On string vibrations influenced by a smooth obstacle at one of the endpoints

In this paper, the vibrations of a string are considered. At one end of the string, a smooth obstacle is placed and the other end of the string is attached to a fixed point. The contact between the string and the obstacle varies in time, and leads to a linear, moving boundary value problem for the string vibrations. By applying a boundary...

On the multiple time-scales perturbation method for differential-delay equations

In this paper, we present a new approach on how the multiple time-scales perturbation method can be applied to differential-delay equations such that approximations of the solutions can be obtained which are accurate on long time-scales. It will be shown how approximations can be constructed which branch off from solutions of differential...

On resonances and transverse and longitudinal oscillations in a hoisting system due to boundary excitations

In this paper, we study transverse and longitudinal oscillations and resonances in a hoisting system induced by boundary disturbances. The dynamics can be described by an initial-boundary value problem for a coupled system of nonlinear wave equations on a slowly time-varying spatial domain. It will be shown how the boundary excitations and...

On resonances in transversally vibrating strings induced by an external force and a time-dependent coefficient in a Robin boundary condition

In this paper an initial–boundary value problem on a bounded, fixed interval is considered for a one-dimensional and forced string equation subjected to a Dirichlet boundary condition at one end of the string and a Robin boundary condition with a slowly varying time-dependent coefficient at the other end of the string. This problem may serve...

On constructing a Green’s function for a semi-infinite beam with boundary damping

The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical...

On the Transverse Vibrations of Strings and Beams on Semi-Infinite Domains

In this paper, we study the transverse vibrations of a string and of a beam which are infinitely long in one direction. These vibration problems can be used as a toy model for rain-wind induced oscillations of cables. In order to suppress undesired vibrations in the string (or beam), dampers are used at the boundary. The main aim of this paper...

On a Rayleigh wave equation with boundary damping

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by the dashpot is assumed to be small. This problem can be...

On approximations of first integrals for a strongly nonlinear forced oscillator

On the periodic solutions of a generalized nonlinear Van der Pol oscillator

On a characteristic layer problem for a weakly damped string

In this paper initial-boundary value problems for a string (a wave) equation are considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by the dashpot system is assumed to be small, and is assumed to be proportional to the vertical and the angular...

On approximations of first integrals for a system of weakly nonlinear, coupled harmonic oscillators

In this paper a system of weakly nonlinear, coupled harmonic oscillators will be studied. It will be shown that the recently developed perturbation method based on integrating vectors can be used to approximate rst integrals and periodic solutions. To show how this perturbation method works the method is applied to a system of weakly nonlinear,...

On boundary damping for a weakly nonlinear wave equation

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a spring-mass-dashpot system, where the damping generated by the dashpot is assumed to be small. This...