Nonlinear nonuniform torsional vibrations of bars by the boundary element method

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross-section taking into account the effect of geometrical nonlinearity. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. The transverse displacement components are expressed so as to be valid for large twisting rotations (finite displacement-small strain theory), thus the arising governing differential equations and boundary conditions are in general nonlinear. The resulting coupling effect between twisting and axial displacement components is considered and torsional vibration analysis is performed in both the torsional pre- or post-buckled state. A distributed mass model system is employed, taking into account the warping, rotatory and axial inertia, leading to the formulation of a coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an “average” axial displacement of the cross-section of the bar. The numerical solution of the aforementioned initial boundary value problem is performed using the analog equation method, a BEM based method, leading to a system of nonlinear differential-algebraic equations (DAE), which is solved using an efficient time discretization scheme. Additionally, for the free vibrations case, a nonlinear generalized eigenvalue problem is formulated with respect to the fundamental mode shape at the points of reversal of motion after ignoring the axial inertia to verify the accuracy of the proposed method. The problem is solved using the direct iteration technique (DIT), with a geometrically linear fundamental mode shape as a starting vector. The validity of negligible axial inertia assumption is examined for the problem at hand.     

Free vibrations of a beam-mass system with elastically restrained ends

The vibration problem of a beam with an arbitrarily placed concentrated mass and elastically restrained against rotation at either end is solved by using Laplace transforms. The effects on eigenfrequencies of the system produced by varying the ratios of the concentrated mass to the mass of the beam, stiffness of the end spring to the stiffness of the beam and position of the mass to the total length of the beam are presented. The effect of neglecting the mass of the beam is considered.     

Transient response in flexure to general uni-directional loads of variable cross-section beam with concentrated tip inertias immersed in a fluid

This paper presents a method for solving problems of transient response in flexure due to general unidirectional dynamic loads of beams of variable cross section with tip inertias. An elastodynamic theory which includes effects of continuous mass and rigidity of the beam has been applied. In the analysis the general dynamic load is expanded into a Fourier series and the beam is divided into many small uniform thickness segments. The equation of motion of each segment is mapped onto the complex domain by use of the Laplace transform method. The solutions of each set of adjoining segments are related to each other at the boundaries by the use of the transfer matrix method. The displacement, the bending slope, the bending moment and the shearing force at each boundary and at arbitrary time are obtained from the Laplace transform inversion integral by using the residue theorem. The theoretical results given in this paper are applicable to problems of dynamic response due to arbitrary loads varying with time of beams of arbitrary shape with concentrated tip inertias. As applications of the present theoretical results, numerical calculations have been carried out for two cases: a uniform beam with a tip inertia and a non-uniform beam (a truncated cone) with a tip inertia. Both are immersed in a fluid and subjected to large waves such as cnoidal waves.     

Scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems

This paper presents a solution for the displacement of a uniform elastic thin plate with an arbitrary cavity, modelled using the biharmonic plate equation. The problem is formulated as a system of boundary integral equations by factorizing the biharmonic equation, with the unknown boundary values expanded in terms of a Fourier series. At the edge of the cavity we consider free-edge, simply-supported and clamped boundary conditions. Methods to suppress ill-conditioning which occurs at certain frequencies are discussed, and the combined boundary integral equation method is implemented to control this problem. A connection is made between the problem of an infinite plate with an arbitrary cavity and the vibration problem of an arbitrarily shaped finite plate, using the jump discontinuity present in single-layer distributions at the boundary. The first few frequencies and modes of displacement are computed for circular and elliptic cavities, which provide a check on our numerics, and results for the displacement of an infinite plate are given for four specific cavity geometries and various boundary conditions.     

Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads

In this paper, a boundary element method is developed for the geometrically nonlinear response of shear deformable beams of simply or multiply connected constant cross-section, traversed by moving loads, resting on tensionless nonlinear three-parameter viscoelastic foundation, undergoing moderate large deflections under general boundary conditions. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse moving loading as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the transverse displacement, to the axial displacement and to a stress functions and solved using the Analog Equation Method, a Boundary Element based method. Application of the boundary element technique yields a system of nonlinear Differential-Algebraic Equations, which is solved using an efficient time discretization scheme, from which the transverse and axial displacements are computed. The evaluation of the shear deformation coefficient is accomplished from the aforementioned stress function using only boundary integration. Analyses are performed to illustrate, wherever possible, the accuracy of the developed method, to investigate the effects of various parameters, such as the load velocity, load frequency, shear deformation, foundation nonlinearity, damping, on the beam displacements and stress resultants and to examine how the consideration of shear and axial compression affects the response of the system.     

The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass

New frequency equations for transverse vibrations of Timoshenko beams carrying a concentrated mass at an arbitrary point along the beam are given. Normal mode equations for the hinged-hinged beam are given and the orthogonality condition is presented for beams with hinged, clamped or free ends. A numerical example is given and frequency charts show the effects of varying the size and location of the concentrated mass.     

Beam vibrations with time-dependent boundary conditions

A procedure is outlined for the solution of the vibration problem of a Bernoulli-Euler beam with time-dependent boundary conditions. The solution is greatly simplified if the dependent variable in the original partial differential equation can be changed to produce homogeneous boundary conditions and at the same time maintain a homogeneous differential equation. A method for making such a change is given and illustrated by solving a cantilever beam problem with a time-dependent tip displacement.     

外伸梁差分离散系统的模态反问题

研究外伸梁模态的反问题,即由两组位移模态或应变模态及相应的频率构造外伸梁差分离散模型物理参数的方法,讨论反问题解存在的充要条件.提出算法并用算例验证.结果表明,由应变模态构造外伸梁的物理参数比用位移模态构造梁的物理参数精度要高.     

Non-linear vibrations of a flat plate with initial stresses

Non-linear free vibrations of a simply supported rectangular elastic plate are examined, by using stress equations of free flexural motions of plates with moderately large amplitudes derived by Herrmann. A modal expansion is used for the normal displacement that satisfies the boundary conditions exactly, but the in-plane displacements are satisfied approximately by an averaging technique. Galerkin technique is used to reduce the problem to a system of coupled non-linear ordinary differential equations for the modal amplitudes. These nonlinear differential equations are solved for arbitrary initial conditions by using the multiple-time-scaling technique. Explicit values of the coefficients that appear in the forementioned Galerkin system of equations are given, in terms of non-dimensional parameters characterizing the plate geometry and material properties, for a four-mode case, for which results for specific initial conditions are presented. A comparison of the results with those obtained in previous studies of the problem is presented. In addition, effects of prescribed edge loadings are examined for the four-mode case.     

Investigating RF and microwave nonlinear magnetoelastic interactions in a three-layer structure

The problem of hypersound excitation in a structure consisting of three magnetic layers is considered. Motion equations and boundary conditions for the components of magnetization and elastic displacement when there is an arbitrary angle of magnetization vector precession are obtained. The development of oscillations over time in an alternating field is considered.     

Diffraction reflection of x-ray radiation with a two-dimensionally bounded wavefront from perfect crystals

The problem of dynamical diffraction of x-ray radiation with a two-dimensionally bounded wavefront is solved in the Bragg and Laue geometries in a crystal with an arbitrary thickness and an arbitrary reflection asymmetry parameter. An analysis of the wavefront deformation during the diffraction and subsequent propagation is carried out. It is shown that the most favorable conditions for the reflected beam to retain its shape are accomplished in a crystal whose thickness is less than the extinction depth.     

Prediction capabilities of classical and shear deformable beam models excited by a moving mass

In this paper, a comprehensive assessment of design parameters for various beam theories subjected to a moving mass is investigated under different boundary conditions. The design parameters are adopted as the maximum dynamic deflection and bending moment of the beam. To this end, discrete equations of motion for classical Euler-Bernoulli, Timoshenko and higher-order beams under a moving mass are derived based on Hamilton's principle. The reproducing kernel particle method (RKPM) and extended Newmark-β method are utilized for spatial and time discretization of the problem, correspondingly. The design parameter spectra in terms of the beam slenderness, mass weight and velocity of the moving mass are introduced for the mentioned beam theories as well as various boundary conditions. The results indicate the existence of a critical beam slenderness mostly as a function of beam boundary condition, in which, for slenderness lower than this so-called critical one, the application of Euler-Bernoulli or even Timoshenko beam theories would underestimate the real dynamic response of the system. Moreover, there would be a roughly linear relation between the weight of the moving mass and the design parameters for a certain value of the moving mass velocity in most cases of boundary conditions.     

Quasibound states in semiconductor quantum well structures

We present a study on quasibound states in multiple quantum well structures using a finite element model (FEM). The FEM is implemented for solving the effective mass Schrödinger equation in arbitrary layered semiconductor nanostructures with an arbitrary applied potential. The model also includes nonparabolicity effects by using an energy dependent effective mass, where the resulting nonlinear eigenvalue problem was solved using an iterative approach. We focus on quasibound/continuum states above the barrier potential and show that such states can be determined using cyclic boundary conditions. This new method enables the determination of both bound and quasibound states simultaneously, making it more efficient than other methods where different boundary conditions have to be used in extracting the relevant states. Furthermore, the new method lifted the problem of quasibound state divergence commonly seen with many other methods of calculation. Hence enabling accurate determination of dipole matrix elements involving both bound and quasibound states. Such calculations are vital in the design of intersubband optoelectronic devices and reveal the interesting properties of quasibound states above the potential barriers.     

Self-consistent field theory of a helix traveling wave tubeamplifier

A self-consistent relativistic field theory of a helix traveling wave tube (TWT) is presented for a configuration in which a thin annular beam propagates through a sheath helix enclosed within a loss-free wall. A linear analysis of the interaction is carried out, subject to the boundary conditions imposed by the beam, helix, and wall. A detrimental dispersion equation is obtained which implicitly includes beam space-charge effects without recourse to a heuristic model of the space-charge field. The equation is valid for arbitrary azimuthal mode number and is solved numerically for the azimuthally symmetric case. The coupled-wave Pierce theory is recovered in the near-resonant limit. Numerical comparisons between the complete dispersion equation and the Pierce model are described. A discrepancy is found between the Pierce and the field theory even for low currents in the nominally ballistic regime, owing to the dielectric effect of the beam on the helix modes     

State of stress at a crystal surface deformed by a concentrated load in the presence of a wedge twin

A procedure for the calculation of the stress fields near a wedge twin located at a crystal surface is developed using a dislocation macroscopic model. The problem is solved for the case of a concentrated load applied to the surface of a crystal with a twin. The concentrated load is found to increase the level of stresses near the wedge twin and to cause their localization at the twin boundary that is closer to the load application point.     

Non-linear forced vibrations of a beam carrying concentrated mass under gravity

The non-linear dynamic behavior of a simply supported beam, with ends restrained to remain a fixed distance apart, carrying a concentrated mass and subjected to a harmonic exciting force at an arbitrary point under the influence of gravity is analysed. By using the one mode approximation and applying Galerkin's method, the governing equation of motion is reduced to the well known Duffing type equation. The harmonic balance method is applied to solve the equation and the dynamic response of a concentrated mass is derived. The effects of the weight, the location, and the vibratory amplitude of the concentrated mass on the natural frequency are also discussed.     

Diffraction of a plane electromagnetic wave by a slot in a conducting screen with a transverse dielectric layer

The problem of diffraction of a plane electromagnetic wave by a slot in a planar perfectly conducting arbitrary thick screen with an infinite planar dielectric layer passing through the slot transversely to the screen is solved rigorously. In each of the field existence domains (two domains on either side of the screen and the interior of the slot), the solution is represented as an expansion in piecewise harmonic or exponential modes that allow for reflection and refraction at the boundaries of the dielectric layer. It is found that a set of functions describing such modes is complete enough to construct a solution satisfying all boundary conditions of the diffraction problem. The procedures of solution construction for the case at hand and for the same diffraction structure without the dielectric layer are compared.     

AN EXACT METHOD FOR THE FREE VIBRATION ANALYSIS OF TIMOSHENKO-KELVIN BEAMS WITH OSCILLATORS

In this paper, a modern exact method is proposed for solving the problem of free vibrations of a Timoshenko-type viscoelastic beam with discrete rigid bodies, connected to the beam by means of viscoelastic constraints. The phenomenon of free vibrations of this discrete-continuous system is described by a set of three partial and two subsystem ordinary differential equations with generalized boundary conditions and initial conditions. Vector notation of the equations allows one to identify the self-adjoint linear operators of inertia, stiffness and damping. In this case, these operators are not homothetic hence a separation of variables in this set of equations is possible only in a complex Hilbert space. Such separation of variables leads to ordinary differential equations of motion with respect to time as well as to a set of three ordinary differential equations with respect to a spatial variable and two subsystem algebraical equations. Solution of the boundary-value problem was carried out in the classical way, but its results are complex conjugated. Using these results and the fundamental principle, describing the orthogonality property of complex eigenvectors, the problem of free vibrations of the system with arbitrary initial conditions has been finally solved exactly.     

A diffusion problem in a two-medium system

A solution is obtained to the boundary-value diffusion-equation problem in a system of two anisotropic semiinfinite media, separated by a movable boundary region. The solution is obtained for arbitrary initial values and concentration on the boundary region. Thus, this class of boundary-value problems is completely solved in the given range.     

Vibrations of an elastic strip with a rough boundary

A plane problem of steady-state forced vibrations of an elastic strip whose lower boundary contains a rough segment is considered. Using Green’s functions for a strip, the problem is reduced to a system of integral equations with integrals over the rough boundary, which is solved by the boundary-element method. The inverse problem of determining the shape of the rough boundary segment from the data on the displacement field of a certain part of the upper boundary is formulated. By the linearization procedure, the inverse problem is reduced to a Fredholm integral equation of the first kind with a smooth kernel, which is solved by Tikhonov’s regularization method.