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 fixing transformation, the problem is transformed from a linear problem with a moving boundary, to a nonlinear problem with fixed boundaries. It is assumed that the vibrations around the stationary position of the string are small. Explicit approximations of the solution are obtained by using a multiple time-scales perturbation method. Depending on the parameters in the problem, it turns out that three different cases for the obstacle boundary condition have to be considered, that is, Dirichlet, or Neumann, or Robin type of boundary conditions. To avoid an infinite-dimensional system of ordinary differential equations that occurs in the analysis of the modal interactions of the string vibrations, characteristic coordinates are used together with a multiple time-scales approach to analyze the string dynamics in terms of traveling waves in opposite directions. A comparison between a direct numerical integration of the PDE problem and the results obtained by using the aforementioned perturbation approach shows an excellent agreement in the results.

In this paper, the oscillations of an actuated, simply supported microbeam are studied for which it is assumed that the electric load is composed of a small DC polarization voltage and a small, harmonic AC voltage. Bending stiffness and mid-plane stretching are taken into account as well as small viscous or structural damping. No tensile axial force is assumed to be present. By using a multiple time-scales perturbation method, approximations of the solutions of the initial-boundary value problem for the microbeam equation are constructed. This analysis is performed without truncating the infinite series representation in advance as is usually done in the existing literature. It is shown in which cases truncation is allowed for this problem. Moreover, accurate and explicit approximations of the natural frequencies up to order ε³ of the actuated microbeam are also obtained. Intriguing and new modal vibrations are found when the frequency of the harmonic AC voltage is (near) half or twice a natural frequency of the microbeam, i.e., near a superharmonic or a subharmonic resonance.