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 beam equation (with string effect) 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. In this paper, the belt is assumed to move with a time-varying velocity . Formal asymptotic approximations of the solutions are constructed to show the complicated dynamical behaviour of the belt. Complicated dynamical behaviour of the belt system occurs when the frequency Ω is the sum or difference of any two natural frequencies of the system with zero belt velocity. For special values of the belt parameters these sum type and difference type of internal resonances coincide giving rise to even more complicated dynamical behaviour. Some examples (including detuning cases) have been studied in detail.     

On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-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 Kirchhoff's 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.