Nonlinear free vibration analysis of strings by the Galerkin method
Abstract
A new, more accurate, system of coupled, nonlinear partial differential equations of motion describing the largeamplitude, planar, free vibration of the elastic stretched string is carefully developed taking into account the nonlinear coupling between the transverse and the longitudinal motions. The only assumption made in the derivation is that the string material remains linearly elastic during the motion so that it always obeys Hooke's law. The developed equations are then solved numerically by applying the Galerkin method both in space and time using an incremental approach to march in time. The assumed solution is written in terms of a series of coordinate functions which are sinusoidal in space and polynomial in time. Over each time step, the assumed solution satisfies the boundary conditions all the time and initial conditions at the beginning of the time step. The validity of the method is established by comparison with exact solutions for three classical problems: (1) damped free vibration of a particle, (2) large (nonlinear) oscillation of a pendulum, and (3) hard spring, nonlinear vibration governed by the Duffing equation. Numerical results of applying the incremental Galerkin method to the developed equations are presented in detail for three example problems: (1) a string subjected to moderate initial tension and moderate amplitude, (2) a string subjected to moderate initial tension and large amplitude, and (3) a string subjected to very small initial tension and moderate amplitude. Up to three transverse and five longitudinal symmetrical modes are included in the analysis. For the last two problems, the results are found to be significantly different from those of the Carrier integrodifferential equation. A parametric study is presented showing the effect of varying the initial tension and the amplitude of the motion on the fundamental frequency of the string. The effect of varying the initial longitudinal displacement, and the generation of (typically higher) modes absent from the initial configuration are also demonstrated. Validity of results is verified by independent finite difference solutions.
 Publication:

Ph.D. Thesis
 Pub Date:
 1992
 Bibcode:
 1992PhDT........40S
 Keywords:

 Dynamic Structural Analysis;
 Equations Of Motion;
 Free Vibration;
 Galerkin Method;
 Nonlinear Equations;
 Nonlinear Systems;
 Partial Differential Equations;
 Strings;
 Duffing Differential Equation;
 Hookes Law;
 Integral Equations;
 Pendulums;
 Time Marching;
 Physics (General)