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: engcristianhansen@gmail.com, sbshiki@gmail.com, vicente@dem.feis.unesp.br, samuel@dem.feis.unesp.br

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

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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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