Rayleigh-Type Waves in a Rotating Fiber-Reinforced Half Space under the Action of Magnetic Field, Gravity and Surface Stress

The aim of the present article is to analyze the propagation of Rayleigh waves in a rotating fiber-reinforced electrically conducting elastic solid medium under the influence of surface stress, magnetic field and gravity. The magnetic field is applied in such a direction that the problem can be considered as a two dimensional one. The wave velocity equation for Rayleigh waves has been obtained. In the absence of gravity field, surface stress, rotation and fiberreinforcement, the frequency equation is in complete agreement with the corresponding classical results. The effects on various subjects of interest are discussed and shown graphically. Comparisons are made with the corresponding results in absence of surface stress

EIGENFREQUENCIES OF OSCILLATIONS OF A PLATE WHICH SEPARATES A TWO-LAYER IDEAL FLUID WITH A FREE SURFACE IN A RECTANGULAR CHANNEL

The frequency spectrum of plane vibrations of an elastic plate separating a two-layer ideal fluid with a free surface in a rectangular channel is investigated analytically and numerically. For an arbitrary fixing of the contours of a rectangular plate, it is shown that the frequency spectrum of the problem under consideration consists of two sets of frequencies describing the vibrations of the free surface of the liquid and the elastic plate. The equations of coupled vibrations of the plate and the fluid are presented using a system of integro-differential equations with the boundary conditions for fixing the contours of the plate and the condition for the conservation of the volume of the fluid. When solving a boundary value problem for eigenvalues, the shape of the plate deflection is represented by the sum of the fundamental solutions of a homogeneous equation for a loose plate and a partial solution of an inhomogeneous equation by expanding in terms of eigenfunctions of oscillations of an ideal fluid in a rectangular channel. The frequency equation of free compatible vibrations of a plate and a liquid is obtained in the form of a fourth-order determinant. In the case of a clamped plate, its simplification is made and detailed numerical studies of the first and second sets of frequencies from the main mechanical parameters of the system are carried out. A weak interaction of plate vibrations on vibrations of the free surface and vice versa is noted. It is shown that with a decrease in the mass of the plate, the frequencies of the second set increase and take the greatest value for inertialess plates or membranes. A decrease in the frequencies of the second set occurs with an increase in the filling depth of the upper liquid or a decrease in the filling depth of the lower liquid. Taking into account two terms of the series in the frequency equation, approximate formulas for the second set of frequencies are obtained and their efficiency is shown. With an increase in the number of terms in the series of the frequency equation, the previous roots of the first and second sets are refined and new ones appear.

A Method for Predicting the Influences of Bearing Support Stiffness and Position on the Vibrations of a Flexible Rotor System

The bearing support stiffness and position can greatly affect the vibrations of flexible rotor systems (FRSs). However, most previous works only focused on the effect of the bearing support stiffness on the critical speeds or modal characteristics including the natural frequencies and mode shapes of rigid rotor systems (RRSs). The previous studies missed the combined effects of the bearing support stiffness and position. To overcome this issue, an analytical method of a FRS based on the finite element (FE) method is proposed. Our model considers the bearing support stiffness and rotational inertia of FRS. The frequency equation of FRS is established for solving the critical speeds. The critical speeds and modal deformations of FRS from our model and the numerical model based on a commercial software are compared to verify the effectiveness of the presented method. The effects of the bearing support stiffness and position on the critical speeds of FRS are analyzed. The results show that the critical speeds are positively correlated with the bearing support stiffness. The critical speeds of FRS are also greatly affected by the bearing position. This study can provide some guidance for the optimization design method of bearing support stiffness and position in FRSs.

Stability and Modal Conversion Phenomenon of Pipe-in-Pipe Structures with Arbitrary Boundary Conditions by Means of Green’s Functions

In this paper, a closed-form frequency equation of the pipe-in-pipe (PIP) structure with arbitrary boundaries is obtained. The frequency equation is derived from Green’s function of the transverse forced vibration of the PIP structure and takes into account the effects of internal two-phase flow and axial pressure. The reliability of the method in this paper is proved by comparison with the published literature. In the numerical discussion part, the PIP structures with clamped-clamped, clamped-free, and elastic boundary conditions are used as examples to discuss. The effects of equivalent stiffness coefficient, internal flow velocity, and gas volume fraction on the stability of PIP structure are studied. The results show that the stability of the PIP structure is better than that of the single-pipe structure, and the greater the equivalent stiffness coefficient of the elastic layer, the higher the critical flow velocity of the structure. In addition, a modal conversion phenomenon existing in the PIP structure is discovered. There are different forms of modal conversion for different boundary conditions, and the modal conversion makes the order of instability of the PIP structure different from that of a single-pipe. The conclusion of this paper has positive significance for the dynamic research of PIP structure.

Propagation of Surface Waves in a Rotating Coated Viscoelastic Half-Space under the Influence of Magnetic Field and Gravitational Forces

The present manuscript focuses on the study of surface wave propagation in a rotating coated viscoelastic half-space and its response to external forces comprised of the magnetic field and gravitational forces. A celebrated normal mode analysis procedure is adopted as the methodology of interest for its high level of efficiency in the literature. The analytically obtained frequency equation is analyzed for certain scenarios of curiosity, in addition to the determination of the resulting displacements and stresses. Moreover, certain physical data of relevance with the viscoelasticity index of unity are considered for the numerical simulations. As for the findings, the presented graphical illustrations showed that both the magnetic field and rotation positively accelerated the dispersion of surface waves in the coated half-space, while the obtained approximate fields in the half-space are found to be oscillatory as they steadily move towards the limiting point.

Initial stress and heterogeneity effects on torsional waves in dry sand half-space between reinforced half-space and poroelastic medium

PurposeThe purpose of this paper is to investigate the effect of initial stress and heterogeneity on the propagation of torsional waves in dissipative medium. The problem consists of dry sand poroelastic half-space embedded between heterogeneous self-reinforced half-space and poroelastic medium. The frequency equation is derived in the framework of Biot's theory with some variants.Design/methodology/approachTorsional wave propagation in dry sand poroelastic half-space embedded between self-reinforced half-space and poroelastic medium. All the constituents here are assumed to be dissipative, heterogeneous and initial stressed.FindingsPhase velocity and attenuation are computed against wavenumber for various values of self-reinforcement parameter, inhomogeneity parameter and initial stress. Particular cases are discussed in absence of dissipation. The numerical results are presented graphically.Originality/valueInitial stress and heterogeneity effects on torsional waves in dry sand half-space between reinforced half-space and poroelastic medium are investigated. The frequency equation is derived, and which intern gives the phase velocity and attenuation coefficient for various values of initial stress, self-reinforcement parameter and heterogeneity parameter. From the numerical results, it is clear that as wavenumber varies phase velocity and attenuation are periodic in nature for all the cases. Particular cases are discussed in absence of dissipation. This kind of analysis can be extended to any elastic solid by taking magnetic, thermo and piezoelectric effects into account.

Dimensionless wave numbers evolution of a three spans simply supported beam when the intermediate supports are moving along the whole beam

Continuous beams simply supported with several intermediate supports are very common in engineering achievements everywhere. The paper shows the evolution of the dimensionless wave number in 3D format, respectively of the eigenfrequencies for a continuous beam with three openings when the intermediate supports take any position inside the beam. The frequency equation for calculating the dimensionless wave number is presented and the modal function is given with an example for the case where the eigenfrequency has the maximum value at fist vibration mode.

Frequency Analysis for Functionally Graded Material Cylindrical Shells: A Significant Case Study

Cylindrical shells play an important role for the construction of functionally graded materials (FGMs). Functionally graded materials are valuable in order to develop durable materials. They are made of two or more materials such as nickel, stainless steel, zirconia, and alumina. They are extremely beneficial for the manufacturing of structural elements. Functionally graded materials are broadly used in several fields such as chemistry, biomedicine, optics, and electronics. In the present research, vibrations of natural frequencies are investigated for different layered cylindrical shells, those constructed from FGMs. The behavior of shell vibration is based on different parameters of geometrical material. The problem of the shell is expressed from the constitutive relations of strain and stress with displacement, as well as it is adopted from Love’s shell theory. Vibrations of natural frequencies (NFs) are calculated for simply supported-simply supported (SS-SS) and clamped-free (C-F) edge conditions. The Rayleigh–Ritz technique is employed to obtain the shell frequency equation. The shell equation is solved by MATLAB software.

Accurate Results for Free Vibration of Doubly Curved Shallow Shells of Rectangular Planform (Part.1)

A method is presented for determining the free vibration frequencies of doubly curved, isotropic shallow shells under general edge conditions and is used to obtain accurate natural frequencies for wide range of geometric parameters. Based on the shallow shell theory applicable to thin thickness shells, a method of Ritz is extended to derive a frequency equation wherein the displacement functions are modified to accommodate arbitrary sets of edge conditions for both in-plane and out-of-plane motions. In numerical computation, convergence is tested against series terms and comparison study is made with existing results by other authors. Twenty one sets of frequency parameters are tabulated for a wide range of shell shape and curvature ratio to serve as data for future comparison and practical design purpose.