PSI - Issue 14

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

239

6

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



ˆ

 mx(t)+cx(t)+ g[x(t)] = f(t) with g[x(t)] 2 

 3 4 = g +c kx +c kx +c kx +c kx ; x(0)= x(0)= 0 4 0 1 2 3

(19)

Where g 0 indicates the constant term, k indicates the uncracked stiffness and c 1 kx represents the linear component and the other high order terms indicate the nonlinear components of the approximated polynomial nonlinear system. In practice, the term g 0 given in Eqn. (4.31) can be neglected when no static force is applied to the structure. The coefficients, c 1 , c 2 , c 3 , c 4 are evaluated by minimizing the error function E between the actual and approximated polynomial nonlinear system using the above-mentioned Weierstrass approximation theorem in the domain [-X, X] as

( 1 2 3 4 E c ,c ,c ,c )= g x t -g x t X ò{

E

¶

2

ˆ [ ( )] [ ( )]} ; dx

(20)

0,

fo r i=1,2,3,4

=

c i

¶

-X

By applying the above equation, the coefficients are obtained as

(1+ α )

105(1 - α )

105(1 - α )

c = , c = -

, c = 0, c 4 =

1

2

3

3

(21)

2

128X

256X

The accuracy of the approximated polynomial model is already evaluated by the authors and more details can be found in Prawin et.al.,2016 &2018, Peng et.al., 2007. These details are avoided in this paper as it deviates from the scope of the present work.

2x10 ‐4

2.5x10 ‐15

2.0x10 ‐15

1x10 ‐4

1.5x10 ‐15

0

1.0x10 ‐15

‐1x10 ‐4 Response(m)

5.0x10 ‐16 PSD Amplitude

‐2x10 ‐4

20 40 60 80 100 120 140 0.0

0.00 0.20 0.40 0.60 0.80 1.00

Frequency(Hz)

Time (sec)

(a)

(b)

Fig. 2. (a) Displacement time history response; (b) Power spectrum of breathing crack

The actual and approximated polynomial of the considered example is given below r    x(t)+ 23.5619x(t)+ 3.5e4 α x(t) = 1cos(188.414t); α = 0.75; x(0) = x(0) = 0

(22)

 3 4 x(t)+ 23.5619x(t)+3.5e04k x(t)+3.5e04k x (t)+3.5e04k x (t)+3.5e04k x (t) = 1cos(188.414t); x(0) = x(0) = 0 4 1 2 3   2

(23)

The displacement time history response obtained using Runge-Kutta (RK) integration scheme and the corresponding power spectrum (PSD) of the actual system are shown in Figure 2. It can be observed from Figure.2 (b) that the spectrum exhibits peaks at the excitation frequency and at second and fourth order super-harmonics.