Curve Veering: Inherent Behaviour of Some Vibrating Systems

The plots of variation of eigenvalues of vibrating structures with a system parameter often cross each other or abruptly veer away avoiding the crossing. The phenomenon is termed as curve veering and has been observed both in approximate solutions as well as in exact solutions associated with vibration of some vibrating systems. An explanation to such behavior is provided and illustrated by solving a simple example. The curve veering behavior is induced into a membrane by introducing a parameter that can change the mathematical model from an exact form to an approximate one. Approximate deflection functions such as those used in Galerkin's method or the Rayleigh Ritz method invariably create an approximate or a ficticious system model in lieu of the actual system. The ficticious system may exhibit curve veering while the corresponding real system has no such behaviour. When the ficticious nature of the system is minimized by using large number of terms in the approximate techniques or by discretisation of the domain as in finite difference or by assuming spline type deflection functions, the curve veering behavior subsides and in some cases almost vanishes