FREE VIBRATION ANALYSIS OF CANTILEVER PLATES WITH STEP DISCONTINUITIES IN PROPERTIES BY THE METHOD OF SUPERPOSITION

Utilizing the superposition method, an analytical type solution is obtained for the free vibration eigenvalues and mode shapes of a cantilever plate with step discontinuities in plate properties. Property discontinuity lines run parallel to the clamped edge of the plate. Verification tests are performed for limiting cases by comparing computed eigenvalues with known eigenvalues for plates with uniform properties. Very good agreement is also obtained when computed results are compared with those obtained experimentally utilizing a test plate with discontinuities in thickness. Computed eigenvalues and mode shapes are presented for the benefit of other researchers. Besides the general interest, the problem has an application in the modelling of certain multi-story buildings during seismic studies.     

A note on vibration of a cantilever plate immersed in water

The added mass of the fluid surrounding it plats an important role in the dynamic behaviour of a submerged structure. The first few mode shapes and the respective natural frequencies of a submerged cantilever plate are found by using a finite element procedure, eigenvalues being obtained by a simultaneous iteration technique. The influence of the water depth below the plate and also of the water's lateral extent is considered, in order to test the convergency of the results. Results on the effects of the depth of immersion on the natural frequencies and mode shapes of the cantilever plate for different aspect ratios are presented.     

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.     

Exact solutions of in-plane natural vibration of a circular plate with outer edge restrained elastically

We solve exactly the equations of in-plane natural vibration for a circular plate whose outer edge is restrained elastically. The mode shapes are represented by trigonometric functions with a number of nodal diameters in the circumferential direction and mode functions in the radial direction. We present the exact frequency equations and mode functions and tabulate the frequency parameters satisfying the frequency equations. The corresponding mode functions and two-dimensional mode shapes are illustrated when both radial and tangential stiffness are zero (free edge), infinity (clamped edge), or medium. Comparisons with previous reported results confirm the accuracy of the present work.     

Nodal line optimization and its application to violin top plate design

In the literature, most problems of structural vibration have been formulated to adjust a specific natural frequency: for example, to maximize the first natural frequency. In musical instruments like a violin; however, mode shapes are equally important because they are related to sound quality in the way that natural frequencies are related to the octave. The shapes of nodal lines, which represent the natural mode shapes, are generally known to have a unique feature for good violins. Among the few studies on mode shape optimization, one typical study addresses the optimization of nodal point location for reducing vibration in a one-dimensional beam structure. However, nodal line optimization, which is required in violin plate design, has not yet been considered. In this paper, the central idea of controlling the shape of the nodal lines is proposed and then applied to violin top plate design. Finite element model for a violin top plate was constructed using shell elements. Then, optimization was performed to minimize the square sum of the displacement of selected nodes located along the target nodal lines by varying the thicknesses of the top plate. We conducted nodal line optimization for the second and the fifth modes together at the same time, and the results showed that the nodal lines obtained match well with the target nodal lines. The information on plate thickness distribution from nodal line optimization would be valuable for tailored trimming of a violin top plate for the given performances.     

Determination of the steady state response of viscoelastically point-supported rectangular specially orthotropic plates by an energy-based finite difference method

A method based on a variational procedure in conjunction with a finite difference method is used to examine the free vibration characteristics and steady state response to a sinusoidally varying force applied at the center of a viscoelastically point-supported orthotropic elastic plate of rectangular shape. Using the energy-based finite difference method, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, and of the damping of the supports to the mode shapes and to the steady state response of viscoelastically point-supported rectangular plates is investigated numerically for a concentrated load at the center for various values of the mechanical properties characterizing the anisotropy of the plate material and for various damping ratios. The results are given for the frequencies and mode shapes of the first three symmetrical modes. Convergence studies are made. The validity of the present approach is demonstrated by comparing the results with other solutions based on the Kirchhoff-Love plate theory.     

Damage localisation in plate like-structures using the two-dimensional polynomial annihilation edge detection method

The topic of non-destructively detecting localised damage in plates is addressed in this article. Since the presence of a crack or a delamination causes a discontinuity in the mode shape first derivatives, a numerical method for detecting discontinuities in smooth piecewise functions and their derivatives, based on a polynomial-annihilation technique is presented. The method, already proposed for beam-type structures, has been extended to enable the detection and localisation of damage in plate-like structures for which only post-damage mode shapes are available. Applying finite element analysis, the mode shapes of an isotropic plate with a saw-cut and a multi-layered composite plate with a delamination have been calculated and the performance of the approach evaluated for increasing amounts of noise. Encouraging results indicate that further development of the technique for non-destructive testing of plate-like structures would be highly worthwhile.     

Index to volume 95

The linear and non-linear (large amplitude), axisymmetric free vibration of a circular plate of variable thickness, with immovable edges, is analyzed by applying the transfer matrix method. Two types of circular plate are studied: the stepped thickness plate and the continuously variable thickness plate. Numerical calculations are carried out for these two types of plate, with both simply supported and clamped edges, and the backbone curves and mode shapes are determined. The results are compared with those of other authors.     

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.     

A note on the damping of large amplitude beam vibrations

This paper presents a new series-type method for solving the eigenvalue problems of irregularly shaped plates clamped at all edges. An irregularly shaped plate is formed on a simply supported rectangular plate by rigidly fixing several segments. With the reaction forces and moments acting on all edges of an actual plate of irregular shape regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force and moment distributions along the edges are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to a cross-shaped, an I-shaped and an L-shaped plate clamped at all edges, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the shape is discussed.     

Free vibration of skew Mindlin plates by p-version of F.E.M.

Accurate natural frequencies and mode shapes of skew plates with and without cutouts are determined by p-version finite element method using integrals of Legendre polynomials for p=1-14. The hierarchical plate element is formulated based on Mindlin's plate theory including rotatory inertia effects and based on a skew co-ordinate system. Non-dimensional frequency parameter and mode shapes are presented for a range of skew angle (β), aspect ratio (a/b), thickness-width ratio (h/b), cutout dimensions and different boundary conditions. The results were verified by comparison with those available in the open literature.     

Free vibration of three-dimensional multilayered magneto-electro-elastic plates under combined clamped/free boundary conditions

In this paper, we study the free vibration of multilayered magneto-electro-elastic plates under combined clamped/free lateral boundary conditions using a semi-analytical discrete-layer approach. More specifically, we use piecewise continuous approximations for the field variables in the thickness direction and continuous polynomial approximations for those within the plane of the plate. Group theory is further used to isolate the nature of the vibrational modes to reduce the computational cost. As numerical examples, two cases of the lateral boundary conditions combined with the clamped and free edges are considered. The non-dimensional frequencies and mode shapes of elastic displacements, electric and magnetic potentials are presented. Our numerical results clearly illustrate the effect of the stacking sequences and magneto-electric coupling on the frequencies and mode shapes of the anisotropic magneto-electro-elastic plate, and should be useful in future vibration study and design of multilayered magneto-electro-elastic plates.     

Free vibration of a circular plate elastically restrained along some radial segments

An analysis is presented for the free vibration of a circular plate restrained against deflection along radial segments. With the reaction forces acting on the segments regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force distributions along the segments are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the supports. The natural frequencies and the mode shapes of the plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to circular plates supported along several radial segments located at equal angular intervals, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the supports is discussed.     

Free vibration analysis of stiffened plates using finite difference method

Free vibration characteristics of rectangular stiffened plates having a single stiffener have been examined by using the finite difference method. A variational technique has been used to minimize the total energy of the stiffened plate and the derivatives appearing in the energy functional are replaced by finite difference equations. The energy functional is minimized with respect to discretized displacement components and natural frequencies and mode shapes of the stiffened plate have been determined as the solutions of a linear algebraic eigenvalue problem. The analysis takes into consideration inplane deformation of the plate and the stiffener and the effect of inplane inertia on the natural frequencies and mode shapes. The effect of the ratio of stiffener depth to plate thickness on the natural frequencies of the stiffened plate has also been examined.     

Flexural vibrations of a rectangular plate for the lower normal modes

Theoretical and experimental results for flexural waves of a rectangular plate with free ends are obtained. Both the natural frequencies and mode shapes are analyzed for the lower normal modes. To take into account the boundary conditions, a plane wave expansion method is used to solve the thin plate theory also known as the 2D Kirchhoff-Love equation. The excitation and detection of the normal modes of the out-of-plane waves are performed using non-contact electromagnetic-acoustic transducers. We conclude that this experimental technique is highly reliable due to the good agreement between theory and experiment.     

Structural–acoustic model of a rectangular plate–cavity system with an attached distributed mass and internal sound source: Theory and experiment

In this paper three approaches are combined to develop a structural–acoustic model of a rectangular plate–cavity system with an attached distributed mass and internal sound source. The first approach results from a recently presented analysis based on the Rayleigh–Ritz method and is used to circumvent the difficulties in obtaining the natural frequencies and mode shapes of a plate with an attached, distributed mass. Furthermore, different plate boundary conditions can be accommodated. The resulting mode shapes are defined as continuous functions; this is advantageous as they can be directly used in the second approach, i.e., the classic modal-interaction approach in order to obtain the coupled equations of the system. Finally, in the third approach a group of point sources emitting a pressure pulse in the time domain is used to model an internal sound source. For the validation of the developed model an experiment was conducted in two configurations using a simply supported aluminium plate and a clamped plate coupled with a plexiglas box containing a loudspeaker. Good agreement was found between the analytical and experimental data.     

Vibrations of a square plate with parabolically varying thickness

Vibration results for a square plate with a parabolically varying thickness distribution and built-in edges are presented. Frequency and mode shape predictions obtained from a finite element analysis are compared with measurements made with real-time laser holography. In general, the agreement between the two is very good except for a few of the lower modes where the predicted frequencies are about 15% high. So far, a satisfactory explanation for this has not been found. The vibration mode shapes for the plate exhibit a striking blend of radial and square symmetries that results from the axisymmetric thickness distribution and the square symmetry of the boundary frame.     

Axially symmetric stability of a completely free circular plate subjected to a non-conservative edge load

This paper presents a study on the behaviour of the vibration and stability of a two dimensional structure: i.e., a completely free circular plate subjected to non-conservative radial loading. The eigencurves and mode shapes of the circular plate are presented for various values of the non-conservativeness parameter. Some interesting conclusions concerning the behaviour of a completely free plate are drawn from the analytical investigation of the solution of the problem and from the numerical calculations as well.     

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.     

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.