LARGE-AMPLITUDE VIBRATIONS OF STRINGS

The study of nonlinear, free, undamped transverse vibrations of a stre tched string may be traced back to an equation of motion developed by Kirchhoff more than a century ago. Subsequent studies have typically a ssumed small slopes and done little to incorporate longitudinal coupli ng effects. In the present work a new set of coupled equations is deri ved. These highly nonlinear equations are solved by means of a newly d eveloped Galerkin procedure which determines variations of the transve rse and longitudinal displacements in both space and time. Time variat ion is determined by incremental use of polynomials. Validity of solut ions is verified by independent finite difference solutions. Numerical results are presented for three example problems wherein the string i s displaced transversely into a half-sine wave, and released from rest . The problems include: (1) moderate strain (epsilon(o) = 0.005) in th e straight line equilibrium position and moderate transverse displacem ent (delta = 0.1L), (2) moderate strain and large displacement (delta = 0.4L), and (3) small strain (epsilon(o) = 10(-5)) and moderate displ acement. Plots of transverse and longitudinal displacement with time a re shown for all three examples. From these plots it is seen that high er modes are generated when only a single mode is initiated, and that the motion is definitely nonperiodic.