ABSTRACT: The consideration of nonlinearities in mechanical structures is a question of high importance because several
common features as joints, large displacements and backlash may give rise to these kinds of phenomena. However, nonlinear
tools for the area of structural dynamics are still not consolidated and need further research effort. In this sense, the Volterra
series is an interesting mathematical framework to deal with nonlinear dynamics since it is a clear generalization of the linear
convolution for weakly nonlinear systems. Unfortunately, the main drawback of this non-parametric model is the need of a large
number of terms for accurately identifying the system, but it can be overcomed by expanding the Volterra kernels with
orthonormal basis functions. In this paper, this technique is used to identify a Volterra model of a nonlinear buckled beam and
the kernels are used for the detection of the nonlinear behavior of the structure. The main advantages and drawbacks of the
proposed methodology are highlighted in the final remarks of the paper.
KEY WORDS: Volterra series, detection of nonlinearities, nonlinear buckled beam, orthonormal basis function
1 INTRODUCTION
In real-world engineering structures it can be necessary to take
into account the nonlinearities depending on the level of
excitation, type of material, joints, level of displacement, and
applied load [1,2]. Different experimental effects can be
observed analyzing the response of the system operating in
nonlinear conditions, as jumps, harmonics, limit cycle,
discontinuities, etc. [3].
Unfortunately, even with the recent development, the
nonlinear identification approach is not mature as the classical
linear techniques. Thus, there is a growing search for general
and easy methods able to treat the most common
nonlinearities in engineering structures, mainly the
polynomial stiffness effect [4]. Several methods have been
used for detection, identification and analysis of nonlinear
behavior in structural systems, as for example: nonlinear time
series [5], nonlinear normal modes [6], harmonic probing
method using Volterra series [7], principal component
analysis [8], extended constitutive relation error [9], frequency
subspace methods [10], harmonic balance method [11], and
others. However, all of these methods have drawbacks in
some aspects. So, there is no general approach for the analysis
nonlinear systems.
Among these methods the Volterra series is very powerful
because it is a clear generalization of the convolution of the
input force signal with the impulse response function and
several properties of the linear systems can be extended for
nonlinear systems [12,13]. However, problems associated
with overparameterization, ill-posed convergence, etc. have
motivated criticism to this approach [14]. Partially this is a
true statement; although, there exist some techniques to
overcome this inconvenience. In order to make easier the
practical application of this technique for engineers, the
present work shows the use of discrete-time version of the
Volterra series expanded with orthonormal functions [15].
The main contributions of this paper is to detect and
identify, in the same time and with the same experimental
data, the nonlinear behavior in function of the level of input
signal applied in a geometrically nonlinear buckled beam. The
paper is organized in four sections. First of all, a background
review in the Volterra series expanded in orthonormal
functions is discussed. Next the description of the test rig and
the experimental tests performed are described in full details.
Nonlinear features detection using classical metrics and the
Volterra kernels contribution in the experimental output are
also presented. Finally, the concluding remarks are presented.
2 DISCRETE-TIME VOLTERRA SERIES
The output of a system can be described by a combination of
linear and nonlinear contributions given by [12]:
1 2 3( ) ( ) ( ) ( )y k y k y k y k (1)
where 1( )y k is the linear term of the output ( )y k and
2 3( ) ( )y k y k are the nonlinear terms of the response.
Each th order component can be written using a
multidimensional convolution:
1
1
1
0 0 1
( , , ) )( ) (m
NN
i
n n i
n n u ky nk
(2)
where 1,( ),n n is the th Volterra kernel
considering 1, ,N N the memory length of each kernel and
( )u k is the input signal. The order of the model can
generally be truncated in 3 to represent most part of structural
nonlinearities with smooth behavior.
Non-parametric identification of a non-linear buckled beam using discrete-time
Volterra Series
Cristian Hansen1, Sidney Bruce Shiki
1, Vicente Lopes Jr
1, Samuel da Silva
1
1Departamento de Engenharia Mecânica, Faculdade de Engenharia de Ilha Solteira, UNESP - Univ Estadual Paulista, Av.
Brasil 56, 15385-000, Ilha Solteira, SP, Brasil
email: [email protected], [email protected], [email protected], [email protected]
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4
2013
Unfortunately, there are several problems associated with
ill-posed problems of convergence due to the high number of
terms 1, ,N N . An effective way to overcome these
problems is to reduce the number of terms in the Volterra
kernels by using orthonormal Kautz functions [16]. Thus, the
Volterra kernels can be described by:
1
1
1 1
1 1 1
( , , ) , , ( )j
JJ
i j
i i j
n n i i n
(3)
where 1, ,J J is the number of samples in each
orthonormal projection of the Volterra kernels 1, ,i i
and ( )ji in are the Kautz functions that are applied in the
representation of underdamped oscillatory systems. Details
about the application of Kautz functions can be found in [17]
and [18]. In this sense, the Volterra series can be rewritten as:
1
1
1
1 1 1
( ) , , ( )j
JJ
i
i i j
y k i i l k
(4)
which is now a multidimensional convolution between the
orthonormal kernel 1, ,i i and ( )jil k which is the input
signal ( )u k filtered by the Kautz function ( )ji in :
1
0
( ) ( ) ( )j j
i
V
i i i i
n
l k n u k n
(5)
where 1max{ , , }V J J . Since the Volterra series is
linear with relationship to the parameters of the model, the
terms of the orthonormal kernels can be grouped in a vector
and can be estimated by a classical least-squares
approximation:
1( )T T y (6)
where the matrix contains the filtered input ( )jil k and
(1), , ( )y y Ky . The estimation of the orthonormal
kernels can also be done in two steps since the linear and
nonlinear components of the response are represented in a
separated way. It is possible to estimate the linear kernel using
a dataset where it is known that the system behaves in a linear
way. A second dataset can then be used to estimate the
nonlinear kernels, considering that the time series shows some
relevant nonlinear behavior. This two step methodology is
applied in this work in order to ensure a coherent model.
3 EXPERIMENTAL APPLICATION IN A NONLINEAR
BEAM
This section shows the main results of the application of
Volterra series in a test rig. The results show the applicability
of this technique in the identification of nonlinear structures.
3.1 Buckled beam description
The test structure is composed by an aluminum beam with
dimensions of 460×18×2 mm in a clamped configuration with
a preload applied in the top. The structure can be axially
loaded by a compression mechanism until the buckling of the
beam. An electrodynamic shaker is attached 65 mm from the
clamped end of the beam. A laser vibrometer is focused in the
center of the beam measuring the velocities in this position.
Four accelerometers were also used for monitoring the
vibration in the length of the beam, but these signals are not
used in this work. Figures 1 and 2 shows the experimental
setup and a schematic figure of the test rig respectively.
Figure 1. View of the test rig.
Figure 2. Schematic figure showing the test rig.
This structure was tested in growing levels of input
amplitude configured in the signal generator in order to
observe the nonlinear behavior of the system. It was used a
chirp input sweeping up the frequency range from 20 to 50 Hz
and a white noise sequence filtered in the same range of
frequency in order to excite essentially the first mode of the
system (around 37 Hz).
A sampling frequency of 1024 Hz was used and 4096
samples were recorded. Stepped sine and random excitation
were also applied in order to observe some interesting
nonlinear phenomena in the studied system. The next section
shows the effects of the nonlinear behavior.
3.2 Detection of nonlinear behavior
The signal data measured in the tests can be analyzed in order
to detect some nonlinear behavior through the violation of the
superposition principle. The presence of harmonic frequencies
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
2014
in the response of the system is the simplest way to do it [1].
Figure 3 shows the spectrogram of the response of the
structure when a chirp excitation is applied. In the time from 0
to 2 seconds it is possible to observe a stronger diagonal line
that follows the input frequency. However, some parallel lines
starting around 1 s can be also observed representing the
second and third harmonics of the input frequency. This
clearly shows the nonlinear behavior of the system.
Figure 3. Spectrogram of the output signal to a chirp input
(input at 0.1 V).
Figure 4 shows the frequency response function (FRF) in
three different levels of excitation for the chirp signal. The
levels of input correspond to 0.01 V (low input), 0.05 V
(medium input) and 0.1 V (high input) applied in the signal
generator. The natural frequency of the system presents a
stiffening effect since it moves to the right. This is another
violation on the superposition principle since for a linear
system the ratio between the response and the excitation is
kept constant for different levels of the input signal. One can
also observe differences in the peak amplitude of the FRF.
Figure 4. FRF of the system in different levels with a chirp
input.
The random excitation can also be used to estimate the FRF.
This is a common test done in the industry. However, as
showed in Fig. 5 there is no clear effect of nonlinear behavior
caused by changes in the amplitude of excitation. The random
excitation is usually not very appropriate to excite
nonlinearities of the system since the input energy is spread in
a broadband of frequency [1].
Figure 5. FRF of the system in different levels with a random
input.
Another test performed was done exciting the structure with
stepped up sinusoidal excitation. The big advantage of this
kind of input signal is that it concentrates the energy in a
single frequency and in this case it is possible to clearly
observe some nonlinear phenomena like the jump [18]. Figure
6 shows the frequency response using a frequency resolution
of 0.1 Hz in three diferent levels of stepped sines with
comparison to the FRF based on the random excitation
(linear). The frequency response of the stepped sine with a
low input stepped sine is very close to the one calculated with
random excitation. For higher amplitude levels, it is possible
to observe the change in the natural frequency of the system
and a sudden drop in the amplitude of the FRF characterizing
the jump phenomenon.
Figure 6. Frequency response of the stepped sine testing in
different amplitude levels.
All this results demonstrated the nonlinear effects of the test
rig in different conditions of the excitation signal. The
presented behavior show some features available in systems
with the restoring force of the stiffness with nonlinear
polynomial shape.
3.3 Volterra kernel identification
In this section the Volterra kernels are computed in order to
extract a model describing the behavior of the first mode of
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
2015
the structure when the nonlinearity is activated. The procedure
can also be used to detect the level of nonlinearity presented.
A discrete-time Volterra model is truncated until the third
kernel to consider even and odd harmonics that are presented
in the response of the system as shown in Fig. 3. All datasets
used were based on a chirp input and the corresponding
response measured. The random input signal was not
employed for this task since this kind of signal is not
appropriate to excite the nonlinear behavior of the structure as
shown in Fig. 5. The low chirp input dataset (0.01 V) was
used to calculate the first Volterra kernel and the high input
dataset (0.1 V) was used to estimate the second and third
kernels using the two step methodology previously described.
One pair of Kautz functions was used in order to represent
each Volterra kernel. These functions have complex conjugate
parameters representing the dynamics of the kernel. These
parameters can be represented in terms of the frequency ( )
and the damping ratio ( ). These values are generally close
to the values describing the linear dynamics of the system.
Methodologies to find these parameters are available [18] but
they are generally very limited since previous knowledge of
the shape of the Volterra kernel is needed. In this paper the
frequencies and damping ratios of the first three kernels were
defined using an optimization algorithm with the objective to
minimize the prediction error of the model. The algorithm was
based on a first global search by a genetic algorithm [20]
followed by a local search using sequential quadratic
programming [21]. Table 1 shows the results of the
optimization algorithm. As stated before, the frequency of the
third pole is close to the frequency of the first pole and the
second frequency has a value around two times the first one.
Table 1. Kautz parameters of the orthonormal filters.
1 [Hz] 1 [%]
2 [Hz] 2 [%]
3 [Hz] 3 [%]
35.74 1.04 78.33 1.76 35.90 0.14
Figure 7. 1
st Volterra kernel
1 1( )n .
Now it is possible to identify the orthonormal kernels
through eq. (6). Figures 7, 8 and 9 show the Volterra kernels
1 1( )n , 2 1 2( , )n n and 3 1 2 3( ,, )n n n . The first kernel 1 1( )n
is equal to a vector representing the impulse response function
of the system while the second kernel 2 1 2( , )n n is a matrix
and the third kernel 3 1 2 3( ,, )n n n is a tridimensional matrix.
This last kernel is illustrated through the main diagonal of the
matrix since the full representation is not possible in a
conventional way.
Figure 8. 2nd
Volterra kernel 2 1 2( , )n n .
Figure 9. Main diagonal of the 3rd
kernel 3 1 2 3( ,, )n n n .
Figure 10. Response to chirp excitation of the Volterrra model
(input at 0.1 V).
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
2016
The identified Volterra kernels can be used to predict the
output obtained by multiple convolutions using eq. (4). The
predicted response to a chirp excitation is shown in Fig. 10.
A powerful feature of the Volterra series is that it represents
the response of the system separating the linear contribution
and many different levels of nonlinear components as showed
previously in eq. (1). This can be an interesting tool for the
analysis of nonlinear systems. Figures 11 and 12 shows the
separated responses of the system to a chirp input in a low
(0.01 V) and high level (0.1 V) respectively.
Clearly the nonlinear components of the response increase
with the input amplitude. While 2y and
3y are negligible
compared to 1y at a low level of input amplitude as seen in
Fig. 11, these components have a relevant effect in the
response with a higher level (Fig. 12). This effect is a
consequence of the polynomial nature of the stiffness in this
kind of structure that was also seen analyzing Fig. 3. It can
also be observed that the third kernel has more influence than
the second kernel since the contribution of 2y is lower
compared to 3y .
Figure 11. Components of the response of the Volterra model
to a low level chirp input (0.01 V).
The system was also excited in the natural frequency of the
system (around 37 Hz). Figure 13 shows the comparison
between the power spectral density (PSD) of the responses of
the model and from the experimental data. Some even and odd
harmonics appear in the measured response and it is possible
to observe that the second and third harmonics are well
represented by the Volterra model until the third kernel.
Furthermore, the second kernel represents the second
harmonics of the response and the third kernel has the third
harmonics. This fact shows that the representation of the
harmonic content of the response of nonlinear systems can
only be achieved using an appropriate nonlinear
representation. The fact that the th Volterra kernel
represents the th harmonic frequency of the response is
directly related to the functional form of the model since the
the th component of the response has a power of order
in the input. This feature was also observed in [17] in a
numerical Duffing oscillator.
(a) Linear component of the response.
(b) Second order component of the response.
(c) Third order component of the response.
Figure 12. Components of the response of the Volterra model
for a high level chirp input (0.1 V).
Figure 13. PSD of the measured response in comparison to the
one estimated by the Volterra model.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
2017
The results showed the capabilities of the Volterra series to
model the nonlinear dynamics of a buckled beam considering
different levels of nonlinear behavior. The separation of the
response of the system in linear and nonlinear components
was exploited in this paper and it was shown that it has some
physical sense with the level of excitation. This means that
with a good model it is possible to observe the degree of
nonlinearity in the response of the system by evaluating the
response of the model. However, one of the main drawbacks
of this technique is the limitation in the representation of
effects like jumps and bifurcations that are common in this
kind of nonlinear system [22]. Other possible limitation of the
method is in the representation of discontinuous nonlinearities
(e.g. Coulomb friction, clearance, gaps and others). However,
some authors claim that abrupt nonlinearities can be easily
approximated using polynomials enabling the Volterra
representation with a few terms [7]. Parametric identification
(model updating) can be also performed based on the
objective function using the identified Volterra kernels [14].
4 FINAL REMARKS
This work showed that discrete-time Volterra series
expanded in orthonormal Kautz functions can be used
effectively to detect and identify nonlinear features. The
approach was validated using experimental the dataset of a
buckled beam which gave similar conclusions comparing with
classical tools to detect the nonlinear behavior. Additionally,
the feature of the separation of the response of the system in
linear and nonlinear components is a powerful aspect found in
the results. With this property, one can identify and analyze
separately the part of the model that corresponds to linear and
nonlinear response in a further parametric identification.
ACKNOWLEDGMENTS
The authors acknowledge the financial support provided by
the following funding agencies: São Paulo Research
Foundation (FAPESP) by grant number 12/09135-3 and the
National Council for Scientific and Technological
Development (CNPq) grant number 47058/2012-0. The first
and second authors acknowledge FAPESP for their
scholarships by grant number 13/09008-4 and 12/04757-6
respectively. The authors also thank the CNPq and FAPEMIG
for partially funding the present research work through the
National Institute of Science and Technology in Smart
Structures in Engineering (INCT-EIE).
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
2018
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