In this paper, an analytical method is presented to study an initial-boundary value problem describing the transverse displacements of a vertically moving beam under boundary excitation. The length of the beam is linearly varying in time, i.e., the axial, vertical velocity of the beam is assumed to be constant. The bending stiffness of the beam is assumed to be small. This problem may be regarded as a model describing the lateral vibrations of an elevator cable excited at its boundaries by the wind-induced building sway. Slow variation of the cable length leads to a singular perturbation problem which is expressed in slowly changing, time-dependent coefficients in the governing differential equation. By providing an interior layer analysis, infinitely many resonance manifolds are detected. Further, the initial-boundary value problem is studied in detail using a three-timescales perturbation method. The constructed formal approximations of the solutions are in agreement with the numerical results.

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The free, linear, transverse vibrations of a semi-infinite axially moving string are considered in order to study the reflection of an incident wave and the damping properties of different types of boundary conditions such as free, fixed, spring–dashpot, and mass–spring–dashpot. To obtain the response to the initial conditions, the method of d׳Alembert is applied. Furthermore, analytical expressions for the time-rate of change of the total mechanical energy of the system are derived. The obtained results give insight into the most efficient way of placing a boundary support depending on the direction of the transport velocity. Moreover, for nonclassical boundary conditions, the dynamics of the string is described by the relative values of the system parameters.@en

In this paper, the free transverse vibrations of a vertically moving string with a harmonically time-varying length are studied. The string length variations are assumed to be small. By using the multiple‐timescales perturbation method in conjunction with a Fourier series approach, we determine the resonance frequencies and derive the non-secularity conditions in the form of an infinite dimensional system of coupled ordinary differential equations. This system describes the long time behavior of the amplitudes of the oscillations. Then, the eigenvalues of the obtained system are studied by the Galerkin truncation method, and applicability of this method is discussed. Apart from this, the dynamic stability of the solution is investigated by an energy analysis. Additionally, resonance detuning is considered.@en

In this paper, the free lateral vibrations of a vertically translating continuum modeled as a taut string with variable length are studied. The time-varying length of the cable is described by a harmonically varying function about a constant mean length. This model can be used to describe the lateral vibrations of an elevator cable with changing length. A Fourier series approach is used to predict resonant frequencies arising because of the length fluctuation. The amplitudes of vibrations are represented by the infinitely dimensional system of coupled ordinary differential equations. This system is studied by Galerkin’s truncation method for the lowest resonant frequency. The obtained results are in agreement with the energy analysis, but the truncation technique leads to inaccurate approximations on long timescales. Hence, an alternative analytical method is proposed promising an accurate approximation of the response on long timescales for the general resonant frequency.@en