Non-linear output frequency response functions for ... - Semantic Scholar

International Journal of Control Vol. 80, No. 6, June 2007, 843–855

Non-linear output frequency response functions for multi-input non-linear Volterra systems Z. K. PENG*, Z. Q. LANG and S. A. BILLINGS Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK (Received 26 October 2006; in final form 20 December 2006) The concept of non-linear output frequency response functions (NOFRFs) is extended to the non-linear systems that can be described by a multi-input Volterra series model. A new algorithm is also developed to determine the output frequency range of non-linear systems from the frequency range of the inputs. These results allow the concept of NOFRFs to be applied to a wide range of engineering systems. The phenomenon of the energy transfer in a two degree of freedom non-linear system is studied using the new concepts to demonstrate the significance of the new results.

1. Introduction Linear systems, which have been widely studied by practitioners in many different fields, have provided a basis for the development of the majority of control system synthesis, mechanical system analysis and design, and signal processing methods. However, there are certain types of qualitative behaviour encountered in engineering, which cannot be produced by linear models (Pearson 1994), for example, the generation of harmonics and inter-modulation behaviour. In cases where these effects are dominant or significant non-linear behaviours exist, non-linear models are required to describe the system, and non-linear system analysis methods have to be applied to investigate the system dynamics. The Volterra series approach (Worden et al. 1997) is a powerful tool for the analysis of non-linear 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) (Lang and Billings 1996), or generalized frequency response functions (GFRFs), and these provide a convenient tool for analyzing non-linear systems in the *Corresponding author. Email: [email protected]

frequency domain. If a differential equation or discretetime model is available for a system, the GFRFs can be determined using the algorithm in Billings and Tsang (1989), Peyton Jones and Billings (1989), Billings and Peyton Jones (1990). The GFRFs can be regarded as the extension of the classical frequency response function (FRF) of linear systems to the non-linear case. However, the GFRFs are much more complicated than the FRF. GFRFs are multidimensional functions (Zhang and Billings 1993, 1994), which can be difficult to measure, display and interpret in practice. Recently, the novel concept of non-linear output frequency response functions (NOFRFs) was proposed by Lang and Billings (2005). The concept can be considered to be an alternative extension of the FRF to the non-linear case. NOFRFs are one dimensional functions of frequency, which allow the analysis of non-linear 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 nonlinear behaviours. The NOFRF concept was recently used to investigate the sub-resonance phenomena for a class of non-linear systems (Peng et al. 2006). The results revealed the existence of resonances at frequencies different from the frequencies at the input excitation in this class of oscillators. The objective of this paper is to extend the concept of NOFRFs to multi-input non-linear Volterra systems so

International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online ß 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207170601185038

844

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that the concept of NOFRFs can be applied to MIMO systems which allow the basic idea to be used in more complicated systems. The phenomenon of energy transfer in a 2DOF non-linear system is investigated using the extended concept to demonstrate the effectiveness and significance of the results obtained in the present study.

2. The concept of non-linear output frequency response functions NOFRFs were recently proposed and used to investigate the behaviour of structures with polynomial-type non-linearities. The definition of NOFRFs is based on the Volterra series theory of non-linear systems. Consider the class of non-linear systems which are stable at zero equilibrium and which can be described in the neighbourhood of the equilibrium by the Volterra series Z1 N Z 1 n Y X hn ð1 , . . . , n Þ uðt i Þdi ; ð1Þ yðtÞ ¼ n¼1

1

1

i¼1

where y(t) and x(t) are the output and input of the system, hð1 , . . . , n Þ is the nth order Volterra kernel, and N denotes the maximum order of the system non-linearity. Lang and Billings (1996) have derived an expression for the output frequency response of this class of non-linear systems to a general input. The result is 8 N X > > > Yn ð j!Þ for 8 ! Yð j!Þ ¼ > > > > n¼1 > > pffiffiffi Z < 1= n Hn ð j!1 , . . . , j!n Þ ð2Þ Yn ð j!Þ ¼ > ð2Þn1 !1 þþ!n ¼! > > > > n Y > > > > Xð j!i Þ dn! : : i¼1

This expression is just a different form description for non-linear system output frequency responses. Compared with other descriptions, equation (2) provides a more physically meaningful insight into the composition of non-linear output frequency responses and reveals how non-linear mechanisms operate on the input spectra to produce the system output frequency response. In (2), Yð j!Þ and Xð j!Þ are the spectra of the system output and input respectively, Yn ð j!Þ represents the nth order output frequency response of the system, Z1 Z1 ... hn ð1 , . . . , n Þ Hn ð j!1 , . . . , j!n Þ ¼ 1 1 ð!1 1 þþ!n n Þj

e

d1 . . . dn

ð3Þ

is the definition of the generalized frequency response function (GFRF), and Z n Y Hn ð j!1 , . . . , j!n Þ Xð j!i Þdn! !1 þþ!n ¼!

i¼1

Q denotes the integration of Hn ð j!1 , . . . , j!n Þ ni¼1 Xð j!i Þ over the n-dimensional hyper-plane !1 þ þ !n ¼ !. Equation (2) is a natural extension of the well-known linear relationship Yð j!Þ ¼ H1 ð j!ÞXð j!Þ to the nonlinear case. For linear systems, the possible output frequencies are the same as the frequencies in the input. For non-linear systems described by equation (1), however, the relationship between the input and output frequencies is more complicated. Given the frequency range of the input, the output frequencies of system (1) can be determined using an explicit expression derived by Lang and Billings (1996). Based on the above results for output frequency responses of non-linear systems, a new concept known as the non-linear output frequency response function (NOFRF) was recently introduced by Lang and Billings (2005). The concept was defined as R !1 þþ!n ¼!

Gn ð j!Þ ¼

Hn ð j!1 , . . . , j!n Þ

n Q

Xð j!i Þdn!

i¼1

R !1 þþ!n ¼!

n Q

Xð j!i Þdn!

i¼1

ð4Þ under the condition that Z Un ð j!Þ ¼

n Y

Xð j!i Þdn! 6¼ 0:

ð5Þ

!1 þþ!n ¼! i¼1

Note that Gn ð j!Þ is valid over the frequency range of Un ð j!Þ, which can be determined using the algorithm in Lang and Billings (1996). By introducing the NOFRFs Gn ð j!Þ, n ¼ 1, . . . , N, equation (4) can be written as Yð j!Þ ¼

N X n¼1

Yn ð j!Þ ¼

N X

Gn ð j!ÞUn ð j!Þ

ð6Þ

n¼1

which is similar to the description of the output frequency response of linear systems. For a linear system, the relationship between Yð j!Þ and Xð j!Þ can be illustrated as in figure 1. Similarly, the non-linear system input and output relationship of equation (1) can be illustrated in figure 2. The NOFRFs reflect a combined contribution of the system and the input to the frequency domain output behaviour. It can be seen from equation (4) that Gn ð j!Þ depends not only on Hn (n ¼ 1, . . . , N) but also on the input Xð j!Þ. For a non-linear system, the dynamical

845

Functions for multi-input non-linear Volterra systems X( jw)

where

Y( jw) H( jw)=G1( jw)

Figure 1.

The output frequency response of a linear system.

X

yiðnÞ ðtÞ ¼

Z

Z

þ1

þ1

N1 þþNm ¼n 1

1

hði,ðnÞP1 ¼N1 ...Pm ¼Nm Þ

ð1 , . . . , n Þx1 ðt 1 Þ x1 ðt N1 Þ UN( jw) U2( jw) U1( jw)

Figure 2. system.

GN( jw) G2( jw) G1( jw)

YN( jw)

x2 ðt N1 þ1 Þ x2 ðt N1 þN2 Þ xm ðt N1 þþNm1 þ1 Þ xm ðt n Þ d1 . . . dn

Y2( jw) Y1( jw)

ð8Þ

Y( jw)

hði,ðnÞP1 ¼N1 Pm ¼Nm Þ

The output frequency response of a non-linear

properties are determined by the GFRFs Hn (n ¼ 1, . . . , N). However, from equation (3) it can be seen that a GFRF is multidimensional (Zhang and Billings 1993, 1994), and may become difficult to measure, display and interpret in practice. Feijoo et al. (2004, 2005) 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 equation (4), the NOFRF Gn ð j!Þ is a weighted sum of Hn ð j!1 , . . . , j!n Þ over !1 þ þ !n ¼ ! with the weights depending on the input. Therefore Gn ð j!Þ can be used as an alternative representation of the structural dynamical characteristics described by Hn. The most important property of the NOFRF Gn ð j!Þ is that it is one dimensional, and thus allows the analysis of non-linear systems to be implemented in a convenient manner similar to the analysis of linear systems. Moreover, there is an effective algorithm Lang and Billings (2005) available which allows the evaluation of the NOFRFs to be implemented directly using system input output data.

and represents the nth order kernel associated with the ith output and N1th input x1 ðtÞ, N2th input x2 ðtÞ, . . . , Nmth input xm ðtÞ. Equation (8) can be rewritten as Z þ1 Z þ1 X ðnÞ hði,ðnÞP1 ¼N1 ...m ¼Nm Þ ð1 , . . . , n Þ yi ¼ N1 þþNm ¼n 1

1

xðN1 ...Nm Þ ð1 , . . . , n Þd1 . . . dn ,

ð9Þ

where xðN1 ,..., Nm Þ ð1 , . . . , n Þ ¼ x1 ðt 1 Þ x1 ðt N1 Þ x2 ðt N1 þ1 Þ x2 ðt N1 þN2 Þ xm ðt N1 þþNm1 þ1 Þ xm ðt n Þ:

ð10Þ

In the single-input case, the Volterra series has only one kernel for each order of nonlinearity, for example, hð1 , . . . , n Þ is the second order kernel. It can be seen, however, that in the multi-input case more kernels are involved for each order of non-linearity. For example, for a two input system there are three 2nd order kernels for the ith output which are hði,ð2ÞP1 ¼2, P2 ¼0Þ ð1 , 2 Þ, hði,ð2ÞP1 ¼0, P2 ¼2Þ ð1 , 2 Þ and hði,ð2ÞP1 ¼1, P2 ¼1Þ ð1 , 2 Þ: The frequency domain description of (7) and (8) can be expressed as Yi ð j!Þ ¼

N X

yiðnÞ ð j!Þ

ð11Þ

n¼1

3. NOFRFS for multi-input non-linear Volterra systems YiðnÞ ð j!Þ ¼

3.1 Multi-input non-linear Volterra systems

1

Multi-input multi-output non-linear Volterra systems can be expressed so that each output can to be modeled as a multi-input Volterra series. The extension of the single input single output Volterra series representation (1) to this more general case is as follows: yi ðtÞ ¼

N X n¼1

yiðnÞ ðtÞ,

X 1 n1 2 N þþN

ð7Þ

ð j!1 , ...,j!n Þ

Z !1 þþ!n ¼! m ¼n

N1 Y i¼1

n Y

X1 ð j!i Þ

Hði,ðnÞP1 ¼N1 ...Pm ¼Nm Þ

NY 1 þN2

X2 ð j!i Þ

i¼N1 þ1

Xm ð j!i Þdn! :

ð12Þ

i¼N1 þþNm1 þ1

This is an extension of equation (12) for the single-input case to the multi-input case. Define N0 ¼ 0, then

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Z. K. Peng et al.

equation (12) can be written as

where

YiðnÞ ð j!Þ

Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ 0 ðnÞ 1 Hði, P1 ¼N1 ,..., Pm ¼Nm Þ ð j!1 , . . . , j!n Þ B C R B Q Cdn! N0 þþN m Q j !1 þþ!n ¼! @ A Xj ð j!i Þ

X 1 n1 1 pffiffiffi ¼ 2 n N þþN 1

ð j!1 , . . . , j!n Þ

Z m ¼n

m Y

Hði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ

!1 þþ!n ¼!

¼

N0 þþN Y j

Xj ð j!i Þdn! :

ð13Þ

j¼1 i¼N0 þþNj1 þ1


R !1 þþ!n ¼!

N0 þþN Q j

m Q

Xj ð j!i Þdn!


ð18Þ

For a given set of N1, N2, . . . , Nm, define Yði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ

1 n1 1 pffiffiffi ¼ 2 n

Z

ðnÞ Hði,P ð j!1 ,... ,j!n Þ 1 ¼N1 ,...,Pm ¼Nm Þ

!1 þþ!n ¼!

N1 þþNm ¼n

N0 þþN Y j

m Y

will be referred to as the non-linear output frequency response function for multi-input non-linear Volterra systems, and is a natural extension of equation (14) to more general case. Substituting (18) into (15) yields X Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ YiðnÞ ð j!Þ ¼

Xj ð j!i Þ dn!

ð14Þ


then equation (13) can be written in a compact form X YiðnÞ ð j!Þ ¼ Yði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ: ð15Þ N1 þþNm ¼n

3.2 Definition of the NOFRFs for multi-input non-linear Volterra systems Define ðnÞ UðP ð j!Þ 1 ¼N1 ,..., Pm ¼Nm Þ n1 Z N0 þþN m Y Y j 1 1 pffiffiffi ¼ Xj ð j!i Þdn! 2 n !1 þþ!n ¼! j¼1 i¼N þþN þ1 0

j1

ðnÞ ð j!Þ UðP 1 ¼N1 ,..., Pm ¼Nm Þ

ð19Þ

It is easy to verify that Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ has the following important properties. (i) Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ allows Yði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ to be described in a manner similar to the description for the output frequency response of linear systems. is valid over a (ii) Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ ðnÞ ð j!Þ ¼ 6 0. frequency range where UðP 1 ¼N1 ,..., Pm ¼Nm Þ (iii) Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ is insensitive to the change of the input spectra by a constant gain, that is, Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!ÞX1 ð j!Þ¼X 1 ð j!Þ .. . Xm ð j!Þ¼X m ð j!Þ

¼ Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!ÞX1 ð j!Þ¼X 1 ð j!Þ :

ð20Þ

.. . Xm ð j!Þ¼X m ð j!Þ

ð16Þ then (14) can be rewritten as Yði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ 0 1 Hði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!1 ,. ..,j!n Þ B C R B C N þþN C dn! 0 j m !1 þþ!n ¼! B Q @ Q A X ð j! Þ j

i


¼

R !1 þþ!n ¼!

m Q

N0 þþN Q j

Xj ð j!i Þdn!


Z N0 þþN m Y Y j 1 n1 1 pffiffiffi Xj ð j!i Þdn! 2 n !1 þþ!n ¼! j¼1 i¼N þþN þ1 0

3.3 Determination of the output frequency range of multi-input non-linear Volterra systems For a non-linear system that can be modelled as a singleinput Volterra series, given the frequency range of the input, Lang and Billings (1996) derived an explicit expression for the output frequency range. In the following, a method will be derived to determine the output frequency range of multi-input non-linear Volterra systems. Obviously, the frequency range of ðnÞ ð j!Þ is given as the range of UðP 1 ¼N1 ,..., Pm ¼Nm Þ

j1

ðnÞ ðnÞ ¼ Gði,P ð j!ÞUðP ð j!Þ 1 ¼N1 ,...,Pm ¼Nm Þ 1 ¼N1 ,v, Pm ¼Nm Þ

ð17Þ

!¼

m X k¼1

!ðkÞ ,

ð21Þ

847

Functions for multi-input non-linear Volterra systems where !ðkÞ is associated to the kth input xk(t) of order Nk and !ðkÞ ¼ !N0 þþNk1 þ1 þ þ !N0 þþNk :

ð22Þ

Define the frequency range of the kth input xk(t) (k ¼ 1, . . . , m) as ak [ ½ ak

½ bk

Finally, according to equation (11), the frequency range of Yi ð j!Þ is given by

bk :

Now, assume Lk of Nk components are located in ½ bk ak , and the remainder are located ½ ak bk , in this case, the frequency range is

fyi ¼

N [

fiðnÞ :

ð30Þ

n¼1

Therefore, given the frequency ranges of the inputs xk(t) (k ¼ 1, . . . , m) as [ ½ ak bk , ½ bk ak

ðNk Lk Þbk Lk ak

ð23Þ

the output frequency range can be determined by equations (28)–(30). The validity of this method will be verified by numerical studies in x 3.

½Nk ak Lk ðbk þ ak Þ Nk bk Lk ðak þ bk Þ:

ð24Þ

3.4 Evaluation of the NOFRFs for multi-input systems

½ðNk Lk Þak Lk bk that is

Therefore the range of ! Nk [

ðkÞ

is

½Nk ak Lk ðbk þ ak Þ

Nk bk Lk ðak þ bk Þ

ð25Þ

Lk ¼0

that is !ðkÞ 2

Nk [

½Nk ak Lk ðbk þ ak Þ Nk bk Lk ðak þ bk Þ:

Lk ¼0

ð26Þ This can easily be extended to case of !ð1Þ þ þ !ðmÞ ð1Þ

! ¼ ð!

ðmÞ

þ þ !

Þ2

N1 [

L1 ¼0

m X

Li ðai þ bi Þ

m X

i¼1

Nm [

"

m X

Lm ¼0

N i bi

m X

i¼1

Ni ai

i¼1

# Li ðai þ bi Þ :

ð27Þ

ðnÞ ¼ fðP 1 ¼N1 ,..., Pm ¼Nm Þ

L1 ¼0

m X

Nm [ Lm ¼0

N i bi

i¼1

yi ð j!Þ ¼

N X

Yði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ:

ð31Þ

n¼1 N1 þþNm ¼n

"

m X i¼1

m X

N i ai

m X

Equation (31) can be further written in the polynomial form yi ð j!Þ ¼

Li ðai þ bi Þ

m X m X Gð1Þ ð j!Þ U ð j!Þ þ G2k1 k2 ð j!Þ k 1 k1

k1 ¼1

k1 ¼1 k2 ¼k1

Uk1 ð j!ÞUk2 ð j!Þ þ

i¼1

# Li ðai þ bi Þ :

m X

ð28Þ

i¼1

þ

m X k1 ¼1

From equation (17), it can be shown that ðnÞ ð j!Þ have Yði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ and UðP 1 ¼N1 ,..., Pm ¼Nm Þ the same frequency range. Furthermore, according to equation (15), it can easily be shown that the frequency range of YiðnÞ ð j!Þ is [ ðnÞ fiðnÞ ¼ fðP : ð29Þ 1 ¼N1 ,..., Pm ¼Nm Þ N1 þþNm ¼n

X

i¼1

ðnÞ Therefore, the frequency range of UðP ð j!Þ 1 ¼N1 ,..., Pm ¼Nm Þ can be expressed N1 [

For single-input non-linear systems, Lang and Billings (2005) derived an effective algorithm for the estimation of the NOFRFs, which can be implemented directly using system input output data. To estimate the NOFRFs up to Nth order, the algorithm generally requires experimental or simulation results for the system under N different input signal excitations, which have the same waveforms but different intensities. This algorithm can be extended to estimate the NOFRFs in the multi-input case. As a multi-input system of non-linearity up to Nth order involves more than N NOFRFs, more than N experiments or simulations under different signal excitations are needed to estimate the NOFRFs. Combining equations (11) and (19) yields

...

m X

GðNÞ k1 ...kN ð j!Þ Uk1 ð j!Þ...UkN ð j!Þ ,

kN ¼kN1

ð32Þ where GðNÞ k1 ...kN ð j!Þ represents a specific Gði,ðnÞP1 ¼N1 ,..., Pm ¼Nm Þ ð j!Þ ¼ G1 . . . 1 2 . . . 2... m . . . m ð j!Þ |fflffl{zfflffl} |fflffl{zfflffl} |fflfflffl{zfflfflffl} N1

N2

Nm

ð33Þ

848

Z. K. Peng et al. when, xi ðtÞ ¼ xi ðtÞ, i ¼ 1, . . . , m, where is a constant and xi ðtÞ, i ¼ 1, . . . , m are the input signals under which the NOFRFs of the system are to be evaluated,

with N1 þ þ Nm ¼ n, and n ¼ 1, . . . , N, and " ðnÞ ð j!Þ ¼ U1 ð j!Þ . . . U1 ð j!Þ U2 ð j!Þ2 ð j!Þ UðP 1 ¼N1 ,..., Pm ¼Nm Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} N1

ðnÞ ð j!Þ UðP 1 ¼N1 ,...,Pm ¼Nm Þ n1 Z N0 þþN m Y Y j 1 1 pffiffiffi ¼ n X ð j!i Þdn! 2 n !1 þþ!n ¼! j¼1 i¼N þþN þ1 j

N2

# . . . Um ð j!Þ . . . Um ð j!Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð34Þ

Nm

0

¼

The number of terms contained in equation (32) can easily be calculated using the method given in Guo and Billings (2005), as

k ¼ 1 : Cðn, mÞ ,

ð35Þ

0

In this case, from (40), equation (31) can be written as h i ð1Þ ð1Þ N ðNÞ N ðNÞ . ..UðC ... U . .. U yi ð j!Þ ¼ Uð1Þ ð1Þ ðCðN:mÞ Þ Gi , ð1,mÞ Þ

ð36Þ

ð43Þ ½Gði,ð1Þ 1Þ

Cðn, mÞ ¼ ðm þ n 1Þ . . . ðm þ 1Þm=n!:

ð37Þ

k ¼ 1 : Cðn, mÞ ,

xi ðtÞ ¼ j xi ðtÞ,

ð38Þ

.. .

. . . Gði,ðNÞ 1Þ

T . . . Gði,ðNÞ CðN, mÞ Þ

i ¼ 1 : m,

and

j ¼ 1 : CðN, mÞ

CN, m > CN, m 1 > > 1 > 0

then equation (31) can be rewritten as h i ð1Þ ðNÞ ðNÞ yi ð j!Þ ¼ Uð1Þ ð1Þ . . . UðCð1, mÞ Þ . . . Uð1Þ . . . UðCðN:mÞ Þ ½Gi , ð39Þ

ð1Þ 1 Uð1Þ 6 .. Yi ð j!Þ ¼ 6 4 . ð1Þ CðN, mÞ Uð1Þ

. . . Gði,ð1Þ Cð1, mÞ Þ

where ½Gi ¼ are the NOFRFs to be evaluated. Excite the system CðN, mÞ ¼ Cð1, mÞ þ þ CðN, mÞ with the input signals

Denoting the corresponding [Uk1 ð j!Þ . . . Ukn ð j!Þ] as

2

j1

ð42Þ

where

ðnÞ UðkÞ ð j!Þ,

ð41Þ

ðnÞ ð j!Þ UðP 1 ¼N1 ,...,Pm ¼Nm Þ n1 Z N0 þþN m Y Y j 1 1 pffiffiffi ¼ X ð j!i Þdn! : 2 n !1 þþ!n ¼! j¼1 i¼N þþN þ1 j

It can be seen that there are ðm þ n 1Þ ðm þ 1Þm=n! terms for the nth order NOFRFs. Sorting all GðNÞ k1 kN ð j!Þ, ki ¼ ki1 , . . . , m, i ¼ 1, . . . , n, k0 ¼ 1 as a series, and labelling them as Gði,ðnÞkÞ ð j!Þ,

j1

ðnÞ UðP ð j!Þ, 1 ¼N1 ,..., Pm ¼Nm Þ

where

CðN, mÞ ¼ m þ ðm þ 1Þm=2! þ þ ðm þ N 1Þ . . . ðm þ 1Þm=N!:

n

to generate C(N, m) output frequency responses Yki ð j!Þ, k ¼ 1: C(N, m). From (43), these output frequency responses can be written as

ð1Þ 1 UðC ð1, mÞ Þ .. . ð1Þ CðN, mÞ UðC ð1, mÞ Þ

3 ðNÞ N 1 UðCðN, mÞ Þ 7 .. .. 7 Gi , 5 . . ðNÞ N CðN, mÞ UðCðN, mÞ Þ

ðNÞ N 1 Uð1Þ .. .. . . ðNÞ N CðN, mÞ Uð1Þ

where

ð44Þ

where

h iT ð1Þ ðNÞ ðNÞ ½Gi ¼ Gð1Þ . . . G . . . G . . . G ði, 1Þ ði, Cð1, mÞ Þ ði, 1Þ ði, CðN, mÞ Þ

h iT mÞ ð j!Þ : Yi ð j!Þ ¼ Y1i ð j!Þ . . . YCðN, i

ð40Þ

ð45Þ

Moreover, defining 2 6 6 AU1,..., CðN, mÞ ð j!Þ ¼ 6 4

ð1Þ 1 Uð1Þ

ð1Þ 1 UðC ð1, mÞ Þ

ðNÞ N 1 Uð1Þ

ðNÞ N 1 UðCðN, mÞ Þ

.. .

.. .

.. .

.. .

.. .

.. .

.. .

ð1Þ CðN, mÞ Uð1Þ

ð1Þ CðN, mÞ UðC ð1, mÞ Þ

ðNÞ N CðN, mÞ Uð1Þ

ðNÞ N CðN, mÞ UðCðN, mÞ Þ

3 7 7 7 5

ð46Þ

849

Functions for multi-input non-linear Volterra systems yields Yi ð j!Þ ¼ AU1,..., CðN, mÞ ð j!Þ½Gi

ð47Þ

ð1Þ From equation (47), ½Gi ¼ ½Gði,ð1Þ 1Þ . . . Gði, Cð1, mÞ Þ . . . ðNÞ ðNÞ T Gði, 1Þ . . . Gði, CðN, mÞ Þ can then be determined using an Least Square based approach to yield T i1 h 1,..., CðN, mÞ Gi ¼ AU ð j!Þ AU1,..., CðN, mÞ ð j!Þ T AU1,..., CðN, mÞ ð j!Þ Yi ð j!Þ: ð48Þ

From equation (35), it is known that the number of NOFRF terms will increase with the number of system inputs. For instance, a single input non-linear system (m ¼ 1) with up to 4th order non-linearity (N ¼ 4) has only 4 NOFRF terms; however, a nonlinear system of N ¼ 4 and m ¼ 2 will have 14 NOFRF terms. This implies that 14 different signal excitations are needed to generate the data of the output spectra to estimate these NOFRFs.

4. Energy transfer phenomena of a multi-input non-linear system In this section, the concept of NOFRFs for multi-input non-linear systems is applied to investigate the energy transfer phenomena in a 2-DOF non-linear system (Worden et al. 1997). The differential equation of the considered non-linear system is given by

where y1(t), y2(t) are the two outputs of the system, m1, m2, c11, c12, c22, c2, c3, k11, k12, k22, k2, k3 are the system parameters: mass, damping and stiffness respectively. The non-linear system can be illustrated as a mechanical oscillator shown in figure 3. In the following study, the values of all the parameters used are m1 ¼ m2 ¼ 1 kg, c11 ¼ c12 ¼ c22 ¼ 20 N/m/s, c2 ¼ 1 500 N(m/s)2, c3 ¼ 1 104 N(m/s)3, k11 ¼ k12 ¼ k2 ¼ 1 107 N/m2, k3 ¼ 5 109 k22 ¼ 1 104 N/m, 3 N/m , and the two input excitations are u1 ðtÞ ¼

u2 ðtÞ ¼

3 sinð2 35 tÞ sinð2 10 tÞ 2 t 10 sec t 10 sec ð50Þ 3 sinð2 100 tÞ sinð2 85 tÞ 2 t 10 sec t 10 sec : ð51Þ

The frequency ranges of the first input and the second input are [10 35] [ [10 35] Hz and [85 100] [ [85 100] Hz respectively. These spectra are shown in figure 4. According to equation (27), it can be ð2Þ ð j!Þ shown that the frequency range of UðP 1 ¼2, P2 ¼0Þ is [2 10 2 35] [ [10 35 35 10] [ [2 352 10] ¼ [70 70] Hz. Similarly, it can be deduced that

k22

m1 y€1 ðtÞ þ ðc11 þ c12 Þy_1 ðtÞ c12 y_ 2 ðtÞ þ ðk11 þ k12 Þy1 ðtÞ

k3, k2, k11

k12

k12 y2 ðtÞ þ c2 ðy_1 ðtÞ y_ 2 ðtÞÞ2 þ c3 ðy_1 ðtÞ y_ 2 ðtÞÞ3 m2

þ k2 y21 ðtÞ þ k3 y31 ðtÞ ¼ u1 ðtÞ m2 y€2 ðtÞ þ ðc12 þ c22 Þy_2 ðtÞ c12 y_ 1 ðtÞ þ ðk12 þ k22 Þy2 ðtÞ c22

k12 y1 ðtÞ c2 ðy_1 ðtÞ y_ 2 ðtÞÞ2 c3 ðy_ 1 ðtÞ y_2 ðtÞÞ3 ¼ u2 ðtÞ,

ð49Þ

c3, c2, c12

Figure 3.

U1

c11

A 2-DOF non-linear system.

U2 0.15

Amplitude / m

0.15

Amplitude / m

m1

0.1

0.05

0

0.1

0.05

0 0

10

20

30

Frequency / Hz

Figure 4.

40

50

0

50

100

Frequency / Hz

The spectra of the two inputs for the system in equation (49).

150

850

Z. K. Peng et al.

ð2Þ the frequency range of UðP ð j!Þ is [200170] [ 1 ¼0, P2 ¼2Þ [15 15] [ [170 200] Hz. According to equation (29), it can be shown that the frequency range of ð2Þ ð j!Þ is [135 95] [ [90 50] [ [50 90] [ UðP 1 ¼1, P2 ¼1Þ [95 135] Hz. These results are verified by the spectra of u22 ðtÞ and u1 ðtÞu2 ðtÞ shown in figure 5. Furthermore, using equation (31), the possible frequency range of the output can be calculated to be [200 170] [ [135 135] [ [170 200] Hz. The forced response of the system is obtained through integrating equation (49) using a fourth-order Runge-Kutta method, and the results over (3 t 3) are shown in figure 6. The initial conditions are y1(0) ¼ y2(0) ¼ 0 and y_1 ð0Þ ¼ y_ 2 ð0Þ ¼ 0. Figure 7 shows the spectra of the outputs, which clearly indicate the two outputs have the same frequency range over [0 135] [ [170 200] Hz, and this frequency range is the same as that determined using the analysis result by equation (30). From figure 7, it can be seen that considerable input energy is transferred by the system from the input frequency band [10 35] [ [85 100] Hz to the other frequency ranges [0 10) [ (35 70] Hz.

The NOFRFs of system (49) under the excitation (50) and (51) have been evaluated up to second order over the frequency range [0 135] [ [170 200] Hz. According to equation (36), to evaluate the NOFRFs of a 2-DOF non-linear system up to the second order, generally, five different signal excitations are needed. However, ð1Þ from the frequency ranges of Uð1Þ ðP1 ¼1Þ ð j!Þ, UðP2 ¼1Þ ð j!Þ, ð2Þ ð2Þ ð2Þ UðP1 ¼2, P2 ¼0Þ ð j!Þ, UðP1 ¼0, P2 ¼2Þ ð j!Þ and UðP1 ¼1, P2 ¼1Þ ð j!Þ, the output frequency responses in equation (32) can be simplified as ð2Þ ð j!Þ yi ð j!Þ ¼ Gði,ð2ÞP1 ¼2, P2 ¼0Þ ð j!ÞUðP 1 ¼2, P2 ¼0Þ ð2Þ þ Gði,ð2ÞP1 ¼0, P2 ¼2Þ ð j!ÞUðP ð j!Þ 1 ¼0, P2 ¼2Þ

! 2 ½0 10 Hz yi ð j!Þ

yi ð j!Þ

U21

U22 6

10

5

8

Amplitude / m

Amplitude / m

ðaÞ

ð1Þ ð2Þ ¼ Gð1Þ ði, P1 ¼1Þ ð j!ÞUðP1 ¼1Þ ð j!Þ þ Gði, P1 ¼2, P2 ¼0Þ ð ð2Þ UðP ð j!Þ þ Gði,ð2ÞP1 ¼0, P2 ¼2Þ ð j!Þ 1 ¼2, P2 ¼0Þ ð2Þ UðP ð j!Þ ! 2 ½10 15 Hz 1 ¼0, P2 ¼2Þ ð1Þ ð2Þ ¼ Gði, P1 ¼1Þ ð j!ÞUð1Þ ðP1 ¼1Þ ð j!Þ þ Gði, P1 ¼2, P2 ¼0Þ ð ð2Þ UðP ð j!Þ ! 2 ð15 35 Hz 1 ¼2, P2 ¼0Þ

6 4 2

4 3 2 1

0

0 0

20

40

60

80

0

50

Frequency / Hz

100

150

200

250

Frequency / Hz U1U2

3

Amplitude / m

2.5 2 1.5 1 0.5 0 0

50

100

150

200

Frequency / Hz

Figure 5.

The spectra of u21 ðtÞ, u22 ðtÞ and u1 ðtÞu2 ðtÞ for the system in equation (49).

j!Þ

ðbÞ j!Þ ðcÞ

851

Functions for multi-input non-linear Volterra systems ð2Þ yi ð j!Þ ¼ Gði,ð2ÞP1 ¼2, P2 ¼0Þ ð j!ÞUðP ð j!Þ 1 ¼2, P2 ¼0Þ

ð1Þ yi ð j!Þ ¼ Gð1Þ ði, P2 ¼1Þ ð j!ÞUðP2 ¼1Þ ð j!Þ ! 2 ð90 95Þ Hz

! 2 ð35 50Þ Hz yi ð j!Þ

ðdÞ

ð2Þ ¼ Gði,ð2ÞP1 ¼2, P2 ¼0Þ ð j!ÞUðP ð j!Þ 1 ¼2, P2 ¼0Þ ð2Þ ð2Þ þ Gði, P1 ¼1, P2 ¼1Þ ð j!ÞUðP1 ¼1, P2 ¼1Þ ð j!Þ

ð2Þ j!ÞUðP ð 1 ¼1, P2 ¼1Þ

j!Þ

! 2 ð70 85Þ [ ð100 135Þ Hz

ðfÞ

ð1Þ ð2Þ Gð1Þ ði, P2 ¼1Þ ð j!ÞUðP2 ¼1Þ ð j!Þ þ Gði, P1 ¼1, P2 ¼1Þ ð j!Þ ð2Þ UðP ð j!Þ ! 2 ½85 90 [ ½95 100 Hz 1 ¼1, P2 ¼1Þ

ðgÞ

15

ðiÞ

i ¼ 1, 2:

ð52Þ

Equation (52) indicates that, to estimate the NOFRFs up to the second order, three different excitations are enough. Equations (52(a)–(i)) also clearly show how the energy transfer happens in the non-linear system (49) when subjected to the inputs (50) and (51). For example, from equation (52(a)), it is clearly that it is the 2nd order ð2Þ ð2Þ NOFRFs Gð1, P1 ¼2, P2 ¼0Þ ð j!Þ, Gð1, P1 ¼0, P2 ¼2Þ ð j!Þ which transfer the energy from the frequency bands of the

× 10−3

y1 / m

10 5 0 −5 −3

−2

−1

0

1

2

3

1

2

3

Time / Sec 0.01

y2 / m

0.005 0 −0.005 −0.01 −3

−2

−1

0 Time / Sec

Figure 6.

7

The output response of the system in equation (49).

Spectrum of y1

× 10−5

Spectrum of y2

× 10−5 7

6

6

5

Amplitude / m

Amplitude / m

yi ð j!Þ ¼

for

ðeÞ

yi ð j!Þ ¼ Gði,ð2ÞP1 ¼1, P2 ¼1Þ ð

ð2Þ yi ð j!Þ ¼ Gði,ð2ÞP1 ¼0, P2 ¼2Þ ð j!ÞUðP ð j!Þ 1 ¼0, P2 ¼2Þ

! 2 ½170 200 Hz

! 2 ½50 70 Hz

4 3

5 4 3

2

2

1

1

0

0 0

50

100 150 Frequency / Hz

Figure 7.

200

250

ðhÞ

0

50


The output spectra of the system in equation (49).

200

250

852

Z. K. Peng et al. × 10−4

× 10−7

G1110

2.5 1st NOFRF Input: U1 For Output y1

1.5

1st NOFRF Input: U2 For Output y1

1.5 Amplitude / m

2 Amplitude / m

G1 10 1

2

1

1

0.5

0.5 0

0 0

10

20 30 40 Frequency / Hz

50

0

G1210

× 10−4

50

100 Frequency / Hz

150

G1201

× 10−6 3.5 1st NOFRF Input: U1 For Output y2

1st NOFRF Input: U2 For Output y2

3 Amplitude / m

Amplitude / m

2

1

2.5 2 1.5 1 0.5

0

0

10


Figure 8.

40

50

ð2Þ Gð1, P1 ¼0, P2 ¼2Þ ð

ð2Þ ð2Þ Gð1, P1 ¼2, P2 ¼0Þ ð j!ÞUðP1 ¼2, P2 ¼0Þ ð j!Þ, ð2Þ j!ÞUðP ð 1 ¼0, P2 ¼2Þ

0

50

100 Frequency / Hz

150

ð1Þ ð1Þ The first order NOFRFs Gði, P1 ¼1Þ ðj!Þ, GðP2 ¼1Þ ðj!Þ (i ¼ 1, 2).

first input ([10 35] Hz) and the second input ([85 100] Hz) respectively to the frequency band [0 10) Hz in the output. The evaluated NOFRFs are shown in figures 8 and 9. As the spectra in figure 7 show, most of the output energy is located in the frequency range [0 70] Hz. From figures 8 and 9, it can be seen that, in the frequency range of [0 70] Hz, the first input dependent ð2Þ NOFRFs, such as Gð1Þ ði, P1 ¼1Þ ð j!Þ and Gði, P1 ¼2, P2 ¼0Þ ð j!Þ, are much bigger than the other NOFRFs. This implies that the output energy in this frequency range is mainly contributed by the first input. For example, according to equation (52(b)), the first output response at 14 Hz is contributed by the three terms Gð1Þ ð1, P1 ¼1Þ ð j!Þ Uð1Þ ðP1 ¼1Þ ð j!Þ,

0

and

j!Þ. The contributions from these terms to the output are given in table 1. From table 1, it can be seen that the contribution of ð2Þ ð2Þ Gð1, P1 ¼0, P2 ¼2Þ ð j!ÞUðP1 ¼0, P2 ¼2Þ ð j!Þ is so small that it can

be ignored. The output response at other frequencies can be analysed in a similar way. Comparing equations (14) and (18), the main difference between the NOFRFs of the single-input and the multi-input non-linear systems is that the multi-input NOFRFs have more cross-NOFRF terms, for instance, Gði,ð2ÞP1 ¼1, P2 ¼1Þ ð j!Þ, (i ¼ 1, 2) for the second order NOFRF. From equations (52(e–g)), it can be seen that, in this study, Gði,ð2ÞP1 ¼1, P2 ¼1Þ ð j!Þ, (i ¼ 1, 2) only influence the components at [50 90] [ [95 135] Hz. Equation (52(f)) also indicates that the output responses at (70 85) [ (100 135) Hz are only determined by Gði,ð2ÞP1 ¼1, P2 ¼1Þ ð j!Þ, (i ¼ 1, 2) and have a very small amplitude. At other frequency ranges, Gði,ð2ÞP1 ¼1, P2 ¼1Þ ð j!Þ, (i ¼ 1, 2) will influence the output response together with other NOFRF terms, for example with Gði,ð2ÞP1 ¼2, P2 ¼0Þ ð j!Þ, (i ¼ 1, 2) at [50 70] Hz. For the first output response at 55 Hz, ð2Þ the contributions by Gð1, and P1 ¼2, P2 ¼0Þ ð j!Þ ð2Þ Gð1, P1 ¼1, P2 ¼1Þ ð j!Þ are given in below table 2.

853

Functions for multi-input non-linear Volterra systems × 10−5

G2120 8 2nd NOFRF Input: U12 For Output y1

1

Amplitude / m

Amplitude / m

1.5

× 10−7

G2102 2nd NOFRF Input: U12 For Output y1

6

4

0.5 2

0

0 0

20

40

60

80

0

50

× 10−5

G2220 8

1.4 2nd NOFRF Input: U12 For Output y2

1

150

× 10−7

200

250

G2202 2nd NOFRF Input: U22 For Output y2

6 Amplitude / m

Amplitude / m

1.2

100

Frequency / Hz

Frequency / Hz

0.8 0.6 0.4

4

2

0.2 0

0 0

20

40

60

0

80

50

× 10−7

0.8 0.6 0.4 0.2

200

250

G2211 2nd NOFRF Input: U1U2 For Output y2

0.8 0.6 0.4 0.2

0

0 0

50

100

150

200

Frequency / Hz

Figure 9.

150

1

2nd NOFRF Input: U1U2 For Output y1

Amplitude / m

Amplitude / m

× 10−7

G2111

1

100

Frequency / Hz

Frequency / Hz

0

50

100

150

200

Frequency / Hz

ð2Þ ð2Þ The second order NOFRFs jGð2Þ ði, P1 ¼2, P2 ¼0Þ ðj!Þj, jGði, P1 ¼0, P2 ¼2Þ ðj!Þj and jGði, P1 ¼1, P2 ¼1Þ ðj!Þj, (i ¼ 1,2).

The results in table 2 show that, compared ð2Þ ð2Þ with Gð1, P1 ¼2, P2 ¼0Þ ð j!ÞUðP1 ¼2, P2 ¼0Þ ð j!Þ, the contribution ð2Þ ð2Þ of Gð1, P1 ¼1, P2 ¼1Þ ð j!ÞUðP1 ¼1, P2 ¼1Þ ð j!Þ to the output response at 55 Hz is very small and can be ignored. Similarly, it can be found that the contribution of the cross-NOFRF to the output responses at other frequencies is also very small. To a certain degree, this implies

that the influence of the cross-NOFRFs on the output responses can be ignored in this specific case. The results shown in figures 8 and 9 indicate that the maximum gains in the NOFRFs of jGði,ð2ÞP1 ¼2, P2 ¼0Þ ð j!Þj appear near 16 Hz and 28 Hz, (i ¼ 1, 2). This means that, at these frequencies, the energy transfer through these NOFRFs becomes more efficient, and the frequency

854

Z. K. Peng et al. Table 1.

The contributions of different terms to the output response at frequency of 14 Hz.

Terms

Values

Modulus

(0.1496 þ 0.0066i) (1.5899e-004-1.1215e-004i)

2.9135e-005

ð2Þ ð2Þ Gð1, P1 ¼2, P2 ¼0Þ ðj!ÞUðP1 ¼2, P2 ¼0Þ ðj!Þ

(4.9330 þ 0.2171i) (6.0092e-006 þ 5.1081e-006i)

3.8944e-005

ð2Þ ð2Þ Gð1, P1 ¼0, P2 ¼2Þ ðj!ÞUðP1 ¼0, P2 ¼2Þ ðj!Þ

(0.3888 þ 0.0171i) (4.8131e-007-3.9035e-007i)

2.4117e-007

ð1Þ ð1Þ Gð1, P1 ¼1Þ ðj!ÞUðP1 ¼1Þ ðj!Þ

Table 2.

The contributions of different terms to the output response at frequency of 55 Hz.

Terms ð2Þ ð2Þ Gð1, P1 ¼2, P2 ¼0Þ ðj!ÞUðP1 ¼2, P2 ¼0Þ ðj!Þ ð2Þ ð2Þ Gð1, P1 ¼1, P2 ¼1Þ ðj!ÞUðP1 ¼1, P2 ¼1Þ ðj!Þ

Values

Modulus

(3.3131 þ 0.5782i) (9.0729e-007 þ 8.2225e-008i)

3.0639e-006

(1.0498 þ 0.1832i) (8.5484e-008-6.8922e-008i)

1.1702e-007

components at these frequencies will become significant in the output spectra. This can be confirmed by the output spectra shown in figure 7 where some significant components can be found at these frequencies. The above qualitative analysis gives a clear interpretation regarding why and how the generation of new frequencies happens in a multi-input non-linear system, and extends the procedure for the same analysis for single-input non-linear systems to the more general multi-input non-linear system case.

5. Conclusions and remarks The Volterra series is a powerful tool that can be used to describe a wide class of non-linear systems which are stable at zero equilibrium. Generally, this class of nonlinear systems can describe the non-linear phenomena including secondary resonance, the generation of superharmonics, and inter-modulations, et al., but excluding the chaos and bifurcation. In our previous study Lang and Billings (2005), based on the Volterra series, a new concept called NOFRF was proposed to study the energy transfer properties of non-linear systems in the frequency domain. In the present study, the concept of NOFRFs has been extended from the single-input nonlinear system case to the multi-input non-linear system case. Given the frequency range of the inputs, a new method was also developed to determine the output frequency range. The phenomenon of energy transfer (Nayfeh and Mook 1979) in a 2DOF non-linear system subjected to two input excitations was

investigated using the concept of NOFRFs for multi-input systems. Multi-input systems are important in many engineering systems and structures. For example, multi-degree of freedom mechanical structures are a typical example of this category of systems. Therefore, the extension of the NOFRF concept to the more general multi-input case of non-linear systems is important for the potential applications of the NOFRF concept to a wide range of engineering areas.

Acknowledgements The authors gratefully acknowledge the support of the Engineering and Physical Science Research Council, UK, for this work.

References S.A. Billings and J.C. Peyton Jones, ‘‘Mapping nonlinear integrodifferential equations into the frequency domain’’, Int. J. Cont., 52, pp. 863–879, 1990. S.A. Billings and K.M. Tsang, ‘‘Spectral analysis for nonlinear system, part I: parametric non-linear spectral analysis’’, Mech. Sys. Sig. Process., 3, pp. 319–339, 1989. L.Z. Guo and S.A. Billings, ‘‘A comparison of polynomial and wavelet expansions for the identification of chaotic coupled map lattices’’, Int. J. Bifurcat. Chaos, 9, pp. 2927–2938, 2005. Z.Q. Lang and S.A. Billings, ‘‘Output frequency characteristics of nonlinear system’’, Int. J. Cont., 64, pp. 1049–1067, 1996. Z.Q. Lang and S.A. Billings, ‘‘Energy transfer properties of nonlinear systems in the frequency domain’’, Int. J. Cont., 78, pp. 354–362, 2005. A.H. Nayfeh and D.T. Mook, Nonlin. Oscillat, New York: John Wiley and Sons, 1979.

Functions for multi-input non-linear Volterra systems R.K. Pearson, Discrete Time Dynamic Models, New York: Oxford University Press, 1994. Z.K. Peng, Z.Q. Lang and S.A. Billings, ‘‘Resonances and resonant frequencies for a class of nonlinear systems’’, J. Sound and Vibration, 300, pp. 993–1014, 2007. J.C. Peyton Jones and S.A. Billings, ‘‘A recursive algorithm for the computing the frequency response of a class of nonlinear difference equation models’’, Int. J. Cont., 50, pp. 1925–1940, 1989. J.A. Vazquez Feijoo, K. Worden and R. Stanway, ‘‘System identification using associated linear equations’’, Mech. Sys. Sig. Process., 18, pp. 431–455, 2004.

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J.A. Vazquez Feijoo, K. Worden and R. Stanway, ‘‘Associated linear equations for Volterra operators’’, Mech. Sys. Sig. Process., 19, pp. 57–69, 2005. K. Worden, G. Manson and G.R. Tomlinson, ‘‘A harmonic probing algorithm for the multi-input Volterra series’’, J. Sound and Vibration, 201, pp. 67–84, 1997. H. Zhang and S.A. Billings, ‘‘Analysing non-linear systems in the frequency domain, I: the transfer function’’, Mech. Sys. Sig. Process., 7, pp. 531–550, 1993. H. Zhang and S.A. Billings, ‘‘Analysing nonlinear systems in the frequency domain, II: the phase response’’, Mech. Sys. Sig. Process., 8, pp. 45–62, 1994.