Journal of Solid Mechanics
Volume:13 Issue: 4, Autumn 2021

In industry applications, planetary gear systems are widely used in power transmission systems. In planetary gears, dynamic loads, noise and reduction the structural life are produced by system vibrations. For gear transmission systems, the parametric excitation which introduced by the periodically time–varying mesh stiffness of each gear oscillation is the main source of vibration. Generally, there are two methods to evaluate the gear mesh stiffnesses, the finite element method and the analytical method. In this wok, the periodically time–varying mesh stiffness of planetary gears is investigated. The influence of pressure angles on mesh stiffness of meshing gears is shown and the dynamic model of planetary gear sets is studied. When planets of the single–stage spur planetary gear system are meshed by new planets, the system is converted to special type of system with meshed planets. Vibration for geometrical structures (symmetric and anti–symmetric) of planetary system with meshed planets is investigated. Mesh stiffness of meshing gears by estimation function is obtained and numerical results of natural frequencies and vibration modes are derived.

This paper presents the buckling and out-of-plane free vibration response of a sandwich panel with flexible core for the different boundary condition. In the desired configuration of the sandwich panel, the top and bottom plates are made of magneto-electro-elastic (MEE) plates. Moreover, the in-plane electric and magnetic potential fields are neglected for the derivation of the required relations. The sandwich structure is subjected to axial force in both longitudinal and transverse directions; in addition, and the top and bottom plates are exposed to electric and magnetic fields. The governing equations of motion for MEE sandwich panel with a flexible core are derived based on the first-order shear deformation theory by neglecting the displacement of the mid-plate and using the Hamilton’s principle. Furthermore, the derived partial differential equations (PDEs) are solved. According to the obtained numerical results, the core thickness, variation of electric field, variation of magnetic field and plate length are introduced as the most influential parameters on the free vibration response of the panel as well as the critical force of buckling.  As one of the results, the electric potential is inversely related to the natural frequency and buckling load, so that with increasing the electric potential, the natural frequency and critical load of the structure is also increased.Moreover, the magnetic potential is directly related to the natural frequency and buckling load of the system, and increasing trends of natural frequency and critical load are observed by increasing the magnetic potential.

The aim of this paper is to investigates the existence of the dispersion of SH-wave in a heterogeneous orthotropic layer lying over a heterogeneous elastic half-space and underlying an inhomogeneous semi-infinite medium. Hyperbolic variation in upper semi-infinite associated with directional rigidities and density has been considered while linear variation in the intermediate layer associated with initial stress, density, shear moduli and lower half-space associated with rigidity and density has been considered. The dispersion equation of SH-wave has been obtained in a closed form by using variable separation method. The effects of inhomogeneities of the assumed media are illustrated by figures using MATLAB programming. The Earth's composition is heterogeneous that incorporates extremely hard layers. The propagation of SH-wave across crustal layer of the Earth very much depends upon heterogeneity and orthotropic properties. In fact, the observation reveals that the phase velocity of SH-wave is directly proportionate to inhomogeneity parameter, orthotropic parameter and heterogeneity parameter. That means as inhomogeneity parameter and heterogeneity orthotropic parameter increases, the phase velocity of SH-wave increases proportionately. Moreover, the obtained dispersion equation of SH-wave coincides with the classical result of Love wave as initial stress, inhomogeneities, and the upper semi-infinite medium is neglected. This analysis may be helpful to expound the nature of the dispersion of seismic waves in elastic media.

Using Galerkin vector approach closed-form analytic expressions for the displacements and stresses caused by a doublet source buried in a homogenous, isotropic, perfectly elastic half-space have been obtained. Further, the viscoelastic deformation field has been obtained by applying the correspondence principle of linear viscoelasticity, assuming the medium to be elastic in dilatation and Kelvin, Maxwell, or SLS (Standard linear solid) type viscoelastic in distortion. The effect of Poisson’s ratio on the deformation field due to a doublet source is examined in elastic half-space. The effect of relaxation time on displacement and stress fields is studied due to a doublet source in viscoelastic half-space. The variation of the displacements and stresses with the epicentral distance is studied graphically using MATLAB software. Stresses for a doublet with axis parallel to x-axis attain minimum value for Poissonian half- space. Viscoelastic displacements and stresses attain maxima for the Maxwell model and minima for the Kelvin model.

Dynamic vibration absorbers (DVAs) play an important role in the energy dissipation of a vibrating system. Undesirable vibrations of structures can be reduced by using the absorbers. This paper investigates the effect of an attached energy sink on the energy dissipation of a simply supported beam subjected to harmonic excitation. The aim is to design an optimal linear energy sink (LES) and a nonlinear energy sink (NES) and then compare them with each other. Each absorber includes a spring, a mass, and a damper. For each absorber, the optimum mass, stiffness, and damping coefficients are obtained in order to minimize the beam’s maximum amplitude at the resonant frequencies. The optimization problem is minimizing the maximum amplitude of the beam subjected to an arbitrary harmonic force excitation. For consideration of the effects of rotary inertia and shear deformation, the Timoshenko beam theory is used. The mathematical model of beam with DVA is verified by using the ANSYS WORKBENCH software. Finally, by considering the uncertainty on the DVA parameters it was observed that the LES is more robust than the NES.

In this work, a nonlinear investigation of non-homogeneous varying thickness circular plates resting on elastic foundations under the influence of the magnetic fieldis investigated. The non-homogeneity of the circular plates’ material is presumed to occur due to linear and parabolic changes in Young’s modulus likewise the density along the radial direction in a unique manner. The geometric Von Kármán equations are used in modelling the governing differential equations. The transverse deflection is approximated using an assumed single term mode shape while the central deflection in form of Duffing’s equation is obtained using the Galerkin method.  Subsequently, the semi-analytical solutions are provided using the Optimal Homotopy Asymptotic Method (OHAM), the analytical solutions are used for parametric investigation. The results in this work are in good harmony with past results in the literature. From the results, it is realized that the nonlinear frequency of the circular plate increases with an increase in the linear elastic foundation. Also, the results showed that clamped edge and simply supported edge condition produced the same hardening nonlinearity. However, varying taper and non-homogeneity lower the nonlinear frequency ratio. Also, maximum deflection occurs when excitation force is zero, and attenuation of deflection is observed due to the presence of a magnetic field, varying thickness, homogeneity, and elastic foundation. It is anticipated that the discoveries from this research will boost the design of structures subjected to vibration.

In the present paper, a finite element nonlinear coupled thermoelasticity formulation is presented for analysis of the wave propagation, reflection, and mixing phenomena in the finite length isotropic solids. The governing equations are derived based on the second Piola-Kirchhoff stress and the full form of Green’s strain-displacement tensors to account for the large deformations and finite strains. In contrast to the available researches, the assumption of very small temperature changes compared to the reference temperature is released in the present research. Galerkin’s method, a weak formulation, and cubic elements are employed to obtain the time-dependent non-linear finite element governing equations. The proposed solution procedure to the resulting highly nonlinear and time-dependent governing equations employs an updating algorithm and Newmark’s numerical time integration method. The wave propagation and reflection phenomena are investigated for both the mechanical and thermal shocks and time variations of distributions of the resulting displacements, temperature rises, and stresses are illustrated graphically and discussed comprehensively. Furthermore, the effects of the non-linear terms are discussed comprehensively. Results reveal that in the non-linear analysis, no fixed speed of wave propagation can be defined.

This work is a study of the elastic fields’ effect (stresses and displacements) caused by dislocations networks at a heterostructure interface of a InAs / GaAs semiconductors thin system in the cases of isotropic and anisotropic elasticity. The numerical study of this type of heterostructure aims to predict the behavior of the interface with respect to these elastic fields satisfying the boundary conditions. The method used is based on a development in Fourier series. The deformation near the dislocation is greater than the other locations far from the dislocation.