Page 484 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 14.5 Perturbation Method 471

development in the neighborhood of the solution of the linearized problem. If the
solution of the linearized problem is periodic, and if ¡jl is small, we can expect the
perturbed solution to be periodic also. We can reason from the phase plane that
the periodic solution must represent a closed trajectory. The period, which
depends on the initial conditions, is then a function of the amplitude of vibration.
Consider the free oscillation of a mass on a nonlinear spring, which is defined
by the equation
X + (o^x + ixx^ = 0 (14.5-1)
with initial conditions x(0) = A and i(0) = 0. When ¡jl = 0, the frequency of
oscillation is that of the linear system, co^ = ^Jk/m .
We seek a solution in the form of an infinite series of the perturbation
parameter ¡jl as follows:
=-^o(0 + M-ï|(0 + + ■■■ (14.5-2)
Furthermore, we know that the frequency of the nonlinear oscillation will depend
on the amplitude of oscillation as well as on ¡jl. We express this fact also in terms
of a series in /x:
(o^ = jLtaj -h ‘ ‘ ■ (14.5-3)
where the a, are as yet undefined functions of the amplitude, and o) is the
frequency of the nonlinear oscillations.
We consider only the first two terms of Eqs. (14.5-2) and (14.5-3), which will
adequately illustrate the procedure. Substituting these into Eq. (14.5-1), we obtain
Xq + jLtij + — /xai)(X() + /xx,)-h)Lt(xQ+ 3)UJCyJc, + •'•) = 0 ( 14.5-4)
Because the perturbation parameter fi could have been chosen arbitrarily, the
coefficients of the various powers of ¡x must be equated to zero. This leads to a
system of equations that can be solved successively:
in -h iO^Xn = 0
, , (14.5-5)
i j -h oj x^ = ajX() —X5

The solution to the first equation, subject to the initial conditions jr(0) = A,
and i(0) = 0 is

jCq = A cos cot (14.5-6)
which is called the generating solution. Substituting this into the right side of the
second equation in Eq. (14.5-5), we obtain
ij + co^x^ = a^A cos cot —A^ cos^ cot

(14.5-7)
(- - M A cos cot----cos 3cot
cot = I | cos coi + |cos3a)i has been used. We note here that the
where cos^ cot =
forcing term cos cot would lead to a secular term t cos cot in the solution for Xj