
2.2.4 Non-linear parameters extraction: Modal methods ...................................... 31 
2.2.4 Model updating for non-linear systems .......................................................... 33 
2.3 Non-linear response prediction............................................................................... 35 
2.4 Non-linear structural dynamics: Real life structures ............................................ 39 
2.5 Summary of the literature review ........................................................................... 41 
Chapter 3 ............................................................................................................................. 43 
Non-linearity type identification using a ‘footprint library’ .................................... 43 
3.1 Introduction ................................................................................................................ 43 
3.2 Generation of non-linear force curves ................................................................... 44 
3.3 Footprints of different non-linearities ..................................................................... 46 
3.3.1 Cubic stiffness non-linearity ............................................................................. 47 
3.3.2 Clearance non-linearity..................................................................................... 48 
3.3.3 Friction non-linearity .......................................................................................... 49 
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3.3.4 Combination of different non-linearities ......................................................... 50 
3.4 Quantitative comparison using shape-matching algorithm ................................ 55 
3.4.1 Generation of footprint curves ......................................................................... 55 
3.4.2 Shape-matching algorithm ............................................................................... 57 
3.4.3 Universal applicability of the footprint comparison ....................................... 59 
3.4.4 Numerical examples ......................................................................................... 64 
3.5 Concluding remarks ................................................................................................. 68 
Chapter 4 ............................................................................................................................. 71 
Spatial method for non-linear parameter identification .......................................... 71 
4.1 Introduction ................................................................................................................ 71 
4.2 Theoretical formulation ............................................................................................ 72 
4.2.1 Non-linearity detection and characterization ................................................. 73 
4.2.2 Formulation of the optimization problem ....................................................... 73 
4.2.3 Search for the non-linear parameters using a genetic algorithm ............... 75 
4.2.4 Use of binary multipliers to improve the efficiency ....................................... 77 
4.2.5 Non-linear parameter identification without full measurement set at non-
linear DOFs .................................................................................................................. 78 
4.3 Numerical examples ................................................................................................. 79 
4.3.1 Effect of the number of measurements on parameter estimation .............. 80 
4.3.2 Effect of measurement noise on parameter estimation ............................... 83 
4.3.3 Effect of binary multipliers on computational efficiency ............................... 85 
4.3.4 Effect of the error in the FE model on parameter estimation ...................... 87 
4.3.5 Non-linear parameter identification in absence of measurements at non-
linear DOFs .................................................................................................................. 89 
4.4 Concluding remarks ................................................................................................. 90 
Chapter 5 ............................................................................................................................. 92 
Non-linear identification using modal models and response models ................ 92 
5.1 Introduction ................................................................................................................ 92 
5.2 Improved hybrid modal technique (I-HMT) ........................................................... 93 
5.2.1 Theoretical formulation ..................................................................................... 93 
5.2.2 Quantification of the level of non-linearity ..................................................... 96 
5.2.3 Decoupling the non-linear modal vector ........................................................ 97 
5.2.4 Comments on the non-linear residual ............................................................ 99 
5.2.5 Numerical validation of I-HMT method ......................................................... 102 
5.3 FRF-based method ................................................................................................ 107 
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5.3.1 Theoretical formulation ................................................................................... 107 
5.3.2 Conditioning of the FRF matrix ..................................................................... 109 
5.3.3 Numerical validation of the FRF based method ......................................... 111 
5.4 Concluding remarks ............................................................................................... 115 
5.4.1 Comments on the I-HMT method ................................................................. 115 
5.4.2 Comments on the FRF-based method ......................................................... 115 
Chapter 6 ........................................................................................................................... 117 
Comparison of non-linear parameter identification methods ............................. 117 
6.1 Introduction .............................................................................................................. 117 
6.2 Test cases for the comparison ............................................................................. 119 
6.3 Pre-processing of the input data .......................................................................... 120 
6.3.1 Non-linear displacement measurements on the structure ........................ 120 
6.3.2 Input models for different methods ............................................................... 122 
6.4 Results of non-linear parameter identification for Case A ................................ 123 
6.4 Results of non-linear parameter identification for Case B ................................ 128 
6.5 Discussion and concluding remarks .................................................................... 134 
Chapter 7 ........................................................................................................................... 137 
Experimental investigation of non-linearities in MACE structure ...................... 137 
7.1 Introduction .............................................................................................................. 137 
7.1.1 Choice of the structure ................................................................................... 137 
7.1.2 Description of joints in Sub-assembly 3 ....................................................... 138 
7.1.3 Objectives of the exercise .............................................................................. 139 
7.2 Validation of the FE models .................................................................................. 140 
7.3 Detection of non-linear behaviour ........................................................................ 144 
7.3.1 Problem of force-drop near resonance ........................................................ 144 
7.3.2 Non-linearity detection with constant-force step-sine tests ...................... 147 
7.4 Characterization of the non-linearity .................................................................... 150 
7.4.1 Identifying the location non-linearity ............................................................. 151 
7.4.2 Identifying the type of non-linearity ............................................................... 152 
7.4.3 Estimation of non-linear parameters ............................................................ 156 
7.5 Concluding remarks ............................................................................................... 160 
Chapter 8 ........................................................................................................................... 161 
Non-linear parameter identification for base-excited structures ....................... 161 
8.1 Introduction .............................................................................................................. 161 
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