In this paper, it is shown how characteristic coordinates, or equivalently how the well-known formula of d'Alembert, can be used to solve initial-boundary value problems for wave equations on fixed, bounded intervals involving Robin type of boundary conditions with time-dependent coefficients. A Robin boundary condition is a condition that specifies a linear combination of the dependent variable and its first order space-derivative on a boundary of the interval. Analytical methods, such as the method of separation of variables (SOV) or the Laplace transform method, are not applicable to those types of problems. The obtained analytical results by applying the proposed method, are in complete agreement with those obtained by using the numerical, finite difference method. For problems with time-independent coefficients in the Robin boundary condition(s), the results of the proposed method also completely agree with those as for instance obtained by the method of separation of variables, or by the finite difference method.

In this paper, a modelling method and an accurate numerical procedure are presented to simulate the dynamical responses of a multi-cable driven parallel suspension platform system. For such systems, the cables might become slack due to external excitations and due to the fact that cables can become tensionless when been pushed in longitudinal direction. In lateral and torsional directions, the constraint forces between the cables and platform can be positive as well as negative. This paper will deal with the non-smooth cable vibrations (in longitudinal, lateral and torsional directions) by taking into account the slackness of the cables. Firstly, the Lagrange equation with constraints is used to derive the equations of motion of the multi-cable suspension platform. Then, by expressing the equations of motion and constraint equations at velocity level, a non-smooth algorithm is used to numerically solve the equations. Finally, the numerical results are compared with an ADAMS simulation, and the two results agree well with each other. Moreover, the results in this paper significantly improve the numerical results used in the analysis of the dynamics for multi-cable systems which usually neglect the lateral properties of the cables.