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

else
CEðHnðjo1; ...; jonÞÞ ¼ 0,
which can be summarized as
CEðHnðjo1; ...; jonÞÞ ¼ c
ðn1Þ= pþq 1 ð Þ d
n 1
p þ q 1
n 1
p þ q 1
Proof. The results are directly followed from Propositions 1 and 2. &
ð1dðpÞposðn qÞÞ.
(13)
Corollary 1 provides a more special case of nonlinear system (1) or (2). There are only several nonlinear
parameters of the same nonlinear type and degree in the considered system. This result will be used in the example
of Section 5. The following results can be achieved for the output frequency response.
Corollary 2. Consider only the nonlinear parameter Cp,q ¼ c. The parametric characteristic vector of the output
spectrum in (6) with respect to the parameter c can be written as
N
N 1
pþq 1 dðpÞ posðN qÞd
N 1
pþq 1
N 1
pþq 1 .
c ¼ CEðYðjoÞÞ ¼ CEðHnð ÞÞ ¼
n¼1
1; c; c 2 ; ...; c
If all the other degree and type of nonlinear parameters are zero except that Cp,q ¼ c is non-zero (p+q41).
Then the parametric characteristic vector of the output spectrum in (6) with respect to the parameter c is
if p ¼ 0
c ¼ CEðYðjoÞÞ ¼ 1 CEðHqð ÞÞð1 posðq NÞÞ ¼ ½1cð1posðq NÞÞŠ,
else
bðN c ¼ CEðYðjoÞÞ ¼
1Þ=
ðpþq1Þc CEðHðpþq 1Þiþ1ð ÞÞ ¼ 1; c; c 2 h
ðN ; ...; cb 1Þ=
ðpþq i
1Þc
.
i¼0
Proof. The results are straightforward from Proposition 2 and Corollary 1. &
The results above involve the computation of c n .Ifc is an I-dimension vector, there will be many repetitive
terms involved in c n . To simplify the computation, the following lemma can be used.
Lemma 1. Let be c ¼ [c1,c2,y,cI] which can also be denoted by c [1:I], and cn ¼ c
Kronecker product defined in Jing et al. [24], nX1 and IX1. Then
c c,‘‘
’’ is the reduced
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
n
c n ¼ c n 1 c1; ...; c n 1 ½sð1Þn sðiÞn þ 1 : sð1ÞnŠci; ...; c n 1 ½sð1ÞnŠcI ,
where sðiÞn ¼ PI
sðjÞn 1, s(.) 1 ¼ 1, and 1pipI. Moreover, DIM(c
j¼i
n n n
) ¼ s(1) n+1, and the location of ci in c is
s(1)n+1–s(i)n+1+1.
Proof. See Appendix C. &
4.2. Analysis based on the OFRF with respect to Cp,q
ARTICLE IN PRESS
X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 109
Based on the result in Corollary 2 and Eq. (6), with respect to a specific parameter c, the OFRF can be
written as
YðjoÞ ¼ ¯j 0ðjoÞþc ¯j 1ðjoÞþc 2 j2ðjoÞþ þc ‘ j ‘ðjoÞþ . (14a)
Since Y(jo) is also a function of c, therefore, Eq. (14a) is rewritten more clearly as
Yðjo; cÞ ¼ ¯j 0ðjoÞþc ¯j 1ðjoÞþc 2 j2ðjoÞþ þc ‘ j ‘ðjoÞþ . (14b)
Y(jo;c) is in fact a series of an infinite order, ‘ is a positive integer which can be determined by Corollary 2,
¯j iðjoÞ is the complex valued function corresponding to the coefficient c i in Eq. (6), which can be obtained by
following the method in Process C. If all the other degree and type of nonlinear parameters are zero