Uniform stability of damped nonlinear vibrations of an elastic string

Abstract

Abstract.Here we are concerned about uniform stability of damped nonlinear transverse vibrations of an elastic string fixed at its two ends. The vibrations governed by nonlinear integro-differential equation of Kirchoff type, is shown to possess energy uniformly bounded by exponentially decaying function of time. The result is achieved by considering an energy-like Lyapunov functional for the system.

Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain inRn

“…2 t) is a solution of the system (4)- (7), then the time derivative of the functional ρ (cf. [6,7,12]) defined by…”

“…Later, Horn [10] discussed the exact controllability and stability of the problem using a bending moment feedback on the boundary, and Shahruz [24] established boundedness of the output displacement subject to a distributed viscous damping, in the planar case. Recently, Gorain and Bose [7] treated the above problem with u = (v, w), to obtain a uniform stability by means of an exponential energy decay estimate. Such estimate has earlier been obtained by Gorain [6] for internally damped wave equation in a bounded domain in R n .…”

“…We now proceed as in [7,16,24] by defining energy E(t) at time t, for every solution of the system (4)-(7), by the functional…”

Sound synthesis of a nonlinear string using Volterra series

“…If the fluid and structural damping forces are included, it is given by (2) where c = is the coefficient of the nonlinearity [m.s −2 ] which takes into account the variation of tension. This model holds under three hypotheses [18]: (H1) the transverse vibrations are confined to a plane; (H2) the string is perfectly flexible (second order equation); (H3) the nonlinear effects are due to the global variation of length.…”

“…(18) and (19), the Volterra kernels of the whole system in Fig. 7 b are given by, for all n ∈ N * , A…”