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.     

EXPERIMENTAL MEASUREMENTS BY AN OPTICAL METHOD OF RESONANT FREQUENCES AND MODE SHAPES FOR SQUARE PLATES WITH ROUNDED CORNERS AND CHAMFERS

Most of the work done on vibration of plates published in the literature includes analytical and numerical studies with few experimental results available. In this paper, an optical system called the amplitude-fluctuation electronic speckle pattern interferometry for the out-of-plane displacement measurement is employed to investigate the vibration behavior of plates with rounded corners and with chamfers. The boundary conditions are traction free along the circumference of the plate. Based on the fact that clear fringe patterns will appear only at resonant frequencies, both resonant frequencies and corresponding mode shapes can be obtained experimentally using the present method. Numerical calculations by finite element method are also performed and the results are compared with the experimental measurements. Good agreements are obtained for both results. It is interesting to note that the mode number sequences for some resonant modes are changed. The transition of mode shapes from the square plate to the circular plate is also discussed.     

Free vibrations of a vessel consisting of circular plates and a shell of revolution having varying meridional curvature

An analytical solution procedure is presented for the free vibration of vessels consisting of a shell of revolution having varying meridional curvature and circular plate lids. The Lagrangian of vibration of the combined system is obtained in quadratic forms of boundary values. The frequency equation and the relations among the boundary values are obtained from minimizing conditions of the Lagrangian with respect to the unknown boundary values. The natural frequencies and the mode shapes of vessels having elliptical and hyperbolical meridians have been obtained by carrying out numerical calculations. Effects of various parameters upon natural frequencies and mode shapes are illustrated in discussions of numerical results.     

The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations

The effects of large vibration amplitudes on the first two axisymmetric mode shapes of clamped thin isotropic circular plates are examined. The theoretical model based on Hamilton's principle and spectral analysis developed previously by Benamar et al. for clamped-clamped beams and fully clamped rectangular plates is adapted to the case of circular plates using a basis of Bessel's functions. The model effectively reduces the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. Numerical results are given for the first and second axisymmetric non-linear mode shapes for a wide range of vibration amplitudes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions defining the non-linear transverse displacement function and the associated non-linear frequency are given. The non-linear frequencies associated to the fundamental non-linear mode shape predicted by the present model were compared with numerical results from the available published literature and a good agreement was found. The non-linear mode shapes exhibit higher bending stresses near to the clamped edge at large deflections, compared with those predicted by linear theory. In order to obtain explicit analytical solutions for the first two non-linear axisymmetric mode shapes of clamped circular plates, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. for beams and rectangular plates, has been adapted to the case of clamped circular plates, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values of 0.5 and 0.44 for the first and second axisymmetric non-linear mode shapes, respectively.     

The effect of different combinations of boundary conditions on the average radiation efficiency of rectangular plates

The boundary conditions of a vibrating plate are known to have an influence on its sound radiation for frequencies below the critical frequency. To investigate this effect in a systematic way, the average radiation efficiency and radiated power are calculated for a rectangular plate set in an infinite baffle using a modal summation approach. Whereas analytical expressions exist for simply supported boundary conditions, a numerical approach is required for other cases. Nine combinations of boundary conditions are considered, consisting of simply supported, clamped and free edges on different plate edges. The structural vibration is approximated by using independent beam functions in orthogonal directions allowing simple approximate formulae for mode shapes and natural frequencies. This assumption is checked against a finite element model and shown to give reliable results. It is shown that a free plate has the lowest radiation efficiency and a clamped plate the highest for most frequencies between the fundamental panel natural frequency and the critical frequency. Other combinations of boundary condition give intermediate results according to the level of constraint introduced. The differences depend on frequency: excluding the extreme case of a fully free plate all the other boundary conditions give results within a range of 8 dB in the middle part of the short-circuiting region, decreasing towards the critical frequency. At low frequency the differences can be even greater, in some cases up to 20 dB. These conclusions are shown to hold for a range of plate thicknesses and dimensions.     

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.     

Free vibration of polar-orthotropic sector plates

The free vibration of ring-shaped polar-orthotropic sector plates is analyzed by the Ritz method using a spline function as an admissible function for the deflection of the plates. For this purpose, the transverse deflection of a sector plate is written in a series of the products of the deflection function of a sectorial beam and that of a circular beam satisfying the boundary conditions. The deflection function of the sectorial beam is approximately expressed by a quintic spline function, which satisfies the equation of flexural vibration of the beam at each point dividing the beam into small elements. The frequency equation of the plate is derived by the conditions for a stationary value of the Lagrangian. The present method is applied to ring-shaped polar-orthotropic sector plates with some combination of boundary conditions, and the natural frequencies and the mode shapes are calculated numerically up to higher modes. This method is very effective for the study of vibration problems of variously shaped anisotropic plates including these sector plates.     

Axisymmetric vibrations of polar orthotropic mindlin annular plates of variable thickness

Analysis and numerical results are presented for the axisymmetric vibrations of polar orthotropic annular plates with linear variation in thickness, according to Mindlin's shear theory of plates. A chebyshev collocation technique has been employed to obtain the frequency equations for the transverse motion of such plates, for three different boundary conditions. Frequencies, mode shapes and moments for the first three modes of vibration have been computed for different plate parameters. A comparison of frequencies with the corresponding values obtained by classical plate theory leads to some interesting conclusions.     

Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions

In this paper, the free and forced vibration analysis of circular cylindrical double-shell structures under arbitrary boundary conditions is presented. This is achieved by employing the improved Fourier series method based on Hamilton’s principle. In the formulation, each displacement component of the cylindrical shells and annular plates is invariantly expanded as the superposition of a standard Fourier series with several supplementary functions introduced to remove the potential discontinuities of the original displacement and its derives at the boundaries. With the introduction of four sets of boundary springs at the coupling interfaces and end boundaries of the shell–plate combination, both elastic and rigid coupling and end boundary conditions can be easily obtained by assigning the stiffnesses of the artificial springs to certain values. The natural frequencies and mode shapes of the structures as well as frequency responses under forced vibration are obtained with the Rayleigh–Ritz procedure. The convergence of the method is validated by comparing the present results with those obtained by the finite element method. Several numerical results including natural frequencies and mode shapes are presented to demonstrate the excellent accuracy and reliability of the current method. Finally, a number of parameter studies concerning various end and coupling boundary conditions, different dimensions of shells and annular plates are also performed.     

Modal analysis of circular plates with radial side cracks and in contact with water on one side based on the Rayleigh–Ritz method

In this paper, free vibrations of the baffled circular plates with radial side cracks and in contact with water on one side are investigated based on the Rayleigh–Ritz method. The completely free, simply supported and completely clamped boundary conditions are considered. Corner functions are introduced to describe the singularities at the crack tip. The motion of water is expressed by the velocity potential and the interaction between the water and the plate is derived in the form of an integral equation including the dynamic deformation of the cracked plate. The convergence studies are carried out and the numerical results show that the distinctions between the dry and wet mode shapes will be increased obviously excluding the first symmetric and antisymmetric modes when cracks appear. When the approximate methods based on the assumption that the wet modes are identical with the dry modes are adopted to calculate the eigenfrequencies, the errors of the results for cracked circular plates are larger than those for intact ones. The influences of the water on the symmetric and antisymmetric modes are different evidently, and the greatest reduction ratio of eigenfrequency and least difference between dry and wet mode are relative to the first symmetric mode. The verifications based on numerical simulation show that the proposed method is adequate for the investigation of free vibration of baffled circular plates with radial side cracks and in contact with water on one side.     

VIBRATION CHARACTERISTICS AND TRANSIENT RESPONSE OF SHEAR-DEFORMABLE FUNCTIONALLY GRADED PLATES IN THERMAL ENVIRONMENTS

Free and forced vibration analyses for initially stressed functionally graded plates in thermal environment are presented. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Theoretical formulations are based on Reddy's higher order shear deformation plate theory and include the thermal effects due to uniform temperature variation. The plate is assumed to be clamped on two opposite edges with the remaining two others either free, simply supported or clamped. One-dimensional differential quadrature technique, Galerkin approach, and the modal superposition method are used to determine the transient response of the plate subjected to lateral dynamic loads. Comprehensive numerical results for silicon nitride/stainless-steel rectangular plates are presented in dimensionless tabular and graphical forms. The roles played by the constituent volume fraction index, temperature rise, shape and duration of dynamic loads, initial membrane stresses as well as the character of boundary conditions are studied. The results reveal that, when thermal effects are considered, functionally graded plates with material properties intermediate to those of isotropic ones do not necessarily have intermediate natural frequencies and dynamic responses.     

Free vibration of three-dimensional anisotropic layered composite nanoplates based on modified couple-stress theory

Based on the modified couple-stress theory, three-dimensional analytical solutions of free vibration of a simply supported, multilayered and anisotropic composite nanoplate are derived by solving an eigenvalue system and using the propagator matrix method. By expanding the solutions of the extended displacements in terms of two-dimensional Fourier series, the final governing equations of motion with modified couple-stress effect are reduced to an eigenvalue system of ordinary differential equations. Analytical expressions for the natural frequencies and mode shapes of multilayered anisotropic composite plates with modified couple-stress effect are then derived via the propagator matrix method. Numerical examples are carried out for homogeneous thick-plates and sandwich composite plates to show the effect of the non-local parameter in different layers and stacking sequence on the mode shapes. The present solutions can serve as benchmarks to various thick-plate theories and numerical methods, and could be further useful for designing layered composite structures involving small scale.     

Vibration of moderately thick square orthotropic stepped thickness plates

Numerous studies that address the vibration of stepped thickness plates are reported in the literature. Predominately, classical plate theory has been used to formulate studies for both isotropic and anisotropic stepped plates. Mindlin plate theory has been employed to obtain results for thick isotropic stepped thickness plates. Exact solutions, Rayleigh-Ritz, differential quadrature and finite element methods have been employed to compute results for frequency of vibration. Results for frequency of vibration for thick orthotropic stepped thickness plates are presented here using orthorhombic material properties of aragonite. The finite element method has been used to compute frequencies and determine mode shapes for simply supported and clamped square Mindlin plates.     

Vibration analysis of annular-like plates

The existence of eccentricity of the central hole for an annular plate results in a significant change in the natural frequencies and mode shapes of the structure. In this paper, the vibration analysis of annular-like plates is presented based on numerical and experimental approaches. Using the finite element analysis code Nastran, the effects of the eccentricity, hole size and boundary condition on vibration modes are investigated systematically through both global and local analyses. The results show that analyses for perfect symmetric conditions can still roughly predict the mode shapes of “recessive” modes of the plate with a slightly eccentric hole. They will, however, lead to erroneous results for “dominant” modes. In addition, the residual displacement mode shape is verified as an effective parameter for identifying damage occurring in plate-like structures. Experimental modal analysis on a clamped-free annular-like plate is performed, and the results obtained reveal good agreement with those obtained by numerical analysis. This study provides guidance on modal analysis, vibration measurement and damage detection of plate-like structures.     

The flexural vibration of rectangular plates with point supports

Orthogonally generated polynomial functions are used in the Lagrangian multiplier method to study the free, flexural vibration problem of point supported, thin, flat, rectangular plates. The analysis applies to isotropic and specially orthotropic plates having any combination of clamped, simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges. Numerical results are presented for a number of specific problems, illustrating the accuracy and versatility of the approach, and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports.     

DSC ANALYSIS OF RECTANGULAR PLATES WITH NON-UNIFORM BOUNDARY CONDITIONS

This paper introduces the discrete singular convolution (DSC) for the vibration analysis of rectangular plates with non-uniform and combined boundary conditions. A systematic scheme is proposed for the treatment of boundary conditions required in the proposed approach. The validity of the DSC approach for plate vibration is tested by using a large number of numerical examples that have a combination of simply supported, clamped and transversely supported (with non-uniform elastic rotational restraint) edges. The present results are in excellent agreement with those in the literature.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

An analytical and experimental study of the vibration response of a clamped ribbed plate

An analytical solution is presented in this paper for the vibration response of a ribbed plate clamped on all its boundary edges by employing a traveling wave solution. A clamped ribbed plate test rig is also assembled in this study for the experimental investigation of the ribbed plate response and to provide verification results to the analytical solution. The dynamic characteristics and mode shapes of the ribbed plate are measured and compared to those obtained from the analytical solution and from finite element analysis (FEA). General good agreements are found between the results. Discrepancies between the computational and experimental results at low and high frequencies are also discussed. Explanations are offered in the study to disclose the mechanism causing the discrepancies. The dependency of the dynamic response of the ribbed plate on the distance between the excitation force and the rib is also investigated experimentally. It confirms the findings disclosed in a previous analytical study [T.R. Lin, J. Pan, A closed form solution for the dynamic response of finite ribbed plates, Journal of the Acoustical Society of America 119 (2006) 917–925] that the vibration response of a clamped ribbed plate due to a point force excitation is controlled by the plate stiffness when the source is more than a quarter plate bending wavelength away from the rib and from the plate boundary. The response is largely affected by the rib stiffness when the source location is less than a quarter bending wavelength away from the rib.     

EXPERIMENTAL AND THEORETICAL INVESTIGATION OF THE LINEAR AND NON-LINEAR DYNAMIC BEHAVIOUR OF A GLARE 3 HYBRID COMPOSITE PANEL

A theoretical model based on Hamilton's principle and spectral analysis is used to study the non-linear free vibration of hybrid composite plates made of Glare 3, a new aircraft structural material. It consists of alternating layers of metal- and fibre-reinforced composites. In previous work, the theoretical model has been used to calculate the first non-linear mode of fully clamped rectangular composite fibre-reinforced plastic (CFRP) laminated plates. This study concerns determination of the linear dynamic properties of the Glare 3 hybrid composite rectangular panel (G3HCRP) such as natural frequencies and mode shapes. The theoretical model is used to calculate the fundamental non-linear mode shape and associated flexural behaviour of the fully clamped G3HCRP. A series of experimental investigations have been conducted using a G3HCRP in order to determine linear dynamic properties. The response due to random excitation was investigated and the experimental measurements are analyzed and discussed. Comparisons are made with finite element predictions and response estimates given by the ESDU method, the latter being a “design guide” approach used by industry. Concerning the non-linear analysis, the results are given for various plate aspect ratios and vibration amplitudes, showing a higher increase of the induced bending stress near the clamps at large deflections. Comparisons between the dynamic behaviour of an isotropic plate and G3HCRP at large vibration amplitudes are presented and good results are obtained.     

Transverse vibration of clamped trapezoidal plates having rectangular orthotropy

This paper deals with the free transverse vibration of orthotropic trapezoidal plates clamped at the edges. A series-type method is applied for obtaining an analytical solution of the problem, and the resulting frequency equation is presented for symmetric trapezoids. In the numerical study, accurate frequency parameters and mode shapes of the plates are calculated for the first several modes, and the effects of the orthotropy are discussed.