Free Vibration Of Rectangular Plates Research Articles

Sensitivity analysis for free vibration of rectangular plate

This paper presents an effective sensitivity analysis algorithm for free vibration of a rectangular plate structure by using the finite element-transfer stiffness coefficient method (FE-TSCM). The basic concept of FE-TSCM combines both the modeling technique of the finite element method (FEM) and the transfer technique of the transfer stiffness coefficient method (TSCM) in order to benefit from the merits of both FEM and TSCM in the dynamic analysis of a structure. From the results computing the sensitivities of eigenvalues and eigenvectors for a simply supported rectangular plate structure by FE-TSCM and Fox's method, we can confirm that FE-TSCM has merits from the viewpoint of computational time and memory in sensitivity analysis for free vibration of the rectangular plate structure with a large number of degrees-of-freedom. In addition, when the design variables are the thicknesses of the plate elements, the results computing the sensitivities of the first eigenvalues for the rectangular plate structures with clamped edges are illustrated in detail by using the computation program introducing the concept of FE-TSCM.

Free Vibration of Rectangular Plates with Attached Discrete Sprung Masses

A direct approach is used to derive the exact solution for the free vibration of thin rectangular plates with discrete sprung masses attached. The plate is simply supported along two opposite edges and elastically supported along the two other edges. The elastic support can represent a range of boundary conditions from free to clamped supports. Considering only the compatibility of the internal forces between the plate and the sprung masses, the equations of the coupled vibration of the plate-spring-mass system are derived. The exact expressions for mode and frequency equations of the coupled vibration of the plate and sprung masses are determined. The solutions converge steadily and monotonically to exact values. The correctness and accuracy of the solutions are demonstrated through comparison with published results. A parametric study is undertaken focusing on the plate with one or two sprung masses. The results can be used as a benchmark for further investigation. The solution provided in the paper is general and includes several special cases, such as the plate with classical boundary conditions, the plate attached with discrete rigid masses, the plate supported by discrete springs and the plate restricted by rigid vertical point-supports.

Free Vibration of Rectangular Plates with Attached Discrete Sprung Masses

A direct approach is used to derive the exact solution for the free vibration of thin rectangular plates with discrete sprung masses attached. The plate is simply supported along two opposite edges and elastically supported along the two other edges. The elastic support can represent a range of boundary conditions from free to clamped supports. Considering only the compatibility of the internal forces between the plate and the sprung masses, the equations of the coupled vibration of the plate-spring-mass system are derived. The exact expressions for mode and frequency equations of the coupled vibration of the plate and sprung masses are determined. The solutions converge steadily and monotonically to exact values. The correctness and accuracy of the solutions are demonstrated through comparison with published results. A parametric study is undertaken focusing on the plate with one or two sprung masses. The results can be used as a benchmark for further investigation.The solution provided in the paper is general and includes several special cases, such as the plate with classical boundary conditions, the plate attached with discrete rigid masses, the plate supported by discrete springs and the plate restricted by rigid vertical point-supports.

Vibrations of rectangular plates with internal cracks or slits

This work applies the famous Ritz method to analyze the free vibrations of rectangular plates with internal cracks or slits. To retain the important and useful feature of the Ritz method providing the upper bounds on exact natural frequencies, the paper proposes a new set of admissible functions that are able to properly describe the stress singularity behaviors near the tips of the crack and meet the discontinuous behaviors of the exact solutions across the crack. The validity of the proposed set of functions is confirmed through comprehensive convergence studies on the frequencies of simply supported square plates with horizontal center cracks having different lengths. The convergent frequencies show excellent agreement with published accurate results obtained by an integration equation technique, and are more accurate than those obtained by a previously published approach using the Ritz method combined with a domain decomposition technique. Finally, the present solution is employed to obtain accurate natural frequencies and mode shapes for simply supported and completely free square plates with internal cracks having various locations, lengths, and angular orientations. Most of the configurations considered here have not been analyzed in the previously published literature. The present results are novel, and are the first published vibration data for completely free rectangular plates with internal cracks and for plates with internal cracks, which are not parallel to the boundaries.

Free Vibration Analysis of Punctured Plate by Matching Polynomials and Constraint Technique

The hallmark of this paper is the adoption of the constraint technique, in combination with a subparametric, triangular-plate bending element of first-order shear deformation, to maintain uniform mesh size and shape even while dealing with cutouts of arbitrary shapes. The evolution of two cutout models in the present investigation is a distinct improvement over the existing practices of cutout analysis. The use of matching polynomials offers the scope of eliminating the hazards of locking and spurious zero energy modes, while solving problems of very thin plates. Benchmark examples, as well as the author’s own problems on free vibration of rectangular plates with different shapes of cutouts, have been solved to exhibit versatility of these models. Mode shapes of plates with different shapes of cutout have also been provided.

Element-free Galerkin method for free vibration of rectangular plates with interior elastic point supports and elastically restrained edges

The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges. Based on the extended Hamilton’s principle for the elastic dynamics system, the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established using the element-free Galerkin method. Through numerical calculation, curves of the natural frequency of thin plates with three edges simply supported and one edge elastically restrained, and three edges clamped and the other edge elastically restrained versus the spring constant, locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained. Effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.

Free Vibration of Rectangular Plate with Delamination

AbstractIn this paper, the effect of delamination on free vibration of a simply supported rectangular homogeneous plate with through-width delamination was investigated by the finite strip method. A constrained model was used and a finite strip with bending and in-plane stiffness was derived for the free vibration analysis. The effects of delamination length, delamination location in the thickness-wise and spanwise directions, and aspect ratio of the plate on the natural frequencies of the plate were presented. Results show that the delamination has considerable effect on the natural frequencies of the plate. The aspect ratio of the plate is also having significant effect on the natural frequency of the plate, especially on the mode 2 frequency of the plate.

Free Vibration of Perforated Plates

A subparametric triangular plate bending element of first order shear deformation has been combined for the first time with the approach of constraints that helps maintain uniform mesh size and shape even while dealing with cutouts of arbitrary shapes. This is a distinct improvement over the existing practices of cutout modeling. Further the formulation being based on the subparametric element offers the scope of achieving matching modes, which enables the model to remain free from problems of locking and spurious zero energy modes while solving problems of very thin plates. Benchmark examples on free vibration of rectangular plates with cutouts have been solved to test the accuracy of the model. The author's own problems are also presented on mode shapes.

Vibration and buckling characteristics of weld-bonded rectangular plates using the flexibility function approach

Structures with a combination of spot welds and adhesive bonding, often referred to as weld-bonded structures, are likely to see increasing usage in automotive and other engineering structures. The present study considers a representative weld-bonded rectangular plate having simple supports on two opposite edges and weld-bonded support conditions with periodic spot welds along the other two edges. The study shows that the flexibility function approach for modeling free edges with point supports [Bapat AV, Venkatramani N, Suryanarayan S. Simulation of classical edge conditions by finite elastic restraints in the vibration analysis of plates. Journal of Sound and Vibration 1988;120(1):127–40; Bapat AV, Venkatramani N, Suryanarayan S. A new approach for the representation of a point support in the analysis of plates. Journal of Sound and Vibration 1988;120(1):107–25; Bapat AV, Venkatramani N, Suryanarayan S. The use of flexibility functions with negative domains in the vibration analysis of asymmetrically point-supported rectangular plates. Journal of Sound and Vibration 1988;124(3):555–76; Bapat AV, Suryanarayan S. Free vibrations of periodically point-supported rectangular plates. Journal of Sound and Vibration 1989;132(3):491–509; Bapat AV, Suryanarayan S. The flexibility function approach to vibration analysis of rectangular plates with arbitrary multiple point supports on the edges. Journal of Sound and Vibration 1989;128(2):203–33; Bapat AV, Suryanarayan S. Free vibrations of rectangular plates with interior point supports. Journal of Sound and Vibration 1989;134(2):291–313; Bapat AV, Suryanarayan S. Importance of satisfaction of point-support compatibility conditions in the simulation of point supports by the flexibility function approach. Journal of Sound and Vibration 1990;137(2):191–207; Bapat AV, Suryanarayan S. A fictitious foundation approach to vibration analysis of plates with interior point. Journal of Sound and Vibration 1992;155(2):325–41; Bapat AV, Suryanarayan S. A theoretical basis for the experimental realization of boundary conditions in the vibration analysis of plates. Journal of Sound and Vibration 1993;163(3):463–78], used in the direct series solution of the governing differential equations, can be employed very effectively to study the vibration and buckling characteristics of the weld-bonded rectangular plates. This is done by using a flexibility function constructed in terms of Fourier components to model the weld-bonded edge that represents the finite uniform flexibility of the adhesively bonded segment of the weld-bonded edge along with zero flexibility at the spot welds modeled as discrete point supports. A detailed convergence study shows that by a proper choice of the number of terms used to represent the flexibility function and the number of terms in the Levy sine series for the solution of the plate displacement, accurate results can be obtained for vibration and buckling characteristics. This paper also includes a parametric study undertaken to show the effect of plate geometry, number of spot welds and adhesive joint parameters. The paper also discusses how such parametric studies can be of use to the designer in arriving at an optimal joint configuration of weld-bonded rectangular plates from linear elastic buckling and free vibration considerations.

Health Monitoring for Damage Initiation and Progression During Mechanical Shock in Electronic Assemblies

Electronic products may be subjected to shock and vibration during shipping, normal usage, and accidental drop. High strain rate transient bending produced by such loads may result in failure of fine pitch electronic interconnects. Current experimental techniques rely on electrical resistance for determination of failure. Significant advantage can be gained by prior knowledge of impending failure for applications where the consequences of system failure may be catastrophic. This research effort focuses on an alternate approach to damage quantification in electronic assemblies subjected to shock and vibration, without testing for electrical continuity. The proposed approach can be extended to monitor product level damage. In this paper, statistical pattern recognition and leading indicators of shock damage have been used to study the damage initiation and progression in shock and drop of electronic assemblies. Statistical pattern recognition is currently being employed in a variety of engineering and scientific disciplines such as biology, psychology, medicine, marketing, artificial intelligence, computer vision, and remote sensing . The application quantification of shock damage in electronic assemblies is new. Previously, free vibration of rectangular plates has been studied by various researchers for development of analytical closed form models. In this paper, closed form models have been developed for the eigen frequencies and mode shapes of electronic assemblies with various boundary conditions and component placement configurations. Model predictions have been validated with experimental data from modal analysis. Pristine configurations have been perturbed to quantify the degradation in confidence values with progression of damage. Sensitivity of leading indicators of shock damage to subtle changes in boundary conditions, effective flexural rigidity, and transient strain response has been quantified. A damage index for experimental damage monitoring has been developed using the failure indicators. The above damage monitoring approach is not based on electrical continuity and hence can be applied to any electronic assembly structure irrespective of the interconnections. The damage index developed provides parametric damage progression data, thus removing the limitation of current failure testing, where the damage progression cannot be monitored. Hence the proposed method does not require the assumption that the failure occurs abruptly after some number of drops and can be extended to product level drops.

Free vibration of rectangular plates with internal column supports

A direct method is proposed to derive the exact solution for the free vibration of a thin rectangular plate with two opposite edges simply supported and with internal column supports. Considering the compatibility of displacements and rotations between the plate and the columns, the coupled vibration of a plate–column system is derived. The accuracy and correctness of the present method are demonstrated through the comparison of the results obtained from the proposed method and from finite-element analysis. Then the effects of column sizes and column models on the natural frequencies of a plate–column system are investigated in detail. A parametric study is focused on fully simply supported rectangular plates with a single internal column. It is shown that the effect of column flexibility in plate–column structures should be considered. The solution provided in the paper is general and includes several particular solutions for fixed point-supports, pinned point-supports and elastic point-supports. Results with high accuracy have been obtained and these can be used as the benchmark for the further investigation and for other approximate methods.

Effect of concentrated masses with rotary inertia on vibrations of rectangular plates

The effect of attached masses on free vibrations of rectangular plates is studied by considering rotary inertia of concentrated masses and geometric imperfections of the plate. Experiments are performed to understand the changes in natural frequencies and mode shapes. In order to validate and better understand the experiments, numerical results are also presented. Boundary conditions at the plate edges are those of simply supported plate with additional rotational springs at the plate edges. By tuning the stiffness of these springs, any boundary conditions comprised between simply supported and clamped edges can be simulated. Numerical results are successfully compared to experiments performed on a stainless-steel plate with an attached mass of different translational and rotary inertia. In particular, it is shown that a small mass placed on the diagonal of a square thin plate is enough to transform the shape of modes with one nodal line parallel to the edge, into diagonal modes, i.e. modes with diagonal nodal lines. Rotary inertia of concentrated masses reduces significantly natural frequencies. Moreover, additional modes due to the presence of large rotary inertia of concentrated masses arise, which do not appear if rotary inertia is neglected. Finally it is shown that masses placed only on one side of the plate or symmetrically (with respect to the mid-plane) placed on both sides give similar results.

Free vibration of rectangular plates with continuously distributed spring-mass

The vibratory characteristics of rectangular plates attached with continuously and uniformly distributed spring-mass in a rectangular region are studied which may represent free vibration of a human–structure system. Firstly, the governing differential equations of a plate with uniformly distributed spring-mass are developed. When the spring-mass fully occupies the plate, the natural frequencies of the coupled system are exactly solved and a relationship between the continuous system and a series of discrete two degrees-of-freedom system is provided. The degree of frequency coupling is defined. Then, the Ritz–Galerkin method is used to derive the approximate solution, when the spring-mass is distributed on a part of the plate, by using the Chebyshev polynomial series to construct the admissible functions. Comparative studies demonstrate the high accuracy and wide applicability of the proposed method. Finally, the frequency and modal characteristics of the plate partially occupied by distributed spring-mass are numerically analysed. It has been observed that both the natural frequencies and the modes appear in pairs. Moreover, a parametric study is performed for rectangular plates with three edges simply supported and one edge free. The effects of occupation size and position of the distributed spring-mass on natural frequencies of the coupled system are studied in detailed. The present investigation provides an improved understanding of human–structure interaction, such as grandstands or floors occupied by a stationary crowd.

Vibration Analysis of Rectangular Plates with Free Corners Using Spline‐Based Differential Quadrature

A spline‐based differential quadrature method (SDQM) is elaborated and applied to the vibration analysis of rectangular plates with free edges. The sextic B‐spline functions are used to construct the pertaining cardinal spline interpolants. Thus, explicit expressions of weighting coefficients for approximation of derivatives are obtained. Free vibrations of rectangular plates with free edges, which pose considerable challenge to the conventional differential quadrature, are dealt with and the results are in excellent agreement with those in the literature, highlighting the effectiveness and potential of the spline‐based differential quadrature.

Exact solutions for the free vibrations of rectangular plates having in-plane moments acting on two opposite simply supported edges

An exact solution procedure is formulated for the free vibration analysis of rectangular plates having two opposite edges simply supported when these edges are subjected to linearly varying normal stresses causing pure in-plane moments. The other two edges may be clamped, simply supported or free, or they may be elastically supported. The transverse displacement ( w) is assumed as sinusoidal in the direction of loading ( x), and a power series is assumed in the lateral ( y) direction (i.e., the method of Frobenius). Applying the boundary conditions yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to obtain adequate convergence for accurate vibration frequencies, as is demonstrated by two convergence tables. Some interesting and useful results for vibration frequencies and contour plots of their mode shapes are presented for plates having all nine possible combinations of clamped, simply supported or free unloaded edges. Particularly interesting is that for some of the edge conditions, applying opposite in-plane edge moments causes the fundamental frequency to increase initially.

Analysis of the free vibration of rectangular plates with central cut-outs using the discrete Ritz method

We present an analysis of the free vibration of plates with internal discontinuities due to central cut-outs. A numerical formulation for a basic L-shaped element which is divided into appropriate sub-domains that are dependent upon the location of the cut-out is used as the basic building element. Trial functions formed to satisfy certain boundary conditions are employed to define the transverse deflection of each sub-domain. Mathematical treatments in terms of the continuities in displacement, slope, moment, and higher derivatives between the adjacent sub-domains are enforced at the interconnecting edges. The energy functional results, from the proper assembly of the coupled strain and kinetic energy contributions of each sub-domain, are minimized via the Ritz procedure to extract the vibration frequencies and mode shapes of the plates. The procedures are demonstrated by considering plates with central cut-outs that are subjected to two types of boundary conditions.

SEMI-ANALYTICAL SHAPE FUNCTIONS IN THE FINITE ELEMENT ANALYSIS OF RECTANGULAR PLATES

This paper presents a method for the dynamic analysis of a thin, elastic, isotropic rectangular plate. The method is a hybrid of finite element theory and classical thin plate theory. The displacement functions are derived from Sanders' thin-shell equations, and are expanded in power series. Expressions for mass and stiffness are determined by precise analytical integration. The free vibrations of rectangular plates, with various boundary conditions, are studied by following this method. The results obtained reveal that the frequencies calculated in this way are in good agreement with those obtained by others.

FREE VIBRATION ANALYSIS OF RECTANGULAR PLATES WITH VARIOUSLY-SHAPED HOLES

An approximate method for analyzing the free vibration of rectangular plates with a hole of different shapes is proposed. The shapes of the holes are circular, semi-circular, elliptic, square, rectangular, triangular, rhombic, etc. These rectangular plates with a hole can be considered ultimately as a kind of rectangular plates with non-uniform thickness. A hole in a plate can be considered as an extremely thin part of the plate. Therefore, the free vibration problem of a plate with a hole can be translated into the free vibration problem of the equivalent rectangular plate with non-uniform thickness. For some plates with different-shaped holes the convergency and accuracy of the numerical solutions calculated by the proposed method are investigated.

Comparative accuracy of DQ and HDQ methods for three-dimensional vibration analysis of rectangular plates

An accuracy study between the Differential Quadrature (DQ) and Harmonic Differential Quadrature (HDQ) methods for three-dimensional elasticity solutions of free vibration of rectangular plates is carried out. The solution capability of the DQ and HDQ methods is first studied. Then the numerical performance of both the methods is compared. It is found that the DQ method displays more superior convergence characteristics over the HDQ method for the lower modes of vibration. However, the HDQ method is advantageous over the DQ method for computing higher modes of vibration. It is also discovered that the DQ and HDQ methods produce better convergent solutions than the Finite Element Method (FEM) when a similar number of discrete points/nodes are used. Copyright © 1999 John Wiley & Sons, Ltd.

THREE-DIMENSIONAL VIBRATION ANALYSIS OF RECTANGULAR PLATES BASED ON DIFFERENTIAL QUADRATURE METHOD

This paper presents the formulation and numerical analysis of the three-dimensional elasticity plate model using the differential quadrature (DQ) method. The governing equations in terms of displacement, stress-displacement relations, and boundary conditions for the three-dimensional plate model are first presented. These equations are then normalized and discretized using the DQ procedure. Example problems on the free vibration of rectangular plates with generic boundary conditions are selected. Two types of mesh pattern, the uniform mesh pattern and the cosine mesh pattern, were used and their convergence characteristics were studied. The cosine mesh was then chosen as the better mesh pattern pertaining to the problems solved. The solutions calculated using the cosine mesh pattern were then compared, where possible, with the exact or the numerical or the analytical solutions. It is found that the differential quadrature method yields accurate results for the plate problems under the current investigation.