Detecting the position of non-linear component in periodic structures from the system responses to dual sinusoidal excitations

TLDR: Based on the nonlinear output frequency response functions (NOFRRFs), a novel method is developed to detect the position of nonlinear components in periodic structures as mentioned in this paper, which requires exciting the non-linear systems twice using two sinusoidal inputs separately.

Abstract: Based on the non-linear output frequency response functions (NOFRFs), a novel method is developed to detect the position of non-linear components in periodic structures The detection procedure requires exciting the non-linear systems twice using two sinusoidal inputs separately The frequencies of the two inputs are different; one frequency is twice as high as the other one The validity of this method is demonstrated by numerical studies Since the position of a non-linear component often corresponds to the location of defect in periodic structures, this new method is of great practical significance in fault diagnosis for mechanical and structural systems

Figures

Table 11, Moduli and (i=1,…,7) )2( 1 1, 1 F

Table 2, the evaluated values of , and )(1,1 F ii jωλ + )(1,3 F

Table 5, Moduli and (i=1,…,7) )2( 1 1, 1 F

Table 1. The evaluated results of , and )(1 F H jG ω )(3 F

Table 3, )2( 1),1( FiF jX ω and (i=1,…,8) )2( 1 1, 1 F

Table 10, )( 2),2( FiF jX ω and (i=1,…,8) )( 2 1, 2 F

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Published paper
Peng, Z.K. and Lang, Z.Q. (2007) Detecting the position of non-linear component
in periodic structures from the system responses to dual sinusoidal excitations,
International Journal of Non-Linear Mechanics, Volume 42 (9), 1074 - 1083.
eprints@whiterose.ac.uk

Detecting the Position of Nonlinear Component in
Periodic Structures from the System Responses to
Dual Sinusoidal Excitations
Z.K. Peng, Z.Q. Lang
Department of Automatic Control and Systems Engineering, University of Sheffield
Mappin Street, Sheffield, S1 3JD, UK
Email: z.peng@sheffield.ac.uk; z.lang@sheffield.ac.uk
Abstract: Based on the Nonlinear Output Frequency Response Functions (NOFRFs), a
novel method is developed to detect the position of nonlinear components in periodic
structures. The detection procedure requires exciting the nonlinear systems twice using
two sinusoidal inputs separately. The frequencies of the two inputs are different; one
frequency is twice as high as the other one. The validity of this method is demonstrated
by numerical studies. Since the position of a nonlinear component often corresponds to
the location of defect in periodic structures, this new method is of great practical
significance in fault diagnosis for mechanical and structural systems.
1 Introduction
Periodic structures are defined as structures consisting of identical substructures
connected to each other in identical manner. The real life systems which can be modelled
as either finite or infinite, one-dimension or multi-dimension periodic structures range
from the simple structures like periodically supported beams [1]~[6], plates[5][6] and
building block [7]. Analysis of the free and forced vibration and the mode analysis of
linear periodic structures are of particular interests [1]-[6]. Mead [8] provides an
excellent review about periodic structure studies.
Attentions also have been paid to the study of nonlinear periodic structures [9]-[13]. Chi
and Rosenberg [9] have studied the existence of classical normal mode motion in one-
dimension non-linear mass-spring-damper systems with many degrees of freedom where
all springs and / or all dampers may be strongly non-linear. Using exact and asymptotic
techniques, Vakakis et al [10] have studied the localized responses of a nonlinear
periodic oscillator chain subjected to harmonic excitations with general spatial
distributions. Royston and Singh [11] have studied the periodic response of mechanical
systems with local nonlinearity using an enhanced Galerkin technique where a new semi-
1

analytical framework for the study of mechanical systems with local nonlinearities has
been presented. Chakraborty and Mallik [12] have investigated the harmonic vibration
propagation in an infinite, non-linear periodic chain using a perturbation approach where
the case of cyclic one-dimension nonlinear chain has also been discussed. Marathe and
Chatterjee [13] have studied the Wave attenuation in one-dimension nonlinear periodic
structures using harmonic balance and multiple scale method. In engineering practices,
there are considerable periodic structures that behave nonlinearly just because one or a
few components have nonlinear properties, and the nonlinear component is often the
component where a fault or abnormal condition occurs. One of the well known examples
is beam structures [14] with breathing cracks, the global nonlinear behaviors of which are
caused only by a few cracked elements. Therefore it is of great significance to effectively
detect the position of the nonlinear component in a periodic structure. The detection of
damage in large periodic structures has been studied by Zhu and Wu [15]. In their studies,
the periodic structure with damage is still considered to be linear and the locations and
magnitude of damage in large mono-coupled periodic systems have been estimated using
measured changes in the natural frequencies. Based on a one-dimensional periodic
structure model, Sakellariou and Fassois [7][16] have used a stochastic output error
vibration-based methodology to detect the damage in structures where the damage
elements were modeled as components of cubic stiffness.
The Volterra series approach [17] is a powerful method for the analysis of nonlinear
systems, and extends the familiar concept of the convolution integral for linear systems to
a series of multi-dimensional convolution integrals. The Fourier transforms of the
Volterra kernels are known as the kernel transforms, Higher-order Frequency Response
Functions (HFRFs) [18], or more usually Generalised Frequency Response Functions
(GFRFs). These provide a convenient concept for analyzing nonlinear systems in the
frequency domain. If a differential equation or discrete-time model is available for a
nonlinear system, the GFRFs can be determined using the algorithm in [19]~[21]. The
GFRFs can be regarded as the extension of the classical frequency response function
(FRF) for linear systems to the nonlinear case. The concept of Nonlinear Output
Frequency Response Functions (NOFRFs) [22] is
an alternative extension of the FRF to
the nonlinear case. NOFRFs are one dimensional functions of frequency, which allow the
analysis of nonlinear systems to be implemented in a manner similar to the analysis of
linear systems, and which provides great insight into the mechanisms which dominate
many important nonlinear behaviors.
2

In this paper, a novel method is derived based on the NOFRF concept to detect the
position of the nonlinear component in a periodic structure. The detection procedure
requires exciting the nonlinear systems twice using two sinusoidal inputs separately. The
frequencies of the two inputs are required to be different; one frequency is twice as high
as the other one. Numerical studies verify the effectiveness of the method. The new
method is of great practical significance in fault diagnosis for mechanical and structural
systems.
The paper is organized as follows. Section 2 gives a brief introduction to the new concept
of NOFRFs. Some important properties of the NOFRFs for locally nonlinear MDOF
systems, which were first revealed in the authors’ recent studies [23], are introduced in
Section 3. The novel method for the nonlinear component position detection is presented
in Section 4. In Section 5, three numerical case studies are used to verify the
effectiveness of the proposed method. Finally conclusions are given in Section 6.
2. Output Frequency Response Functions of Nonlinear Systems
2.1 Output Frequency Response Functions under General Input
The definition of NOFRFs is based on the Volterra series theory of nonlinear systems.
The Volterra series extends the well-known convolution integral description for linear
systems to a series of multi-dimensional convolution integrals, which can be used to
represent a wide class of nonlinear systems [18].
Consider the class of nonlinear systems which are stable at zero equilibrium and which
can be described in the neighbourhood of the equilibrium by the Volterra series
i
n
i
in
N
n
n
dtuhtx
ττττ
)(),...,()(
1
1
1
∏
∑
∫∫
=
=
∞
∞−
∞
∞−
−= L (1)
where x(t) and u(t) are the output and input of the system,
),...,(
n1n
h
τ
τ
is the nth order
Volterra kernel, and N denotes the order of the Volterra series representation. Lang and
Billings [18] derived an expression for the output frequency response of this class of
nonlinear systems to a general input. The result is
⎪
⎪
⎩
⎪
⎪
⎨
⎧
=
∀=
∫
∏
∑
=++
=
−
=
ωωω
ω
σωωω
π
ω
ωωω
n
n
n
i
inn
n
n
N
n
n
djUjjH
n
jX
jXjX
,...,
1
1
1
1
1
)(),...,(
)2(
1
)(
)()( for
(2)
3

This expression reveals how nonlinear mechanisms operate on the input spectra to
produce the system output frequency response. In (2),
)(
ω
jX
is the spectrum of the
system output,
)(
ω
jX
n
represents the nth order output frequency response of the system,
n
j
nnnn
ddehjjH
nn
ττττωω
τωτω
...),...,(...),...,(
1
),...,(
11
11
++−
∞
∞−
∞
∞−
∫∫
=
(3)
is the nth order Generalised Frequency Response Function (GFRF) [18], and
∫
∏
=++
=
ωωω
ω
σωωω
n
n
n
i
inn
djUjjH
,...,
1
1
1
)(),...,(
denotes the integration of
over the n-dimensional hyper-plane
∏
=
n
i
inn
jUjjH
1
1
)(),...,(
ωωω
ω
ω
ω
=++
n
L
1
. Equation (2) is a natural extension of the well-known linear relationship
)()()(
ω
ω
ω
jUjHjX =
, where
)(
ω
jH
is the frequency response function, to the
nonlinear case.
For linear systems, the possible output frequencies are the same as the frequencies in the
input. For nonlinear systems described by equation (1), however, the relationship between
the input and output frequencies is more complicated. Given the frequency range of an
input, the output frequencies of system (1) can be determined using the explicit expression
derived by Lang and Billings in [18][24].
Based on the above results for the output frequency response of nonlinear systems, a new
concept known as the Nonlinear Output Frequency Response Function (NOFRF) was
recently introduced by Lang and Billings [22]. The NOFRF is defined as
∫
∏
∫
∏
=++
=
=++
=
=
ωωω
ω
ωωω
ω
σω
σωωω
ω
n
n
n
n
i
i
n
n
i
inn
n
djU
djUjjH
jG
,...,
1
,...,
1
1
1
1
)(
)(),...,(
)(
(4)
under the condition that
0)(
)2(
1
)(
,...,
1
1
1
≠=
∫
∏
=++
=
−
ωωω
ω
σω
π
ω
n
n
n
i
i
n
n
djU
n
jU
(5)
Notice that
)(
ω
jG
n
is valid over the frequency range of
)(
ω
jU
n
, which can be
determined using the algorithm in [18].
By introducing the NOFRFs
)(
ω
jG
n
,
Nn L,1
=
, equation (2) can be written as
∑∑
==
==
N
n
nn
N
n
n
jUjGjXjX
11
)( )( )()(
ωωωω
(6)
which is similar to the description of the output frequency response for linear systems.
The NOFRFs reflect a combined contribution of the system and the input to the system
4

Frequently asked questions

Q1. What are the contributions in "Detecting the position of nonlinear component in periodic structures from the system responses to dual sinusoidal excitations" ?

In this paper, the authors proposed a nonlinear output frequency response function ( NOFRF ) to detect the position of the nonlinear component in a periodic structure, which can be defined as structures consisting of identical substructures connected to each other in identical manner. 

Q2. What are the future works mentioned in the paper "Detecting the position of nonlinear component in periodic structures from the system responses to dual sinusoidal excitations" ?

It is a more complicated problem the authors are planning to investigate in future studies. 

Q3. What is the significance of the nonlinear component position detection method in periodic structures?

Since the positions of the nonlinear components in periodic structures often correspond to the locations of faults, the nonlinear component position detection method is of practical significance in the fault diagnosis for mechanical and structural systems. 

Q4. What is the definition of a NOFRF?

NOFRFs are one dimensional functions of frequency, which allow the analysis of nonlinear systems to be implemented in a manner similar to the analysis of linear systems, and which provides great insight into the mechanisms which dominate many important nonlinear behaviors. 

Q5. What is the definition of a Volterra series?

The Volterra series extends the well-known convolution integral description for linear systems to a series of multi-dimensional convolution integrals, which can be used to represent a wide class of nonlinear systems [18]. 

Q6. What are the types of periodic structures?

The real life systems which can be modelled as either finite or infinite, one-dimension or multi-dimension periodic structures range from the simple structures like periodically supported beams [1]~[6], plates[5][6] and building block [7]. 

Q7. What are the main topics of the periodic structure study?

Analysis of the free and forced vibration and the mode analysis of linear periodic structures are of particular interests [1]-[6]. 

Q8. What is the definition of periodic structures?

In engineering practices, there are considerable periodic structures that behave nonlinearly just because one or a few components have nonlinear properties, and the nonlinear component is often the component where a fault or abnormal condition occurs. 

Q9. What is the advantage of the method?

The distinct advantage of this method is that it only needs the test data under two sinusoidal input forces, which can be readily carried out in practices. 

Q10. What are the main characteristics of periodic structures?

Marathe and Chatterjee [13] have studied the Wave attenuation in one-dimension nonlinear periodic structures using harmonic balance and multiple scale method. 

Q11. What is the purpose of the paper?

In this paper, a novel method is derived based on the NOFRF concept to detect the position of the nonlinear component in a periodic structure. 

Q12. What is the effect of the second super-harmonics?

The second super-harmonics were used to calculate , which were extracted from the FFT spectra of the )2( 1 1, 1 F ii F jR ω +responses of the system subjected to . 

Q13. What is the main difference between the two types of structures?

It is worthy to note here that in real periodic structures there are always minor non-linearities between each component and, such structures therefore should be better to model as weakly nonlinear chains which have been by Chakraborty and Mallik [12]. 

Q14. What is the definition of the component on the right side of the 6 )2?

According to the detection method in Section 4, it can be known that the component on the right side of the 6 )2( 1 1, 1 F ii F jR ω + ≠ )( 2 1, 2 F ii F jR ω + th mass is the nonlinear one, that is, the 7th spring component. 

Q15. What is the effect of the sinusoidal forces in this case?

The sinusoidal forces used in this case are the same as the ones used in Case 1, but were imposed on the 4th mass of this system, that is J=4, so L=J. 

Q16. What is the maximum frequency of a nonlinear system?

nk ,,0 L=Substituting equations (17) and (18) into (16), it can be derived that the output spectrum )( ωjY of nonlinear systems subjected to a harmonic input can be expressed as[ ]∑ +−= −+−+=2/)1(1 )1(2)1(2 )( )()(kNn FnkFH nkF jkAjkGjkX ωωω ( ) (19) Nk ,,1,0 L=where [·] denotes the operator of taking the integral.