PSI - Issue 14

J. Prawin et al. / Procedia Structural Integrity 14 (2019) 234–241 J. Prawin et.al.,/ Structural Integrity Procedia 00 (2018) 000–000

238

5

( ) T H n = h 0 , h 1 ,....,h M - 1 ,h 0,0 ,h 0,1 ,....h M - 1,....,M - 1 p 1 1 1 2 2 é ù ê ú ë û ( ) ( ) ( ) ( ) ( ) ( )

(17)

Once the Volterra kernel/model coefficients of the total response at each degree of freedom are computed using the proposed adaptive LMS Volterra model, the various order components at each degree of freedom can be easily determined. It is also possible to characterize (classification/type) the nonlinearity present in the system using the structure of the response components (Chatterjee et. al., 2001). The Volterra series expansion for the polynomial type of nonlinearity present in the system exhibits a unique structure i.e., the odd harmonics appear only in odd-order response components and even harmonics appear only in even order response components. If this structured nature is absent in the extracted Volterra components, it can be easily concluded that the type of nonlinearity present in the system is non-polynomial. The absence of even harmonics in the extracted response components confirms the symmetric polynomial form of nonlinearity and vice versa (Chatterjee et. al., 2001). 3. Numerical Investigation The numerical example considered is a beam with a breathing crack and shown in Figure1. In the present work, the bilinear nature of the breathing crack problem is first converted into an equivalent polynomial model and then the individual components are estimated using the proposed adaptive Volterra filter approach. Crack P

L

Fig. 1. Breathing crack problem.

The equation of motion (bilinear oscillator) of the beam with a breathing crack can be written as

ìïï íï ïî

0

x α kx

³

( ) ( ) [ ( )] ( );   mx t +cx t +g x t =f t

( )

;

g x

=

 x(0)=x(0)=0

(18)

0

kx x<

Where g ( x ) is the restoring force and m and c are the mass and damping respectively; x(t) is the displacement; k is the stiffness; α is known as the stiffness ratio (0 ≤ α ≤ 1), f(t) is the external force exciting the system. The stiffness ratio can be viewed as the ratio of squares of the cracked frequency to uncracked frequency. For a polynomial approximation of nonlinearity, the Weierstrass approximation theorem (Jeffreys et. al., 1998) is used for idealization. The Weierstrass approximation theorem states that: If f(x) is a continuous real-valued function on [a, b] and if any ε > 0 is given, then there exists a polynomial P(x) on [a, b] such that |f(x)-P(x)|< ε for all x ε [a, b]. Since the restoring force g(x) is a continuous function of displacement x, it can be well approximated by a polynomial. The approximated polynomial type nonlinear system with assumed fourth order polynomial restoring force for the bilinear oscillator using Weierstrass approximation theorem (Jeffreys et. al., 1998) is given by