PSI - Issue 14

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

235

2

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

1. Volterra series approximation For a Single Input Single Output (SISO) non-linear system, with f(t) and x(t) as the input and output respectively, the response of the system using Volterra series can be expressed as

( ) 1 x t = x t n n= ¥ å

(1)

( );

n x (t) is given by

Where n th order response component

' '  time s ¥ ¥ ò ò ¥ ¥

( , ., ) ( ) .. (

) … f t - τ d τ …d τ

x n (t)= …… - - n

h τ … τ f t - τ

(2)

n

n

n

n

1

1

1

The first term in the series is the impulse response function of a linear system in time domain or called as frequency response function (FRF) in frequency domain and h n (τ 1 ,…,τ n ) is the n th order Volterra kernel and its Fourier transform provides the n th order frequency response function (FRF) or Volterra kernel transform as

n -j ω τ i i

' '  n times ¥ ¥ ò ò ¥ ¥ - -

( , ., H ω … ω = …… h τ … τ ) ( , ., )

. d τ …d τ

e

(3)

n

n

n

n

n



1

1

1

1

i=

The discrete-time counterpart of the continuous time domain single degree of freedom (SDOF) given in Eqn. (1) can be written as [ ] [ ] [ ] [ ] [ ] [ ] M M M 1 1 2 x n = h + h m f n - m + h m , m f n - m f n - m + ... å å å

0

1 1

1

2 1 2

1

2

m = 0 1

m = 0 m = 0

1

2

(4)

M M Mp 1 2 m = 0 m = 0 mn = 0 1 2 å å å

é ê ë

ù ú û

[

] [

]

[

]

+

h m , m , ..., m f n - m f n - m ... f n - m + ...) p p n 1 2 1 2

where h p [m l , m 2, ..., m p ] is known as the p

th order Volterra kernel of the system and M

1 , M 2 ,…, M p are the memory

lengths of the filters. It may be noted that the Volterra kernels are always symmetric in nature. Usually, infinite numbers of Volterra kernels are required to describe the continuous nonlinear system. However, due to practical consideration, only finite numbers of kernels are considered. The accuracy of the response estimated using discrete Volterra series depends on the order ‘p’ and the memory depth chosen. Various experimental and theoretical methods have been proposed in the literature (Tomilson et. al., 1996; Billings et. al., 1999, 2001) to find the truncation order of nonlinear system both in time and frequency domain Volterra kernels. Response Spectrum Map (Billings et. al., 1999) is used in the present work as it is the most effective method as the truncation error associated with a finite Volterra series representation in this approach is a function of both input amplitude and response harmonics (which in turn depends on the excitation frequency).From the response spectrum map, for the given excitation amplitude, the harmonics (either sub or super), that significantly contributes to the system response can be easily identified. It may be appropriate to point out that the kernel order also depends on the type of nonlinearity (symmetric or anti-symmetric) present in the system. Hence, the kernel order thus obtained from response spectrum map can be further fine-tuned by performing higher order spectral analysis (HOSA) (Hickey et. al., 2009 ). For example, a nonlinear system containing an anti-symmetric nonlinearity can be modelled using a finite sum of odd and even ordered Volterra operators, while nonlinear system containing symmetric nonlinearities can be modelled using a finite sum of odd-ordered Volterra operators only. Once the order of the system is chosen, the next step is to select appropriate memory depth. Since Volterra kernels of fading memory systems decay to zero in a finite period of time, the output depends only on the recent inputs. Therefore, the memory depth can be identified by convergence