A. H. P. van der Burgh

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

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

On the dynamics of aeroelastic oscillators with one degree of freedom

u_{tt} - u_{xx} + u + \varepsilon u^3 = 0

Rain-wind-induced vibrations of a simple oscillator

is considered. To construct an

Combined parametrical transverse and in-plane harmonic response of an inclined stretched string

O(\varepsilon )

Higher-order averaging: periodic solutions, linear systems and an application

approximation of the solution of this problem a two-time-scale perturbation method is applied.

Parametric Excitation in Mechanical Systems

In this paper two aeroelastic oscillators in crossflow with one degree of freedom are considered.. The first oscillator is a special mass-spring system that is able to oscillate in crossflow, that is perpendicular to the direction of a one-dimensional uniform flowing medium. The second oscillator is a seesaw-type oscillator in crossflow.The geometry of the oscillators is such that for both oscillators an axis of symmetry can be defined. The interesting difference between the two oscillators is the difference between the dynamical behaviour of this axis. For the first oscillator the slope of the axis of symmetry with the horizontal plane does not change with time, whereas for the seesaw-type oscillator this slope is time-dependent. By using a quasi-steady theory as model equations, a Lienard equation and a generalised Lienard equation are obtained. For the first equation a global analysis is presented, and for the second equation a local analysis is presented resulting in conditions for the existence and u...

Validity of approximations for time periodic solutions of a forced non–linear hyperbolic differential equation

Abstract In this paper a relatively simple mechanical oscillator which may be used to study rain-wind-induced vibrations of stay cables of cable-stayed bridges is considered. In recent publications, mention is made of vibrations of (inclined) stay cables which are excited by a wind field containing rain drops. The rain drops that hit the cables generate a rivulet on the surface of the cable. The presence of flowing water on the cable changes the cross section of the cable experienced by the wind field. A symmetric flow pattern around the cable with circular cross section may become asymmetric due to the presence of the rivulet and may consequently induce a lift force as a mechanism for vibration. During the motion of the cable the position of rivulet(s) may vary as the motion of the cable induces an additional varying aerodynamic force perpendicular to the direction of the wind field. It seems not too easy to model this phenomenon, several author state that there is no model available yet. The idea to model this problem is to consider a horizontal cylinder supported by springs in such a way that only one degree of freedom, i.e. vertical vibration is possible. We consider a ridge on the surface of the cylinder parallel to the axis of the cylinder. Additionally, let the cylinder with ridge be able to oscillate, with small amplitude, around the axis such that the oscillations are excited by an external force. It may be clear that the small amplitude oscillations of the cylinder and hence of the ridge induce a varying lift and drag force. In this approach it is assumed that the motion of the ridge models the dynamics of the rivulet(s) on the cable. By using a quasi-steady approach to model the aerodynamic forces, one arrives at a non-linear second-order equation displaying three different kinds of excitation mechanisms: self-excitation, parametric excitation and ordinary forcing. The first results of the analysis of the equation of motion show that even in a linear approximation for certain values of the parameters involved, stable periodic motions are possible. In the relevant cases where in linear approximation unstable periodic motions are found, results of an analysis of the non-linear equation are presented.

On the Modeling of Rain-Wind Induced Vibrations of a Simple Oscillator

In this paper, a two-point boundary value problem for an integrodifferential equation is studied. This equation describes the dynamics of an inclined stretched string suspended between a fixed support and a vibrating support. Due to the inclination, the string will vibrate under combined parametrical and transversal excitation. The attention will be focused on time-periodic solutions consisting of one mode (semi-trivial solution) generated by transverse (external) excitation and two modes (non-trivial solution) generated by combined parametrical and transverse excitation.

Nonlinear Dynamics of Structures Excited by Flows

Existence and stability of periodic solutions by using second-order averaging when the vector field by first-order averaging vanishes, will be studied in this paper as well as its generalization to higher order. A special averaging algorithm for the computation of higher approximations of the fundamental matrix of linear equations with periodic coefficients is given. As an application the existence and stability of periodic solutions of an inhomogeneous second-order equation with time-dependent damping coefficient are studied.

A Double Pendulum in a Wind Field

Parametric excitation in a mechnical system may occur if a parameter of the system becomes time-dependent. The mathematical model for this type of excitation is characterized by terms in the differential equations which have time-dependent coefficients. A standard example of an equation which displays parametric excitation is the Mathieu equation. In this paper two systems will be considered in more detail: a pendulum and a stretched string both with varying length. The attention will be focused to the (unstability) of the equilibrium position (trivial solution). Also finite amplitude motions will be considered, for the description of which however additional nonlinear terms are taken into account.