Large Amplitude Vibrations of Strings

The study of nonlinear, free, undamped transverse vibrations of a stretched string may be traced back to an equation of motion developed by Kirchhoff more than a century ago. Subsequent studies have typically assumed small slopes and done little to incorporate longitudinal coupling effects. In the present work a new set of coupled equations is derived. These highly nonlinear equations are solved by means of a newly developed Galerkin procedure which determines variations of the transverse and longitudinal displacements in both space and time. Time variation is determined by incremental use of polynomials. Validity of solutions is verified by independent finite difference solutions. Numerical results are presented for three example problems wherein the string is displaced transversely into a half-sine wave, and released from rest. The problems include: (1) moderate strain (ε₀ = 0.005) in the straight line equilibrium position and moderate transverse displacement (δ = 0.1L), (2) moderate strain and large displacement (δ = 0.4L), and (3) small strain (ε₀ = 10 ⁻⁵) and moderate displacement. Plots of transverse and longitudinal displacement with time are shown for all three examples. From these plots it is seen that higher modes are generated when only a single mode is initiated, and that the motion is definitely nonperiodic.