Large-amplitude damped free vibration of a stretched string

Abstract

The equations of motion of a vibrating string are derive, and it is shown that a coupling exists between the longitudinal and transverse modes of vibration. Free transverse vibration of small order, under sinusoidal initial conditions, is analyzed. Under these conditions, the equations are separable. The time-dependent parts of the equations are solved by the method of variable amplitude and phase. It is seen that, when the vibration is nonplanar, part of the energy oscillates between the mutually perpendicular transverse components with a frequency proportional to the nonlinearity parameter a. The path of any point on the string is shown to be an ellipse with slowly rotating and shrinking axes.