Output frequency response function-based analysis for nonlinear ...

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For clarity in discussion, let M ¼ 240, K ¼ 16000, and B ¼ 296, then Eq. (21) can be rewritten as
240 €x ¼ 16000x 296 _x c1 _x 2 x c2 _xx 2
c3x 3
c4 _x 3 þ uðtÞ, (22a)
y ¼ 16000 x, (22b)
Eq. (22a) is a simple case of the NDE model (1) with M ¼ 3, K ¼ 2, c10(2) ¼ 240, c10(1) ¼ 296, c10(0) ¼ 16000,
c30(111) ¼ c4, c30(110) ¼ c1, c30(100) ¼ c2, c30(000) ¼ c3, c01(0) ¼ 1, and all the other parameters are zero.
Therefore, what is interested in for this study is to analyse the effect of the nonlinear terms with coefficients c1, c2, c3 and c4 on the system output frequency response. To achieve this objective, the procedure proposed in
Section 3 are adopted to derive the OFRF of system (22), and the results in Section 4 will be used for the
computation of the parametric characteristic of the OFRF with respect to the nonlinear parameters c1, c2, c3
and c4. Moreover, though the method proposed in this paper is suitable for a general input function u(t), yet
for convenience in discussion, the input of system (22) is considered to be a sinusoidal function
u(t) ¼ 100 sin(8.lt). To illustrate the new results more clearly and conveniently, the parameter c2 under the
case of c1 ¼ c3 ¼ c4 ¼ 0 is studied firstly in detail.
(1) Determine the parametric characteristics of OFRF
Note that all the parameters of interest belong to C30, and the other degrees of nonlinear parameters are all
zero. Thus Corollary 1 and 2 can be utilised directly. Therefore,
n 1
CEðHnðjo1; ...; jonÞÞ ¼ c ð Þ=2 d
n 1
2
n 1
2
ð1dð3ÞposðnÞÞ ¼ c
bðN1Þ= ðpþq1Þc c ¼ CEðYðjoÞÞ ¼
CEðHðpþq 1Þiþ1ð ÞÞ
i¼0
¼ 1; c; c 2 h i
ðN1Þ= ðpþq1Þ ; ...; cb c ¼ 1; c; c 2 h i
ðN1Þ=2 ; ...; cb c ,
n 1 ð Þ=2 d
where c ¼ c2. To derive the detailed form for c, the largest order N should be determined first according to
Process B in Section 3.2. In order to have a larger range in which the parameters can vary, in this case let
c2A(0,10 8 ). Then the magnitude bound of Yn(jo) can be evaluated as mentioned in Process B. However, for
paper limitation, the detailed computation is omitted in this case. It can be verified that N ¼ 23 is enough for
use in this case. Therefore,
c ¼ 1; c; c 2 h i
ð23 1Þ=2
; ...; cb c
(2) Determination of F(jo) for the OFRF
Following Process C, the matrix c ¼½c T
1
¼½1; c2; c 2 2 ; c32 ; c42 ; c52 ; ...; c11 2 Š.
n 1
2
n 1
2
c T
r ŠT should be constructed first. In this case, for any 12
different values of c 2, the matrix c is a Vandermonde matrix and thus non-singular. Note that in many cases,
the parameters may be set to be some large values and cover a large range. This will make the element values in
the matrix c extraordinarily large. Then when the inverse of matrix c is computed, there may be some
computation error involved in Matlab. To overcome this problem, c can be written as
bðN1Þ= ðpþq1Þc c ¼
kCEðHðpþq 1Þiþ1ð ÞÞ=k ¼ 1; kðc=kÞ; k 2 ðc=kÞ 2 ðN1Þ=2 ; ...; kb c h i
ðN1Þ=2 ðc=kÞb c .
i¼0
Then Eq. (6) can be written as
ARTICLE IN PRESS
X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120
Yðjo; cÞ ¼cFðjoÞ T ¼ 1 ðc=kÞ ðc=kÞ 2
ðc=kÞ ‘ j1ðjoÞ; kj2ðjoÞ; ...; k ‘ j ‘ðjoÞ T , (23)
where ‘ ¼ ðN1Þ=2 . Moreover, the range for each parameter can be divided into several sub-range, and the
final result is the combination of these results obtained for each sub-range. In this study, let k ¼ 10 5 , then
¯c2 ¼ c2=k 2½0; 1000Š. Choose ¯c2 to be the following values to construct c ¼½c T
1 c T
r ŠT , i.e.,
0.1,1,50,65,80,100,150,200,250,300,350,400,450,500,550, 600,650,700,750,800,850,900,950,980,1000. The output
frequency response
h
YYðjoÞ ¼ YðjoÞ1; YðjoÞ2; ...; YðjoÞr i
,