518
14 Nonlinear Vibration
Equation (i) is written for free vibration as
¨
y + p
2
0 y + hy
3
= 0
( j )
where p 0 =
√
2T 0 /Lm and h =
AE
mL 3 .
Let the solution be assumed as
y(t) = y 0 (t) + hy 1 (t) + h
2 y 2 (t) + · · ·
(k)
where y 0 (t) is the generating solution, h is the perturbation parameter and y 1 (t),
y 2 (t), etc., are added corrective terms.
The frequency is perturbed as follows:
p
2
= p
2
0 + hα 1 + h
2
α 2 + · · ·
p
2
0 = p
2
− hα 1 − h
2
α 2 − · · ·
(l)
Substituting Eqs. (k) and (l) into Eq. (j), we get
( ¨
y 0 + h ¨
y 1 + h
2
¨
y 2 + · · · ) + ( p
2
− hα 1 − h
2
α 2 − · · · )
× (y 0 + hy 1 + h
2 y 2 + · · · ) + h(y 0 + hy 1 + h
2 y 2 + · · · )
3
= 0
(m)
The coefficients of various powers of h are equated to zero which results in the
following system of equations.
¨
y 0 + p
2 y 0 = 0
( n )
¨
y 1 + p
2 y 1 = α 1 y 0 − y
3
0
. . .
(o)
The generating solution of Eq. (n) is
y 0 = A cos pt + B sin pt
(p)
where A and B are constants dependent on the initial conditions.
Let at t = 0, y 0 = 0, and ˙
y 0 = A
Then
y 0 = A/ p sin pt
(q)
We substitute the value of y 0 from Eq. (q) into Eq. (o), which gives
¨
y 1 + p
2 y 1 = α 1
A
p
sin pt −
A
3
p 3 sin
3 pt