W.T. van Horssen

On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part II: the beam-like case

In this paper initial-boundary-value problems for a linear wave (string) equation are considered. These problems can be used as simple models to describe the vertical vibrations of a conveyor belt, for which the velocity is small with respect to the wave speed and is assumed to move with a time-varying speed. Formal asymptotic approximations of the solutions are constructed to show the complicated dynamical behavior of the conveyor belt. It will also be shown that the truncation method cannot be applied to this problem in order to obtain approximations valid on long time scales.

On the weakly nonlinear, transversal vibrations of a conveyor belt with a low and time-varying velocity

In this paper the weakly nonlinear, transversal vibrations of aconveyor belt will be considered. The belt is assumed to move witha low and time-varying speed. Using Kirchhoffs approach a singleequation of motion will be derived from a coupled system ofpartial differential equations describing the longitudinal andtransversal vibrations of the belt. A two time-scalesperturbation method is then applied to approximate the solutionsof the problem. It will turn out that the frequencies of the belt speed fluctuations play an important role in the dynamic behaviourof the belt. It is well-known in linear systems that instabilitiescan occur if the frequency of the belt speed fluctuations is thesum of two natural frequencies. However, in the weakly nonlinearcase as considered in this paper this is no longer true. It turns out that the weak nonlinearity stabilizes the system.

On Mode Interactions for a Weakly Nonlinear Beam Equation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a Rayleigh perturbation will be studied. It will be shown that the calculations to find internal resonances in this case are much more complicated than and differ substantially from the calculations for the weakly nonlinear wave equation with a Rayleigh perturbation as for instance presented in [3] or [7]. The initial-boundary value problem can be regarded as a simple model describing wind-induced oscillations of flexible structures like suspension bridges or iced overhead transmission lines. Using a two-timescales perturbation method approximations for solutions of this initial-boundary value problem will be constructed.

An asymptotic theory for a weakly nonlinear beam equation with a quadratic perturbation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a quadratic nonlinearity will be studied. The initial-boundary value problem can be regarded as a simple model describing free oscillations of flexible structures like suspension bridges. Using a two-timescales perturbation method an approximation for the solution of this initial-boundary problem will be constructed. For a class of initial-boundary value problems the existence and uniqueness of solutions and the asymptotic validity of approximations on a large timescale are shown. It will be shown that for specific values of the beam parameters complicated internal resonances occur.

A Perturbation Method Based on Integrating Factors

In this paper it will be shown that all integrating factors for a system of n first-order, ordinary differential equations have to satisfy a system of %%

On initial boundary value problems for weakly semilinear telegraph equations: asymptotic theory and application

\frac12n(n+1)

On the Applicability of the Method of Separation of Variables for Partial Difference Equations

\frac{1}{2}n(n+1)

On boundary damping for a weakly nonlinear wave equation

first-order, linear partial differential equations. A perturbation method based on integrating factors will be presented for problems containing a small parameter. When approximations of integrating factors have been obtained an approximation of a first integral (including an error estimate) can be given. To show how this perturbation method works the method is applied to the Van der Pol equation, a forced Duffing equation, and a perturbed Volterra--Lotka system. Not only will asymptotic approximations of first integrals be given, but it will be shown how, in a rather efficient way, the existence and stability of time-periodic solutions can be obtained from these approximations.

On Invariance Factors and Invariance Vectors for Difference Equations

In this paper an asymptotic theory for a class of initial boundary value problems for weakly semilinear telegraph equations is presented. The theory implies the well-posedness of the problem and the validity of formal approximations on long time-scales. As an application of the theory an initial-boundary value problem for the equation