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14 Nonlinear Vibration
Nonlinear vibration problems can be solved by a number of methods:
1. The Method of Direct Integration.
2. Graphical methods such as the Phase Plane Method , Pell’s Method and Isoclinic
Method.
3. Analytical Methods, such as the Method of Iteration, the Method of Perturbation, the Method of Fourier series, the Method of Harmonic Balance and The
Variation of Parameter Method.
4. Numerical Methods such as Step-by-Step Integration, Wilson Method, Runge–
Kutta Method and Newmark’s Method.
In the following, we deal with some of the above methods.
14.2 Perturbation Method
When the parameter associated with the nonlinear term of a differential equation is
small, the perturbation method can be successfully applied. The proposed solution
in this method is expressed in terms of a power series, which consists of a generating
solution and added corrective terms.
Example 14.1 A taut spring with a centrally attached mass m is shown in Fig. 14.1.
The spring has an initial tension T 0 . The tension in the spring is assumed to be linear
with its elongation. Determine the equation of motion of the spring. Also determine
by the perturbation method the response of the system in free vibration.
The tension in the string is
T = T 0 +
AE
L
δ
(a)
where AE is the axial rigidity of the string and δ is its elongation.
From Fig. 14.1,
L + δ =
L 2 + y 2
Fig. 14.1 Example 14.1