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Rayleigh surface wave motion in a homogeneous orthotropic elastic solid half-space under time-harmonic sources is theoretically investigated. In this article, closed-form expressions of the Rayleigh wave field as a result of a time-harmonic concentrated line load are simply determined by using reciprocity theorems. For the verification purpose, analytical predictions are also obtained through Fourier transform technique with the aid of the residue theorem. The solutions found by the two approaches are shown to be mathematically the same. As an example of calculation, surface waves due to a set of buried loadings in the orthotropic half-plane are then examined in detail. A discussion of these results is presented to explore the influence of the loading types on the displacement amplitudes of the generated wave fields.

The dynamic response of an infinite Euler–Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.

In this paper, the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler–Bernoulli beam resonators, due to time harmonic load have been investigated. The axial ends of the beam are assumed to be at either clamped-clamped, simply supported-simply supported or clamped-free conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection. The numerical simulation has been carried out with the help of MATLAB software for magnesium like material. The graphical representations and observations have been discussed for deflection of beam under various boundary conditions and for distinct considered values of time and space as well.

The ideal linear hysteretic damper possesses a non-causal impulse response precursor (IRP). This has been proved in theory and confirmed numerically. Yet the mathematical conciseness of this model is so attractive that several modifications have been tried to restrain this pitfall. In this paper, the modification proposed by Bishop et al., the band-limited hysteretic damper was studied in detail. The non-causal and causal parts of the unitary impulse response function h ( t ) were investigated systematically by the residue theorem. The closed form of h (0) was derived, then its property was studied numerically. The main conclusions are as follows: Firstly, the Bishop’s model is still non-causal. The asymptotic rate of the IRP is of an order t −1 as time approaches negative infinity. Secondly, the IRP is not monotonic, but a damped sinusoidal wave. The frequency of this wave relates to the lower bound of the band-limited damper. Thirdly, h (0) can be zero by appropriately combining the lower and upper bounds of the pass band.