Survey of Nonlinear Detection and Identification Techniques
for Experimental Vibrations
Douglas E. Adams, Research Assistant
Randall J. Allemang, Professor
University of Cincinnati
Structural Dynamics Research Laboratory (SDRL)
Cincinnati, Ohio USA
Abstract
Nonlinear structural dynamic system identification is often more a subjective art than it is a direct application
of some particular method in systems theory. The nonlinear problem is subjective because although there are
many analytical methods from which to choose, there is no general approach to detect, characterise, or model
input-output relationships in nonlinear systems. This paper is a survey of the numerous experimental
nonlinear structural dynamic system identification techniques that have been implemented by vibration
engineers in recent years for SDOF and MDOF systems in both the time and frequency domain. The survey
discusses the following detection and identification methods from an experimental perspective: Spectral
analysis and the reverse-path formulation, Volterra and Wiener series, nonlinear auto-regressive moving
average models, the restoring force method for SDOF and MDOF systems, the describing function methods,
direct parameter estimation, Hilbert transforms and wavelet transforms, neural networks and general
nonlinear experimental testing techniques.
1 Introduction and Motivation
Nonlinear structural dynamic analysis techniques
are necessary because all systems are nonlinear over
certain input-output amplitude ranges. Saturation is
one of the most obvious examples of this in physical
systems. By preventing the input or output from
becoming too large, saturation nonlinearities create
high frequency content in the input or output
spectra. Hardening and softening springs are by far
the most commonly encountered nonlinearities in
the nonlinear structural dynamic literature. These
nonlinear springs are like springs in the real world in
that they cannot provide constant linear stiffness
over an infinite range of displacement; therefore, the
force-deflection characteristic bends to either reduce
displacement (hardening) or allow for more
displacement (softening) as the force grows. The
prototypical example of softening spring stiffness,
the potential energy of a swinging pendulum, causes
the period of oscillation of the pendulum to increase
as the amplitude of oscillation increases.
Other physically important examples of
analytical nonlinear behaviors include jumps, splits,
and modal interaction in the frequency domain, limit
cycles in the phase plane, hysteresis in the
static characteristic of a spring or damper, and
chaotic responses in the time domain. A general
discussion of these and other nonlinear topics is
found in [3].
Linear analysis techniques fail when the system
is nonlinear because the principle of superposition
fails. Input frequencies interact with one another to
produce output frequencies at harmonics or
subharmonics of the input frequencies. Detecting,
classifying, and modelling nonlinear structural
dynamics is difficult because there is no analysis
method that is superior to all other methods for all
systems in all instances. In linear vibration tests,
modal analysis is the most popular method because
it produces a small set of parameters that can
describe the behavior of a system for any input type
and any range of the input. If modal analysis is used
on nonlinear systems, the modal parameters vary for
different inputs and different input amplitudes.
Linear models can actually do deceptively well at
replicating a given nonlinear input-output
relationship because the nonlinear terms are usually
highly correlated with the linear terms.
The
correlation between linear and nonlinear terms is
one of the difficult features of nonlinear systems to
overcome.
For systems that can be modelled with a few
well-separated degrees of freedom (DOF), there are
a variety of popular nonlinear analysis methods that
work well. For systems with highly coupled modes,
modes with high damping and poor separation, the
search for a nonlinear analysis approach may lead to
a dead end. Furthermore, systems with multiple
degrees of freedom (MDOF) are rarely used as test
vehicles to verify that new nonlinear analytical or
experimental analysis techniques work. In fact, it
may not always be possible to extend the single
degree of freedom (SDOF) methods to MDOF
applications. This is a problem because most
systems have many DOF.
This paper presents simplified descriptions of the
most popular nonlinear structural dynamic analysis
methods and discusses the experimental limitations
of the methods in light of a MDOF analytical test
system where appropriate. An intuitive framework
is used throughout the paper to simplify the
discussion. Although limitations on the paper
length prevent a thorough derivation of each
analysis method, the discussion surrounding the test
system is used to highlight the most important
features of each method. The authors recommend
the following references for further review of these
topics: [1,2]. All systems referenced in this report
are assumed to be time-invariant.
Eq 2 is not the typical experimental form of the
differential equation because the motion is not
usually controlled in the test; however, this form of
the input-output equation can be used to develop
powerful methods for analyzing nonlinearities
[7,10]. The reverse path equation is akin to an
impedance, or dynamic stiffness, relation whereas
the FRF is a compliance relation. Section 4.1.4
discusses the merits of this method.
2.2 Nonlinearities as Feedback
Many of the effects that nonlinearities have on
system response can be more easily understood if
the output nonlinearity is moved to the right-handside of the equation and treated as a feedback
forcing term. For example, a SDOF system with a
cubic stiffness nonlinearity can be represented by
the following equation:
m&x& + cx& + kx = f (t ) − μx 3
(3)
Eq 3 indicates that for sinusoidal inputs, the
response will feedback into the system and create
responses of decreasing amplitude at higher
harmonics of the forcing frequency.
This is
precisely what is observed in systems with nonlinear
stiffness (see Figure 1).
2 Features of Nonlinear Systems
2.1 Common Output Nonlinearities
Output nonlinearities are the terms in the inputoutput differential equation that are nonlinear
functions of the output only. Experimentally, this
distinction is important because the input-output
model of the system must be both accurate and
physically obtainable. Consider a SDOF system
with Coulomb damping that can be represented by
the following differential equation:
m&x& + c ⋅ sgn( x& ) + kx = f (t )
(1)
Although the nonlinearity is on the output in this
equation, measurements on the input and output
could be swapped to give a reverse path nonlinear
equation:
f (t ) = m&x& + c ⋅ sgn( x& ) + kx
(2)
Figure 1: Windowed FFT of Sinusoidal Input
and Response of System with Cubic Stiffness
2.3 Time Invariance vs. Nonlinearity
In this report and in much of the nonlinear structural
dynamic literature, one of the ways in which
nonlinearities are characterized is the change in
frequency or modal damping with amplitude of the
response or input. Since the change in the modal
parameter is brought on by a change in an
amplitude, the nonlinear system is still time
invariant even though the amplitude is changing
with time. This is a conceptually important point
because so many nonlinear systems are
characterized with this type of technique.
detecting, classifying, and locating nonlinearities in
structural dynamic systems. After applying one or
more of these three steps, system identification can
be carried out to generate a working model of the
system. Figure 2 illustrates the nonlinear analysis
cycle for a structural dynamic system.
2.4 Non-Additive Noise
Dynamic response and input measurements always
contain noise. In linear vibration analysis, the noise
is modelled as additive on both the input and output
data channels, and because the model of the system
is linear, the noise propagates through the model as
an additive term, which is uncorrelated with the
data. One of the difficulties in experimentally
modelling nonlinear vibrating systems is that even if
the noise is modelled as additive on the input and
output channels, the measured noise propagates
through the model and creates cross-product noise
that is correlated with the measured data. This fact
contributes to the bias errors of the estimated
nonlinear model parameters.
3 Aspects of Nonlinear Analysis
Figure 2: Aspects of Nonlinear Analysis
3.1 Characterizing Nonlinearities
There are three sequential steps that help to
characterize nonlinearities: detection, classification,
and location. Detection decides if there are any
nonlinearities, classification determines their type,
and location determines where they are. For
instance, if a cubic stiffness nonlinearity causes a
measured frequency response function (FRF) to
bend towards higher or lower frequencies, then the
FRF detects the nonlinearity and the shape of the
FRF classifies the nonlinearity as a cubic-like
stiffness. Since nonlinearities are non-unique by
nature, i.e. one class of nonlinearity can behave like
another in a certain input-output amplitude range,
the shape of the FRF is not conclusive evidence of a
cubic nonlinearity.
Locating the nonlinear link in a discrete model
can be even more challenging. Without prior
knowledge of the relative motion between the
measurement DOF in certain frequency ranges of
interest, trial-and-error methods must be used to
locate the nonlinearity in a particular model
structure. Section 4.1 discusses several methods for
3.2 System Identification
The goal of experimental system identification in
nonlinear analysis is to generate a mathematical
model of a system from measurements of the
input(s) and output(s) of the system. If there is
some prior knowledge of a suitable model of the
system, a parametric model is used to generate a set
of parameters for describing the input-output
behavior of the system. Modal analysis in linear
systems goes one step further by condensing the
parameters in the mathematical model into a
minimum set of modal parameters, the modal
frequencies, modal vectors, and modal scaling.
Since the frequencies and mode shapes of a
nonlinear system generally change with the input
and/or response amplitude, nonlinear system
identification usually produces a larger set of
parameters than the corresponding linear problem.
Direct parameter estimation is an example of a
parametric method for modelling systems. Several
of the most popular methods for estimating
nonlinear system parameters are described in section
4.2.
There are also system identification methods that
do not automatically produce parameters for
modelling structural dynamic systems.
The
nonparametric methods do not make any prior
assumptions that systems have a particular model
structure. Instead, these methods use functional
relationships of some kind to describe the system
input-output dynamic behavior.
Volterra and
Wiener series are the most well known
nonparametric identification methods.
The first step in modelling a nonlinear system
should always be characterization since this helps to
define the components in the model. In a linear
modal model, the model order and the number of
references are the important choices because they
determine the number of modal frequencies in the
model. In nonlinear models, there are many more
choices that go into designing a model structure.
Characterization and system identification are both
described in the next section.
4 Review of Analysis Methods
4.1 Detection, Classification, Location
Detecting, classifying, and locating nonlinearities
can be accomplished in either the time domain or in
a frequency transform domain. The MDOF mass,
stiffness, and damper test system that is shown in
the following figure will be used to illustrate the key
features of the various nonlinear analysis methods.
Although many methods will be discussed in the
following review, there will undoubtedly be
omissions for which the authors apologize in
advance. Note that detection, classification, and
location have been collectively referred to as
'structure detection' by other researchers [5].
4.1.1
First Order FRF Methods
If a stepped sine excitation is used to generate an
input-output FRF, the shape of the FRF relative to
the shape of the FRF for the underlying linear
system can be used to detect, classify, and locate the
Figure 3: Three DOF Analytical Test System
nonlinearities in the system. Impact excitations can
also indicate that there are nonlinearities, but they
make it more difficult to classify the nonlinearity
(i.e. quadratic, cubic, sin, etc.). Random inputs do
not work as well either because they require many
averages to reduce the variance in the measurement,
and averages do their best to hide the nonlinearity,
which is treated as just another noise source in the
system.
An excellent discussion of the principles behind
this technique are given in [6]. For a system with an
output nonlinearity, the basic idea is this: if the
response amplitude is held constant while the force
is varied, a linear FRF generated. If an entire family
of linear FRFs are produced for different constant
output amplitudes, then the true FRF of the system
can be thought of as a combination of the family of
linear FRF. The resulting FRF bends because of the
nonlinearity, and it is this warping of the FRF that
not only detects but may also classify the
nonlinearity.
Figure 4 is an illustration of several FRFs of the
three DOF test system for varying stiffness values.
The nonlinearity is a cubic stiffness to ground at the
first DOF. The FRFs clearly bend to the right.
Although stepped-sine FRFs are a relatively easy
way to detect simple nonlinearities, the data
acquisition time for measuring them is long, the
detection process is very subjective, and harmonics
and subharmonics do not show up in the first order
FRF estimates. Neural networks have been used to
eliminate subjective errors with some success, but
the amount of training data that would be required to
classify all types of nonlinearities is mammoth; this
is a problem with implementing these networks [8].
The transfer of input energy to frequencies other
than the driving frequency is a defining behavior of
nonlinear systems, and limits the usefulness of the
linear FRF estimate for describing nonlinear systems
by making the FRF dependent on input amplitude.
Higher order FRFs remove this limitation by
accounting for the energy transfer to higher or lower
harmonics.
Figure 4: Stepped-sine FRFs for three DOF with
Varying Cubic Stiffness to Ground at First DOF
4.1.2
Figure 5: Detection of Cubic Nonlinearity with
the Least Squares Coherence Function Estimate
Least Squares Coherence Functions
If a random input is used instead of stepped-sine, the
least squares FRF estimate falls somewhere in
between the FRF of the underlying linear system
and the stepped-sine FRF. Since it is usually
difficult to detect nonlinearity in the FRF estimate
for a random input, the coherence function is used
instead. Recall that the coherence can be low if
there are uncorrelated inputs, leakage or other signal
processing errors, or stimulated nonlinearities in the
system [9]. If the other two possibilities are
eliminated by good testing procedure, the
experimental coherence function can sometimes be
used to detect nonlinearities.
Figure 5 shows that even though the nonlinear
FRF does not clearly detect the nonlinearity, the
coherence function does detect the cubic stiffness in
the first two modes of vibration. The problems
with using coherence functions for nonlinear
analysis include: all measurement sources of low
coherence are not always eliminated completely,
less severe nonlinearities do not cause very large
drops in the coherence, and classification is usually
not possible with coherence functions.
4.1.3
Higher order FRFs can be computed directly using
multi-dimensional Fourier analysis; however, this is
involved for all orders above two, so they are
usually estimated in the same way as first order
FRFs: a sinusoidal input is used to drive the system
and the response at the driving frequency is
Higher Order FRF Methods
measured. The difference between the first and
higher order methods is that the higher order FRFs
keep track of the harmonics that are produced by a
nonlinear system for a sinusoidal input, whereas the
first order FRFs do not. There are two important
manifestations of this difference: 1) the first order
FRF changes with the input amplitude, whereas the
(theoretical) higher order FRF is the same for any
input amplitude; and 2) the higher order FRF is a
combination (product) of powers of the first order
FRF evaluated at multiple frequency variables. The
Volterra and Wiener series solutions are the key to
computing the higher dimensional FRFs.
An
excellent reference on experimentally estimating
higher order FRFs is found in [11]. The most
complete reference on Volterra and Wiener series is
[12].
Higher order FRFs are derived from the Volterra
model of a nonlinear input-output relationship. This
model uses a functional power series of the input to
generate the output as follows:
∞
x(t ) = ∑Vn [ f (t )]
n =1
(4)
where the Vn[•] are the Volterra integral operators.
The theory is based entirely on multiple integrals,
and the main results are the forms of the solution for
a complex exponential and a sinusoid. These are
obtained by probing the system with single and
multiple harmonic inputs [13]. For the complex
exponential, the solution is,
x(t ) = H1 (ω ) Fe jωt + H 2 (ω ,ω ) F 2e j 2ωt + L (5)
and for the sinusoid (cos), the solution is,
x(t ) = H1 (ω )
they are evaluated along a n-dimensional line of
equal frequencies.
The problems with higher order FRFs are that they
are difficult to compute and measure, difficult to
interpret for MDOF experimental data, and do not
do as well characterizing non-polynomial
nonlinearities because the Volterra series only converge for polynomial nonlinearities (e.g. Volterra
theory cannot deal with hysteretic nonlinearities).
The Wiener theory uses orthogonal polynomials to
extend the capabilities of functional series to nonpolynomial nonlinearities.
F jωt
F
e + H1 (−ω ) e − jωt + L (6)
2
2
Since these two expressions are different, and
because a complex exponential forcing function is
not experimentally realizable, the FRFs that can be
measured are not the same as the theoretical higher
order FRFs. This is the reason why the first order
measured FRF bends as in Figure 4; the extra terms
that are produced by the experimentally applied
stepped-sine input contaminate the theoretical FRF.
This will be demonstrated using the SDOF Duffing
oscillator:
m&x& + cx& + kx + μx 3 = f (t )
(7)
The higher dimensional FRFs for this system to
third order are given by:
1
− mω + jωc + k
H 2 (ω ) = 0
H1 (ω ) =
2
(8)
H 3 (ω ) = − μH13 (ω ) H1 (3ω )
The presence of the third order subharmonic in
the third order FRF in Eq [8] is obvious; the
theoretical and measured FRF for complex
exponential and sinusoidal stepped inputs,
respectively, are shown in Figure 6. Note that the
measured first and second order FRFs are
characteristically
distorted
towards
higher
frequencies. Also note that only the 'diagonal' FRFs
were measured and plotted using Eq 8 because the
'off-diagonal' terms are too difficult to measure or
interpret. The FRFs are called 'diagonal' because
Figure 6: Measured and Theoretical FRFs for
System with Cubic Stiffness Nonlinearity
4.1.4 First Order Dynamic Stiffness
By estimating the inverted FRF, or dynamic
stiffness function (DSF), the stiffness and damping
are uncoupled by the real and imaginary parts,
respectively [6]. Even though the nonlinearity
bleeds into both the real and imaginary parts of the
DSF, the real part contains all of the pertinent
information for analysis if there is nonlinear
stiffness, and the imaginary part is the relevant
function to examine if there is nonlinear damping.
Characterization of the nonlinearity is carried out by
plotting the real part of the DSF as a function of the
square of the frequency-squared or the imaginary
part as a function of linear frequency. If the real
part is a straight line near resonance, the system is
linear; however, if the line is not straight, the shape
detects and can be used to classify the nonlinearity.
Similar rules apply for the imaginary part of the
DSF.
A SDOF system with cubic stiffness will be used
to demonstrate the technique. The two plots in
Figure 7 show the characteristic that the cubic
stiffness produces in the real and imaginary parts of
the DSF.
amplitudes and frequencies are computed by taking
the magnitude of the analytic signal and the
derivative of the instantaneous phase of the signal.
Only a brief account of the important formulas will
be given in this paper A more complete account of
the theory is found in several textbooks and papers
(e.g. [14,15,16,17]).
The Hilbert transform shifts all frequency
components in a signal by ± π/2 rad; the formula is,
H [ x(t )] =
1
π∫
∞
−∞
x(τ )
−1
dτ =
∗ x(t )
τ −t
πt
(9)
The resulting generalized analytic signal, y(t), that is
used in all Hilbert transform time domain analysis
methods is,
y (t ) = x(t ) + jH [ x(t )]
Figure 7: Measured Dynamic Stiffness Function
for SDOF System with Cubic Stiffness
The
problem
issues
with
the
DSF
characterization method are similar to those of the
first order linear FRF estimate.
Because the
characterization process is so subjective and because
the DSF is dependent on input for nonlinear
systems, this technique may not be the best way to
characterize nonlinearities and is certainly not
appropriate for modelling nonlinear systems over
wide input-output amplitude ranges.
4.1.5
Hilbert Transform - Time Domain
One of the main features of nonlinear vibrations is
an amplitude dependent period of oscillation, for
which the swinging pendulum is most famous. By
tracking the changes in frequency (period) with
amplitude, the nonlinearity in a system can often be
detected, classified, and located. The plot of
amplitude versus frequency is the system backbone,
the main by-product of Hilbert transform time
domain techniques.
The actual time response of a system is just one
of many possible responses that could have been
produced by a phase shifted version of the input.
The Hilbert transform uses the time response to
create a kind of rotating phasor, or generalized
analytic signal, from which important signal
characteristics are derived.
Instantaneous
(10)
Lastly, the instantaneous amplitude, A(t), and
frequency, ω(t), are compute directly from Eq 10:
A(t ) = y (t ) =
ω (t ) =
x 2 (t ) + H 2 [ x(t )]
d⎛
H [ x(t )] ⎞
⎟
⎜⎜ arctan
dt ⎝
x(t ) ⎟⎠
(11)
The second of these formulas implies that the
measured time data should contain a single (timevarying) frequency component for the instantaneous
frequency to be useful (i.e. more than one frequency
cannot be tracked simultaneously).
The two
formulas together also imply something else about
the nature of the nonlinear signals: the change in
amplitude must be slow relative to the change in
phase, otherwise the change in frequency with
amplitude will be aliased.
The backbone curves for the three modes in
the first response of the three DOF test system
(Figure 3) are shown in Figure 8 (impact excitation).
The first and second plots show a characteristic
cubic arc in the backbone for increasing response
amplitude because the cubic stiffness is active in
those two modes. The third backbone plot indicates
that the third modal frequency is constant with
increases in amplitude because the nonlinear
element is not active in that mode.
Extravagant filtering efforts were needed to
produce all of these backbone curves. The filtering
is needed to remove all but one of the modes from
the data and to smooth the instantaneous frequency
record. Note that even with several low, high, and
bandpass filters, the second backbone curve is still
oscillatory. In this simple problem, the oscillations
do not prevent a conclusive check for the cubic
nonlinearity, but highly coupled modes can make it
very difficult to come to any conclusion at all. The
same types of analysis and comments hold for
damping
nonlinearities;
however,
damping
parameters would be plotted versus amplitude in
these cases.
The
problems
with
Hilbert
transform
(instantaneous frequency and damping) techniques
are that they are very subjective, require heroic
filtering and smoothing efforts to obtain discernible
backbone curves, do not work very well for systems
with many modes of vibration, and only work when
the response data is asymptotic.
H [Im(FRF (ω ))] = − Re( FRF (ω ))
H [Re( FRF (ω ))] = Im(FRF (ω ))
(12)
When these equalities fail, the system has
nonlinearities.
Since the Hilbert transform
integration is over all frequencies, FRFs that are not
bandlimited have truncation errors in their Hilbert
transforms. Formulas for correction terms have
been derived in [17] to reduce the truncation errors;
however, these formulas make a special assumption
that the system is lightly damped. The formulas for
the real and imaginary parts of the Hilbert transform
are,
ω
Re( H ) =
2ω 0 Re( FRF (ω ))
dω
π ∫0 ω 2 − ω 02
ω
2 ω Im(FRF (ω ))
Im(H ) = ∫
dω
π 0 ω 2 − ω 02
(13)
The Hilbert transform of the imaginary part of an
FRF for the three DOF test system with a cubic
nonlinearity to ground is shown in Figure 9. Since
the first two modes are affected by the cubic
stiffness, the Hilbert transform diverges from the
FRF near those modes. The FRF and the Hilbert
transform are equal at the third mode because the
nonlinearity does not affect that mode.
Figure 8: Backbone Curves for MDOF Test
System with Cubic Nonlinearity to Ground
4.1.6
Hilbert Transform
Domain
-
Frequency
The Hilbert transform frequency domain techniques
use the fact that the real and imaginary parts of a
linear, causal, time invariant FRF are related to one
another. The relationship is simple and is derived
by combining the definition of the Hilbert
transform, Cauchy's Integral Theorem, and complex
integration:
The problems with the Hilbert transform
frequency domain characterization technique are
that it is very subjective, it does not work well on
highly damped (coupled) FRFs, makes it difficult to
characterize weaker nonlinearities, and is not
conducive for classifying nonlinearities in MDOF
systems.
Figure 9: Hilbert Transform of Imaginary Part of
FRF for Nonlinear MDOF Test System
correlations are not in general used to classify
nonlinearities.
4.2 System Identification and Modelling
4.1.7
Wavelet Transform
Wavelet transforms use specialized basis function to
decompose signals into a set of coefficients. It is the
same type of operation as the Fourier transform
except that the kernels (basis functions) for the
wavelet
transformation
are
not
complex
exponentials.
Wavelets use localized basis
functions that decay quickly to locate parts of a
signal that occur over short time intervals. This
localization property of wavelets makes them a
natural choice for analyzing non-stationary systems
and detecting structural dynamic faults. Fourier
transforms cannot locate and analyze short portions
of a signal because the entire time record is
processed at each frequency. The properties and use
of discrete wavelets are discussed in [18].
Nonlinear characterization with wavelets is
similar to the technique that is used with Hilbert
transforms.
Instead of using instantaneous
amplitudes and frequencies, wavelet analysis uses
ridges and skeletons to generate a backbone curve of
the system response [19].
4.1.8
Other Characterization Methods
Two other types of characterization methods are the
restoring force method and time domain correlation
methods. Also called force state mapping, the
restoring force method fits a two dimensional
surface to the velocity-displacement responses of a
SDOF [20] system.
Nonlinearities in MDOF
systems can also be characterized using the modal
vectors of the system in combination with an
expression for the restoring force in modal
coordinates [21]. Force state maps enable quick
visualization of nonlinearities in a SDOF system,
but they have the disadvantage of being
cumbersome to apply and interpret for MDOF
systems. A problem with the MDOF restoring
force technique is its reliance on the modal matrix,
since the modal vectors can change with amplitude
in a nonlinear system.
Time domain correlation functions have been
used to detect nonlinearities prior to implementing
nonlinear difference equation models of a system
[22]. Although they can detect nonlinearities,
Linear models (e.g. modal models and impedance
models) fall apart when the system is nonlinear over
some input-output amplitude range. When the
principle of superposition fails, other means of
system identification that do not rely on
superposition are sought to describe the input-output
behavior. The most popular of these methods are
discussed in this section.
After detecting,
classifying, and locating the nonlinearities in the
system, system identification can be used to
generate a model of the input-output measurement.
Time and frequency domain nonlinear modelling
techniques are both in use by researchers and
engineers. Time data seems to be preferred because
short lived, or time discrete, events in the time
record or directly available in raw form. Frequency
domain techniques transform the entire time record
at once, so the discrete time events are smeared
across the frequency range of the transform. Each
modelling
domain
has
advantages
and
disadvantages.
Since modal analysis is the most popular method
for analyzing vibrating systems, it makes sense to
view the nonlinear modelling techniques in this
section in a framework that is consistent with modal
analysis. For instance, characterizing nonlinearities
is directly analogous to choosing the number of
spatial references and model order in a modal
model. Moreover, the process of throwing away
computational poles in modal parameter estimation
is similar to the process of discarding nonlinear
terms in a model because they are poor contributors
to the response.
4.2.1 Direct Parameter Estimation Models
Direct parameter estimation (DPE) is conceptually
the easiest time domain method to apply for
modelling nonlinear systems. The method begins
by assuming that there are linear and nonlinear
elements in certain locations throughout a discrete
model of a system. Then the constant parameters in
the model are estimated by curve fitting the data
with the assumed model using a least squares
technique. For the three DOF test system, the
assumed model with a single input applied at the
first DOF is given by,
m1&x&1 + (c10 + c12 + c13 ) x&1 + (k10 + k12 + k13 ) x1 −
c12 x&2 − k12 x2 − c13 x&3 − k13 x3 + μkn10 x13 = f1 (t )
m2 &x&2 + (c20 + c12 + c23 ) x&2 + (k20 + k12 + k23 ) x2 −
- c12 x&1 − k12 x1 − c23 x&3 − k23 x3 = 0
m3 &x&3 + (c30 + c13 + c23 ) x&3 + (k30 + k13 + k23 ) x3 −
- c13 x&1 − k13 x1 − c23 x&2 − k23 x2 = 0
(14)
The techniques in section 3.1 are used to obtain a
guess of the form of the system equations in Eq 14.
A solution is obtained as in [23,24] for a single force
by estimating the parameters in the forced equation,
feeding the common terms forward into the next
unforced equation, estimating the parameters in that
unforced equation, and then continuing onto the
other unforced equations.
This feedforward
estimation technique is not necessary if there are
forces applied at all the DOF.
This method works well as long as the model
structure is known beforehand. Since the velocity
and position are each needed to solve the system of
equations for the parameters, the input is designed to
excite the system in the range where there are modal
frequencies. There will be a errors in the parameters
if there are modes outside the frequency range of the
input; this is a disadvantage of DPE since there will
always be some contribution from out-of-band
modes when the modal density is high near the edge
of the frequency window. Note that the position and
velocity estimates are obtained by integrating the
measured accelerations; the DC integration errors
(constants of integrations) are removed by running
the data in through the front and back of the filter to
prevent phase changes.
The level of filtering that is needed for real
experimental data with low and high frequency
modes can be prohibitive to the application of DPE.
If there is any delay between the acceleration
samples at the system DOFs due to multiplexing, the
delay must be removed in order to have consistent
data for parameter estimation. DPE is extremely
sensitive to errors in amplitude and phase.
4.2.2
system identification [25]. The idea is to 'regress'
the current response sample of a system on the past
response and input samples with a nonlinear
regression function. The moving average part of the
model uses an unknown sampled noise sequence to
reduce the bias error in the predicted response [2]
and the exogenous inputs are just the past samples
of the input time record. Since polynomials may
only approximate nonpolynomial nonlinearities in
certain amplitude ranges, the range of the
NARMAX model is extended by using other types
of
nonlinear
regression
functions
(e.g.
transcendental), the so called 'extended model set
[28].
Although the theory seems complex, the idea is
very simple: the input and output measurements
from the past are used to generate the current output.
The NARMAX model is essentially a direct
parameter estimation sampled-data model of the
system. The general form of the model will be
illustrated with the MDOF test system. After
substituting difference approximations for the first
and second derivatives in Eq 14, the system
difference equations can be rearranged to yield Eq
15. The equation says that the current outputs are a
linear combination of the past outputs and input; the
nonlinear term is treated as an input. This model is
unrealistic because it assumes that the measurements
of the response are noise-free. The more realistic
model includes the measurement noise, n(k). The
measured output (denoted with a hat) is the sum of
the true output and an uncorrelated noise term (Eq
16). Note that when this expression is substituted
into Eq 15, multiplicative, correlated noise terms are
generated.
⎧ x1 (k − 1) ⎫
⎪ x (k − 2) ⎪
⎪ 1
⎪
⎪ x2 (k − 1) ⎪
⎧ x1 (k ) ⎫
⎪
⎪
⎪
⎪
⎪ x2 (k − 2) ⎪
(
)
=
x
k
C
[
]
⎨ 2 ⎬
⎨
⎬
⎪ x (k ) ⎪
⎪ x3 (k − 1) ⎪
⎩ 3 ⎭
⎪ x3 (k − 2) ⎪
⎪ 3
⎪
⎪ x1 (k − 2)⎪
⎪ f (k ) ⎪
⎩ 1
⎭
(15)
NARMAX Models (Discrete Time)
Nonlinear autoregressive moving average models
with exogenous inputs are a derivative of the
general ARMA models that are used in linear
xˆi (k ) = xi (k ) + ni (k ), i = 1,2,3
(16)
Figure 10: NARMAX Model Output
Generalization for 3 DOF Test System
Eq 15 is written once for each sampling instant
and the resultant set of overdetermined equations is
solved. The important thing to remember is that if
an orthogonal least squares solution does not
provide consistent estimates of the parameters, the
correlated noise may be responsible. The solution to
this problem is to model the noise source itself.
Many different quality measures are used to
check the model parameters for accuracy. These
measures check to see if the model, 1) can replicate
the input-output data that was used to estimate the
parameters, 2) can respond correctly to a new input
(generalize outside dataset), and 3) produces errors
in the output estimates that are uncorrelated with
the input. The three DOF test system was simulated
with 8% uncorrelated measurement noise on each
response. The ability of the model to generalize to
inputs that are outside the dataset is illustrated in
Figure 10. The input used to create the dataset for
parameter estimation was a random input with zero
mean. The input in the generalization test was also
random. Note that the NARMAX procedure works
for any type of input.
There are a few difficulties with the NARMAX
technique for experimental applications: first, the
number of parameters in the general NARMAX
model can be prohibitively large, even after the
relative contribution of the terms is evaluated [29];
second, error models for correlated noise terms can
be difficult to obtain. Despite these problems, the
NARMAX approach seems to be the most practical
and useful of the time domain nonlinear modelling
methods.
4.2.3 Reverse Path Stiffness Models
When linear spectral analysis is used to analyze
nonlinear systems, and first order FRFs are
estimated, the nonlinear effects are disguised as
sources of correlated noise in the problem. Reverse
path models extract the nonlinear noise sources by
looking at dynamic stiffness functions (DSF) instead
of FRFs. This reversal provides access to the
nonlinear terms through simple mathematical
operations on the output [7].
The MDOF test system will be used to describe
the procedure. First, the system is flipped so that
the force becomes the output and the response
becomes the force. Then a linear path and a
nonlinear path are used to connect the linear input
force and the nonlinear input force (response) to the
output (force). The resulting model for the MDOF
test system is shown in Figure 11. Note that there
are four effective inputs and one output. Moreover,
the nonlinear single input multiple output problem
has been transformed into a multiple input single
output linear problem. By conditioning the four
inputs on one another, the inputs become
uncorrelated, and standard MIMO FRF estimation
techniques can be used [9]. Note however that the
FRFs will be DSFs in this problem, which are then
used to characterize and model the nonlinear system.
Figure 11: Reverse Path Model for MDOF Test
System with Cubic Stiffness to Ground
Since the nonlinear response terms can be highly
correlated with the measured responses, the spectral
problem may be poorly conditioned in some
instances. A sufficient number of averages with a
random excitation will help to reduce the correlation
between the two signals. The interpetation of the
DSFs can be very subjective, and the quality of the
DSF in the nonlinear path can be rather noisy and
difficult to interpret. Note that the major difficulty
in modelling the statistics of the system in Figure 11
is that the inputs are no longer Gaussian because of
the nonlinearity.
4.2.4
Other Modelling Techniques
Other modelling techniques that seem to work in
certain applications are the experimental describing
function method [30], where all harmonics of the
driving frequency are discarded in favor of the
driving frequency itself, and neural networks for
sampled-data systems [31], in which a black box of
neurons and activation functions are used to
replicate and generalize on a given set of inputoutput behaviors.
5 Conclusions
There are a vast collection of nonlinear vibration
analysis tools that can be used to analyze many
types of common nonlinearities. Although many of
these techniques are targeted to SDOF systems,
many can be extended to MDOF systems with some
effort. The steps involved in characterizing and
modelling a structural dynamic nonlinear inputoutput relationship are similar to those in linear
modal analysis. An approach to nonlinear analysis
that begins with detecting, classifying, and locating
the nonlinearity is necessary since many of the
existing nonlinear modelling techniques assume that
the model structure is known beforehand.
Acknowledgements
The authors acknowledge all of the researchers that
were referenced in this report, and apologize to
those who were not mentioned, for their
contributions to this exciting area of experimental
vibration analysis.
References
1. Vinh, T. and Liu, H., Nonlinear Structural
Dynamics
by
Nonparametric
Method,
Identificazione Strutturale, Seriate, Oct. 1992.
2. Worden, K., Characterization of Nonlinear
Systems Using Time Data, Identificazione
Strutturale, Seriate, Oct. 1992.
3. Jordan, D. W. and Smith, P., Nonlinear
Ordinary Differential Equations, Second
Edition, Clarendon Press, Oxford, 1987.
4. Nayfeh, A. H. and Mook, D. T., Nonlinear
Oscillations, John Wiley and Sons, New York,
1979.
5. Billings, S. A., and Voon, W. S. F., Least
Squares Parameter Estimation Algorithms for
Nonlinear Systems, International Journal of
Systems Science, 1984, Vol. 15, No. 6, 601615.
6. He, J., and Ewins, D. J., A Simple Method of
Interpretation for the Modal Analysis of
Nonlinear Systems, 1987 International Modal
Analysis Conference, pg. 626-634.
7. Bendat, J. S., Nonlinear Systems Techniques
and Applications, John Wiley & Sons, New
York, 1998.
8. Worden, K., Wardle, R., and King, N. E.,
Classification of Nonlinearities Using Neural
Networks, 1997 International Modal Analysis
Conference, 980-986.
9. Bendat, Julius S., Piersol, Allan G., Engineering
Applications of Correlation and Spectral
Analysis, John Wiley & Sons, New York, 1980.
10. Rice, H. J., and Fitzpatrick, J. A., A Procedure
for the Identification of Linear and Non-Linear
Multi-Degree-of-Freedom Systems, Journal of
Sound and Vibration (1991), 149(3), 397-411.
11. Storer, D. M., and Tomlinson, G. R., Recent
Developments in the Measurement and
Interpretation of Higher Order Transfer
Functions
from
Non-Linear
Structures,
Mechanical Systems and Signal Processing
(1993), 7(2), 173-189.
12. Schetzen, M., The Volterra and Wiener
Theories of Nonlinear Systems, Krieger
Publishing Company, Malabar, 1989.
13. Bedrossan, E., and Rice, S. O., The Output
Properties of Volterra Systems Driven by
Harmonic and Gaussian Inputs, Proceedings of
the IEEE (1971), 59, 1688-1707.
14. Bendat, J. S., and Piersol, A. G., Random Data:
Analysis and Measurement Procedures, Second
Edition, John Wiley & Sons, New York, 1986.
15. Bracewell, R. N., The Fourier Transform and
Its Applications, Second Edition, McGraw Hill,
Boston, MA, 1986.
16. Feldman, M. S. and Braun, S., Identification of
Non-Linear System Parameters Via the
Instantaneous Frequency: Application of the
Hilbert
Transform
and
Wigner-Ville
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
Techniques, 1995 International Modal Analysis
Conference, 637-642.
Simon, M., and Tomlinson, G. R., Use of the
Hilbert Transform in Modal Analysis of Linear
and Non-Linear Structures, Journal of Sound
and Vibration (1984), 96(4), 421-436.
Newland, D. E., Random Vibrations, Spectral &
Wavelet Analysis, Third Edition, Longman
Publishers, 1993.
Staszewski, W. J., and Chance, J. E.,
Identification of Nonlinear Systems Using
Wavelets - Experimental Study, 1997
International Modal Analysis Conference, 10121016.
Masri, S. F., and Caughey, T. K., A
Nonparametric Identification Technique for
Nonlinear Dynamic Systems, Journal of Applied
Mechanics (1979), 46, 433-447.
Masri, S. F., Sassi, H., and Caughey, T. K.,
Nonparametric
Identification
of
Nearly
Arbitrary Nonlinear Systems, Journal of Applied
Mechanics (1982), 49, 619-627.
Billings, S. A., and Voon, W. S. F., 1983,
Proceedings of the Institute of Electronics
Engineers, 130, 193.
Mohammad, K. S., Worden, K., and Tomlinson,
G. R., Direct Parameter Estimation for Linear
and Non-Linear Structures, Journal of Sound
and Vibration (1992), 152(3), 471-499.
Worden, K. and Atkins, P., Identification of a
Multi-Degree-of-Freedom Nonlinear System,
1997 International Modal Analysis Conference,
1023-1028.
Chen, S., and Billings, S. A., Representations of
Nonlinear Systems: The NARMAX Model,
International Journal of Control (1989), 49(3),
1012-1032.
Leontaritis, I. J., and Billings, S. A., InputOutput Parametric Models for Nonlinear
Systems Part I:
Deterministic Nonlinear
Systems, International Journal of Control
(1985), 41(2), 303-328.
Leontaritis, I. J., and Billings, S. A., InputOutput Parametric Models for Nonlinear
Systems Part II: Stochastic Nonlinear Systems,
International Journal of Control (1985), 41(2),
329-344.
Billings, S. A., and Chen, S., Extended Model
Set, Global Data, and Threshold Model
Identification of Severely Nonlinear Systems,
International Journal of Control (1989), 50(5),
1897-1923.
Chen, S., Billings, S. A., Cowan, C. F., and
Grant, P. M., Practical Identification of
NARMAX Models Using Radial Basis
Functions, International Journal of Control,
(1990), 52(6), 1327-1350.
30. Benhafsi, Y., Penny, J. E., and Friswell, M. I.,
Identification of Damping Parameters of
Vibrating Systems with Cubic Stiffness
Nonlinearity, 1995 International Modal
Analysis Conference, 623-629.
31. Chen, S., Billings, S. A., and Grant, S. Chen,
Nonlinear System Identification Using Neural
Networks, International Journal of Control
(1990), 51(6), 1191-1214.