PSI - Issue 14

J. Prawin et al. / Procedia Structural Integrity 14 (2019) 234–241

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J. Prawin et.al.,/ Structural Integrity Procedia 00 (2018) 000–000

coefficients in the Volterra kernels as well as in the force vector is M p . The first order input vector for a memory depth of 3 (M=3) can be written as (1) T F = [f(n) f(n - 1) f(n - 2)]

(11)

Similarly, the second order input matrix F (2) , can be obtained by multiplying the first order input vector F (1) with its transpose.

é ê ê ê ê ê ë

ù ú ú ú ú ú ú û

f 2 (n) f(n)f(n - 1)

f(n)f(n - 2)

T

(2) (1) (1) F = F * F ........ ê

2

(12)

f (n - 1) f(n - 1)f(n - 2) 2 f (n - 2) ........

........

Even though the number of coefficients in second order input force vector is M 2 =9, it reduces to 6 due to symmetry. It can be written in the vector form as (2) T 2 2 2 F = [f (n) f(n)f(n - 1) f(n)f(n - 2) f (n - 1) f(n - 1)f(n - 2) f (n - 2)] (13) The third order is generated by multiplying the second order input vector with the elements of the first order input vector. Therefore, the third order input results in M number of MxM matrices and can be written as

3 2 F * f(n) = ........ f(n)f (n - 1) f(n)f(n - 1)f(n - 2) ; 2 ........ ........ f(n)f (n - 2) ........ ........ ........ (2) 3 2 F * f(n - 1) = ........ f (n - 1) f (n - 1)f(n - 2) ........ ........ f( é ù ê ú ê ú ê ú ê ú ê ú ë û ; n - 1)f 2 (n - 2) ........ ........ ........ F (2) * f(n - 2) = ........ ........ ........ 3 ........ ........ f (n - 2) é ù ê ú ê ú ê ú ê ú ë û é ù ê ú ê ú ê ú ê ú ë û 2 2 f (n) f (n)f(n - 1) f (n)f(n - 2) (2)

(14)

Due to symmetry, the number of kernel coefficients reduces to 10 and it can be written in the vector form as

(3) T 3 2 2 2 F = [f (n) f (n)f(n - 1) f (n)f(n - 2) f(n)f (n - 1) f(n)f(n - 1)f(n - 2) f(n)f (n - 2) 3 2 2 3 f (n - 1) f (n - 1)f(n - 2) f(n - 1)f (n - 2) f (n - 2)] 2

(15)

In a generalized form, the force vector with memory depth M and order ‘p’ can be written as

T

p

( ) ( ) ( ) 2 f n = f n , f n - 1 , ...., f n - M +1 , f é (

ù ú û

) ( ) ( ) ( )

(

)

(16)

n , f n f n - 1 , ...., f

n - M +1

e

ê ë

The corresponding expanded filter coefficients vector H(n) can be written as