DETECTION OF STRUCTURAL NONLINEARITIES USING THE FREQUENCY
RESPONSE AND COHERENCE FUNCTIONS
A thesis submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
in the Department of Mechanical Engineering
of the College of Engineering
2000
by
Thomas Roscher
Dipl.-Ing. (FH), Hochschule fuer Technik und Wirtschaft, Dresden, 1999
Committee Chair: Dr. Randall J. Allemang
ABSTRACT
It has been observed during modal tests of a structure, that high forcing level cause the Frequency
Response Function (FRF) estimates to show non-coherent behavior over certain frequency bands due to
non-linearities. This investigation on a multiple degree of freedom (MDOF), multi-connected simulation
model is intended to find corresponding non-linearities for the above-mentioned problems, whereas FRFs
and Coherence (COH) functions are the objects in focus. Special attention is paid to the typical testing
case, where broadband techniques are used and the presence of noise has to be assumed. In detail, these
testing cases include single input multiple output (SIMO) and multiple input multiple output (MIMO)
situations. The effects of cubic, softening, deadzone, hardening/softening, softening/hardening
displacement related non-linearities will be analyzed. Furthermore, velocity related non-linearities like
quadratic, non-symmetric and coulomb damping terms are applied to the simulation model. In addition, a
technique for lumped mass systems will be introduced which is capable of eliminating the effects of
non-linear motion between two DOFs, making it possible to separate structural non-linearity induced
distortions from measurement noise and digital signal processing errors.
ACKNOWLEDGEMENT
This work was done at the University of Cincinnati Structural Dynamics Research Laboratory (UC-SDRL).
I would like to thank all members of the UC-SDRL for their help and support throughout my work. I
appreciated all formal and informal discussions (because that is, what keeps progress alive); all suggestions
and honest criticism, which offered different points of view on the subject and by these means, shaped the
thesis in a crucial way.
I would also like to address my special thanks to my advisor and committee chair Dr. Randall Allemang.
His guidance and council were invaluable, even before I was taking on my thesis work and continued all
along the entire study. Without him, and that is true, this project would not have been possible. I also want
to express my gratitude to all members of my thesis committee, Dr. Edward Berger, Dr. Dave Brown and
Dr. Allyn Phillips. Their ideas, comments and suggestions provided a significant share to the final result.
Let me express my feelings in only one sentence: I consider myself lucky to have met all of you.
TABLE OF CONTENTS
1
MOTIVATION AND OVERVIEW .............................................................................................. 1
2
LINEAR AND NON-LINEAR VIBRATION............................................................................... 1
2.1
2.2
3
Linear Theory and Modal Analysis .............................................................................................. 1
Non-Linear Overview................................................................................................................... 1
SIMULATION CONSIDERATIONS ........................................................................................... 1
3.1
3.2
4
Simulation Model ......................................................................................................................... 1
Numerical Precision...................................................................................................................... 1
APPLICATION OF NON-LINEAR INTERACTIONS.............................................................. 1
4.1
4.2
Combined Coherence Function .................................................................................................... 1
Characteristic Effects of Non-Linearities ..................................................................................... 1
4.2.1 Hardening Stiffness.............................................................................................................. 1
4.2.2 Softening Stiffness ............................................................................................................... 1
4.2.3 Non-Symmetric Stiffness ..................................................................................................... 1
4.2.4 Deadzone/ Play..................................................................................................................... 1
4.2.5 Non-Linear Damping ........................................................................................................... 1
4.3 Non-Linear Effects in the Presence of Noise................................................................................ 1
5
SUMMARY/ FUTURE WORKS................................................................................................... 1
6
REFERENCE LIST ........................................................................................................................ 1
7
APPENDIX ...................................................................................................................................... 1
7.1
7.2
7.3
SIMULINK® Model ..................................................................................................................... 1
Definition of Physical Parameters, 4-DOF-Model, Linear ........................................................... 1
List of Cases in Section 4.2 .......................................................................................................... 1
I
LIST OF FIGURES
Figure 2-1: Transfer Function ........................................................................................................................1
Figure 2-2: Superposition Principle ...............................................................................................................1
Figure 2-3: Types of Non-Linearities ............................................................................................................1
Figure 2-4: Reduction of Linear Stiffness Factors in Serial Connection .......................................................1
Figure 3-1: 4-DOF-Model, a) Physical Scheme, b) Connection Scheme ......................................................1
Figure 3-2: Frequency Response Function, Linear Model.............................................................................1
Figure 3-3: Frequency Response H32 and Error Estimate (|tol| = 1⋅10-5/1⋅10-8m) ..........................................1
Figure 3-4: Response x4(t) for different Tolerances and Error Estimation, One Ensemble ...........................1
Figure 3-5: Error Estimate RMS(dx4) ............................................................................................................1
Figure 4-1: a) MDOF System, b) Internal and External Forces on MDOF System.......................................1
Figure 4-2: a) SDOF System, b) Linearizing Concept for Cubic Stiffness....................................................1
Figure 4-3: FRFs and COH, Hardening Stiffness, Inp. 3, k13 .......................................................................1
Figure 4-4: Coherence and Combined Coherence, Hardening Stiffness, Inp. 3, k13.....................................1
Figure 4-5: FRFs and COH, Hardening Stiffness, Inp. 2, k13 .......................................................................1
Figure 4-6: Coherence and Combined Coherence, Hardening Stiffness, k13................................................1
Figure 4-7: FRFs and COH, Hardening Stiffness, Inp. 2, k13 .......................................................................1
Figure 4-8: Correlation between {H12} and {H32} .........................................................................................1
Figure 4-9: Coherence and Combined Coherence, Hardening, Inp. 1, k23 ...................................................1
Figure 4-10: FRFs and COH, Hardening Stiffness, Inp. 1, k23 .....................................................................1
Figure 4-11: FRFs and MCOH, Hardening Stiffness, Inp. 1+2, k23 .............................................................1
Figure 4-12: Multiple Coherence and Combined Coherence, Hardening Stiffness, k23 ...............................1
Figure 4-13: FRFs H21 and H31, MCOH, Hardening Stiffness, Inp. 1+2, k23 ...............................................1
Figure 4-14: a) SDOF System, b) Linearizing Concept for Softening Stiffness............................................1
Figure 4-15: Ordinary and Combined Coherence, Softening Stiffness, Inp. 1, k23 ......................................1
Figure 4-16: FRFs and COH, Softening Stiffness, Inp. 1, k23 ......................................................................1
Figure 4-17: FRFs and MCOH, Softening Stiffness, Inp. 1+2, k23...............................................................1
Figure 4-18: Multiple Coherence and Combined Coherence, Softening Stiffness, Inp. 1+2, k23 .................1
Figure 4-19: FRFs and Coherence, Softening Stiffness, Inp. 3, k3all............................................................1
Figure 4-20: Ordinary and Combined Coherence, Softening Stiffness, Inp. 3, k3all ....................................1
Figure 4-21: Linearizing Concept for Non-Symmetric Stiffness ...................................................................1
II
Figure 4-22: SDOF System with Hardening/Softening Stiffness ..................................................................1
Figure 4-23: FRFs and COH, Hardening/Softening Stiffness, Inp. 2, k13 ....................................................1
Figure 4-24: Ordinary and Combined Coherence, Hardening/Softening Stiffness, Inp. 2, k13.....................1
Figure 4-25: FRFs and MCOH, Hardening/Softening Stiffness, Inp. 1+2, k23.............................................1
Figure 4-26: Multiple and Combined Coherence, Hardening/Softening, Inp 1+2, k23 .................................1
Figure 4-27: FRFs and COH, Hardening/Softening Stiffness, Inp. 3, k3all ..................................................1
Figure 4-28: Ordinary and Combined Coherence, Hardening/Softening Stiffness, Inp. 3, k3all...................1
Figure 4-29: a) SDOF System, b) Linearizing Concept.................................................................................1
Figure 4-30: FRFs and COH, Deadzone, Inp. 2, k13 .....................................................................................1
Figure 4-31: Distortion Correlation for Deadzone, k13 .................................................................................1
Figure 4-32: Ordinary and Combined Coherence, Deadzone, Inp. 2, k13 .....................................................1
Figure 4-33: FRFs and COH, Deadzone, Inp. 3, k3all...................................................................................1
Figure 4-34: Ordinary and Combined Coherence, Deadzone, Inp. 3, k3all...................................................1
Figure 4-35: FRFs and COH, Quadratic Damping, Inp. 1, c13 .....................................................................1
Figure 4-36: FRFs and COH, Softening/Hardening Damping, Inp 1, c13.....................................................1
Figure 4-37: Ordinary and Combined Coherence, Coulomb Friction, Inp. 1, c13.........................................1
Figure 4-38: FRFs and COH, Coulomb Friction, Inp. 1, c13 ........................................................................1
Figure 4-39: FRF Estimation as Function of Spectral Averages....................................................................1
Figure 4-40: H21 and H31 as Function of Averages, Softening Stiffness, Inp. 1 ............................................1
Figure 4-41: Ordinary and Combined Coherence, Softening Stiffness, Inp. 1 ..............................................1
III
NOMENCLATURE
a
.........................................................
C, c
..........................................
f
.........................................................
Force (time domain), Frequency
fd, fr
.........................................................
Damping, Restoring Force
Δf
.........................................................
Frequency Resolution
F
.........................................................
Force (frequency domain)
GFF
.........................................................
Power Spectrum, Input
GXF
.........................................................
Cross Power Spectrum
GXX
.........................................................
Power Spectrum, Output
h
.........................................................
Stepsize
H
............................
j
.........................................................
K, k
......................................................... Stiffness Matrix, Stiffness Element
M, m
.........................................................
Mass Matrix, Mass Element
p
.........................................................
Output Location
q
.........................................................
Input Location
r
.........................................................
Correlation Coefficient
s
.........................................................
Non-linear Scaling Factor
Δt
.........................................................
Sample Time
T
.........................................................
Time Period
Δv
.........................................................
Relative Velocity
x
.........................................................
Displacement
x&
.........................................................
Velocity
&x&
.........................................................
Acceleration
X
.........................................................
Response (frequency domain)
Δx
.........................................................
Relative Displacement
ε
.........................................................
Non-linear Scaling Factor
ω
.........................................................
circular Frequency
IV
Non-linear Scaling Factor
Damping Matrix, Damping Element
Frequency Response between Input q and Output p
Index, Imaginary Unit
COH
.........................................................
Ordinary Coherence Function
CCOH
.........................................................
Combined Coherence Function
DOF
.........................................................
Degree of Freedom
FRF
.........................................................
Frequency Response Function
MCOH
.........................................................
Multiple Coherence Function
MCCOH
..........................................
MDOF
.........................................................
Multi Degree of Freedom
MIMO
.........................................................
Multiple Input, Multiple Output
SDOF
.........................................................
Single Degree of Freedom
SIMO
.........................................................
Single Input, Multiple Output
RMS
.........................................................
Root Mean Square
V
Multiple Combined Coherence Function
Detection of Non-Linearities
Motivation and Overview
1 MOTIVATION AND OVERVIEW
Experimental and analytical modal analysis, as known today, is based on several assumptions, which all
apply the powerful theory of linear algebra. One of the most important assumptions to be aware of is
linearity. Only when the structure exhibits linearity in the frequency range of excitation will the theory of
linear algebra be applicable and yield valid information. If, on the other hand, the structure does not behave
according to the linearity assumption, serious errors will result. Even though the frequency response
function may appear very smooth after many averages are taken, the corresponding coherence function will
show drops over a range of frequencies, which are believed to be caused by structural non-linearities. The
intention of this thesis is to investigate the effects of structural non-linearities on the frequency response
and coherence functions. This thesis will also investigate methods for distinguishing between errors
induced by digital signal processing errors (e.g. leakage) and the effects of structural non-linearities. A
study will be performed by means of a simulation model with several degrees of freedom, which provides
the advantage of knowing the parameters of the structure under consideration exactly and therefore
allowing an interpretation of results.
The next Chapter briefly reviews linear theory and the application of linear theory in modal analysis. In
addition, a short introduction to non-linear vibration and phenomena will be given. Chapter 3 focuses on
the set up of the simulation model and analyzes the issue of numerical precision. Chapter 4 introduces a
method to detect structural non-linearities by eliminating the effects of non-linear motion in the
measurements between degrees of freedom. Also, in Chapter 4 non-linear elements are applied to the
system in order to investigate their characteristic effects on the frequency response and coherence
functions, first in a noise-free environment and second when the presence of measurement noise has to be
1
Detection of Non-Linearities
Motivation and Overview
assumed. Chapter 5 concludes and summarizes the work, giving recommendations for future work. A
reference list is provided in Chapter 6 and Chapter 7, as appendix, contains pertinent information to
complete the thesis.
2
Detection of Non-Linearities
Linear and Non-Linear Vibration
2 LINEAR AND NON-LINEAR VIBRATION
2.1
Linear Theory and Modal Analysis
For mechanical systems, as for all other systems, the input-output relationship is usually of strong interest
and is foremost in determining system performance under given service conditions. In the field of
vibrations, the typical inputs applied are forces and moments, whereas the system response will be
measured in accelerations, velocities or displacements. A schematic picture of a dynamic system is shown
in Figure 2-1. Note, that no restriction in terms of system behavior is assumed.
{F}
[H]
{X}
Input
System
Output
Figure 2-1: Transfer Function
In most practical, real-world cases the analytical approach to determining the system properties, without
knowledge of outputs versus given inputs, will not lead to successfully identifying the system parameters.
Therefore, corresponding inputs and system responses will be used to estimate the structure’s behavior
indirectly. In order to proceed with that indirect technique, referred to as modal analysis, some restrictions
have to be applied. Specifically, the structure is assumed to obey linearity, reciprocity, observability and
time invariance.
3
Detection of Non-Linearities
Linear and Non-Linear Vibration
If time invariance is assumed, the same test performed at a later time should provide the same system
properties as the first test. Typically, the critical conditions for a structure, with respect to time invariance,
are the property changing effects of different temperature, humidity and life cycle. Since it is known that
most structures are indeed dependent on the surrounding conditions, the tests to determine the unknown
system properties are usually performed under the anticipated operating conditions. Therefore time
invariance is a reasonable and often valid assumption.
Observability is satisfied if enough measurements are taken to describe the motion of the structure
completely. The level of observability necessary usually depends on the model chosen to describe the
structure and on the frequency range of interest. It requires knowledge of the information wanted,
sometimes a pretest and sufficient experience are necessary to choose the right number of measurements to
be acquired during the modal analysis to conform to the assumption of observability.
The important principal of reciprocity expresses the independence of path, stating that the relationship, or
transfer function, between input q and output p on a given structure will yield the exact same relationship,
or transfer function, as if point p is used as input and point q as output.
Furthermore, in order to apply the techniques of modal analysis, linearity has to hold true (besides the
assumptions of time invariance, observability, and reciprocity). Even if non-linear behavior of all real
structures is well accepted, the linearity assumption can be made for most of the testing conditions since
the error introduced by that assumption will be small and therefore negligible. In order to confirm linearity,
not only a reciprocity check has to be performed but also the principle of superposition must be true. If one
imagines exciting a system with a certain input F1 at point q which will result in a response X1 at point p,
furthermore exciting point q in a separate test by F2 resulting in X2 at point p, superposition then states, that
4
Detection of Non-Linearities
Linear and Non-Linear Vibration
input (F1 + F2) at point q yields (X1 + X2) at point p for a given structure. Figure 2-2 shows a graphical
interpretation of the principle of superposition.
X2
X1
p
(X1 + X2)
p
p
+
q
q
F2
F1
=
q
(F1 + F2)
Figure 2-2: Superposition Principle
In order to model a system, if the assumption of linearity is valid, the equations of motion for a given
multi-degree of freedom (MDOF) structure can be written, in the time-domain, as:
[M ]{&x&(t )}+ [C ]{x& (t )}+ [K ]{x(t )} = { f (t )}
(2-1)
Taking the Fourier Transform of Equation (2-1) and therefore mapping a set of differential equations from
the time domain to an equivalent set of algebraic equations in the frequency domain will yield:
− ω 2 [M ]{X (ω )} + jω [C ]{X (ω )} + [K ]{X (ω )} = {F (ω )}
(2-2)
It is important to realize, that response vector {X} and reference vector {F} in Equation (2-2) are frequency
and not time dependent. Exploiting the algebraic advantages one can simplify Equation (2-2) to:
[− ω
2
[M ] + jω [C ] + [K ]]{X (ω )} = {F (ω )}
[B(ω )]{X (ω )} = {F (ω )}
5
(2-3)
Detection of Non-Linearities
Linear and Non-Linear Vibration
Matrix [B] in Equation (2-3), the so called impedance matrix, contains all system information and relates
responses {X} to external forces {F}. Since one is usually more interested in relating applied external forces
to the corresponding system responses, the inverse of the impedance matrix has to be found:
{ X (ω )} = [ B (ω )] −1 {F (ω )}
{ X (ω )} = [ H (ω )]{F (ω )}
(2-4)
As mentioned above, the system properties for a chosen model are generally unknown in practical
applications and an analytical solution for the components of matrix [H], the frequency response function
(FRF), in Equation (2-4) can not be found. Therefore, using an indirect approach, the FRF matrix [H] can
be estimated by exciting the system and measuring corresponding responses. In this sense, Equation (2-4)
will be rewritten so that measured responses {X} will be normalized by the input vector {F}, yielding the
frequency response [H]:
[ H (ω )] =
{ X (ω )}
{F (ω )}
(2-5)
It has to be emphasized, that the outlined theory is only valid under the assumption of linearity, since the
equations of motion in Eq. (2-1) used as starting point, a force equilibrium formulation, only express linear
relationships.
In real world experiments, a more practical approach is used to determine the frequency response function.
The next section, therefore, discusses modal analysis techniques and signal processing issues regarding the
estimation of the frequency response function (FRF) and coherence function (COH). Since these
techniques are well known and described, the review will be concise. For more information, the reader is
6
Detection of Non-Linearities
Linear and Non-Linear Vibration
pointed to the appropriate literature, references [6], [9], [10], [12] and [24] should be considered as
examples.
The excitations used in modal analysis can be grouped primarily into pseudo harmonic, random, and
transient excitation. Pseudo harmonic signals, as for example slow swept sine, will be employed to
especially investigate non-linear behavior of structures, where the extended time needed for this excitation
is accepted. Impact testing is very popular because its easy application, whereas the analysis associated
with impacting proves very sophisticated. In this study, the focus will rely on random excitation, since
these signals are widely used to determine the frequency response function for a given structure and
random excitation is very likely to be applied to non-linear structures, assumed to obey linearity.
Therefore, evidence of non-linearity is very likely to appear in the FRFs computed using random
excitations.
When using random data for frequency response estimation it must be acknowledged that obtaining only
one (1) ensemble of measurements may not contain all the information needed and therefore several
spectral averages should be taken in order to yield reliable, confident results. Utilizing a number of spectral
averages to compute auto and cross power spectra, yields
[GFF (ω )] =
[GXX (ω )] =
[GXF (ω )] =
1
N Avg
1
N Avg
1
N Avg
N Avg
∑ {F (ω )}{F (ω )}H
(2-6)
1
N Avg
∑ {X (ω )}{X (ω )}H
(2-7)
1
N Avg
∑ {X (ω )}{F (ω )}H
1
7
(2-8)
Detection of Non-Linearities
Linear and Non-Linear Vibration
where {F} and {X} denote the Fourier Transform of inputs, and outputs and {F}H, and {X}H denote the
complex conjugate . The frequency response function [H] can be computed by:
[H (ω )] = [GXF (ω )][GFF (ω )]−1
(2-9)
The formulation in Equation (2-9) is referred to as an H1 estimation and minimizes the noise on the
responses. Using an H2 algorithm will minimize the noise on the input and an HV estimation is designed to
minimize the noise on both input and output measurements in a least squares sense ([6],[24]).
After computing the frequency response, the coherence function (COH) is used to determine the level of
linearity between inputs and responses as function of frequency. The ordinary coherence for the single
reference case, between input q and response p can be formulated as:
COH pq (ω ) =
GXF pq (ω )
2
GFFqq (ω ) ⋅ GXX pp (ω )
(2-10)
To understand the physical significance of the coherence function, one might imagine that the ratio
between output and input is evaluated for each spectral average and at each spectral line. Coherence values
of one (1) then correspond to a constant ratio for each spectral average and indicate that the output is linear
with respect to the input. Reasons for low ordinary coherence values are structural non-linearities, signal
processing errors (e.g. leakage) and multiple inputs. To account for multiple input cases, the multiple
coherence function (MCOH) is defined as:
8
Detection of Non-Linearities
MCOH p (ω ) =
Linear and Non-Linear Vibration
Ni Ni
∑∑
H pq (ω ) ⋅ GFFqt (ω ) ⋅ H ∗pt (ω )
GXX pp (ω )
q =1 t =1
(2-11)
When applying the appropriate coherence function, only unmeasured inputs and signal or structural
non-linearities remain as source for low coherence values. Before turning to the effects of structural
non-linearities, a brief discussion of digital signal processing errors will be pursued.
As it is well known, by choosing a sample time (Δt) and a observation period (T) for a certain test, the
maximum frequency (fmax) to be observed (Shannon’s Sampling Theorem) and the frequency resolution
(Δf) possible (Rayleigh’s Criteria) are determined. If the measurements contain frequencies higher than fmax
an error called aliasing will result. Aliasing is induced in the measurements since the magnitude of
frequencies higher than fmax are “mistaken” by the signal processing for magnitudes of frequencies within
the legitimate frequency range defined by Δt. The only means of reducing the influence of aliasing, when
that error is expected, is the application of low-pass filters and/or drastic oversampling ([12],[24]).
Another digital signal processing error, introduced by violating the assumption of periodicity for the
Fourier Transform, is called leakage ([6],[24]). This error occurs when the time data has content whose
frequencies are not integer multiples of Δf. The magnitude of the frequencies that are not integer multiples
of Δf will “leak” into nearby frequencies which are periodic with respect to T and therefore cause a
distortion in the associated frequency spectrum. If leakage is expected, which is always the case for pure
random excitation, increasing the frequency resolution and applying a windowing concept are the common
approaches used to reduce the influence of leakage. As was already mentioned above, the effects of
leakage will be “perceived” as non-linearities by the coherence function estimate, since part of the
magnitude of the response, caused by leakage, at a certain spectral line can not be accounted for by the
corresponding input magnitude. It is also known, that in lightly damped systems, resonances and
9
Detection of Non-Linearities
Linear and Non-Linear Vibration
anti-resonances are very sensitive to leakage which results in low coherence values in those frequency
regions.
2.2
Non-Linear Overview
Even though the assumption of linearity introduces convenient analyzing methods it has to be realized that
all real structures will sooner or later exhibit non-linear behavior. Depending on the forcing level, the
response of the system may be well approximated by linear motion and the error introduced by using linear
techniques will be insignificant. On the other hand, if the motion of the system is not well described by
linear approximation, linearizing techniques yield large and unacceptable errors. One familiar example of
the importance of response levels is the linearized pendulum equation, which approximates small angle
response reasonable but does not sufficiently describe large amplitude oscillation.
There are many ways of grouping the different types of non-linearities. One way is to recognize the
response variable that the non-linearity is acting upon. By formulating the general equation of motion with
constant mass,
m&x&(t ) + u (x& (t )) + v(x(t )) = f (t )
(2-12)
the velocity related function u and displacement related function v can be defined. Displacement
non-linearities can often be characterized as a hardening stiffness (frequently assumed to be cubic)
softening stiffness, hardening/softening stiffness, and softening/hardening terms (see Figure 2-3). Another
displacement related non-linearity is known as deadzone or backlash where no restoring force is present for
a certain range of displacement values. If one looks at velocity related non-linearities practical experience
shows that quadratic damping, softening/hardening damping (car-shocks), and coulomb friction are
10
Detection of Non-Linearities
Linear and Non-Linear Vibration
possible relationships besides linear damping. In most cases, the assumed non-linearity will only be a
rough approximation of the actual relationship, but will better describe the true relationship than a linear
assumption.
Departing from linear vibration will give rise to a variety of new and often unexpected phenomena. One of
the more easy to accept, but nonetheless very important properties of non-linear systems, is the failure of
the principle of superposition. Twice the input force will not produce twice the response, but will deviate
from the contemplated linear relationship depending on the specific type of non-linearity. Also, the shifting
of natural frequencies for different excitation levels will indicate non-linear behavior and the direction of
shift can help identify the underlying non-linearity. The non-linear regime can also be represented by
effects not as easily explained and understood, such as the jump-phenomenon or the occurrence of sub- or
super-harmonics and secondary resonances.
As diverse and numerous the effects of non-linear relationships in dynamics are, there exists an even larger
number of techniques and methods to approach non-linear systems and extract the needed information.
Historically, the starting point was the formulation and analysis of SDOF systems and today the focus of
many techniques is still based on a SDOF system. This fact underlines the complexity of non-linear
motion, ranging from the question of stability to the advent of chaos. Many of these analysis techniques
require highly sophisticated and advanced mathematics and are very hard to apply to real world structures
in order to analyze their non-linear behavior. Still, there is a great demand on understanding non-linear
behavior of practical structures and experimental methods have been developed to investigate and evaluate
non-linear relations. Methods especially designed to study the effects of non-linearities are, for example,
techniques using the distribution of time histories (Bendat, [7],[8]), power spectral analysis at different
force levels, higher order frequency estimations, Hilbert transforms, Nyquist plots, and sinusoidal
excitation for each spectral line.
11
Detection of Non-Linearities
Linear and Non-Linear Vibration
Displacement Related
Velocity Related
F
F
Δx
Δv
Hardening
Quadratic with Sign
F
F
Δv
Δx
Soft/ Hardening
Softening
F
F
Δv
Δx
Softening/ Hardening
Coulomb
F
Δx
Hardening/ Softening
F
Δx
Deadzone
Figure 2-3: Types of Non-Linearities
It has to be understood that for practical MDOF systems the non-linear effects will not show up as clearly
as in non-linear theory and will often seem to be just measurement distortions to the examiner.
Furthermore, for MDOF systems, the non-linear evidence will be a function of relative response level
between certain DOFs and since this relative motion in turn is a function of frequency, the non-linear
distortion will also be a function of frequency. So, in order to apply analysis tools, the first step will be
determining whether the distortions seen in real measurements are in fact caused by structural
12
Detection of Non-Linearities
Linear and Non-Linear Vibration
non-linearities. Only after completing the detection step can analysis tools be applied to investigate the
non-linear relationship observed.
It has to be emphasized that this simulation study is mainly intended to detect and analyze the systematic
effects of non-linearities on the frequency response and coherence function and preferably find differences
in the non-linear effects as compared to system noise. Therefore, little attention will be paid to techniques
or excitations especially designed or intended to investigate non-linear effects on given structures. The aim
is to set up a modal analysis procedure, as in real experiments, but using the advantage of knowing the
model and relationships between certain degrees of freedom. Furthermore, since it is not reasonable (or
feasible) to cover all possible non-linear combinations applicable to the simulation model, the study
concentrates on cases of practical importance considering both the most likely non-linear effects and
typical testing conditions.
If non-linear system behavior is likely to occur, test setup in terms of types of excitation, excitation level,
location of excitation and points of response measurements have to be considered very carefully. For
example, it has already been shown (Adams, [20]), that it is essential to use spatial and temporal
information in order to successfully identify and analyze non-linear motions. Therefore, SIMO and MIMO
testing procedures should be preferred when expecting non-linear relations.
Also, the issue of observability, already a crucial subject in linear vibration analysis, becomes more
important when dealing with non-linear structures. Imagine two separate points on a complex structure,
where a serial connection of stiffness factors relates the motion of these two points (see Figure 2-4). If this
structure is excited at a linear level, the corresponding stiffness for serial connections will replace the
different stiffnesses between these two degrees of freedom, no distortion will result and measurements
acquired from these two points will reflect a linear system. If, on the other hand, the excitation level is
13
Detection of Non-Linearities
Linear and Non-Linear Vibration
chosen to be higher and one of the serial stiffnesses between these two points now exhibits non-linear
behavior, the measurements will very likely show distortions caused by that non-linearity. In order to
locate or analyze this non-linear stiffness between these two points, more measurements have to be made.
As can be seen, this fact suggests that the level of observability of a structure varies according to the
system behavior and since the system behavior is determined by the excitation level, observability will vary
with changed forcing levels.
DOF j
k3
k2
⎛1
1
1 ⎞
ktotal = ⎜⎜ +
+ ⎟⎟
k
k
k
2
3⎠
⎝ 1
k1
−1
DOF i
Figure 2-4: Reduction of Linear Stiffness Factors in Serial Connection
In the same sense, observability has to be considered more carefully in non-linear structures, the subject of
energy distribution becomes more important when investigating non-linearities. Using SIMO testing
procedures might induce non-linear behavior in the vicinity of the input location for high forcing levels and
furthermore may not excite the structure well at remote points. Therefore not only the excitation level but
also the number and location of inputs will determine if the structure responds in a non-linear regime.
14
Detection of Non-Linearities
Simulation Considerations
3 SIMULATION CONSIDERATIONS
3.1
Simulation Model
This section will describe the setup procedure of the simulation models using a linear 4-DOF model as an
example. Further, even when models with different DOFs are used or a particular type of non-linearity is
applied to the physical model, the corresponding simulation model can still be developed in the same way
as this example. Consider the multi-connected model given in Figure 3-1. Spring connections are marked
by kij and the velocity proportional damping connections by cij, where indices i and j refer to the degrees of
freedom they are connecting (i < j by definition and for clarity).
4. DOF
k24
c34
c24
k34
1. DOF
3. DOF
2. DOF
4. DOF
3. DOF
k23
c23
2. DOF
c13
c12
c14
k14
Ground
k13
k12
1. DOF
c01
k01
a)
b)
Figure 3-1: 4-DOF-Model, a) Physical Scheme, b) Connection Scheme
15
Detection of Non-Linearities
Simulation Considerations
As the first step towards a simulation-model, the equations of motion have to be found. By using Newton’s
Method for each degree of freedom and rearrange the equations, the equations of motion take the familiar
form of:
m1 &x&1 + (c 01 + c12 + c13 + c14 )x&1 − c12 x& 2 − c13 x& 3 − c14 x& 4 K
K + (k 01 + k12 + k13 + k14 )x1 − k12 x 2 − k13 x 3 − k14 x 4 = f 1
m 2 &x&2 − c12 x&1 + (c12 + c 23 + c 24 )x& 2 − c 23 x& 3 − c 24 x& 4 K
K − k12 x1 + (k12 + k 23 + k 24 )x 2 − k 23 x3 − k 24 x 4 = f 2
m3 &x&3 − c13 x&1 − c 23 x& 2 + (c13 + c 23 + c34 )x& 3 − c34 x& 4 K
K − k13 x1 − k 23 x 2 + (k13 + k 23 + k 34 )x3 − k 34 x 4 = f 3
m 4 &x&4 − c14 x&1 − c 24 x& 2 − c34 x& 3 + (c14 + c 24 + c34 )x& 4 K
K − k14 x1 − k 24 x 2 − k 34 x3 + (k14 + k 24 + k 34 )x 4 = f 4
( 3-1.1)
( 3-1.2)
( 3-1.3)
( 3-1.4)
It is worthwhile to note, that each equation of motion describes a state of dynamic equilibrium that
balances the internal and external forces acting on each degree of freedom. Internal forces are caused by
the existence of inertia, damping and stiffness and are a function of acceleration, relative velocity, and
relative displacement whereas external terms are caused by the external forces applied to the corresponding
degree of freedom. Equations (3-1) can now be rewritten in the form of Eq. (3-2), required by integration:
&x&n =
1
g ( f n , {x&}, {x}, c, k )
mn
( 3-2)
Therefore, the equations of motion for the 4-DOF-Model, in a slightly concise form, become:
16
Detection of Non-Linearities
⎛
⎜
1 ⎜
&x&1 =
⎜ f1 + {x&1
m1 ⎜
⎜
⎝
⎛
⎜
1 ⎜
&x&2 =
⎜ f 2 + {x&1
m2 ⎜
⎜
⎝
⎛
⎜
1 ⎜
&x&3 =
⎜ f 3 + {x&1
m3 ⎜
⎜
⎝
⎛
⎜
1 ⎜
&x&4 =
⎜ f 4 + {x&1
m4 ⎜
⎜
⎝
Simulation Considerations
∑
x& 2
x& 2
x& 2
x& 2
x& 3
⎧− ci1 ⎫
⎪
⎪
⎪
⎪ c
x& 4 }⋅ ⎨ 12 ⎬ + {x1
c
⎪ 13 ⎪
⎪
⎪ c
⎩ 14 ⎭
x& 3
⎧ c12 ⎫
⎪− c ⎪
⎪
⎪
x& 4 }⋅ ⎨ ∑ i 2 ⎬ + {x1
c
⎪ 23 ⎪
⎪⎩ c 24 ⎪⎭
x& 3
⎧ c13 ⎫
⎪ c
⎪
⎪
⎪
x& 4 }⋅ ⎨ 23 ⎬ + {x1
−
c
3
i
⎪ ∑ ⎪
⎪⎩ c 34 ⎪⎭
x& 3
⎧ c14 ⎫
⎪ c
⎪
⎪ 24 ⎪
x& 4 } ⋅ ⎨
⎬ + {x1
⎪ c 34 ⎪
⎪− ∑ c i 4 ⎪
⎩
⎭
∑
x2
x2
x2
x2
⎧− k i1 ⎫ ⎞
⎪⎟
⎪
⎪ k12 ⎪ ⎟
x 4 }⋅ ⎨
⎬⎟
⎪ k13 ⎪ ⎟
⎪⎟
⎪ k
⎩ 14 ⎭ ⎠
x3
(3-3.a)
x3
⎧ k12 ⎫ ⎞
⎪−
⎪⎟
⎪ ∑ ki2 ⎪⎟
x 4 }⋅ ⎨
⎬⎟
⎪ k 23 ⎪ ⎟
⎪⎩ k 24 ⎪⎭ ⎟
⎠
(3-3.b)
x3
⎧ k13 ⎫ ⎞
⎪ k
⎪⎟
⎪ 23 ⎪ ⎟
x 4 }⋅ ⎨
⎬⎟
⎪− ∑ k i 3 ⎪ ⎟
⎪⎩ k 34 ⎪⎭ ⎟
⎠
(3-3.c)
x3
⎧ k14 ⎫ ⎞
⎪ k
⎪⎟
⎪
⎪⎟
24
x4 }⋅ ⎨
⎬⎟
⎪ k 34 ⎪ ⎟
⎪− ∑ k i 4 ⎪ ⎟
⎩
⎭⎠
(3-3.d)
Equations (3-3) can now be used to build the simulation model in SIMULINK®, a MATLAB®
implemented simulation software. SIMULINK® is based on the idea of describing the governing equations
of motion by block diagrams. Each mathematical operation necessary for modeling the equations of
motion, such as summation, multiplication, or integration, can be expressed by appropriate blocks.
Appendix 7.1 shows the complete SIMULINK® model for the linear 4-DOF-System, a sub-block of a DOF
(DOF_1) and a sub-block (spring-damper-unit, SDU_1_2), modeling the stiffness-damping connection
between degrees of freedom 1 and 2.
It should be mentioned that the governing differential equations could also be integrated by using
MATLAB® alone, but employing SIMULINK® with its graphical description of the system makes the
formulation of differential equations easier, user-friendly and more resistant to errors introduced by
implementing the equations of motion in the conventional way. Furthermore, the application of changes to
an existing model can be done quickly and more efficiently.
17
Detection of Non-Linearities
Simulation Considerations
After the simulation model is set up, a verification check is performed, validating the correct performance
of the SIMULINK® model. Using the linear parameters given in Appendix 7.2, the eigenvalues,
eigenvectors, and frequency response functions (FRFs) of the 4-DOF model are computed theoretically.
Applying the same parameters to the simulation model and comparing the FRFs estimated by the
simulation to the theoretical FRFs, confirmed the behavior of the SIMULINK® simulation model.
Figure 3-2 shows the normalized responses of all 4 DOFs for an input applied at DOF 2.
1,2,3,4/2 FRF - Magnitude and Phase , Sim ulation (s) vs. The ory (d)
-2
10
-3
10
H in [m/N]
-4
10
-5
10
-6
10
-7
10
0
5
10
15
20
25
30
0
5
10
15
Frequenc y in Hz
20
25
30
Phase in [deg]
200
100
0
-100
-200
Figure 3-2: Frequency Response Function, Linear Model
3.2
Numerical Precision
18
Detection of Non-Linearities
Simulation Considerations
When conducting simulation studies, one of the most essential issues is numerical precision. This part of
the thesis will discuss the considerations to be made regarding numerical precision, by first looking at the
ordinary differential equation solver employed and then using a particular example. Since often no
analytical solution for the non-linear equations of motion can be found, numerical integration has to be
employed. As a consequence of using numerical integration, an approximating technique, it must be
understood that the results will not be exact, but must be accurate enough to allow valid conclusions.
There exist numerous algorithms for the numerical integration of sets of ordinary differential equations
(ode), from the easy, straightforward methods, such as Eulers or Heun’s Method without error control, to
more advanced algorithms, such as Runge-Kutta Methods (RK) of different orders. Usually the more
sophisticated methods become more accurate but they are, at the same time, more computationally
demanding. SIMULINK® offers several implemented ode-solvers based on fixed step algorithms (e.g.
Euler [ode1], Heun [ode2], Bogacki-Shampine [ode3], Runge-Kutta [ode4]) and variable step algorithms
(e.g. Bogacki-Shampine [ode23], Runge-Kutta [ode45]). For this study, only the class of variable step
solvers is of interest due to their capability of providing error control during the integration process.
The ode45-solver, which implements a Runge-Kutta algorithm of fourth and fifth order, deserves special
attention. The RK 45 algorithm has been applied to a wide range of problems, realizing good accuracy and
reasonable computational time. For a detailed discussion of this and other integration methods, the
interested reader is pointed to references [15], [16] and [18]. Still, the properties of the ode45 solver
necessary to understanding the decisions on numerical precision are pertinent here and should therefore be
examined briefly. For any solver to be able to integrate the problem, the governing differential equations
have to be in a first order form. Since the equations of motion for a mechanical system are in general of
second order, the familiar technique of state space expansion has to be applied. After doing so, the
differential equations will have the form:
19
Detection of Non-Linearities
Simulation Considerations
y& = f (x, y )
( 3-4)
It should be mentioned that in Equation (3-4) quantities x and y do not have to be scalars but could be
vectors, each entry representing one degree of freedom. The initial condition task of numerical integration
now becomes, to determine a good estimate of y(n+1), let that estimate be called ~
y (n + 1) , from given x(n)
and y(n):
y& (n) = f (x(n ), y (n ))
~
y (n + 1) = ?
( 3-5)
In Equation (3-5), n and (n+1) correspond to time t and (t+h) respectively, where h is usually referred to as
stepsize. It must be noted, that (t+h) does not necessarily coincide with the desired output times at
increments of Δt, since meeting the accuracy requirements often forces h to be much smaller than Δt. The
Runge-Kutta algorithms determine the estimate of the new state (n+1) by computing several auxiliary
y (n + 1) .
slopes for values of y in between t and (t+h), weighting them differently and finally calculating ~
Depending on how many auxiliary slopes are used, the order of the RK method is defined. For example, if
four slopes are calculated the order of that particular RK algorithm is said to be four, likewise for any other
number of intermediate gradients.
As mentioned above and indicated in the last paragraph, by integrating the equations of motion
numerically, a truncation error will be admitted into the result. In order to reduce the integration error, the
MATLAB® employed RK-method ode45 uses the following technique to determine a reasonable local
y 5 (n + 1) .
error estimate. An RK algorithm of order five is used to calculate an estimate of y(n+1), called ~
Since the fourth order coefficients are embedded in the fifth order algorithm, it is easy and time efficient to
20
Detection of Non-Linearities
Simulation Considerations
y 4 (n + 1) . An error estimation, not
retrieve an estimate of y(n+1) based on the fourth order method, called ~
the true error (!), can then be defined as:
e~45 (n + 1) = ~
y 4 (n + 1) − ~
y 5 (n + 1)
( 3-6)
In order to control the error, tolerance settings must be used. Using a smaller stepsize h, if the error
becomes too large (violating a given tolerance setting) and increasing h when the state is changing slowly.
This way, the accuracy needed can be achieved but at the same time computational effort can be saved by
doing time effective integration. SIMULINK® offers error control by letting the user choose relative and
absolute tolerance values. For each integration step the error estimate from Equation (3-6) has to fulfill the
inequality shown in Equation (3-7):
e~ ≤ max (tol _ rel ⋅ y , tol _ abs )
( 3-7)
Depending upon the magnitude of the function at a particular point in time, either the relative or the
absolute tolerance will be active. As can be seen from Equation (3-7), this formulation is set up from the
point of view of most effective integration, since only the rougher tolerance applicable is considered (max).
Therefore, choosing the relative and absolute tolerance values for integration should be made carefully in
order to yield valid results. This will be depicted in the next part of this section by a simulation example
discussing the issue of numerical precision by integrating the equations of motion several times with
different error tolerances.
For this demonstration, the vibration behavior of the linear 4-DOF model shown in Figure 3-1 will be
simulated by putting in a pure random force at DOF 2, recording the input and computing all responses.
The pure random force is set up to yield the maximum frequency to be observed with fmax = 102.4Hz and a
desired frequency resolution of Δf = 0.05Hz.
21
Detection of Non-Linearities
Simulation Considerations
In general, the critical question involves the values of relative and absolute tolerance values that should be
chosen for the integration process. In order to do so, the equations of motion must be integrated first with
default settings for the tolerances to establish the order of magnitude of the vibration time histories. With
the structure’s given physical properties and an input RMS-force level of F = 100N, the RMS values of the
outputs become:
{x}rms
⎧0.0136⎫
⎪0.0172⎪
⎪
⎪
=⎨
⎬m
0
.
0173
⎪
⎪
⎪⎩0.0176⎪⎭
As can be seen, all output motions are displaying the same order of magnitude which is 1⋅10-2 m. Now, for
example, allowing an absolute tolerance of tol_abs = 1⋅10-5m will result in four significant (or valid) digits
after the decimal point. In order to check if the absolute tolerance of tol_abs = 1⋅10-5m already produces
accurate results, the equations of motion are integrated again three more times with smaller absolute
tolerances, namely
tol _ abs = 1 ⋅ 10 −6 / 1 ⋅ 10 −7 / 1 ⋅ 10 −8 m
providing the same input force for each tolerance setting. The relative tolerance remains unchanged during
this test at a level of tol_rel = 1⋅10-6. By looking at Equation 3-7 it will be realized that the relative
tolerance hardly will be activated during the integration because of the small magnitude of the time
histories. This way the only changes in the time histories will be caused by the tighter absolute tolerance
settings.
22
Detection of Non-Linearities
Simulation Considerations
Figure 3-3: Frequency Response H32 and Error Estimate (|tol| = 1⋅10-5/1⋅10-8m)
In Figure 3-3 H32 is shown, both, for the absolute tolerance values of tol_abs = 1⋅10-5m and
tol_abs = 1⋅10-8m. As can be seen, no significant difference is visible and H32(1⋅10-5m) and H32(1⋅10-8m)
seem identical. Furthermore, the magnitude of the estimate of the absolute and relative error between the
two FRFs is plotted, using the more accurate FRF estimation (tol_abs = 1⋅10-8m) as reference. In the
frequency range of interest (0 to 40Hz) the absolute error is roughly four to five decades below the
magnitude of vibration, with slightly higher values of error at the resonance frequencies. At higher
frequencies on the other hand (above 50Hz), the absolute error remains constant, resulting in an increasing
relative error due to the decreasing magnitude of vibration for that frequency range. Overall, it should be
noted that the accuracy gained by an absolute tolerance of tol_abs = 1⋅10-5m seems sufficient and does not
improve significantly by using a smaller tolerance.
If one is integrating a particularly long time history, one might suppose, that the error made from one
integration step to the next accumulates and therefore the end portion of the integration record might not be
as reliable as the beginning. Even though this will be especially true for diverging states, it is not
23
Detection of Non-Linearities
Simulation Considerations
necessarily true for the vibration histories under investigation. In Figure 3-4, one ensemble of response
x4(t) is plotted for two absolute tolerance values (tol_abs = 1·10-5 m/1·10-8 m), where an error estimate is
also shown. Note, that the order of magnitude for the error estimation is 1·10-7 m, whereas the order of
vibration is 1·10-2 m. It can be furthermore noticed, that the error does not increase as integration time
progresses.
Tim e History x (t)
4
0.04
x 4 in [m ]
|tol| = 1e-05m
0.02
0
-0.02
-0.04
0
2
4
6
8
10
12
14
16
18
20
10
12
14
16
18
20
16
18
20
0.04
x 4 in [m]
|tol| = 1e-08m
0.02
0
-0.02
-0.04
0
2
4
6
8
-7
5
x 10
Error Estimate
0
-5
0
2
4
6
8
10
T ime in [s]
12
14
Figure 3-4: Response x4(t) for different Tolerances and Error Estimation, One Ensemble
In general, a large number of consecutive ensembles are integrated. In order to investigate the development
of error, the RMS values of the difference between corresponding time histories (error estimate) for each
ensemble and absolute tolerance value are computed. Figure 3-5 shows the error estimate RMS(dx) for x4(t)
as a function of ensembles, where different plots correspond to different absolute error tolerance settings.
Two important facts might be deduced by analyzing the information provided in Figure 3-5. First, that
indeed the error does not accumulate when integration time progresses and secondly, as might be
suspected, the reduction of absolute tolerance by one order reduces the absolute error by one order as well.
24
Detection of Non-Linearities
Simulation Considerations
Figure 3-5: Error Estimate RMS(dx4)
To summarize the results from this section concerned with numerical precision, the following conclusions
can be drawn. The one-step ordinary differential equation solver ode45 produces accurate and valid results
for the system in focus. Although, in order to set appropriate, accuracy yielding tolerances for the
integration, the magnitude of vibration should be known for all degrees of freedom. If the order of
magnitude is the same for all time histories, only one absolute tolerance for all time histories will be
sufficient; if not, different absolute tolerances should be chosen. Furthermore, since in this example only
linear motion was present, a non-linear case, especially those with discontinuous derivatives, should be
treated more carefully. This can be done by checking the results (as in the linear case), by integrating the
time histories with a smaller tolerance and comparing the computed frequency response and/or coherence
functions with respect to their convergence.
25
Detection of Non-Linearities
Application of Non-Linear Interactions
4 APPLICATION OF NON-LINEAR INTERACTIONS
4.1
Combined Coherence Function
One is often faced with the question, are low coherence values in a certain frequency range caused by
signal processing errors or are structural non-linearities the reason for the distortions? This part of the
thesis proposes a method, which can be used as check for whether non-linear motion is present in the
response histories and whether structural non-linearities are in fact the source of low coherence. Therefore,
a detection method for the presence of structural non-linearities will be provided.
a)
b)
F
F
Figure 4-1: a) MDOF System, b) Internal and External Forces on MDOF System
When building a model of a real structure, one often discretizes the object and thinks of it as a number of
degrees of freedom (lumped masses) connected by stiffness and damping terms. Figure 4-1 a) shows the
26
Detection of Non-Linearities
Application of Non-Linear Interactions
general scheme of a MDOF system. It should be noted that the connections between the DOFs drawn
depict generic relations and are not subject to any linear assumption.
Formulating the dynamic force balance for any DOF of that structure, the sum of all inertia, stiffness,
damping related and external forces acting on that particular DOF vanishes. At this point, the assumption
of constant mass is made and all forces caused by stiffness and damping will be called internal forces. In
Figure 4-1 b) the structure is shown with all internal forces and one external force acting upon it. The force
balance in terms of acceleration, usually referred to as the equation of motion, for each DOF now becomes:
&x&n =
1
⋅
mn
(∑ f ext + ∑ f int )n
(4-1)
In a physical sense, Equation (4-1) states, that the acceleration of every DOF is the sum of all internal and
external forces acting on that degree of freedom scaled by the associated mass.
Deriving Equation (4-1) for the acceleration present at DOFs i and j will yield:
&x&i =
[
1
⋅ f i − v1i (Δx i1 ) − K − v (i −1)i (Δx i (i −1) ) + v i (i +1) (Δx (i +1)i ) + K + v ij (Δx ji ) + K + v in (Δx ni )
mi
]
(4-2.a)
&x& j =
[
1
⋅ f j − v1 j (Δx j1 ) − K − v ij (Δx ji ) − K − v ( j −1) j (Δx j ( j −1) ) + v j ( j +1) (Δx ( j +1) j ) + K + v jn (Δx nj )
mj
]
(4-2.b)
In Equations (4-2.a) and (4-2.b) v(Δx) denotes force interactions between different DOFs as a function of
relative motion Δx. As can be realized, by applying Newton’s law, force vij(Δxji) acting on DOF i because
of relative motion Δxji is the same magnitude but opposite direction to the force acting on DOF j because of
27
Detection of Non-Linearities
Application of Non-Linear Interactions
the relative motion between DOFs i and j. It is important to note, that this is true, independent of the nature
of the relation between the two DOFs. Therefore, by adding Equation (4-2.a) to (4-2.b),
∑ &x&ij = &x&i + &x& j
(4-3)
and under the condition that mi ≈ mj, the contribution of the interaction force between DOFs i and j to the
motion can be minimized or even eliminated. It has to be admitted, that by creating the virtual coordinate
&x&ij , a loss of information about connection ij will result. But if low coherence is caused by non-linear
behavior between DOFs i and j, adding motions &x&i and &x& j , and computing the coherence for the sum of
motions versus a given input will result in drastically improved coherence values (here defined as
combined coherence (CCOH)). In this way, a detection method for non-linear relationships is possible and
is indicated by higher combined coherence values.
This summation is limited to time domain histories (and their Fourier Transforms) but not to acceleration
only. Applying the summation to Fourier Transforms the only difference is that the summation has to be
done at each spectral line, and not at each time point. Since in a discrete sense displacement and velocity
are only scaled by the corresponding sample time (time domain) or frequency (frequency domain), above
arguments are still valid. Furthermore, leakage is not to be affected by this technique because of its digital
signal processing nature and therefore low coherence caused by leakage or aliasing should not improve.
As can be seen in the derivation, the crucial condition for that technique is the approximate equality of the
associated masses of DOFs i and j. If this does not hold true, this method will fail and not improve the
coherence function. On the other hand, if the condition mi ≈ mj is not valid but one has knowledge of the
relationship between mi and mj, it is possible to scale the motions according to the mass ratio and correct
the relationship. After scaling the motions, the CCOH can be applied and will indicate non-linear motion, if
28
Detection of Non-Linearities
Application of Non-Linear Interactions
present. This concludes the introduction of the CCOH function. Examples will be discussed in Sections 4.2
and 4.3 where this technique is actually applied.
4.2
Characteristic Effects of Non-Linearities
This section is intended to analyze the effects of non-linear connections upon the frequency response and
coherence function without the presence of measurement noise. In this way, it should be possible to find
the characteristic effects for particular non-linearities, which will allow a detection and determination of
the actual structural non-linearity by only recognizing the effects on FRFs and COH. Also, the proposed
method of combined coherence will be examined. The 4-DOF system developed in Section 3.1 will serve
as simulation example.
4.2.1
Hardening Stiffness
The hardening stiffness will be generated by a cubic offset to the linear term. Therefore the restoring force
can be formulated by:
(
f r = k lin ⋅ Δx + ε ⋅ Δx 3
)
(4-4)
In Equation (4-4) ε is used to adjust the severity of the non-linear effect and note additionally, that for
small motion (small excitation) the relationship converges towards the linear case. As a first approach to
identification, the hardening stiffness is applied to a SDOF system in order to characterize its effects.
Figure 4-2 a) shows the FRF and COH for increasing level of excitation.
29
Detection of Non-Linearities
10
Application of Non-Linear Interactions
FRF - Magnitude and Coherence, Cubic Stiffness, 10N (s), 20N(d), 40N (dd)
-3
F
F = f(Δx)
10
k_lin 2
-4
k_lin 1
F2
10
Δx
F1
-5
0
5
10
15
20
25
30
Hardening Nonlinearity
1
0.5
0
0
5
10
15
Frequency in Hz
20
25
30
b)
a)
Figure 4-2: a) SDOF System, b) Linearizing Concept for Cubic Stiffness
When increasing the force level for the given SDOF system, it can be observed that the resonance
frequency shifts to higher values, the resonance amplitude decreases, and the distortion effects become
more apparent. The frequency shift can be explained by the “linearizing” concept of the FRF estimation,
depicted in Figure 4-2 b). By forcing at higher levels, resulting in higher motion, the “linearized” stiffness
k_lin2 is greater than k_lin1 and therefore, the resonance frequency will be shifted to higher values when
forced at higher levels.
I)
Non-linearity between DOF 1 and 3, SIMO, Input at DOF 3
This case simulates a SIMO situation where the stiffness connection between DOF 1 and DOF 3 becomes
non-linear (ε = 50000). Compared to real world testing considerations this is a valid assumption, since
single input configurations might induce non-linear behavior in the vicinity of the input location. The
excitation chosen is pure random (RMS: F = 30N). Figure 4-3 shows the FRF estimation (H1 algorithm)
and ordinary coherence for all outputs versus input at DOF 3.
30
Detection of Non-Linearities
Application of Non-Linear Interactions
It can be seen from Figure 4-3 that, compared to the linear case, frequency responses H13 and H33 are most
affected by the non-linearity. This could be expected since the non-linear relationship is placed between
DOFs 1 and 3. Furthermore, one can notice a shift and a decrease in magnitude of the third resonance
frequency. Again, this behavior could be expected when applying this type of non-linearity and was
explained in the SDOF example above.
Looking at the coherence function for all 4 responses the leakage-caused coherence drop at the first, very
lightly damped resonance frequency can be noticed. The driving point measurement H33 also shows a
distinct coherence drop at the anti-resonance. As mentioned above, these low coherence values can be
attributed to leakage and are not caused by the applied structural non-linearity. On the other hand, the sharp
coherence drop at f ≈ 10Hz, noticeable in COH13 and COH33, is not associated with any resonance or
anti-resonance. It turns out that this drop in coherence occurs at exactly 3 times the first resonance and is
therefore evidence of a secondary resonance in the system, a typical non-linear phenomena. The low
coherence in the frequency range f ≈ 13 – 17Hz at COH13, COH23 and COH33 can be explained by the large
relative motion between DOF 1 and 3 in that frequency range. Besides the mentioned effects, there is also a
noise-like distortion at the high frequencies showing up in H13 and resulting in low values for COH13. In
summary it can be said, that the effects of the cubic non-linearity are dominantly present at the responses to
which the non-linearity is connected and have only minor influence on responses 2 and 4.
31
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 3 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 3 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 3 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 3 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-3: FRFs and COH, Hardening Stiffness, Inp. 3, k13
32
Detection of Non-Linearities
Application of Non-Linear Interactions
As a next step, the combined coherence (CCOH) will be computed for responses 1 and 3 in order to
eliminate the effects of the non-linear relation. In order to do so, responses x1 and x3 are added and the
coherence is calculated for this combined response. Figure 4-4 shows the CCOH(13)3 in comparison with
ordinary coherence COH13 and COH33.
COH 1/3 , COH 3/3 and CCOH (1+3)/3, F = 30N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequency in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-4: Coherence and Combined Coherence, Hardening Stiffness, Inp. 3, k13
It can be seen, that the effects of the secondary resonance at f ≈ 10Hz is completely eliminated and the high
frequency range (f > 20Hz) also shows combined coherence values of one (1). In the frequency range of
f ≈ 14 – 17Hz the CCOH does not show improved values, supposedly due to the limitations of the
technique of combined coherence. First, the difference in mass and therefore the scaling difference has to
be recognized (m1 = 12kg, m3 = 9kg) and second since the MDOF system is a multi-connected structure,
the non-linear effects not only enter directly but will be still present even if the effects of the direct
33
Detection of Non-Linearities
Application of Non-Linear Interactions
connection between DOF 1 and 3 are removed. The influence of leakage remains unaffected by the
summation of the responses, which was expected.
In summary, this case showed the following non-linear effects: shifting resonance frequencies according to
the hardening stiffness term; secondary resonance effects correlated with the first resonance; a distortion of
the frequency response in the range where the relative motion between DOF 1 and 3 is large and therefore
the non-linear effects are large too; and a noise-like distortion of the driving point frequency response at
high frequencies.
II)
Non-linearity between DOF 1 and 3, SIMO, Input at DOF 2
This SIMO case is designed to simulate the situation where the non-linearity is somewhere in the system
and the input is applied to a DOF which is not associated with the non-linearity. As under Case I), the
excitation is a pure random signal with RMS value of F = 30N, and the severity of the non-linear offset is
held constant (ε = 50000). Frequency response and coherence function for all responses are shown in
Figure 4-5.
34
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 3 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 3 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 3 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 3 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-5: FRFs and COH, Hardening Stiffness, Inp. 2, k13
35
Detection of Non-Linearities
Application of Non-Linear Interactions
As can be seen from Figure 4-5, the FRFs corresponding to a response connected to the non-linearity
(1 and 3) are affected mostly by the cubic stiffness. As in Case I) a secondary resonance shows clearly at a
frequency of f ≈ 10Hz. The frequency shift appears to be not as large as in Case I), due to the fact that the
excitation is not placed directly at the non-linear relation and therefore the relative motion between DOF 1
and 3 is not as large as in Case I). One can also notice a periodic repeat of low and high coherence parts in
COH12 and COH32. Focusing on the frequency range of these low coherence values (and FRF distortion), it
shows, that they are correlated with the first resonance frequency at f ≈ 3.3Hz by odd multiples of the first
resonance. In fact, distortions occur at f ≈ 5⋅3.3Hz = 15.9Hz, f ≈ 7⋅3.3Hz = 23.1Hz, f ≈ 9⋅3.3Hz = 29.7Hz
and so forth. As the secondary resonance at f ≈ 10Hz, these distortions appear to be secondary resonances
too, justified by the cubic nature of the non-linearity.
COH 1/2 , COH 3/2 and CCOH (1+3)/2, F = 30N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-6: Coherence and Combined Coherence, Hardening Stiffness, k13
The combined coherence CCOH of response 1 and 3 versus input 2 is computed and plotted in comparison
to COH12 and COH32, see Figure 4-6. Except for leakage, the distortion in the coherence functions is
36
Detection of Non-Linearities
Application of Non-Linear Interactions
almost totally removed and CCOH(13)2 indicates pure linear relation between (x1 + x3) and reference at
DOF 2. This allows the conclusion that the low coherence values in COH12 and COH32 are caused by
non-linear motion and not by measurement noise and signal processing errors.
It has also been noticed that the distortions caused by secondary resonance effects seem to be correlated.
Figure 4-7 shows H12 and H32 in a frequency range of f = 20 – 25Hz. The frequency distortions appear to
behave in an out of phase magnitude manner, which should not be expected when random noise in the data
is present.
1/2 and 3/2 F RF - M agni tude and Coherence, F = 30N
-5
10
-6
10
20
20.5
21
21.5
22
20.5
21
21.5
22
22.5
23
23.5
24
24.5
25
22.5
23
Frequenc y in Hz
23.5
24
24.5
25
1
0.5
0
20
Figure 4-7: FRFs and COH, Hardening Stiffness, Inp. 2, k13
Trying to quantify the correlation between H12 and H32, the correlation coefficient r has been computed for
log(|H12|) and log(|H32|), where the linear trend is removed from log(|H12|) and log(|H32|) before calculating
the correlation in order to emphasize the distortion and not the decaying effect of the system response. The
correlation coefficient for the frequency range f = 20 – 25Hz is determined to be r(H12,H32) = -0.76, in fact
indicating an out of phase correlation. For ease of writing, the notation {Hpq} will be used for log(|Hpq|),
37
Detection of Non-Linearities
Application of Non-Linear Interactions
where the linear trend is removed from log(|Hpq|). Figure 4-8 shows a plot of {H12} versus {H32}, visually
representing the correlation in the frequency range of distortion.
F RF Correl ati on, 1/2 vs. 3/2, f = 20 to 25 Hz
0.1
Corr. = -0.75895
0.05
0
log |H3 2 |
-0.05
-0.1
-0.15
-0.2
-0.25
-0.25
-0.2
-0.15
-0.1
-0.05
0
log |H1 2 |
0.05
0.1
0.15
0.2
0.25
Figure 4-8: Correlation between {H12} and {H32}
In summary, for this case, the presence of secondary resonances and the typical frequency shift has been
observed. The distortion because of large relative motion is decreased in comparison to Case I) since the
input is not directly applied at the non-linear location.
III)
Non-linearity between DOF 2 and 3, SIMO, Input at DOF 1
In this case the location of the cubic stiffness in the 4-DOF model is moved to the connection between
DOF 2 and 3. All other connections remain unchanged and express linear relation. The scaling factor is
chosen to be ε = 50000 and a pure random signal (F = 100N) is used to excite the system at DOF 1.
Figure 4-10 contains the FRFs and coherence functions for this particular SIMO case.
38
Detection of Non-Linearities
Application of Non-Linear Interactions
COH 2/1 , COH 3/1 and CCOH (2+3)/1, F = 100N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-9: Coherence and Combined Coherence, Hardening, Inp. 1, k23
The non-linear stiffness at k23 only affects the responses to which the non-linearity is connected, namely
response 2 and 3. FRFs H11 and H41 do not show any significant distortion caused by the applied
non-linearity. By looking at H21 and H31, it can be noticed that the secondary resonances associated with
the first resonance frequency are not present in the data. Frequency distortion in the range f ≈ 15 – 20Hz
can be explained by the large relative motion between DOF 2 and DOF 3 in that frequency range and is not
caused by a secondary resonance effect. Furthermore, in this case the frequency shifting effect has changed
from affecting mainly the third resonance in Cases I) and II) to influence the fourth resonance.
39
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-10: FRFs and COH, Hardening Stiffness, Inp. 1, k23
40
Detection of Non-Linearities
Application of Non-Linear Interactions
As for Cases I) and II) the combined coherence CCOH(23)1 is computed and plotted in Figure 4-9 versus the
ordinary coherence functions COH21 and COH31.
In summary, this case, as Cases I) and II), has shown that only the responses and therefore the frequency
response functions in fact connected with the non-linear behavior will be significantly effected by the
non-linearity. Furthermore, one has seen, even though secondary resonances are typical non-linear
phenomena, they do not necessarily have to show up.
IV)
Non-linearity between DOF 2 and 3, MIMO, Input at DOF 1 and 2
This set up is chosen to investigate the effects of a MIMO testing situation and a more uniform energy
distribution in the system is realized. It should be expected that the non-linear behavior will be more
apparent because of the aforementioned reason. At each input (DOF 1 and 2), the excitation signal applied
has an RMS value of F = 50N, which is half the magnitude at each input compared to the SIMO Case III),
where an RMS value of F = 100N was used for the single input. The severity of the cubic stiffness, the
location of the non-linearity, and all other conditions remain unchanged with respect to Case III).
Figure 4-11 shows the FRFs and multiple coherence functions computed.
41
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
1 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
2 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
2 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
3 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
3 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
4 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
4 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
Figure 4-11: FRFs and MCOH, Hardening Stiffness, Inp. 1+2, k23
42
Detection of Non-Linearities
Application of Non-Linear Interactions
It can be seen in Figure 4-11 that the FRFs associated with responses 2 and 3 are affected most by the
non-linearity. Since the cubic stiffness is at the same location as in Case III), the shift of resonance
frequency also influences the fourth resonance. Besides the frequency shift, a distortion is noticed, more or
less at all FRFs in the frequency range f ≈ 15 – 20Hz, which is caused by the fact that the relative motion
between DOF 2 and 3 in that frequency range is especially large. Even in H41 and H11, which do not have a
DOF in common with the non-linear connection, this influence is still apparent.
M COH 2 , M COH 3 and M CCOH (2+3)
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-12: Multiple Coherence and Combined Coherence, Hardening Stiffness, k23
In Figure 4-12, the multiple combined coherence for (x2 + x3) MCCOH(23) is plotted along with MCOH2
and MCOH3. The fact that no complete clear up of the combined coherence can be accomplished, is
probably caused by the fact, that non-linear motion enters x2 and x3 not only through the non-linear relation
directly in between these two DOFs. Motions x1 and x4 exhibit non-linear vibration too, which will be
transmitted to x2 and x3. Therefore the total elimination of the cubic stiffness effect will not be possible.
43
Detection of Non-Linearities
Application of Non-Linear Interactions
Upon a closer look at the high frequency distortions of H21 and H31, a correlation can be recognized, as in
Case II). A zoom of the FRFs H21 and H31 to the range f ≈ 25 – 30Hz is shown in Figure 4-13. Also the
correlation coefficient between {H21} and {H31} has been computed to r({H21},{H31}) = -0.73 which
indicates a strong out of phase magnitude correlation.
2/1 and 3/1 F RF - M agni tude and M Coherence, F = 50N
-5
10
-6
10
-7
10
-8
10
25
25.5
26
26.5
27
25.5
26
26.5
27
27.5
28
28.5
29
29.5
30
27.5
28
Frequenc y in Hz
28.5
29
29.5
30
1
0.5
0
25
Figure 4-13: FRFs H21 and H31, MCOH, Hardening Stiffness, Inp. 1+2, k23
4.2.2
Softening Stiffness
The softening stiffness non-linearity will be formulated by the following equation:
[
]
a
f r = sig (Δx ) ⋅ k lin ⋅ ⋅ ln (s ⋅ Δx + a ) − ln (a ) (4-5)
s
As can be seen from Equation (4-5), the derivative at Δx = 0 is the linear stiffness factor klin and therefore
the restoring force will converge towards the linear case for small excitation. Scaling factors a and s are
used to adjust the severity of the non-linear effect.
44
Detection of Non-Linearities
Application of Non-Linear Interactions
Before applying the softening stiffness to the 4-DOF system, the effects of this non-linearity are
investigated on a simple SDOF model. The frequency response and the coherence function for increasing
excitation levels are shown in Figure 4-14 a). For higher excitation forces, the resonance frequency
decreases and the distortion effects becoming more dominant. The frequency shift can be explained by
looking at Figure 4-14 b). When the system is forced at a higher level, the linearized stiffness k_lin2 is not
as big as the linearized stiffness k_lin1, which corresponds to the smaller forcing level F1.
FRF - Magnitude and Coherence, SoftStiff, 10N (s), 20N (d), 40N (d)
-3
10
F
k_lin 2
k_lin 1
-4
10
F = f( Δx)
Δx
F2
-5
F1
10
0
5
10
15
20
25
30
Softening Nonlinearity
1
0.5
0
0
5
10
15
Frequency in Hz
20
25
30
b)
a)
Figure 4-14: a) SDOF System, b) Linearizing Concept for Softening Stiffness
45
Detection of Non-Linearities
V)
Application of Non-Linear Interactions
Non-linearity between DOF 2 and 3, SIMO, Input at DOF 1
This case uses Equation (4-5) to formulate the restoring force between DOF 2 and 3. The RMS value of
pure random excitation at DOF 1 is F = 100N, scaling values s and a are chosen to s = 2500 and a = 10,
which relates to an approximate 30% deviation from the linear restoring force at RMS(Δx23).
As learned from the hardening stiffness simulations, the responses directly connected to the non-linearity
are most affected (i.e. H21 and H31, see Figure 4-16). The frequency shift of the fourth resonance and the
distortion in the range of large relative motion (f ≈ 15 – 20Hz) are most noticeable in H21 and H31. It can
also be mentioned that the first anti-resonance appears to be very sensitive with respect to the softening
stiffness. COH21 and COH31 show a drop in the region of the first anti-resonance, which could not be
justified by the fact of leakage (no driving point measurement and not a very steep drop in FRFs).
Computing the combined coherence for responses x2 and x3 proves that the drop at the anti-resonance is not
caused by leakage but instead is a non-linear effect, see Figure 4-15.
COH 2/1 , COH 3/1 and CCOH (2+3)/1, F = 100N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-15: Ordinary and Combined Coherence, Softening Stiffness, Inp. 1, k23
46
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-16: FRFs and COH, Softening Stiffness, Inp. 1, k23
47
Detection of Non-Linearities
Application of Non-Linear Interactions
As in the hardening stiffness case, a correlation of the high frequency distortion can be recognized. For
example the correlation coefficient for {H21} and {H31} in the frequency range of f = 25 – 30Hz is
determined to be r({H21},{H31}) = -0.83, indicating a strong out of phase correlation.
VI)
Non-linearity between DOF 2 and 3, MIMO, Inputs at DOF 1 and 2
This MIMO case is intended to excite the softening system more uniformly and therefore emphasize the
non-linear relation. Input locations are chosen to be at DOF 1 and 2, the RMS value of each pure random
excitation is F = 50N. All system parameters and the parameters of the non-linearity remain unchanged
with respect to Case V). Figure 4-17 shows all eight frequency response functions. The effect of the
non-linearity, again, is largest at the responses directly connected to the softening stiffness. It can be seen
that the fourth resonance frequency, in comparison with the linear case, has been shifted to a lower value.
As noted already, it appears that the anti-resonances are sensitive to the softening stiffness, noticeable in
H31 and H32. On the other hand, by using the multiple coherence approach in characterizing the linear
relationship between inputs and output, the driving point anti-resonances (H11 and H22) do not result in a
coherence drop.
48
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 1 0 0 N
-2
1 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 1 0 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
2 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 1 0 0 N
-2
2 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 1 0 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
3 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 1 0 0 N
-2
3 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 1 0 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
4 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 1 0 0 N
-2
4 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 1 0 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
Figure 4-17: FRFs and MCOH, Softening Stiffness, Inp. 1+2, k23
49
Detection of Non-Linearities
Application of Non-Linear Interactions
The noise-like distortion in H21 and H31 in the frequency range f = 20 – 40Hz was checked for correlation
and in fact the correlation coefficient between {H21} and {H31} was determined to be
r({H21},{H31}) = -0.96 (f = 25 – 30Hz). Using {H22} and {H32} in the same frequency range, a correlation
coefficient r({H22},{H32}) = 0.7 was found. As can be seen, in the first case an out of phase and in the
second, an in phase correlation is present. Therefore it appears that no prediction beyond the correlation
between two frequency responses can be made. One more comment should be made at this point. If, for
this example, no noise was apparent and only linear motion was observed (hence: smooth frequency
responses), a correlation coefficient computed between two FRFs in a frequency range f = 25 – 30Hz
would of course show a strong correlation. But, if the FRFs are distorted in a noise-like manner, a strong
correlation between these two frequency responses does not point towards random errors but to systematic
effects. Therefore finding a correlation, random error can be excluded as possible source.
MCOH 2 , MCOH 3 and MCCOH (2+3), F = 50N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequency in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-18: Multiple Coherence and Combined Coherence, Softening Stiffness, Inp. 1+2, k23
50
Detection of Non-Linearities
Application of Non-Linear Interactions
Figure 4-18 shows the combined coherence for this example, where it appears hard to tell if an
improvement has been made in comparison with MCOH2 and MCOH3. It is questionable if the combined
coherence would be applied to MCOH2 and MCOH3 in the first place, because the MCOHs do not indicate
a non-linear relationship between DOF 2 and 3 (e.g. corresponding drops in coherence).
VII)
Non-linearity between DOF 3 and 1/2/4, SIMO, Input at DOF 3
This simulation setup is intended to create the excitation at a softening element of the model and therefore
introduce non-linear relationships in the vicinity of the reference. The input is chosen to be applied at
DOF 3 and all connections from DOF 3 to the DOFs 1/2/4 will behave non-linear. Equation (4-5) is
employed to determine the softening restoring forces, with s13 = 2000, s23 = 2500, s34 = 5000 and a = 10 for
all three non-linear connections. The RMS excitation level for the pure random signal is F = 50N. From the
FRFs in Figure 4-19, it can be seen, that all resonance frequencies, except the rigid body mode, are
decreasing in frequency and that all the FRFs are now affected since multiple non-linearities are present.
Secondary resonance effects (H13), distortions in the frequency range of large relative motion between
DOFs (e.g. H43), high frequency noise-like distortions (H13, H43) and a sensitivity to anti-resonances can be
found (H33).
Furthermore, since multiple non-linearities are present, applying the combined coherence does not show
any significant improvement (e.g. for x2 and x3, see Figure 4-20). This is a clear indicator that the distortion
in motion is not only caused by one (1) dominant non-linear interaction.
51
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-19: FRFs and Coherence, Softening Stiffness, Inp. 3, k3all
52
Detection of Non-Linearities
Application of Non-Linear Interactions
COH 2/3 , COH 3/3 and CCOH (2+3)/3, F = 50N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-20: Ordinary and Combined Coherence, Softening Stiffness, Inp. 3, k3all
This concludes the discussion of softening stiffness. In summary, it could be found that the softening effect
is evident in decreasing resonance frequencies when a higher forcing level is applied. It also has been
found that the anti-resonances are sensitive to softening stiffness terms in the system and even though it
was not shown it should be mentioned that a correlation in distortions between frequency responses was
observed.
4.2.3
Non-Symmetric Stiffness
Non-symmetric stiffness should be defined as hardening/softening or softening/hardening restoring force,
the first resisting compression more than pull and the latter having a smaller reaction force when
compressed in comparison with pulled. Since causing similar effects in the frequency and coherence
53
Detection of Non-Linearities
Application of Non-Linear Interactions
function these two non-linear interactions can be discussed in one section. The restoring force for both
characteristics can be described in a functional form by:
f r = k lin ⋅
(
)
1
⋅ e ± s⋅Δx − 1
±s
(4-6)
Scaling factor s in Equation (4-6) is used to adjust the severity of the non-linear behavior for given
displacements and the sign models the particular non-symmetric stiffness (+ = softening/hardening,
- = hardening/softening). Figure 4-21 shows both non-linearities sketching the linearizing concept used to
describe the effects on the frequency response.
F
F
k_lin2
k_lin2
k_lin1
F = f(Δx)
F2
Δx
Δx
F2
F1
F1
F = f(Δx)
Hard/ Softening Nonlinearity
Soft/ Hardening Nonlinearity
k_lin1
Figure 4-21: Linearizing Concept for Non-Symmetric Stiffness
For the hardening/softening stiffness, forcing at a higher level the “linearized stiffness” k_lin2 “seen” by
the frequency response estimation has a slightly smaller value than k_lin1. Therefore, a larger excitation
will result in a slight frequency shift to smaller values. The opposite will happen when forcing a system
with softening/hardening elements. It should also be mentioned that for comparable severity of
non-linearities and similar displacements the frequency shift effect of a non-symmetric stiffness will not be
as significant as for a pure hardening or softening stiffness term. Figure 4-22 shows the frequency response
54
Detection of Non-Linearities
Application of Non-Linear Interactions
for a SDOF system with a hardening/softening stiffness element. It can be seen that the frequency shift
effects are not as significant when increasing the force level but a secondary resonance evidence is present
at frequency close to twice the linear resonance. Furthermore, clear distortions are apparent in the low
frequency range, which are caused by the non-symmetric behavior of the non-linearity.
FRF - Magnitude and Cohe re nce , Hard/SoftStiff, 10N (s), 20N (d), 40N (dd)
-3
10
-4
10
-5
10
0
5
10
15
20
25
30
0
5
10
15
Frequenc y in Hz
20
25
30
1
0.5
0
Figure 4-22: SDOF System with Hardening/Softening Stiffness
VIII)
Non-linearity between DOF 1 and 3, SIMO, Input at DOF2
For this case, a hardening/softening stiffness is placed between DOF 1 and 3 in the 4-DOF simulation
model, choosing scaling factor to s13 = 100. The input is applied at DOF 2 and RMS excitation level for the
pure random signal is F = 50N. Figure 4-23 displays all four frequency responses.
55
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-23: FRFs and COH, Hardening/Softening Stiffness, Inp. 2, k13
56
Detection of Non-Linearities
Application of Non-Linear Interactions
Most affected by the non-linearity are frequency responses H12 and H32, since DOFs 1 and 3 are directly
connected to the hardening/softening stiffness. It is interesting to note, that no significant frequency shift
occurs, even though it is known (hardening stiffness at k13) that the stiffness at this location strongly
influences the third resonance. Besides small secondary resonances of order three, a secondary resonance
at twice the rigid body resonance of f1 = 3.3Hz at f ≈ 6.6Hz can be found. Distortions at f ≈ 13Hz (4),
f ≈ 20Hz (6) and f ≈ 33Hz (10) are also believed to be caused by secondary resonances corresponding to
even multiples (in brackets) of the first resonance frequency.
COH 1/2 , COH 3/2 and CCOH (1+3)/2, F = 50N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-24: Ordinary and Combined Coherence, Hardening/Softening Stiffness, Inp. 2, k13
As in the cases discussed before, the combined coherence will be determined. This technique will still
work for a non-symmetric stiffness, since the method is based on a force equality principle and not
dependent on symmetry of the displacement-force relation. Computing CCOH for x1 and x3, a drastic
57
Detection of Non-Linearities
Application of Non-Linear Interactions
improvement in comparison to COH12 and COH32 can be recognized, proving that the drops in ordinary
coherence are in fact caused by non-linear relationship between DOF 1 and 3 (see Figure 4-24).
Since the same effects on the FRFs and coherence functions have been observed when applying a
softening/hardening stiffness of equal severity in equivalent testing situation, the results are not shown
here.
IX)
Non-linearity between DOF 2 and 3, MIMO, Inputs at DOF 1 and 2
This case simulates a MIMO situation, intended to excite the hardening/softening stiffness in between
DOFs 2 and 3 well and investigating the effects of this particular non-linearity in a multi-reference set up.
Scaling factor s is chosen as s23 = 300 and a pure random signal is applied to DOFs 1 and 2, with an RMS
value of F = 50N each input. All eight frequency responses are estimated and shown in Figure 4-25.
Responses of DOFs 1 and 4 seem not affected by the non-linearity and FRFs H11, H12, H41, and H42 appear
to be measurements of a linear system. Frequency responses H21, H22, H31, and H32 on the other hand
clearly show distortions caused by the hardening/softening stiffness term. As in Case VIII), it should be
remarked that no significant shift of resonance frequencies can be observed, a characteristic of that type of
non-linearity. Besides not perfect coherence values at low frequencies, caused by the non-symmetric nature
of the hardening/softening stiffness, a distorted peak can be noted in a frequency of f ≈ 33Hz. It is believed
that this effect is caused by a secondary resonance. A check of correlation between {H21} and {H31} in the
frequency range of that peak (f = 30 – 35Hz) does not reveal a systematic relation (r({H21},{H31}) = -0.07).
If, on the other hand, the correlation coefficient is computed for the frequency range f = 20 – 25Hz, it is
found
that
{H21}
and
{H31}
seem
to
58
be
dependent
(r({H21},{H31}) = -0.85).
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
1 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
2 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
2 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
3 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
3 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
4 /1 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
4 /2 F R F - M a g n i t u d e a n d M C o h e r e n c e , F = 5 0 N
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
Figure 4-25: FRFs and MCOH, Hardening/Softening Stiffness, Inp. 1+2, k23
59
Detection of Non-Linearities
Application of Non-Linear Interactions
Looking at the significant improvement showing combined coherence, Figure 4-26, it can be stated, that
the low coherence values of MCOH2 and MCOH3 are in fact caused by non-linear relationship between
DOFs 2 and 3.
M COH 2 , M COH 3 and M CCOH (2+3)
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-26: Multiple and Combined Coherence, Hardening/Softening, Inp 1+2, k23
For this testing set up, the same results and characteristics were obtained when applying a
softening/hardening stiffness instead of the hardening/softening stiffness used in this example.
X)
Non-linearity between DOF 3 and 1/2/4, SIMO, Input at DOF 3
In this case, all stiffness elements connecting DOF 3 with the other three DOFs in the 4-DOF model
become non-linear, expressing a hardening/softening dependence upon the relative displacement.
Non-linear parameters in Equation (4-6) are chosen as s13 = 100, s23 = 300 and s34 = 200. When exciting at
DOF 3, this will be a valid simulation of a system with a local non-linearity induced by the SIMO situation.
60
Detection of Non-Linearities
Application of Non-Linear Interactions
The RMS level of the pure random excitation is F = 50N. The resulting frequency responses are shown in
Figure 4-27. As can be seen in Figure 4-27, all FRFs now have evidence of non-linear behavior, whereas
no significant frequency shift has been observed, as could be expected. In frequency responses H13 and
H33, the drop caused by a secondary resonance of order two with respect to the first resonance is clearly
identified. It should also be noted that in the region of resonances all coherences express relatively high
values (except the leakage-caused drop at first resonance). Furthermore, all FRFs, except the driving point
measurement H33, show poor coherence at higher frequencies. Also, a correlation between the distortions in
the FRFs, as done in previous cases, could not be established. This might be due to the fact that multiple
non-linearities are acting and a systematic separation might not be possible.
The combined coherences CCOH(13)3, CCOH(23)3, and CCOH(34)3 are computed, see Figure 4-28, where
some improvements are recognizable but a thorough interpretation appears infeasible because of the
multiple influences of the applied non-linearities. For example, CCOH(13)3 shows a clear increase in the
region of the secondary resonance (f ≈ 6.6Hz) in comparison to COH13 and COH33 and also in a frequency
range of about f ≈ 12Hz or f ≈ 17Hz but at the same time decreases in a frequency range of f ≈ 10Hz, where
COH13 and COH33 displayed mostly coherent behavior.
Concluding to the discussion of the effects of non-symmetric stiffness terms, it can be stated that for this
type of non-linearity frequency shifts are not significant when increasing the forcing level, secondary
resonances of even multiples with respect to actual resonances are possible and due to the nature of the
non-linearity the occurrence of non-coherent behavior at very low frequencies is likely. Since similar
effects occur in the frequency response and coherence functions, it will be hard to identify or separate
hardening/softening versus softening/hardening stiffness relations by only analyzing FRFs or coherence.
61
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-27: FRFs and COH, Hardening/Softening Stiffness, Inp. 3, k3all
62
Detection of Non-Linearities
Application of Non-Linear Interactions
COH 1/3 , COH 3/3 and CCOH (1+3)/3
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequency in Hz
25
30
35
40
1
0.5
0
1
0.5
0
COH 2/3 , COH 3/3 and CCOH (2+3)/3
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequency in Hz
25
30
35
40
1
0.5
0
1
0.5
0
COH 3/3 , COH 4/3 and CCOH (3+4)/3
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequency in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-28: Ordinary and Combined Coherence, Hardening/Softening Stiffness, Inp. 3, k3all
63
Detection of Non-Linearities
4.2.4
Application of Non-Linear Interactions
Deadzone/ Play
If, for a certain relative motion, no restoring force is present, one speaks about play or deadzone. This
non-linearity can be modeled as in part linear and zero in the region of play. The following formulation can
be used:
f r = k lin ⋅ (Δx + g )
Δx < -g
fr = 0
-g ≤ Δx ≤ g
f r = k lin ⋅ (Δx − g )
g < Δx
( 4-7)
In Equation (4-7), g denotes the play, where 2g is the total region of deadzone and klin the linear stiffness
factor. There are two main reasons why this non-linearity represents an interesting case to analyze. First,
because the deadzone relation has a non-continuous derivative, challenging the integration algorithm and
second, this non-linearity will behave more linear when forced at increasing forcing levels, as opposed to
all non-linear cases dealt with up to now. Figure 4-29 a) shows the frequency response and coherence for a
SDOF system with a deadzone stiffness term and Figure 4-29 b) demonstrates the linearizing concept of
this particular relation, which helps to understand the effects of increasing force level on the frequency
response and the coherence function.
For small excitation and therefore small relative motion, a given deadzone will result in large distortions
where on the other hand large excitation and large motion let the given deadzone appear small in relation to
the experienced displacement, making the distortion effects smaller as a result. One should expect better
FRF estimation and improved coherence for that type of non-linearity, when forcing at a higher level.
64
Detection of Non-Linearities
Application of Non-Linear Interactions
FRF - Magnitude and Coherence, Deadzone, 10N (s), 20N (d), 40N (dd)
-3
10
F
F = f(Δx)
k_lin2
-4
10
k_lin1
F2
-5
Δx
10
F1
0
5
10
15
20
25
30
Deadzone Nonlinearity
1
0.5
0
0
5
10
15
Frequency in Hz
20
25
30
b)
a)
Figure 4-29: a) SDOF System, b) Linearizing Concept
XI)
Non-Linearity between DOF 1 and 3, SIMO, Input 2
In this case, a deadzone stiffness is placed between DOFs 1 and 3 of the 4-DOF simulation model.
Non-linear parameter g is chosen as g = 0.0015m, representing half of the actual play. Input location is
DOF 2, exciting the system with a pure random signal characterized by a RMS value of F = 50N.
Estimated frequency responses and coherences are shown in Figure 4-30.
65
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /2 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-30: FRFs and COH, Deadzone, Inp. 2, k13
66
Detection of Non-Linearities
Application of Non-Linear Interactions
As can be seen from the FRFs and as it could be expected, the influence is greatest in H12 and H32, because
the non-linearity is directly connected to DOFs 1 and 3. All other stiffnesses in the model are behaving
linear. Also, the leakage caused drop in coherence is visible at all FRFs and at the deep anti-resonance in
the driving point measurement H22. Looking at H12 and H32, the motion in the frequency region of the
resonance shows mostly coherent behavior, not so around f ≈ 9 – 12Hz, where a secondary resonance and a
small relative motion between DOFs 1 and 3 are believed to be the reason for low coherence values.
Distortions in the high frequency range of H12 and H32 seem also caused by secondary resonances, but of
higher orders.
F RF Correl ati on, 1/2 vs. 3/2, f = 25 to 30 Hz
0.1
Corr. = -0.90709
0.05
log |H3 2 |
0
-0.05
-0.1
-0.15
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
log |H1 2 |
Figure 4-31: Distortion Correlation for Deadzone, k13
As mentioned before, in this case also, the high frequency distortions have been found to be correlated. In
fact the correlation factor between {H12} and {H32} has been computed to be r({H12},{H32}) = -0.61
(f = 20 - 25Hz), r({H12},{H32}) = -0.91 (f = 25 - 30Hz), r({H12},{H32}) = -0.94 (f = 30 - 35Hz), and
67
Detection of Non-Linearities
Application of Non-Linear Interactions
r({H12},{H32}) = -0.98 (f = 35- 40Hz). In Figure 4-31, {H32} is plotted versus {H12} for a frequency range
of f = 25 - 30Hz.
By computing the combined coherence function for x1 and x3, and therefore removing the direct interaction
between DOF 1 and 3, a drastic improvement is noticeable versus the ordinary coherence functions COH12
and COH32 (see Figure 4-32).
COH 1/2 , COH 3/2 and CCOH (1+3)/2, F = 50N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-32: Ordinary and Combined Coherence, Deadzone, Inp. 2, k13
XII)
Non-Linearity between DOF 3 and 1/2/4, SIMO, Input at DOF 3
In this case, the connections from DOF 3 to all other DOFs exhibit a play of 2g = 0.00075m, input location
is at DOF 3 and the pure excitation signal has a RMS value of F = 50N. Compared to Case XI), the
severity of the deadzone non-linearity has increased, since multiple non-linearities instead of only one
non-linearity are present. In Figure 4-33, FRFs and coherence functions are plotted.
68
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /3 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 5 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-33: FRFs and COH, Deadzone, Inp. 3, k3all
69
Detection of Non-Linearities
Application of Non-Linear Interactions
From Figure 4-33 it can be seen, that all except the first resonance frequency are at significantly lower
frequencies compared to the linear case. The linear frequency responses would be the asymptotic limits
when increasing the forcing level. Every FRF is clearly affected by the non-linearities applied, which is
evident in the distortion of the FRFs and in low coherence values. Non-coherent behavior is dominant from
a frequencies f > 20Hz, where better coherence can be noticed at the driving point measurement H33, which
could be explained by the direct input at DOF 3. Furthermore, for this case a correlation check between
distortions in FRFs was performed but did not yield solid results. It appears that the a strong correlation
might only be apparent when one dominant non-linearity is present.
COH i /3 and CCOH (1+2+3+4)
1
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0
1
0
1
0
1
0
Figure 4-34: Ordinary and Combined Coherence, Deadzone, Inp. 3, k3all
As in the cases before, the combined coherence function was computed for (x3 + x1), (x3 + x2) and (x3 + x4)
but no significant improvement in CCOH versus the ordinary coherence was found. This can be explained
by noting that adding only two motions together will not eliminate all non-linear effects since there are
multiple non-linearities within the system. Therefore, the combined coherence for the sum of all responses
70
Detection of Non-Linearities
Application of Non-Linear Interactions
(x1 + x2 + x3 + x4) versus input at DOF 3 has been estimated, see Figure 4-34. The combined coherence for
all responses shows in fact higher values, especially for high frequencies.
This concludes the discussion of non-linear stiffness effects upon the frequency response and coherence
function. It has been shown, that non-linear stiffness terms within the system alter frequency response and
consequently influence the coherence function, depending upon the nature of non-linearity. Basically four
different effects could be found: Shifted resonance frequencies; distortions in the frequency range of large
relative motion between certain DOFs; appearance of secondary resonances; and random-like distortions
for high frequencies. Furthermore, depending upon the type of non-linearity, distinct effects are apparent,
making the identification of particular non-linear stiffness dependence possible.
4.2.5
Non-Linear Damping
In this section the effects of non-linear damping will be analyzed, where three different types of non-linear
damping are considered: quadratic, non-symmetric, and coulomb damping.
XIII)
Quadratic Damping between DOF 1 and 3, SIMO, Input at DOF 1
This case simulates a quadratic offset to the linear relation between DOF 1 and 3 and which will be
formulated by the following equation:
f d = c lin ⋅ (Δx& + ε ⋅ Δx& ⋅ Δx& )
71
( 4-8)
Detection of Non-Linearities
Application of Non-Linear Interactions
It has been found, that using the lightly damped system as reference and developing the non-linear cases
from this baseline did not show significant effects on the frequency responses and coherence functions.
Therefore a higher damping value for the anticipated non-linear damping connection between DOF 1 and 3
was chosen, c13 = 200Ns/m as oppose to originally c13 = 8Ns/m. Determining the severity of the non-linear
effect, ε was fixed as ε = 10. The system was excited by a pure random signal with a RMS value of
F = 100N.
Figure 4-35 shows the estimated frequency response functions. As can be seen, the most affected
frequency responses are H11 and H31, since DOFs 1 and 3 are directly connected to the quadratic damping
connection. Since non-linear damping is being examined, no significant frequency shifts are visible and
should not be expected. On the other hand, a magnitude change between resonances can be observed, due
to the non-linear damping. As it turns out, the non-linear connection with c13 = 200Ns/m and ε = 10
modeled by Equation (4-8) behaves mainly as a linear system with a higher linear damping value
(c13 ≈ 400Ns/m). So, for a higher excitation level, the non-linear system would behave as an even larger
damped linear system. Furthermore, ordinary coherence does not necessarily reflect the non-linear
behavior of the system, only for COH31 is minor distortion in the high frequency range noticeable.
72
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-35: FRFs and COH, Quadratic Damping, Inp. 1, c13
73
Detection of Non-Linearities
XIV)
Application of Non-Linear Interactions
Softening/Hardening Damping between DOF 1 and 3, SIMO, Input at DOF 1
This case simulates a softening/hardening damping relation between DOFs 1 and 3. In the real world, this
damping characteristic can, for example, be found in car shocks. To formulate this particular type of
non-linearity, the same relation as for the softening/hardening stiffness is used, utilizing relative velocity as
the independent variable, and not displacement:
(
)
1
f d = c lin ⋅ ⋅ e s⋅Δx& − 1
s
( 4-9)
It can be seen, that Equation (4-9) converges towards the linear case for small relative velocity. As in the
case for quadratic damping, clin is chosen as c13 = 200Ns/m and scaling parameter s to s = 5. The 4-DOF
system is excited at DOF 1 by a pure random signal, RMS value of F = 100N.
All frequency responses are shown in Figure 4-36 and it can be seen, that no significant distortion can be
found in the FRFs and compared to the linear case (c13 = 200Ns/m) no noticeable deviation is apparent.
Analyzing the ordinary coherence functions, the only remarkable evidence is found in high frequency
distortions of COH31, all other coherences express mainly linear behavior. This fact might be due to the
nature of the non-symmetric non-linear damping term, making the “linearized” damping about the same
value as the linear damping value. A detailed explanation was given when non-symmetric stiffness was
discussed, so refer to Section 4.2.3.
74
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /1 F R F - M a g n i t u d e a n d C o h e r e n c e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-36: FRFs and COH, Softening/Hardening Damping, Inp 1, c13
75
Detection of Non-Linearities
XV)
Application of Non-Linear Interactions
Coulomb Damping between DOF 1 and 3, SIMO, Input at DOF 1
The third case, modeling a non-linear damping, will simulate coulomb friction between DOFs 1 and 3,
which can be described by:
f d = sig (Δx& ) ⋅ c13
( 4-10)
Equation (4-10) simply states, that depending on the direction of velocity, a constant damping force will
counteract the motion. This case can not be derived from the linear damping and it should be noted, that
this reaction force has a non-continuous derivative making it especially hard for the integration algorithm.
In order to make an integration possible in reasonable time, the error tolerance had to be lifted but, in order
to guarantee reliable results, the integration was repeated with smaller, but still feasible tolerances until
convergence was observed. For the actual simulation model, the damping factor was chosen to
c13 = 200Ns/m as in Cases XIII) and XIV), forcing the system with a pure random signal at DOF 1
characterized by a RMS value of F = 100N.
From the frequency responses, it can be seen (Figure 4-38), that especially for H11 and H31 the difference
between the linear (c13 = 200Ns/m) and the non-linear solution is significant. While the resonance
frequencies remain basically constant, the magnitude in between resonances and the location of
anti-resonances changes. As it shows, the actual system exhibits a much larger damping value between
DOFs 1 and 3 as the nominal damping value of c13 = 200Ns/m expresses. When compared to linear system
responses with a damping value of c13 = 6400Ns/m a much better match was reached. This fact might be
explained by the step-nature of the reaction force, inducing a much higher damping than the steady state
value of c13 = 200Ns/m suggests.
76
Detection of Non-Linearities
Application of Non-Linear Interactions
Looking at the combined coherence CCOH(13)1, it can be seen, that the distortions in x1 and x3 and therefore
in COH11 and COH31 are in fact caused by the non-linear relation between these two degrees of freedom
(see Figure 4-37).
COH 1/1 , COH 3/1 and CCOH (1+3)/1, F = 100N
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequenc y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-37: Ordinary and Combined Coherence, Coulomb Friction, Inp. 1, c13
This concludes the discussions about the influence of non-linear damping upon the frequency response and
coherence function. It has been shown, that a non-linear damping term basically alters the magnitude of the
FRF and corresponding anti-resonance but no significant shift in resonance frequency has been observed.
Furthermore, it becomes hard to specify the particular non-linear damping relation due to the similar effects
of different damping non-linearities.
77
Detection of Non-Linearities
Application of Non-Linear Interactions
1 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
2 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
3 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
4 /1 F R F - M ag n i tude and C o h e r e nc e , F = 1 0 0 N
-2
10
N o n - L in e a r
L in e a r
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
1
0 .5
0
Figure 4-38: FRFs and COH, Coulomb Friction, Inp. 1, c13
78
Detection of Non-Linearities
4.3
Application of Non-Linear Interactions
Non-Linear Effects in the Presence of Noise
When using pure random signals for modal testing it must be acknowledged, that obtaining only one (1) set
of measurements will not excite all frequencies equally well and therefore, for reliable results, several
ensembles must be taken. Furthermore, noise, which usually is assumed to be of random distribution, has
to be expected on the measured time histories. Figure 4-39 shows the same frequency response estimation
of a linear system for different numbers of spectral averages. As can be seen, the variance in system
response due to the variance in excitation will average out and the frequency response will converge to the
true value. It can also be noticed, that when the system response is covered by noise because of small
response magnitude (anti-resonance and in the high frequency range), low coherence values will result.
The effects of leakage at the first resonance and the anti-resonance are visible too.
FRF - Magni tude and Coherence, Avg = 1
-2
F RF - Magni tude and Coherence, Avg = 30
-2
10
10
-3
10
-3
10
-4
10
-4
10
-5
10
-5
10
-6
10
-6
10
-7
10
-7
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0.5
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequency in Hz
25
30
35
40
0
0
5
10
15
20
Frequency in Hz
25
30
35
40
Figure 4-39: FRF Estimation as Function of Spectral Averages
In summary, when employing a pure random excitation to a linear system, frequency response and/or
coherence distortions can have their source in leakage, insufficient number of spectral averages and
measurement noise. If there is non-linear motion within the structure one additional source of frequency
79
Detection of Non-Linearities
Application of Non-Linear Interactions
response and coherence distortion is possible. After looking at the characteristic effects of non-linearities
without the presence of noise in Section 4.2, this portion of the paper tries to analyze the influence of
measurement noise and the dependence on the number of averages.
For example, a softening stiffness non-linearity, discussed in Section 4.2.2, is chosen. The softening term,
formulated by Equation 4-5, is applied to the connection between DOFs 2 and 3 of the 4-DOF model,
where non-linear scaling factors s and a are set to s = 2500 and a = 10. After exciting the system by a pure
random signal with a RMS value of F = 150N at DOF 1 and simulating the response, random noise was
added, both to reference and response time histories (1% noise level on input and output).
Frequency responses H21 and H31 are plotted in comparison to the linear response and as function of
spectral averages (Avg = 1/10/20/30), see Figure 4-40. For only one (1) average the variance in system
response and the influence of measurement noise in a frequency range of f > 20Hz (small signal to noise
ratio) is clearly recognizable. With only one average taken, no statement regarding the linearity relation can
be made since no base line has been established. After ten (10) averages, the FRFs appear smoother and the
influence of variance at different spectral lines decreases. Also, low coherence values are present at high
frequencies (f > 20Hz) and since the corresponding FRFs show corresponding distortions about a constant
magnitude, it could be assumed that the signal goes down into the noise floor. Notice that even though the
frequency responses are clearing up, low coherence values are present in a region of f ≈ 12 - 18Hz. The
same can be said about the frequency estimates for 20 and 30 averages. The low coherence values in a
frequency range of f ≈ 12 - 18Hz can not be accounted for by leakage, measurement noise, or insufficient
averaging.
80
Detection of Non-Linearities
Application of Non-Linear Interactions
2 /1 F R F - M ag n i tude and C o he r e nc e , Avg = 1
-2
3 /1 F R F - M ag n i tude and C o he r e nc e , Avg = 1
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
2 /1 F R F - M a g n i t u d e a n d C o h e r e n c e , A v g = 1 0
-2
3 /1 F R F - M a g n i t u d e a n d C o h e r e n c e , A v g = 1 0
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
2 /1 F R F - M a g n i t u d e a n d C o h e r e n c e , A v g = 2 0
-2
3 /1 F R F - M a g n i t u d e a n d C o h e r e n c e , A v g = 2 0
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
2 /1 F R F - M a g n i t u d e a n d C o h e r e n c e , A v g = 3 0
-2
3 /1 F R F - M a g n i t u d e a n d C o h e r e n c e , A v g = 3 0
-2
10
10
N o n - L in e a r
L in e a r
-3
N o n - L in e a r
L in e a r
-3
10
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
0
5
10
15
20
25
30
35
40
1
1
0 .5
0 .5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
0
0
5
10
15
20
F r e q u e n c y in H z
25
30
35
40
Figure 4-40: H21 and H31 as Function of Averages, Softening Stiffness, Inp. 1
81
Detection of Non-Linearities
Application of Non-Linear Interactions
C O H 2 /1 , C O H 3 /1 an d C C O H ( 2 + 3 ) /1 , Avg = 1 0
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F req u en c y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
C O H 2 /1 , C O H 3 /1 an d C C O H ( 2 + 3 ) /1 , Avg = 2 0
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F req u en c y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
C O H 2 /1 , C O H 3 /1 an d C C O H ( 2 + 3 ) /1 , Avg = 3 0
1
0.5
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
F req u en c y in Hz
25
30
35
40
1
0.5
0
1
0.5
0
Figure 4-41: Ordinary and Combined Coherence, Softening Stiffness, Inp. 1
82
Detection of Non-Linearities
Application of Non-Linear Interactions
Knowing from Section 4.2.2 that these effects are indeed caused by the non-linear stiffness between
DOFs 2 and 3, the combined coherence is computed for (x2 + x3) at 10, 20 and 30 averages (see
Figure 4-41). It can be seen that the combined coherence exhibits higher values in a frequency range of
f ≈ 12 - 18Hz, confirming the assumption that these coherence drops in COH21 and COH31 are caused by a
non-linear relation. Furthermore, when coherence values start to decrease because of measurement noise
(f > 20Hz), the combined coherence indicates slightly higher values compared to ordinary coherence. This
in turn points out, that the drop in coherence is, in part, also induced by the softening stiffness.
Since it has been shown in Section 4.2 that a correlated, non-linearity caused, noise-like distortion in a high
frequency region is possible, the correlation factor between {H21} and {H31} has been computed. But no
evidence of a correlation could be found, for example r({H21},{H31}) = 0.04 (f = 20 –25Hz, Avg. = 10),
r({H21},{H31}) = 0.03 (f = 20 –25Hz, Avg. = 20) or r({H21},{H31}) = 0.12 (f = 20 –25Hz, Avg. = 30).
Therefore, the high frequency distortions in H21 and H31 seem to be in fact mainly caused by random
measurement noise since the deterministic signal is hidden in the noise floor. It should be mentioned, that
as extension to the SIMO test described here, MIMO situations have been applied (for example Inputs at
DOFs 1 and 2) where the distributed energy level realizes a more uniform excitation. For these cases, the
non-linearity between DOFs 2 and 3 is also more engaged in the system response and high frequency,
noise-like distortions are correlated which appears to be evidence of a non-linear relation. This fact
emphasizes the known argument that in order to investigate non-linear effects, the non-linearity itself has
to be excited well enough.
83
Detection of Non-Linearities
Summary/ Future Work
5 SUMMARY/ FUTURE WORK
In this thesis a simulation study was conducted in order to investigate the possibility of detecting the
presence of structural non-linearities by using the frequency response and coherence functions. A MDOF
SIMULINK® model was generated and structural non-linearities (stiffness and damping related) have been
applied at different locations and in different testing situations. It was intended to duplicate the typical
modal analysis procedure (using pure random excitation), which estimates the frequency response
functions by measuring inputs and outputs.
In summary, it can be said, that the effects of structural non-linearities are diverse and dependent on the
type of non-linearity applied. It has been found, that not only large excitation and with that, large relative
motion between degrees of freedom, cause large non-linear effects. The investigation of deadzone stiffness
related non-linearity showed, that especially small excitation levels force the system to exhibit large
non-linear distortions, whereas higher excitation levels let the system appear more linear. If the non-linear
relationship is stiffness related, shifts of resonance frequencies, distortion in the frequency region of
extensive non-linear motion (small or large motion) between degrees of freedom, the occurrence of
secondary resonances, and high frequency distortions can be noticed. Furthermore it can be mentioned, that
these high frequency distortions appear noise-like but are in fact correlated, which may be used to detect
the presence of structural non-linearities. If the non-linear relationship is damping related, it has been
found, that, in order to show non-linear damping effects at all, the severity of the non-linear damping
relation and the damping itself has to be extremely large. Since the very nature of damping, non-linear
damping mainly does not affect the frequencies of natural resonances but the magnitude of vibration in
between resonances and the anti-resonances correlated with the non-linear element.
84
Detection of Non-Linearities
Summary/ Future Work
This thesis also proposes a method, called combined coherence, which is capable of detecting the effects of
non-linear motion within measurements. The combined coherence is based on the idea of removing the
contribution of direct interaction between two degrees of freedom by forming a new coordinate (sum of
motions). A new coherence function (combined coherence, CCOH) between inputs and the new, virtual
coordinates provides a method, whether distortions in the original coherence functions are in fact caused by
structural non-linearities. It has also been shown that distortions in frequency response and coherence
functions caused by measurement noise or digital signal processing errors (leakage and aliasing) will not
be affected by the technique of combined coherence. This way, not only a detection scheme for the effects
of non-linear motion is realized but also the effects of structural non-linearities can be separated from
measurement noise and digital signal processing errors.
While though the combined coherence is developed on lumped mass structures and some knowledge about
the mass distribution is required, future work (after testing the combined coherence function on a
simulation model) should include the application to continuous, real world structures. The model of a
lumped mass structure might still be a valid approximation of the real system and some knowledge about
the mass distribution might be available in these practical testing situations. Furthermore, it should be
investigated, if the combined coherence method, a linear spectra based technique, can be modified in order
to use it as a post-processing tool. In this way, the advantages of structural non-linearity detection, and
computational and memory efficient analysis could be combined.
85
Detection of Non-Linearities
Reference List
6 REFERENCE LIST
[1]
“The Behavior of Nonlinear Vibrating Systems”, Volume I+II
W. Szemplinska-Stupnicka
Kluwer Academic Publishers, 1990
ISBN 0-7923-0368-7/ 0-7923-0369-5
[2]
“Some Engineering Applications in Random Vibrations and Random Structures”
G. Maymon
American Institute of Aeronautics and Astronautics, Inc., 1998
ISBN 1-56347-258-9
[3]
“Applications of Random Vibrations”
N.C. Nigam, S. Narayanan
Springer Verlag, Berlin, Heidelberg, New York, 1994
ISBN 3-540-19861-X
[4]
“Normal Modes and Localization in Nonlinear Systems”
A.F. Vakakis et al.
John Wiley & Sons, Inc., 1996
ISBN 0-471-13319-1
[5]
“Nonlinear Stochastic Dynamic Engineering Systems”
F. Ziegler, G.I. Schueller
Springer Verlag, Berlin, Heidelberg, New York, 1988
ISBN 3-540-18804-5
[6]
“Model Testing: Theory and Practice”
D.J. Ewins
Research Studies Press, Ltd., 1984
ISBN 0-86380-017-3
86
Detection of Non-Linearities
[7]
Reference List
“Nonlinear System Analysis and Identification from Random Data”
J.S. Bendat
John Wiley & Sons, Inc., 1990
ISBN 0-471-60623-5
[8]
“Nonlinear Systems Techniques and Applications”
J.S. Bendat
John Wiley & Sons, Inc., 1998
ISBN 0-471-16576-X
[9]
“Theory of Vibration with Applications”, 5th Edition
W.T. Thomson, M.D. Dahlen
Prentice-Hall, Inc., 1998
ISBN 0-13-651068-X
[10]
“Mechanical Vibrations”, 2nd Edition
A.H. Church
John Wiley and Sons, Inc., 1963
[11]
“The Volterra and Wiener Theories of Nonlinear Systems”
M. Schetzen
John Wiley & Sons, Inc., 1980
ISBN 0-471-04455-5
[12]
“Shock and Vibration Handbook” 4th Edition
C.M. Harris
The McGraw-Hill Companies, Inc., 1996
ISBN 0-07-026920-3
[13]
“Numerical Solution of Ordinary Differential Equations”
L.F. Shampine
Chapman and Hall, Inc., 1994
ISBN 0-412-05151-6
87
Detection of Non-Linearities
[14]
Reference List
“Network Analysis with Applications”, 2nd Edition
W.D. Stanley
Prentice-Hall, Inc., 1997
ISBN 0-13-260910-X
[15]
“Advanced Engineering Mathematics”, 7th Edition
E. Kreyszig
John Wiley & Sons, Inc., 1993
ISBN 0-471-55380-8
[16]
“Applied Numerical Methods for Digital Computation”, 4th Edition
M.L. James, G.M. Smith, J.C. Wolford
HarperCollins College Publisher, 1993
ISBN 0-06-500494-9
[17]
“Simulink, Dynamic System Simulation for Matlab”
The Math Works; 1997
[18]
“Matlab, The Language of Technical Computing”
The Math Works, 1996
[19]
“Taschenbuch Mathematischer Formeln und Moderner Verfahren”
H. Stoecker
Verlag Harri Deutsch, 1993
ISBN 3-8171-1256-4
[20]
“A Spatial Approach to Nonlinear Vibration Analysis”
D.E. Adams
Ph.D. Dissertation, University of Cincinnati, 2000
[21]
“Detection of Nonlinear Dynamic Behaviour of Mechanical Structures”
M. Mertens, et al.
IMAC Conference, p.712-719, 1986
88
Detection of Non-Linearities
[22]
Reference List
“Detection, Identification and Quantification of Nonlinearity in Modal Analysis – A Review”
G.R. Tomlinson
IMAC Conference, p.837-843, 1986
[23]
“Detecting Nonlinearities Utilizing the Multiple Input Frequency Response Function Estimation
Theory”
G.W. Hopton
Master Thesis, University of Cincinnati, 1987
[24]
“Vibrations: Experimental Modal Analysis”
R.J. Allemang
Course Notes, University of Cincinnati, 1995
89
Detection of Non-Linearities
Appendix
7 APPENDIX
7.1
SIMULINK® Model
4 DOF Model, SIMULINK®
90
Detection of Non-Linearities
Appendix
1
x_dot_1
9
2
f_1
x_1
1/m 1
1
1
s
velocity
s
position
Display
1/m _n
S um
x_ d o t_ 0
1
x_dot_0
x_ d o t_ 1
f_ 0 _ 1
x_ 0
2
x_0
x_ 1
S DU_0_1
x_ d o t_ 1
x_dot_2
x_ d o t_ 2
3
f_ 1 _ 2
x_ 1
x_ 2
4
x_2
S DU_1_2
x_ d o t_ 1
x_dot_3
x_ d o t_ 3
5
f_ 1 _ 3
x_ 1
x_ 3
6
x_3
S DU_1_3
x_ d o t_ 1
x_dot_4
x_ d o t_ 4
7
f_ 1 _ 4
x_ 1
x_ 4
8
x_4
S DU_1_4
4 DOF Model, SIMULINK®, Subblock DOF_1
91
Detection of Non-Linearities
Appendix
1
x_dot_1
u*c_1_2
2
c_n_(n+1)
x_dot_2
S um
1
f_1_2
3
S um 2
x_1
u*k_1_2
4
k_n_(n+ 1)
x_2
S um 1
4-DOF-Model, SIMULINK®, DOF_1, SDU_1_2
92
Detection of Non-Linearities
7.2
Appendix
Definition of Physical Parameters, 4-DOF-Model, Linear
Mass in [kg]
DOF
1
1
2
3
4
12
2
3
4
7
9
14
Damping in [Ns/m]
DOF
Ground
1
2
3
4
Ground
1
2
3
4
8
8
9
7
4
5
6
6
8
8
7
9
4
5
Stiffness in [N/m]
DOF
Ground
1
2
3
4
Ground
1
2
3
4
19000
19000
24000
24000
20000
20000
22000
22000
19000
19000
24000
93
24000
20000
20000
Detection of Non-Linearities
Appendix
7.3
List of Cases in Section 4.2
I)
Hardening Stiffness, k13, SIMO, Input at DOF 3 (page 1)
II)
Hardening Stiffness, k13, SIMO, Input at DOF 2 (page 1)
III)
Hardening Stiffness, k23, SIMO, Input at DOF 1 (page 1)
IV)
Hardening Stiffness, k23, MIMO, Inputs at DOFs 1 and 2 (page 1)
V)
Softening Stiffness, k23, SIMO, Input at DOF 1 (page 1)
VI)
Softening Stiffness, k23, MIMO, Inputs at DOFs 1 and 2 (page 1)
VII)
Softening Stiffness, k3all, SIMO, Input at DOF 3 (page 1)
VIII)
Hardening/ Softening Stiffness, k13, SIMO, Input at DOF 2 (page 1)
IX)
Hardening/ Softening Stiffness, k23, MIMO, Inputs at DOFs 1 and 2 (page 1)
X)
Hardening/ Softening Stiffness, k3all, SIMO, Input at DOF 3 (page 1)
XI)
Deadzone Stiffness, k13, SIMO, Input at DOF 2 (page 1)
XII)
Deadzone Stiffness, k3all, SIMO, Input at DOF 3 (page 1)
XIII)
Quadratic Damping, k13, SIMO, Input at DOF 1 (page 1)
XIV)
Softening/ Hardening Damping, k13, SIMO, Input at DOF 1 (page 1)
XV)
Coulomb Friction, k13, SIMO, Input at DOF 1 (page 1)
94