Influence of support properties on the sound radiated from the vibrations of rectangular plates

The control of the forced vibration response of structures through the optimal tuning of its supports is desirable in many applications. Tuning may enhance the dissipation of vibration energy within the supports, thereby reducing fatigue and structure-borne noise. Two different models were developed to calculate the optimal support stiffness that minimizes the velocity response of homogeneous plates. The first model, based on the wave propagation at the edge, yields a good first cut approximation of the optimal properties. The optimal viscous and viscoelastic support stiffness for minimal reflection at the edge was calculated. Maximum absorption of the incident waves occurs when the viscous support stiffness matches the characteristic mechanical impedances of the plate. The second model, based on the Rayleigh-Ritz method, yields more accurate estimates of the optimal support stiffness required to minimize the forced velocity response of the finite rectangular plate. The optimal support properties calculated from the two different methods were in good agreement. This suggested that the modal response of the plate is strongly influenced by the wave reflections at the edges. Finally, the effects of support properties on the sound radiated from the plate were investigated. The optimal support stiffness that minimizes the radiated sound power was found to be smaller than the value that minimizes the velocity response. The results show that both the velocity response and sound radiation are strongly influenced by dissipation of vibration energy at the edges, and demonstrate that support tuning can yield significant noise and vibration reduction.     

Analysis of the flow-induced vibrations of viscoelastically supported rectangular plates

The purpose of this study was to develop a theoretical model for the flow-induced vibration of viscoelastically supported rectangular plates. In particular, the influence of the dynamic mechanical properties of the elements supporting the plate was investigated. The case of a homogeneous rectangular plate supported along all four edges by a complex viscoelastic element was treated. The Rayleigh-Ritz method was used applying beam functions as the trial functions. This approach ensured a fast convergence rate, which is advantageous for vibration analysis of high order modes. The flow-induced vibration of the plate was calculated using the Corcos model for the surface pressure loading. The results suggest that there is an optimal support stiffness that minimizes the flow-induced vibration response of the plate.     

Vibrations of a polygonal plate having orthogonal straight edges by an extended rayleigh-ritz method

An extended Rayleigh-Ritz method is presented for solving vibration problems of a polygonal plate having orthogonal straight edges. The polygonal plate is considered as an assemblage of several rectangular plates. For each element rectangular plate, the transverse displacement is approximated by interpolation functions corresponding to unknown displacements and slopes at the discrete points which are chosen along the edges, and series of trial functions which satisfy homogeneous artificial boundary conditions. By minimizing the energy functional corresponding to the assumed displacement function, the dynamic stiffness matrix of the element rectangular plate, which is similar to that obtained in the finite element method, is derived. The dynamic stiffness matrix of the whole system is obtained by summing up those of the element rectangular plates. Numerical results are presented for the natural frequencies and mode shapes of cantilever L-shaped and T-shaped plates.     

Transverse vibrations of rectangular plates with edges elastically restrained against translation and rotation

The fundamental frequency coefficient for a rectangular plate with edges elastically restrained against both translation and rotation is calculated by using polynomial coordinate functions and the Rayleigh-Ritz method. The approach is simple and straightforward and allows the solution of a rather difficult elastodynamics problem. Complicating factors (orthotropic properties, in-plane forces, concentrated masses, etc.) can also be taken into account without formal difficulties.     

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the superposition method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's superposition method. These are compared with corresponding results based on the modes using the Rayleigh-Ritz method using the ordinary and degenerated free-free beam functions. There is an excellent agreement between the results from both approaches but the superposition method has shown faster convergence and the results may serve as benchmarks for the transient response of completely free plates.     

Vibration analysis of rectangular plates with general elastic boundary supports

In this investigation, the Rayleigh-Ritz method is used to determine the modal characteristics of a rectangular plate with general elastic supports alone its edges. Each of the admissible functions here is composed of a trigonometric function and an arbitrary continuous function that is introduced to ensure the sufficient smoothness of the so-called residual displacement function at the edges. As a result, a drastic improvement of the convergence can be expected of the solution expressed as a series expansion in terms of the admissible functions. Perhaps more importantly, this study has developed a general approach for deriving a complete set of admissible functions that can be universally applied to various boundary conditions. Several numerical examples are given to demonstrate the accuracy and convergence of the current solution.     

Natural frequencies of orthotropic rectangular plates with stepped thickness

It is shown how one can derive an approximate formula for estimating a natural frequency of an orthotropic rectangular plate with stepped thickness by using several natural frequencies of the corresponding isotropic plate reduced from the orthotropic one. To justify the method, an orthotropic two-part rectangular plate with simply supported sides is discussed, and an approximate formula is proposed for estimating the fundamental natural frequency.     

Transverse vibrations of thin,elastic plates with concentrated masses and internal elastic supports

The fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined. The case of an orthotropic, rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the Galerkin method. Then, vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method. Finally the case of a circular plate elastically restrained against translation and rotation is considered.     

On the vibration of an elastic plate on an elastic foundation

The forced vibration of an elastic plate under a time harmonic point force is studied. The plate is infinite in extent and supported by an elastic foundation. This study is made on the basis of the improved (Timoshenko) plate theory. The mathematical problem is to seek a fundamental solution (the Green's function) of the time-reduced plate equation of the improved plate theory. Such a fundamental solution is constructed by the distributional Fourier transform method. From the explicit expressions of the fundamental solution, the behavior of the fundamental singularity as a function of the vibration frequency and the foundation stiffness is examined. Conditions under which plate resonance occurs are also determined.     

Analytical modeling of squeeze film damping for rectangular elastic plates using Green's functions

An analytical method for the solution of squeeze film damping based on Green's function to the nonlinear Reynolds equation considering an elastic plate is presented. This allows calculating the stiffness and damping forces rapidly for various boundary conditions. The elastic plate velocity is applied to the nonlinear Reynolds equation as a forcing term. The nonlinear Reynolds equation is divided into multiple linear nonhomogeneous Helmholtz equations, which then can be solvable using the presented approach. Approximate mode shapes of a rectangular elastic plate are used, enabling the calculation of the damping ratio and frequency shift for the linear case, as well as the complex resistant pressure, for both linear and nonlinear cases.     

Vibrations of rectangular plates with elastically restrained edges

The Rayleigh-Ritz method is used to determined natural frequencies in transverse vibration of rectangular plates with elastically restrained edges. By treating an elastically restrained edge as intermediate between an appropriate pair of classical boundary conditions and using the corresponding vibration mode shapes of beams with classical boundary conditions as assumed functions, a relatively small number of functions is required; consequently only a modest quantity of computation is necessary. The good accuracy of the method is demonstrated by solving test problems. The method can be applied to a wide range of elastic restraint conditions, any aspect ratio and for higher modes in addition to the fundamental. The usefulness and accuracy of existing simplified approaches to the problem are assessed. The effect of in-plane forces on the natural frequencies and the determination of critical loads for plates with these restraint conditions are considered also.     

Vibration characteristics of a cantilever plate with attached spring-mass system

Coupled free vibration analysis has been performed on a cantilever thin plate carrying a spring-mass system attached on an arbitrary point by using Rayleigh-Ritz method. Influence of an attached ‘spring-mass’ system, i.e., attached position, relative values of mass and spring constant, on the coupled vibration characteristics of the system has been clarified comparing with those of uncoupled ones. Optimal attached position to maximize coupled plate natural frequency is also investigated and shown in contour diagrams. The influence of an attached mass has also been investigated, as the limiting case whereby the spring stiffness of the ‘spring-mass’ system approaches infinity.     

周界固定带裂纹薄圆板的振动基频

在将裂纹板挠度表示为正交多项式的基础上，用分区技术的瑞利一里兹方法求解了含不同长度径向裂纹和弧向裂纹的周界固定圆板的振动基频。实验上用小球撞击法测定了裂纹圆板的辐射声场频谱，并得出了谱的基频。此外用有限元法对裂纹板作了理论计算。三项结果基本符合。探究了裂纹长度对圆板基频的影响规律。     

Analysis of structural - acoustic coupling of elastic rectangular enclosure with arbitrary boundary conditions

The structural acoustic coupling characteristics of a rectangular enclosure con- sisting of two elastic supported flexible plates and four rigid plates are analyzed.A general formulation considering the full coupling between the plates and cavity is developed by using Hamiltonian function and Rayleigh-Ritz method.By means of continuous distributions of ar- tificial springs along boundary of flexible plates,a wide variety of boundary conditions and structure joint conditions are considered.To demonstrate the validity of the analytical model, the responses of sound pressure in the cavity and plate velocity are worked out.The analytical results coincides well with Kim's experimental results.The result is satisfactory.Finally,an- alytical results on the structure vibration and the sound field inside the cavity are presented. These results indicate that the coupling of the combined structure is relatively weak,so the internal cavity sound is controlled by plate directly excited,and the translational stiffness affects the sound more than the rotational stiffness does.     

Approximate closed form solutions for free vibration of polar orthotropic circular plates

For free vibrations of polar orthotropic plate, simple approximate closed form solutions for mode shapes and its natural frequencies were obtained using the Rayleigh-Ritz method. Coordinate function satisfying the natural boundary conditions and the predetermined coefficients was adapted, which results in compact expressions and enables to readily calculate symmetric and nonsymmetric natural frequencies for arbitrary values of the elastic constants. The derived formulation can be used in designing of circular plates such as wood disk, which are naturally endowed with material orthotropy as well as fiber reinforced composite materials. The model can easily be used for the evaluation of parametric studies on dynamic behaviors and nondestructive methods during the initial design process.     

A reduction method for problems of vibration of orthotropic plates

It is shown that the problem of vibration of an orthotropic plate can be reduced to that of another orthotropic plate by a simple co-ordinate transformation, and reduction formulae are obtained. To justify the reduction formulae, fundamental natural frequencies of orthotropic rectangular plates with various boundary conditions and of a clamped orthotropic elliptical plate are discussed. As an example, an exact natural frequency of a simply supported generally orthotropic skew plate with special flexural rigidities is obtained from that of a simply supported isotropic rectangular plate.     

Optimal design of beams under flexural vibration

The minimum weight design of a cantilever beam in flexural vibration is considered. The aim is the maximization of a given natural bending frequency (usually the first) for a given beam weight or equivalently the minimization of beam weight for a specified value of a natural frequency. The beams considered are of rectangular section and are subject, in a range of cases presented, to a variety of constraints on lower and upper bounds on the cross-section dimensions or to the specification of a point mass at the end of the beam. Simple bending theory is regarded as applicable to the problem. A variational statement of the problem is made and the necessary conditions for a minimum are obtained as a system of non-linear equations which are solved numerically. Results are given in the form of tables and of figures showing computed optimum profiles. Some experiments on a sample set of beams of equal mass are described briefly. The optimum profile beam was found to have the greatest fundamental frequency, in support of the theoretical predictions.     

悬臂方薄板振动的辐射声场及竖裂纹对其频谱的影响

在将悬臂板挠度表示为正交多项式之和的基础上，利用瑞利-里兹方法求解了悬臂板的前几阶振动模态频率及挠度．进而利用瑞利积分求解了其自由振动辐射声场的分布规律。用分区技术的瑞利-里兹方法求解了竖裂纹对悬臂板低阶模态频率的影响，并将计算结果与有限元结果作了对照。实验上用小球撞击法测定了含竖裂纹悬臂板辐射声场的频谱，谱成分与有限元结果基本符合。     

RITZ VECTOR APPROACH FOR STATIC AND DYNAMIC ANALYSIS OF PLATES WITH EDGE BEAMS

A Ritz vector approach is used to develop new formulations for evaluating the static and the dynamic characteristics of rectangular plates with edge beams. Unlike previous studies in which stiffness coefficients with specified distributions along the plate edges are used to represent the effect of edge restraints, the effect of elastic edge restraints is accounted for by including appropriate integrals for edge beams in the expressions for total kinetic and potential energies in a Rayleigh-Ritz approach. The effect of various types of boundary conditions at the beam ends is accounted for by considering the corresponding Ritz vectors. The contribution of beam mass to the total kinetic energy is also considered in the proposed approach. This effect has often been neglected in the previous studies but can be significant in some applications. The results obtained from the application of the proposed approach to a variety of examples are compared with the corresponding results obtained from the finite element analysis.     

Free vibration analysis of a flat plate using the hierarchical finite element method

The hierarchical finite element method is used to determined the natural frequencies and modes of a flat, rectangular plate. Ten different boundary conditions—including free edges and point supports—are considered in this paper. Extensive results are presented for each case (including the variation of frequency with the aspect ratio and the Poisson ratio), and these are shown to be in very good agreement with the work of other investigators. This confirms both the applicability and accuracy of solution of the HFEM to problem of this type.