Linear parameter estimation for multi-degree-of-freedom nonlinear systems using nonlinear output frequency-response functions

ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 21 (2007) 3108–3122 www.elsevier.com/locate/jnlabr/ymssp

Linear parameter estimation for multi-degree-of-freedom nonlinear systems using nonlinear output frequency-response functions Z.K. Peng, Z.Q. Lang, S.A. Billings Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK Received 28 September 2006; received in revised form 24 April 2007; accepted 25 April 2007 Available online 17 May 2007

Abstract The Volterra series approach has been widely used for the analysis of nonlinear systems. Based on the Volterra series, a novel concept named nonlinear output frequency-response functions (NOFRFs) was proposed by the authors. This concept can be considered as an alternative extension of the classical frequency-response function for linear systems to the nonlinear case. In this study, based on the NOFRFs, a novel algorithm is developed to estimate the linear stiffness and damping parameters of multi-degree-of-freedom (MDOF) nonlinear systems. The validity of this NOFRF-based parameter estimation algorithm is demonstrated by numerical studies. r 2007 Elsevier Ltd. All rights reserved. Keywords: Nonlinear system; Volterra series; Parameter estimation; MDOF system; Nonlinear vibration

1. Introduction Various methods have been developed to estimate the stiffness and damping parameters for linear structures or machines. Most of these are based on modal analysis techniques, which were essentially derived from the frequency-response functions (FRFs) [1–5]. To tackle the problem with finite element model updating, Arruda and Santos [1] estimated the mechanical parameters via curve fitting for measured FRFs using a nonlinear least-squares method. Sunder and Ting [2] used the system parameter estimation method to detect the occurrence and location of damage on steel jacket offshore platforms. Also based on the FRFs, Hwang [3] put forward an identification method for stiffness and damping parameters of connections using test data for a structure attached to another structure via connections. Woodgate [4] studied the problem of identifying a positive semi-definite symmetric stiffness matrix for a stable elastic structure from measurements of its displacement in response to some set of static loads. Most recently, Zˇivanovic´ et al. [5] described a lively fullscale footbridge from its numerical modelling and dynamic testing. Their work is a successful application of the FRFs to system parameter estimation in practice. Corresponding author. Tel.: +44 77233 42361.

E-mail addresses: z.peng@sheffield.ac.uk (Z.K. Peng), z.lang@sheffield.ac.uk (Z.Q. Lang), s.billings@sheffield.ac.uk (S.A. Billings). 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.04.009

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Nomenclature x(t), u(t) the output and input of the nonlinear system X(jo), U(jo) the spectrum of the system output and input hn(t1,y,tn) the nth-order Volterra kernel Hn(jo1,y,jon) the nth-order GFRF Gn(jo) the nth-order NOFRF On the frequency components of the nth-order output of the system subjected to harmonic inputs O the frequency components of the output of the system GH the nth-order NOFRF of the system subjected to harmonic inputs n ðjoÞ M, K, C the system mass, damping and stiffness matrices mi, ki, ci the ith mass, damping and stiffness parameter _ the restoring forces of the nonlinear spring and damper SLS ðDÞ, S LD ðDÞ ri, wi (i ¼ 1,y,P) the nonlinearity-related parameters Non-F the nonlinear force xi(t), Xi(jo) the displacement and the output frequency response of the ith mass h(i,j)(t1,y,tj) the jth-order Volterra kernel associated to the ith mass G(i,l)(jo) the lth-order NOFRF associated to the ith mass li;iþ1 ðjoÞ the ratio between the nth NOFRFs of the ith and (i+1)th masses n W the vector of the unknown parameters to be estimated G(L1,Z)(jo) the term introduced by the nonlinear force Non-F for the Zth-order NOFRF.

However, there are certain types of qualitative behaviour, which cannot be produced by linear models [6], encountered in engineering, for example, the generation of harmonics and inter-modulation behaviours. In cases where these effects are dominant or significant nonlinear behaviours exist, nonlinear models are required to describe the system, and the linear FRFs are no longer suitable to investigate the system dynamics. The Volterra series approach [7] is a powerful tool for the analysis of nonlinear systems, which 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) [8], or generalised frequency-response functions (GFRFs), and these provide a convenient tool for analysing nonlinear systems in the frequency domain. If a differential equation or discrete-time model is available for a system, the GFRFs can be determined using the algorithms in [9–11]. The GFRFs can be regarded as the extension of the classical FRF of linear systems to the nonlinear case. So far only a few researchers have addressed the problem of nonlinear system parameter estimation for nonlinear systems using the GFRFs. Lee proposed a straightforward method to estimate the nonlinear system parameters using the GFRFs [12]. Khan and Vyas [13] employed the relationships between higher-order GFRFs and first-order GFRF to estimate the nonlinear parameters. Later, Chatterjee and Vyas [14] further developed this method by using a method of recursive iteration. In engineering practice, for many mechanical and structural systems, more than one coordinate is needed to sufficiently describe the system dynamics. The result is a multi-degree-of-freedom (MDOF) model. In addition, there are considerable mechanical and structural systems that behave nonlinearly just because one or a few components within the system are nonlinear. One well-known example is beam structures [15] with breathing cracks, the global nonlinear behaviours of which are caused only by the cracked elements. Such nonlinear MDOF systems can be regarded as locally nonlinear MDOF systems. An important fact is that, for such nonlinear systems, the linear stiffness and damping are still the decisive characteristics, which mainly determine the system behaviour. Therefore, a knowledge of the linear stiffness and damping are still of great significance for understanding the whole system dynamical properties. In this paper, a novel method is proposed to estimate the linear stiffness and damping parameters for locally nonlinear MDOF systems. The method is based on the concept of nonlinear output frequency-response functions (NOFRFs) [16], which was recently proposed by the authors and is an alternative extension of the

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FRF to the nonlinear case. NOFRFs are one-dimensional functions of frequency. This allows the analysis of nonlinear systems to be implemented in a manner similar to the analysis of linear systems and provides great insight into the mechanisms, which dominate many important nonlinear behaviours. The paper is organised as follows: Section 2 provides 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 [17], are given in Section 3. The NOFRF-based algorithm for the linear stiffness and damping parameter estimation is presented in Section 4. In Section 5, a numerical study is used to verify the effectiveness of the presented algorithm. Finally, conclusions are given in Section 6. 2. Nonlinear output frequency-response functions 2.1. Nonlinear output frequency-response functions under general inputs 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 [8]. 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 Z 1 N Z 1 n X Y xðtÞ ¼  hn ðt1 ; . . . ; tn Þ uðt  ti Þ dti , (1) n¼1

1

1

i¼1

where x(t) and u(t) are the output and input of the system, hn ðt1 ; . . . ; tn Þ is the nth-order Volterra kernel, and N denotes the maximum order of the system nonlinearity. Lang and Billings [8] derived an expression for the output frequency response of this class of nonlinear systems to a general input. The result is X ðjoÞ ¼

N X

X n ðjoÞ

for

8o,

n¼1

X n ðjoÞ ¼

pffiffiffi Z n Y 1= n H ðjo ; . . . ; jo Þ Uðjoi Þ dsno . n 1 n ð2pÞn1 o1 þþon ¼o i¼1

ð2Þ

This expression reveals how nonlinear mechanisms operate on the input spectra to produce the system output frequency response. In (2), X(jo) and U(jo) are the spectrum of the system output and input, respectively, Xn(jo) represents the nth-order output frequency response of the system Z 1 Z 1 H n ðjo1 ; . . . ; jon Þ ¼  hn ðt1 ; . . . ; tn Þeðo1 t1 þþon tn Þj dt1 . . . dtn (3) 1

1

is the nth-order generalised frequency-response function (GFRF) [8], and Z n Y H n ðjo1 ; . . . ; jon Þ Uðjoi Þ dsno o1 þþon ¼o

i¼1

Q denotes the integration of H n ðjo1 ; . . . ; jon Þ ni¼1 Uðjoi Þ over the n-dimensional hyper-plane o1 þ    þ on ¼ o. Eq. (2) is a natural extension of the well-known linear relationship X ðjoÞ ¼ HðjoÞUðjoÞ, where HðjoÞ is the FRF, 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 Eq. (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 [8].

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Based on the above results for the output frequency response of nonlinear systems, a new concept known as NOFRF was recently introduced by Lang and Billings [16]. The NOFRF is defined as R n o þþon ¼o H n ðjo1 ; . . . ; jon ÞPi¼1 Uðjoi Þ dsno R G n ðjoÞ ¼ 1 (4) n o1 þþon ¼o Pi¼1 Uðjoi Þ dsno under the condition that Z U n ðjoÞ ¼

n Y

Uðjoi Þ dsno a0.

(5)

o1 þþon ¼o i¼1

Note that G n ðjoÞ is valid over the frequency range of U n ðjoÞ, which can be determined using the algorithm in [8]. By introducing the NOFRFs G n ðjoÞ, n ¼ 1; . . . ; N, Eq. (2) can be written as X ðjoÞ ¼

N X

X n ðjoÞ ¼

N X

n¼1

G n ðjoÞU n ðjoÞ,

(6)

n¼1

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 output frequency-response behaviour. It can be seen from Eq. (4) that G n ðjoÞ depends not only on Hn (n ¼ 1,y,N) but also on the input UðjoÞ. For any structure, the dynamical properties are determined by the GFRFs Hn (n ¼ 1,y,N). However, from Eq. (3) it can be seen that the GFRF is multidimensional [18,19], which makes the GFRFs difficult to measure, display and interpret in practice. Feijoo et al. [20,21] demonstrated that the Volterra series can be described by a series of associated linear equations (ALEs) whose corresponding associated frequency-response functions (AFRFs) are easier to analyse and interpret than the GFRFs. According to Eq. (4), the NOFRF Gn ðjoÞ is a weighted sum of H n ðjo1 ; . . . ; jon Þ over o1 þ    þ on ¼ o with the weights depending on the test input. Therefore G n ðjoÞ can be used as an alternative representation of the dynamical properties described by Hn. The most important property of the NOFRF Gn ðjoÞ is that it is one dimensional, and thus allows the analysis of nonlinear systems to be implemented in a convenient manner similar to the analysis of linear systems. Moreover, there is an effective algorithm [16] available which allows the estimation of the NOFRFs to be implemented directly using system input output data. 2.2. Nonlinear output frequency-response functions under harmonic inputs Harmonic inputs are pure sinusoidal signals, which have been widely used for the dynamic testing of many engineering structures. Therefore, it is necessary to extend the NOFRF concept to the harmonic input case. When system (1) is subject to a harmonic input uðtÞ ¼ A cosðoF t þ bÞ.

(7)

Lang and Billings [8] showed that Eq. (2) could be expressed as 0 1 N N X X X 1 @ X n ðjoÞ ¼ H n ðjok1 ; . . . ; jokn ÞAðjok1 Þ . . . Aðjokn ÞA, X ðjoÞ ¼ 2n ok þþok ¼o n¼1 n¼1 1

where

(

Aðjoki Þ ¼

jAjejsignðkÞb 0

if

(8)

n

oki 2 fkoF ; k ¼ 1g; otherwise:

i ¼ 1; . . . ; n;

(9)

Defining the frequency components of the nth-order output of the system as On, then according to Eq. (8), the frequency components in the system output can be expressed as O¼

N [ n¼1

On ,

(10)

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and On is determined by the set of frequencies   o ¼ ok1 þ    þ okn joki ¼ oF ; i ¼ 1; . . . ; n .

(11)

From Eq. (11), it is known that if all ok1 ; . . . ; okn are taken as oF , then o ¼ noF . If k of these are taken as oF , then o ¼ ðn þ 2kÞoF . The maximal k is n. Therefore, the possible frequency components of X n ðjoÞ are   (12) On ¼ ðn þ 2kÞoF ; k ¼ 0; 1; . . . ; n . Moreover, it is easy to deduce that O¼

N [

On ¼ fkoF ;

k ¼ N; . . . ; 1; 0; 1; . . . ; Ng.

(13)

n¼1

Eq. (13) explains why some superharmonic components will be generated when a nonlinear system is subjected to a harmonic excitation. In the following, only those components with positive frequencies will be considered. The NOFRFs defined in Eq. (4) can be extended to the case of harmonic inputs as GH n ðjoÞ ¼

ð1=2n ÞSok1 þþokn ¼o H n ðjok1 ; . . . ; jokn ÞAðjok1 Þ . . . Aðjokn Þ ð1=2n ÞSok1 þþokn ¼o Aðjok1 Þ . . . Aðjokn Þ

n ¼ 1; . . . ; N

(14)

under the condition that X 1 An ðjoÞ ¼ n Aðjok1 Þ . . . Aðjokn Þa0. 2 ok þþokn ¼o

(15)

1

Obviously, G H n ðjoÞ is only valid over On defined by Eq. (12). Consequently, the output spectrum X ðjoÞ of nonlinear systems under a harmonic input can be expressed as X ðjoÞ ¼

N X

X n ðjoÞ ¼

n¼1

N X

GH n ðjoÞAn ðjoÞ,

(16)

n¼1

when k of the n frequencies of ok1 ; . . . ; okn are taken as oF and the remainders are as oF, substituting Eq. (9) into Eq. (15) yields An ðjðn þ 2kÞoF Þ ¼

1 k n jðnþ2kÞb C jAj e . 2n n

(17)

Thus G H n ðjoÞ becomes k

nk

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ ð1=2 ÞH n ðjoF ; . . . ; joF ; joF ; . . . ; joF ÞC kn jAjn ejðnþ2kÞb H Gn ðjðn þ 2kÞoF Þ ¼ ð1=2n ÞC kn jAjn ejðnþ2kÞb n

k

nk

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ ¼ H n ðjoF ; . . . ; joF ; joF ; . . . ; joF Þ

ð18Þ GH n ðjoÞ

in this case, over the nthwhere H n ðjo1 ; . . . ; jon Þ is assumed to be a symmetric function. Therefore,  order output frequency range On ¼ ðn þ 2kÞoF ; k ¼ 0; 1; . . . ; n is equal to the GFRF H n ðjo1 ; . . . ; jon Þ evaluated at o1 ¼    ¼ ok ¼ oF , okþ1 ¼    ¼ on ¼ oF , k ¼ 0; . . . ; n. 3. The NOFRFs of locally nonlinear MDOF systems The locally nonlinear MDOF systems to be investigated are shown in Fig. 1. If all springs and dampers of the systems have linear properties, then the governing motion equation of the MDOF oscillator can be written as M x€ þ C x_ þ Kx ¼ UðtÞ,

(19)

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k1

m1

k2

c1

m2

mn-1

cn xn-1

c2 x2

x1

kn

mn

3113

u(t)

xn

Fig. 1. A multi-degree freedom oscillator.

where M is the system mass matrix 2 3 0 m1 0 . . . 6 0 m2 . . . 0 7 6 7 M¼6 .. 7 . . . 6 . 7. . . . 5 . . 4 . 0 0 . . . mn C and K are the system damping and stiffness matrices, respectively, 3 2 c1 þ c2 c2 0  0 6 .. 7 .. 7 6 . 7 . c2 þ c3 c3 6 c2 7 6 7 6 .. .. .. 6 C¼6 0 . 0 7 . . 7, 7 6 7 6 . .. 6 .. . cn1 cn1 þ cn cn 7 5 4 ...

0 2

k1 þ k2

6 6 6 k2 6 6 K¼6 6 0 6 6 .. 6 . 4

0

cn

cn

k2

0

...

k2 þ k3

k3

..

.

..

.

..

..

.

..

.

kn1

kn1 þ kn

0

kn

...

0

.

0

3

.. 7 7 . 7 7 7 0 7 7, 7 7 kn 7 5 kn

x is the displacement, x ¼ ðx1 ; . . . ; xn Þ0 , and u(t) is the external force acting on the right end of the oscillator, UðtÞ ¼ ð0; . . . ; uðtÞÞ0 . Eq. (19) forms the basis of the modal analysis method, which is a well-established approach for determining dynamic characteristics of engineering structures. In the linear case, the displacements xi ðtÞði ¼ 1; . . . ; nÞ can be expressed as Z þ1 xi ðtÞ ¼ hðiÞ ðt  tÞuðtÞ dt, (20) 1

where hðiÞ ðtÞði ¼ 1; . . . ; nÞ are the impulse response functions that are determined by Eq. (19), the Fourier transforms of which are the well-known FRFs of the system. Assume the characteristics of the Lth spring and damper are nonlinear, and the restoring forces S LS ðDÞ and _ of the spring and damper are the polynomial functions of the deformation D and its derivative D, _ S LD ðDÞ respectively, e.g. SLS ðDÞ ¼

P X i¼1

ri Di ;

SLD ðDÞ ¼

P X i¼1

i wi D_ ,

(21)

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where P is the degree of the polynomials. Without loss of generality, further assume La1; n. Then the motion of the oscillator in Fig. 1 can be described by Eqs. (22)–(26) as follows. For the masses that are not connected to the Lth spring and damper, the governing motion equations are m1 x€ 1 þ ðc1 þ c2 Þx_ 1  c2 x_ 2 þ ðk1 þ k2 Þx1  k2 x2 ¼ 0,

(22)

mi x€ i þ ðci þ ciþ1 Þx_ i  ci x_ i1  ciþ1 x_ iþ1 þ ðki þ kiþ1 Þxi  ki xi1  kiþ1 xiþ1 ¼ 0 ðiaL  1; LÞ,

ð23Þ

mn x€ n þ cn x_ n  cn x_ n1 þ kn xn  kn xn1 ¼ 0.

(24)

Denote kL ¼ r1 and cL ¼ w1 , then for the mass that is connected to the left of the Lth spring and damper, the governing motion equation is mL1 x€ L1 þ ðkL1 þ kL ÞxL1  kL1 xL2  kL xL þ ðcL1 þ cL Þx_ L1  cL1 x_ L2  cL x_ L þ

P X

ri ðxL1  xL Þi þ

P X

i¼2

wi ðx_ L1  x_ L Þi ¼ 0.

ð25Þ

i¼2

For the mass that is connected to the right of the Lth spring and damper, the governing motion equation is mL x€ L þ ðkL þ kLþ1 ÞxL  kL xL1  kLþ1 xLþ1 þ ðcL þ cLþ1 Þx_ L  cL x_ L1  cLþ1 x_ Lþ1 

P X

ri ðxL1  xL Þi 

i¼2

P X

wi ðx_ L1  x_ L Þi ¼ 0.

ð26Þ

i¼2

Denote Non F ¼

P X

wi ðx_ L1  x_ L Þi þ

i¼2

 NF ¼

P X

ri ðxL1  xL Þi ,

(27)

i¼2

L2

zfflffl}|fflffl{ 0...0

nL

Non F

Non F

zfflffl}|fflffl{ 0...0

0 .

(28)

Then, the governing motion equation of the locally nonlinear oscillator can be written as M x€ þ C x_ þ Kx ¼ NF þ UðtÞ.

(29)

The systems described by (27)–(29) are typical locally nonlinear MDOF systems. The Lth nonlinear spring and damper components can lead the whole system to behave nonlinearly. Based on the Volterra series theory of nonlinear systems, the relationships between the displacements xi ðtÞði ¼ 1; . . . ; nÞ and the input force uðtÞ of the MDOF systems are Z 1 j N Z 1 X Y ... hði;jÞ ðt1 ; . . . ; tj Þ uðt  tl Þ dtl ; i ¼ 1; . . . ; n, (30) xi ðtÞ ¼ j¼1

1

1

l¼1

where hði;jÞ ðt1 ; . . . ; tj Þ is the jth-order Volterra kernel associated to the ith mass. In the frequency domain, the relationship (30) can be expressed as X i ðjoÞ ¼

N X l¼1

X ði;lÞ ðjoÞ ¼

N X

Gði;lÞ ðjoÞU l ðjoÞ;

i ¼ 1; . . . ; n,

(31)

l¼1

where G ði;lÞ ðjoÞ is the lth-order NOFRF associated to the ith mass. In a recent study by the authors [17], a series of relationships between the NOFRF G ði;lÞ ðjoÞ, ði ¼ 1; . . . ; n; l ¼ 1; . . . ; NÞ of locally nonlinear MDOF systems were derived, the results reveal, for the first time, very important characteristics of this general class of nonlinear systems. These relationships in [17] give a comprehensive description how the linear system parameters govern the propagation of the nonlinear effect caused by the nonlinear component in the system and how the nonlinear component affects the vibration

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propagation in the system. It has been rigorously proved in [17] that, for any two consecutive masses, the NOFRFs satisfy the following relationships: li;iþ1 ðjoÞ ¼ 2

G ði;2Þ ðjoÞ G ði;NÞ ðjoÞ ¼    ¼ li;iþ1 N ðjoÞ ¼ G ðiþ1;2Þ ðjoÞ G ðiþ1;NÞ ðjoÞ

li;iþ1 ðjoÞ ¼ 1

G ði;1Þ ðjoÞ G ði;ZÞ ðjoÞ ¼ ¼ li;iþ1 Z ðjoÞ ð1pipL  2; 2pZpNÞ G ðiþ1;1Þ ðjoÞ G ðiþ1;ZÞ ðjoÞ

(33)

li;iþ1 ðjoÞ ¼ 1

G ði;1Þ ðjoÞ G ði;ZÞ ðjoÞ a ¼ li;iþ1 Z ðjoÞ ðL  1pipn  1; 2pZpNÞ, G ðiþ1;1Þ ðjoÞ G ðiþ1;ZÞ ðjoÞ

(34)

ð1pipn  1Þ,

(32)

and

where li;iþ1 Z ðjoÞ ¼

mi



o2



þj 1

li1;i Z ðjoÞ

Li;iþ1 Z ðjoÞ 1þ jciþ1 o þ kiþ1

!

jciþ1 o þ kiþ1   ci þ ciþ1 o þ 1  li1;i Z ðjoÞ ki þ k iþ1

ð1pipn  1;

1pZpNÞ

ð35Þ

with l0;1 Z ðoÞ ¼ 0

8 0 > > > > > < GðL1;ZÞ ðjoÞ i;iþ1 G ðL;ZÞ ðjoÞ LZ ðjoÞ ¼ > > G > ðL1;ZÞ ðjoÞ > > :G ðLþ1;ZÞ ðjoÞ

for

Z¼1

or

iaL  1; L;

for

ZX2

and

i ¼ L  1; (36)

for

ZX2

and

i ¼ L;

and GðL1;ZÞ ðjoÞ is a term introduced by the nonlinear force Non F for the Zth-order NOFRF. As the form of GðL1;ZÞ ðjoÞ will not play crucial role in this study, its explicit expression will not be given. 4. The linear stiffness and damping estimation method Based on the results in Section 3, a novel method is developed in this section to estimate the linear stiffness and damping parameters for locally nonlinear MDOF systems. This method requires that the masses mi, ði ¼ 1; . . . ; nÞ of the systems are known a priori. This is a reasonable assumption since the mass distribution of a mechanical structure or machine can usually be predetermined during the design stage and will not change significantly with the change of the structure’s or machine’s conditions even after years’ operation. In addition, the mass distribution can be easily obtained using the FE method or other methods. Consider Z ¼ 1 in (35), it is known that   jo 1  li1;i ðjoÞ li;iþ1 ðjoÞci þ jo li;iþ1 ðjoÞ  1 ciþ1 1 1 1   þ 1  li1;i ðjoÞ li;iþ1 ðjoÞki þ li;iþ1 ðjoÞ  1 kiþ1 ¼ mi li;iþ1 ðjoÞo2 ; 1pipn  1. ð37Þ 1 1 1 1 Denote  Pi1;i;iþ1 ðjoÞ ¼ 1  li1;i ðjoÞ li;iþ1 ðjoÞ; 1 1 1 ðjoÞ Fi1;i;iþ1 1 0  Im Pi1;i;iþ1 ðjoÞ o 1 ¼@  Re Pi1;i;iþ1 ðjoÞ o 1

Pi;iþ1 ðjoÞ ¼ li;iþ1 ðjoÞ  1, 1 1

  Re Pi1;i;iþ1 ðjoÞ Im Pi;iþ1 ðjoÞ o 1 1   Im Pi1;i;iþ1 ðjoÞ Re Pi;iþ1 ðjoÞ o 1 1

 1 Re Pi;iþ1 ðjoÞ 1  A Im Pi;iþ1 ðjoÞ 1

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and

W i;iþ1 ¼ ci

ki

ciþ1

kiþ1

T

.

Eq. (37) can then be written as  # " ðjoÞ o2 mi Re li;iþ1 1 i1;i;iþ1 i;iþ1  ðjoÞW ¼ F1 mi Im li;iþ1 ðjoÞ o2 1

ð1pipn  1Þ,

where W i;iþ1 is the parameter vector to be estimated. Consider Z ¼ 2 in (35), an equation similar to (38) can be obtained for iaL  1; L as  # " ðjoÞ o2 mi Re li;iþ1 2 i1;i;iþ1 i;iþ1  F2 ðjoÞW ¼ ð1pipn  1 and iaL  1; LÞ. mi Im li;iþ1 ðjoÞ o2 2

(38)

(39)

For i ¼ L1 and i ¼ L, the extra terms introduced by the nonlinear force Non F should be taken into account. When i ¼ L1, the result is   jo 1  lL2;L1 ðjoÞ l2L1;L ðjoÞcL1 þ jo l2L1;L ðjoÞ  1 cL 2  þ 1  l2L2;L1 ðjoÞ l2L1;L ðjoÞkL1  þ l2L1;L ðjoÞ  1 kL  L2L1;L ðjoÞ ¼ mL1 lL1;L ðjoÞo2 , ð40Þ 2 which can further be written as 2 3  L1;L 2# " W L1;L   Re l ðjoÞ o m 6 7 L1 1 0 2 L1;L 6 ReðL2 ðjoÞÞ 7 ¼  F2L2;L1;L ðjoÞ . L1;L 5 0 1 4 mL1 Im l2 ðjoÞ o2 ImðLL1;L ðjoÞÞ 2

(41)

For i ¼ L, the result is 2 3  # " W L;Lþ1   mL Re l2L;Lþ1 ðjoÞ o2 7 1 0 6 L;Lþ1 L1;L;Lþ1 6 ReðL2 ðjoÞÞ 7 ¼  ðjoÞ . F2 5 0 1 4 mL Im l2L;Lþ1 ðjoÞ o2 L;Lþ1 ImðL2 ðjoÞÞ

(42)

According to Eq. (36), it can be known LL;Lþ1 ðjoÞ 2 ðjoÞ LL1;L 2

¼

GðL;2Þ ðjoÞ ¼ lL;Lþ1 ðjoÞ. 2 GðLþ1;2Þ ðjoÞ

(43)

Substituting Eq. (43) into Eq. (42) yields 



Re l2L;Lþ1 ðjoÞ L1;L;Lþ1 F2 ðjoÞ  L;Lþ1 ðjoÞ Im l2

2 3  L;Lþ1 !  # " W L;Lþ1 Im l2 ðjoÞ 6 mL Re l2L;Lþ1 ðjoÞ o2 7 L1;L ReðL2 ðjoÞÞ 7 ¼  6  . 4 5 Re l2L;Lþ1 ðjoÞ mL Im l2L;Lþ1 ðjoÞ o2 L1;L ðjoÞÞ ImðL2 (44)

When a sinusoidal input of frequency oF is used to excite the system, considering li;iþ1 ðjoF Þ and li;iþ1 ðj2oF Þ, 1 2 Eqs. (38), (39), (41) and (44) can be written as  # " ðjoF Þ o2F mi Re li;iþ1 1 i1;i;iþ1 i;iþ1  F1 ðjoF ÞW ¼ ð1pipn  1Þ, (45) mi Im li;iþ1 ðjoF Þ o2F 1

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" Fi1;i;iþ1 ðj2oF ÞW i;iþ1 ¼ 2

 1 F2L2;L1;L ðj2oF Þ 0

0



 # ðj2oF Þ o2F 4mi Re li;iþ1 2  ð1pipn  1 and 4mi Im li;iþ1 ðj2oF Þ o2F 2 2

7 0 6 6 ReðLL1;L ðj2oF ÞÞ 7 ¼ 2 4 5 1 ImðLL1;L ðj2oF ÞÞ 2 

@FL1;L;Lþ1 ðj2oF Þ 2

3

W L1;L



Re l2L;Lþ1 ðj2oF Þ  Im l2L;Lþ1 ðj2oF Þ

2

 3 4mL Re l2L;Lþ1 ðj2oF Þ o2F ¼4  5. 4mL Im l2L;Lþ1 ðj2oF Þ o2F

"

iaL  1; LÞ,

 # 4mL1 Re l2L1;L ðj2oF Þ o2F  , 4mL1 Im l2L1;L ðj2oF Þ o2F 1



2

W L;Lþ1

3117

(46)

(47)

3

7 Im l2L;Lþ1 ðj2oF Þ 6 L1;L 6 ðj2oF ÞÞ 7 7  L;Lþ1 A6 ReðL2 4 5 Re l2 ðj2oF Þ L1;L ðj2oF ÞÞ ImðL2 ð48Þ

Consider ReðL2L1;L ðj2oF Þ and ImðLL1;L ðj2oF Þ as unknown parameters to be estimated, then there are 2 totally 2n+2 parameters to be estimated in Eqs. (45)–(48). There are clearly 4(n1) equations in total in (45)–(48), and, obviously, when nX3, the number of equations is sufficient to estimate the unknown parameters. Denote

T W ¼ c1 k1 . . . cL1 kL1 ReðL2L1;L ðj2oF ÞÞ ImðL2L1;L ðj2oF ÞÞ cL kL . . . cn kn and h  1;2 BZ ðjoÞ ¼ o2 m1 Re lZ ðjoÞ

 m1 Im l1;2 ... Z ðjoÞ

 mn1 Im ln1;n ðjoÞ Z

then Eqs. (45)–(48) can be assembled in the following form ! " # F1 ðjoF Þ B1 ðjoF Þ W¼ , F2 ðj2oF Þ B2 ðj2oF Þ

 n1;n iT mn1 Im lZ ðjoÞ ,

(49)

where both F1 ðjoF Þ and F2 ðj2oF Þ are a 2ðn  1Þ  ð2n þ 2Þ matrix that can be constructed using a similar procedure given in Appendix A. From Eq. (49), a least-square-based approach can be used to estimate the parameters in W as below 2 !T !31 !T " # F1 ðjoF Þ F1 ðjoF Þ B1 ðjoF Þ F1 ðjoF Þ 5 W ¼4 . (50) F2 ðj2oF Þ F2 ðj2oF Þ F2 ðj2oF Þ B2 ðj2oF Þ Eq. (50) provides a simple way to estimate the linear stiffness and damping coefficients for a locally nonlinear MDOF system from the system NOFRFs. This algorithm requires the information of the first- and second-order NOFRFs under sinusoidal inputs, which can readily be evaluated using an effective algorithm developed in [16] from the system input–output data. The linear parameter estimation algorithm can be summarised as the following procedures: Step 1: Estimate G ði;1Þ ðjoF Þ and G ði;2Þ ðj2oF Þ ð1pipnÞ under sinusoidal inputs using the algorithm developed in [16]. Step 2: Calculate li;iþ1 ðjoF Þ and li;iþ1 ðj2oF Þ 1pipn  1. 1 2 i1;i;iþ1 Step 3: Calculate PZ ðjoÞ, Pi;iþ1 ðjoÞ, (1pipn  1, Z ¼ 1, 2). Z Step 4: Construct matrixes F1 ðjoF Þ and F2 ðj2oF Þ using the method in Appendix A. Step 5: Estimate the linear parameters using Eq. (50).

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5. Numerical study To verify the effectiveness of the proposed parameter estimation method, a damped 6-DOF oscillator is used, in which the fourth spring is nonlinear. As widely used in modal analysis, the damping is assumed to be proportional damping, e.g., C ¼ mK. The values of the system parameters are m1 ¼ . . . ¼ m6 ¼ 1; m ¼ 0:01;

r1 ¼ k1 ¼ k2 ¼ k3 ¼ 3:6  104 ;

k4 ¼ k5 ¼ k6 ¼ 0:8k1 ,

r21 ;

w2 ¼ 0

r2 ¼ 0:8 

r3 ¼ 0:4 

r31 ;

w1 ¼ mr1 ;

and the input is a harmonic force, uðtÞ ¼ A sinð2p  20tÞ. If only the NOFRFs up to the fourth order are considered, according to Eqs. (16) and (17), the frequency components of the outputs of the 6 masses can be written as H X i ðjoF Þ ¼ G H ði;1Þ ðjoF ÞU 1 ðjoF Þ þ G ði;3Þ ðjoF ÞU 3 ðjoF Þ, H X i ðj2oF Þ ¼ G H ði;2Þ ðj2oF ÞU 2 ðj2oF Þ þ G ði;4Þ ðj2oF ÞU 4 ðj2oF Þ,

X i ðj3oF Þ ¼ G H ði;3Þ ðj3oF ÞU 3 ðj3oF Þ, X i ðj4oF Þ ¼ G H ði;4Þ ðj4oF ÞU 4 ðj4oF Þ ði ¼ 1; . . . ; 6Þ.

ð51Þ

From Eq. (51), it can be seen that two different inputs with the same waveform but different strengths are sufficient to estimate the NOFRFs up to fourth order. This is the basic principle of the algorithm proposed in [16] for the evaluation of the NOFRFs. Therefore, in this numerical study, two different inputs are used, A ¼ 0.8 and 1.0, respectively. The simulation studies were conducted using a fourth-order Runge–Kutta H H method to obtain the forced responses of the system. The evaluated results of GH 1 ðjoF Þ, G 3 ðjoF Þ, G 2 ðj2oF Þ H and G 4 ðj2oF Þ of the six different masses are given in Table 1. From the evaluated NOFRFs given in Table 1, li;iþ1 ðjoF Þ, li;iþ1 ðjoF Þ, li;iþ1 ðj2oF Þ and li;iþ1 ðj2oF Þ 1 3 2 4 (i ¼ 1,2,3,4,5) can be evaluated using Eqs. (32), (33) and (34). Moreover, the theoretical values of li;iþ1 ðjoF Þ, 1 i;iþ1 i;iþ1 li;iþ1 ðjo Þ, l ðj2o Þ and l ðj2o Þ (i ¼ 1,2,3,4,5) can also be calculated using the method in [17]. Both the F F F 3 2 4 i;iþ1 i;iþ1 i;iþ1 evaluated and theoretical values of li;iþ1 ðjo Þ, l ðjo Þ, l ðj2o Þ and l ðj2o Þ (i ¼ 1,2,3,4,5) are given F F F F 1 3 2 4 Table 1 H H H The evaluated results of G H 1 ðjoF Þ, G 3 ðjoF Þ, G 2 ðj2oF Þ and G 4 ðj2oF Þ

Mass1 Mass2 Mass3 Mass4 Mass5 Mass6

6 GH 1 ðjoF Þ (  10 )

7 GH 3 ðjoF Þ (  10 )

8 GH 2 ðj2oF Þ (  10 )

8 GH 4 ðj2oF Þ (  10 )

3.5291+6.0326i 7.7472+10.2849i 12.8459+11.1323i 19.4063+6.3260i 23.58704.9427i 21.432821.4620i

1.29962.3216i 2.87443.9706i 4.80884.3299i 1.7437+2.4212i 0.5327+3.1966i 1.8419+3.4348i

2.924412.3188i 12.572219.9209i 31.212015.1678i 31.6321+11.0973i 5.1398+17.9423i 9.3753+15.5360i

2.93001.4749i 4.26844.3621i 1.95398.7759i 0.9027+8.6380i 4.2148+2.3702i 4.47761.4327i

Table 2 The evaluated and theoretical values of li;iþ1 ðjoF Þ and li;iþ1 ðjoF Þ 1 3 ðjoF Þ li;iþ1 1

i¼1 i¼2 i¼3 i¼4 i¼5

li;iþ1 ðjoF Þ 3

Evaluated

Theoretical

Evaluated

Theoretical

0.53910.0630i 0.74070.1588i 0.76740.3235i 0.73430.4221i 0.66480.4351i

0.53910.0630i 0.74070.1588i 0.76540.3206i 0.73530.4228i 0.66460.4359i

0.53910.0630i 0.74070.1588i 2.11940.4597i 0.6485+0.6536i 0.7874+0.2672i

0.53910.0630i 0.74070.1588i 2.11940.4597i 0.6485+0.6536i 0.7874+0.2672i

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Table 3 The evaluated and theoretical values of li;iþ1 ðj2oF Þ and li;iþ1 ðj2oF Þ 2 4 ðj2oF Þ li;iþ1 2

i¼1 i¼2 i¼3 i¼4 i¼5

li;iþ1 ðj2oF Þ 4

Evaluated

Theoretical

Evaluated

Theoretical

0.50850.1741i 0.57680.3580i 1.0284+0.1187i 1.0383+1.4656i 0.7002+0.7534i

0.50850.1741i 0.57680.3580i 1.0284+0.1187i 1.0383+1.4655i 0.7002+0.7534i

0.50850.1741i 0.57680.3580i 1.0284+0.1187i 1.0383+1.4656i 0.7002+0.7534i

0.50850.1741i 0.57680.3580i 1.0284+0.1187i 1.0383+1.4655i 0.7002+0.7534i

Table 4 The estimated and real values of stiffness and damping

k1 k2 k3 k4 k5 k6

Evaluated (  104)

Real (  104)

3.6001 3.6001 3.6001 2.9208 2.8800 2.8800

3.6000 3.6000 3.6000 2.8800 2.8800 2.8800

c1 c2 c3 c4 c5 c6

Evaluated (  102)

Real (  102)

3.6000 3.6000 3.6000 2.8573 2.8799 2.8799

3.6000 3.6000 3.6000 2.8800 2.8800 2.8800

in Tables 2 and 3. The results in Table 2 and 3 clearly show that properties (32)–(34) are tenable. Obviously, li;iþ1 ðjoF Þali;iþ1 ðjoF Þ for i ¼ 3, 4, 5. 1 3 Using the evaluated results of li;iþ1 ðjoF Þ and li;iþ1 ðj2oF Þ (i ¼ 1,2,3,4,5), the linear stiffness and damping can 1 2 be estimated by the method proposed in the previous section, and the results are given in Table 4. It can be seen that the estimated results match the theoretical results very well except a slightly difference for k4 and c4. This difference mainly comes from the facts that the truncated Volterra series have to be applied in practice and the truncation can raise error only to the estimations of k4 and c4. 6. Conclusions and remarks A new method for the estimation of the linear stiffness and damping parameters of locally nonlinear MDOF systems has been developed. This method is based on the concept of nonlinear output frequency-response functions (NOFRFs), which were derived from the Volterra series approach of nonlinear systems. This method assumes that the system masses and the position of the system nonlinear component are known priori. The masses can be readily obtained at the system design stage. The position of the nonlinear component can be determined using directly the relationships (32)–(34) [17], which can also be applied to detect the crack position in beams or structures. From these results, all linear stiffness and damping parameters of a nonlinear MDOF system can be estimated using the proposed method directly using the system input–output test data. In addition, although the method is demonstrated on the particular case where the linear oscillators are coupled in series and fixed at one end while excited in the other end, this method can be directly applied (without any modification) to the cases where the excitation force is acting at any position of the system, not only limited to the end. It is worthy noting here that the method is only valid when the system response can be described using the Volterra series, and basically, the validity of the Volterra series is dependent on the amplitude of the external force. Regarding this problem determining the domain of validity of the Volterra series, many researchers have made many efforts, including the author ourselves. The existing literature includes [22–26]. Our recent research studies have shown that, for the nonlinear damping system, the validity domain is very wide; on the other hand, the validity domain of the nonlinear stiffness system is relatively narrow, for example, at the region where the jump phenomenon [27] occurs, the Volterra series representation is not valid. However, from

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the system identification viewpoint, because the amplitude of the external force is controllable, we can try to conduct the identification experiments such that the system always works in a region where the Volterra series theory works. In addition, when conducting the parameter estimation, theoretically, the choice of the excitation frequency can be arbitrary. However, according to our experiences, in practical applications, because the method will make use of the nonlinear effect, therefore, it might be better to choose an excitation frequency that can make the nonlinear effect significant, for example, one half of a resonant frequency which can make the second harmonic component reach a maximum. More information about the resonances in nonlinear systems can be found in [28]. It is worth noting here that the GðL1;ZÞ ðjoÞ in Eq. (36) is related to the nonlinear parameters and, therefore, the nonlinear parameters can be estimated from GðL1;ZÞ ðjoÞ. This work about the nonlinear parameter estimation will be reported in detail in a subsequent paper. Acknowledgements The authors gratefully acknowledge the support of the Engineering and Physical Science Research Council, UK, for this work. Appendix A. Construction of U1(jxF) and U2(j2xF) The construction of F1(joF) using this procedure is given as below. For 1pipL  2, i.e., the first 2(L2) rows of F1(joF) F1 ðjoF Þð2i  1 : 2i;

1 : 2i  2Þ ¼ 0,

F1 ðjoF Þð2i  1 : 2i; F1 ðjoF Þð2i  1 : 2i;

2i  1 : 2i þ 2Þ ¼ Fi1;i;iþ1 ðjoF Þ ð1pipL  2Þ, 1 2i þ 3 : 2n þ 2Þ ¼ 0.

ðA:1Þ

For i ¼ L1, F1 ðjoF Þð2ðL  1Þ  1 : 2ðL  1Þ;

1 : 2ðL  1Þ  2Þ ¼ 0,

F1 ðjoF Þð2ðL  1Þ  1 : 2ðL  1Þ;

2ðL  1Þ  1 : 2ðL  1Þ þ 4Þ !

¼

FL1 1;Left ðjoF Þ

0

0

0

0

L1 F1;Right ðjoF Þ ,

F1 ðjoF Þð2ðL  1Þ  1 : 2ðL  1Þ;

2ðL  1Þ þ 5 : 2n þ 2Þ ¼ 0,

ðA:2Þ

where FL1 Z;Left ðjoÞ ¼

FL1 Z;Right ðjoÞ ¼

 L2;L1;L ðjoÞ o Im PZ  ðjoÞ o Re PL2;L1;L Z  L1;L ðjoÞ o Im PZ  L1;L ðjoÞ o Re PZ

 L2;L1;L ! Re PZ ðjoÞ  L2;L1;L , Im PZ ðjoÞ

(A.3)

 L1;L ! Re PZ ðjoÞ  L1;L . Im PZ ðjoÞ

(A.4)

For i ¼ L F1 ðjoF Þð2L  1 : 2L; F1 ðjoF Þð2L  1 : 2L;

1 : 2L  2Þ ¼ 0, 2L  1 : 2L þ 4Þ ¼

F1 ðjoF Þð2ðL  1Þ  1 : 2ðL  1Þ;

0

0

0

0

! F1L1;L;Lþ1 ðjoF Þ

2ðL  1Þ þ 5 : 2n þ 2Þ ¼ 0.

, ðA:5Þ

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For Loipn  1 F1 ðjoF Þð2i  1 : 2i;

1 : 2iÞ ¼ 0,

F1 ðjoF Þð2i  1 : 2i;

2i þ 1 : 2i þ 4Þ ¼ Fi1;i;iþ1 ðjoF Þ ðLoipn  1Þ, 1

F1 ðjoF Þð2i  1 : 2i;

2i þ 5 : 2n þ 2Þ ¼ 0.

ðA:6Þ

The construction of F2 ðj2oF Þ is similar except for the cases where i ¼ L1 and i ¼ L. For i ¼ L1 F2 ðj2oF Þð2ðL  1Þ  1 : 2ðL  1Þ; F2 ðj2oF¯ Þð2ðL  1Þ  1 : 2ðL  1Þ; ¼

FL1 2;Left ðj2oF Þ

1

0

0

1

1 : 2ðL  1Þ  2Þ ¼ 0, 2ðL  1Þ  1 : 2ðL  1Þ þ 4Þ !

FL1 2;Right ðj2oF Þ ,

F2 ðj2oF Þð2ðL  1Þ  1 : 2ðL  1Þ;

2ðL  1Þ þ 5 : 2n þ 2Þ ¼ 0

ðA:7Þ

and for i ¼ L F2 ðj2oF Þð2L  1 : 2L;

1 : 2L  2Þ ¼ 0,

F2 ðj2oF Þð2L  1 : 2L; 2L  1 : 2L þ 4Þ, 0  L;Lþ1  Re l2 ðj2oF Þ Im l2L;Lþ1 ðj2oF Þ ¼ @  L;Lþ1  ðj2oF Þ Re l2L;Lþ1 ðj2oF Þ Im l2 F2 ðj2oF Þð2ðL  1Þ  1 : 2ðL  1Þ;

1 F2L1;L;Lþ1 ðj2oF ÞA

2ðL  1Þ þ 5 : 2n þ 2Þ ¼ 0.

ðA:8Þ

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