On computation of random response of a clamped plate

The free vibrations of elastically connected circular plate systems with elastically restrained edges and initial radial tensions are investigated analytically. By using the equations developed for the general n-plate system, the plate systems consisting of three and two identical plates with identical boundary conditions and a uniform radial tension are treated in detail. Both axisymmetric and non-axisymmetric vibrations are considered. Attention is directed to the influence of the radial tension and the elastic edge constraints on the first nine eigenvalues and the corresponding natural frequencies of the systems.     

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.     

FREE VIBRATIONS OF SIMPLY SUPPORTED AND MULTILAYERED MAGNETO-ELECTRO-ELASTIC PLATES

Analytical solutions are derived for free vibrations of three-dimensional, linear anisotropic, magneto-electro-elastic, and multilayered rectangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism, from which any physical quantities can be solved for given boundary conditions. In particular, the dispersion equation that characterizes the relationship between the natural frequency and wavenumber can be obtained in a simple form. For multilayered plates, we derive the dispersion relation in terms of the propagator matrices. The present solution includes all previous solutions, such as piezoelectric, piezomagnetic, and purely elastic solutions as special cases, and can serve as benchmarks to various thick plate theories and numerical methods used for the modelling of layered composite structures. Typical natural frequencies and mode shapes are presented for sandwich piezoelectric/piezomagnetic plates. It is shown clearly that some of the modes are purely elastic while others are fully coupled with piezoelectric/piezomagnetic quantities, with the latter depending strongly upon the material property and stacking sequence. These frequency and mode shape features could be of particular interest to the analysis and design of various “smart” sensors/actuators constructed from magneto-electro-elastic composite laminates.     

Vibration of a two layered ring on periodic radial supports

Natural frequencies of a two layered elastic ring, on equi-spaced, identical radial supports, are obtained by using a wave approach. Two types of support conditions are investigated. With the outer layer viscoelastic, the theory of forced damped normal modes is used to obtain the resonant frequencies and the modal loss factors of the structure. Results presented show the effect of the thickness ratio on the resonant frequency and the modal loss factor. The effect of rotational constraints at the supports is also reported.     

Flexural edge waves in semi-infinite elastic plates

This paper presents a detailed analysis of the dispersion for flexural edge waves in semi-infinite isotropic elastic plates. A solution to the dynamic equations of motion is constructed by the superposition of two partial solutions, each providing zero shear stresses at the plate faces. A dispersion equation is expressed via the determinant of an infinite system of linear algebraic equations. The system is reduced to a finite one by taking into account the asymptotic behaviour of unknown coefficients. The accuracy of the solution is confirmed by a good agreement with the available experimental data and by a proper satisfaction of the prescribed boundary conditions.A detailed analysis of dispersion properties for the edge wave and corresponding displacements at various frequencies is carried out. In addition to the well-known results it is shown that the plate height does not influence the existence of the edge wave at high frequencies and, as the frequency increases, the phase velocity of the edge wave in a semi-infinite plate asymptotically approaches the velocity of an edge wave in a right-angled wedge. The performed analysis allows evaluating the plate theories such as the Kirchhoff theory or other refined plate theories developed for modeling edge waves in semi-infinite elastic plates at low frequencies.     

Normal modes of a continuous system with quadratic and cubic non-linearities

A power series solution is presented for the free vibrations of simply supported beams resting on elastic foundation having quadratic and cubic non-linearities. The time-dependence is assumed harmonic and the problem is posed as a non-linear eigenvalue problem. The spatial variable is transformed into an independent variable that satisfies the boundary conditions. This permits a power series expansion of the beam motion in terms of the new variable. A recurrence relation is obtained from the governing equation and used in conjunction with the Rayleigh energy principle to compute the natural frequencies. The results show that, for a first order approximation, only the lower frequencies and first mode shape are significantly affected by the cubic non-linearity.     

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.     

Vibrations of bonded beams with a single lap adhesive joint

The natural frequencies and loss factors of the coupled longitudinal and flexural vibrations of a system consisting of a pair of parallel and identical elastic cantilevers which are lap-jointed by viscoelastic material over a length a_c from their free ends have been investigated. A complete set of equations of motion and boundary conditions governing the vibration of the system are derived. The solution of these equations, subject to satisfying the boundary conditions, yields the desired natural frequencies and associated composite loss factors. The numerical results have been compared with those from two other approximate methods.     

Receiving sensitivity and transmitting voltage response of a fluid loaded spherical piezoelectric transducer with an elastic coating

A method is presented to determine the response of a spherical acoustic transducer that consists of a fluid-filled piezoelectric sphere with an elastic coating embedded in infinite fluid to electrical and plane-wave acoustic excitations. The exact spherically symmetric, linear, differential, governing equations are used for the interior and exterior fluids, and elastic and piezoelectric materials. Under acoustic excitation and open circuit boundary condition, the equation governing the piezoelectric sphere is homogeneous and the solution is expressed in terms of Bessel functions. Under electrical excitation, the equation governing the piezoelectric sphere is inhomogeneous and the complementary solution is expressed in terms of Bessel functions and the particular integral is expressed in terms of a power series. Numerical results are presented to illustrate the effect of dimensions of the piezoelectric sphere, fluid loading, elastic coating and internal material losses on the open-circuit receiving sensitivity and transmitting voltage response of the transducer.     

A finite difference method for a coupled model of wave propagation in poroelastic materials

A computational method for time-domain multi-physics simulation of wave propagation in a poroelastic medium is presented. The medium is composed of an elastic matrix saturated with a Newtonian fluid, and the method operates on a digital representation of the medium where a distinct material phase and properties are specified at each volume cell. The dynamic response to an acoustic excitation is modeled mathematically with a coupled system of equations: elastic wave equation in the solid matrix and linearized Navier-Stokes equation in the fluid. Implementation of the solution is simplified by introducing a common numerical form for both solid and fluid cells and using a rotated-staggered-grid which allows stable solutions without explicitly handling the fluid-solid boundary conditions. A stability analysis is presented which can be used to select gridding and time step size as a function of material properties. The numerical results are shown to agree with the analytical solution for an idealized porous medium of periodically alternating solid and fluid layers.     

Is it possible to determine the type of fastening of a vibrating plate from its sounding?

The problem of determining the type of fastening of a circular plate inaccessible to direct observation from the natural frequencies of its symmetric flexural vibrations is considered. The uniqueness theorem for the solution to this inverse problem is proved, and a method for the reconstruction of unknown boundary conditions is indicated. An approximate formula for the determination of unknown boundary conditions from three natural frequencies is obtained. It is assumed that the natural frequencies can be given approximately, within a certain accuracy. The method of an approximate calculation of unknown boundary conditions is illustrated by four examples of different cases of the plate fastening (a free support, an elastic fixing, a floating fixing, and a free edge).     

Semi-analytic solutions for the free in-plane vibrations of confocal annular elliptic plates with elastically restrained edges

A two-dimensional analytical model is developed to describe the free extensional vibrations of thin elastic plates of elliptical planform with or without a confocal cutout under general elastically restrained edge conditions, based on the Navier displacement equation of motion for a state of plane stress. The model has been simplified by invoking the Helmholtz decomposition theorem, and the method of separation of variables in elliptic coordinates is used to solve the resulting uncoupled governing equations in terms of products of (even and odd) angular and radial Mathieu functions. Extensive numerical results are presented in an orderly fashion for the first three anti-symmetric/symmetric natural frequencies of elliptical plates of selected geometries under different combinations of classical (clamped and free) and flexible boundary conditions. Also, the occurrences of “frequency veering” between various modes of the same symmetry group and interchange of the associated mode shapes in the veering region are noted and discussed. Moreover, selected 2D deformed mode shapes are presented in vivid graphical form. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature. The set of data reported herein is believed to be the first rigorous attempt to obtain the in-plane vibration frequencies of solid and annular thin elastic elliptical plates for a wide range of plate eccentricities.     

Analysis of the response of damped cylindrical shells carrying discontinuously constrained beam elements

A simple analysis is presented of the forced vibratory response of a cylindrical shell having a number of axial beams adhered to it by a viscoelastic material layer. The attached beams are identical, closely spaced and distributed around the full circumference of the shell. The excitation is a concentrated vibratory force acting radially at the mid-section on the surface of the shell. The end conditions of the shell and the attached beams are all assumed to be simply supported. The effects of the operational temperature and frequency on the viscoelastic material properties are considered. An experiment was conducted, for comparison, on a damped cylindrical shell suspended in air by lightweight elastic shock cords and driven at the mid-section by an electromechanical vibration shaker. Good correlations between the test data and analytical solutions were obtained over a wide range of frequencies.     

Interaction of a spherical acoustic wave with an elastic spherical shell

The interaction of a spherical acoustic wave with an elastic spherical shell is treated analytically. The solution includes the coupling between the acoustic sound field and vibration of the shell with any degree of fluid loading. The formulation for the far-field acoustic pressure is derived in terms of natural spherical wave functions, the properties of the acoustic medium, and the material constants of the shell. The far acoustic field is computed for a thin aluminum shell and several sound source locations over a large range of ka, where k is the wavenumber, and a is the shell radius. It is shown that the acoustic pressure depends significantly on whether the shell is in air or is submerged in water, particularly when the sound source is very near the surface. In air, the sound field of the shell is nearly identical to that of a rigid sphere but, in water, the shell is more compliant, which results in a damped radiation field that is characterized by vibrational resonances throughout the range of frequencies considered. As the sound sources is moved further away from the surface, however, this resonance response decreases very rapidly, and the sound field corresponds more closely to that of the shell in air.     

ON THE DISCRETIZATION OF AN ELASTIC ROD WITH DISTRIBUTED SLIDING FRICTION

A one-dimensional elastic system with distributed contact under fixed boundary conditions is investigated in order to study dynamic behavior under sliding friction. A partial differential equation of motion is established and its exact solution is presented. Due to the friction the eigenvalue problem is non-self-adjoint. Mathematical methods for handling the non-self-adjoint system, such as the non-self-adjoint eigenvalue problem and the eigenvalue problem with a proper inner product, are reviewed and applied. The exact solution showed that the undamped elastic system under fixed boundary conditions is neutrally stable when the coefficient of friction is a constant. The assumed mode approximation and the lumped-parameter discretization method are evaluated and their solutions are compared with the exact solution. As a cautionary example the assumed modes approximation leads to false conclusions about stability. The lumped-parameter discretization algorithm generates reliable results.     

Complex eigensolutions of coupled flexural and longitudinal modes in a beam with inclined elastic supports with non-proportional damping

Structure borne vibration and noise in an automobile are often explained by representing the full vehicle as a system of elastically coupled beam structures representing the body, engine cradle and body subframe where the engine is often connected to the chassis via inclined viscoelastic supports. To understand more clearly the interactions between a beam structure and isolators, this article examines the flexural and longitudinal motions in an elastic beam with intentionally inclined mounts (viscoelastic end supports). A new analytical solution is derived for the boundary coupled Euler beam and wave equations resulting in complex eigensolutions. This system is demonstrated to be self-adjoint when the support stiffness matrices are symmetric; thus, the modal analysis is used to decouple the equations of motion and solve for the steady state, damped harmonic response. Experimental validation and computational verifications confirm the validity of the proposed formulation. New and interesting phenomena are presented including coupled rigid motions, modal properties for ideal angled roller boundaries, and relationships between coupling and system modal loss factors. The ideal roller boundary conditions when inclined are seen as a limiting case of coupled longitudinal and flexural motions. In particular, the coupled rigid body motions illustrate the influence of support stiffness coupling on the eigenvalues and eigenfunctions. The relative modal strain energy concept is used to distinguish the contribution of longitudinal and flexural deformation modes. Since the beam is assumed to be undamped, the system damping is derived from the viscoelastic supports. The support damping (for a given loss factor) is shown to be redistributed between the system modes due to the inclined coupling mechanisms. Finally, this article provides valuable insight by highlighting some technical issues a real-life designer faces when balancing modeling assumptions such as rigid or elastic formulations, proportional or non-proportional damping, and coupling terms in multidimensional joint properties.     

The scattering of steady-state SH waves in a bi-material half space with multiple cylindrical elastic inclusions

The scattering of steady-state SH waves in a bi-material half space with multiple cylindrical elastic inclusions is presented analytically. Mirror method and multi-polar coordinate systems are developed to solve the complex boundary value problem. Considering the displacement and stress continuity conditions, a series of integral equations for unknown coefficients are obtained and solved by truncation. The solution is used to calculate the dynamic stress concentration factor around the edge of the inclusion, and the analysis and numerical results are discussed. The results show that degree of dynamic stress concentration around the circular inclusion is influenced by the incident angle, the incident wave number, and the parameters of medium.     

Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method

The main aim of this paper is to provide a simple yet efficient solution for the free vibration analysis of functionally graded (FG) conical shells and annular plates. A solution approach based on Haar wavelet is introduced and the first-order shear deformation shell theory is adopted to formulate the theoretical model. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. The separation of variables is first performed; then Haar wavelet discretization is applied with respect to the axial direction and Fourier series is assumed with respect to the circumferential direction. The constants appearing from the integrating process are determined by boundary conditions, and thus the partial differential equations are transformed into algebraic equations. Then natural frequencies of the FG shells are obtained by solving algebraic equations. Accuracy and reliability of the current method are validated by comparing the present results with the existing solutions. Effects of some geometrical and material parameters on the natural frequencies of shells are discussed and some selected mode shapes are given for illustrative purposes. It’s found that accurate frequencies can be obtained by using a small number of collocation points and boundary conditions can be easily achieved. The advantages of this current solution method consist in its simplicity, fast convergence and excellent accuracy.     

Stress and mixed boundary conditions for two-dimensional dodecagonal quasi-crystal plates

For plate bending and stretching problems in two-dimensional (2D) dodecagonal quasi-crystal (QC) media, the reciprocal theorem and the general solution for QCs are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order. The method developed by Gregory and Wan is used to generate necessary conditions which the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate; these decaying state conditions are obtained explicitly for axisymmetric bending and stretching of a circular plate when stress or mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution. For the stress data, our boundary conditions coincide with those obtained in conventional forms of plate theories. More importantly, appropriate boundary conditions with a set of mixed edge-data are obtained for the first time. Furthermore, the corresponding necessary conditions for transversely isotropic elastic plate are obtained directly, and their isotropic elastic counterparts are also obtained.      

Implementation of a constrained Ritz series modeling technique for acoustic cavity-structural systems

A prior study [Ginsberg, J. H. (2010b). J. Acoust. Soc. Am. 127, 2749-2758] used Ritz series in conjunction with Hamilton's principle to derive general equations describing the time domain response of an acoustic cavity bounded by an elastic structure. The equations of motion are supplemented by constraint equations that explicitly enforce velocity continuity at the cavity's surface. These constraints are imposed by the surface traction, which is represented by unknown coefficients of Ritz-type series. The resulting set of equations are differential-algebraic type. Three methods are presented to convert the governing equations to forms that are familiar to structural acoustics, including one that transforms them from differential-algebraic type to the standard ordinary differential equations associated with linear multi-degree-of-freedom vibratory systems. In cases where only the structure is excited, the formulation offers options as to how displacement/velocity boundary conditions on the nonstructural boundary are enforced, as well as whether zero pressure boundary conditions are enforced at all. An example of a one-dimensional waveguide that is closed at one end by an oscillator is used to explore the quality of solutions obtained from each of these options. Results for natural frequencies and mode functions are examined for accuracy and convergence.