On the effect of free,rectangular cut-outs along the edge on the transverse vibrations of rectangular plates

The title problem is tackled using two alternative approaches: a simple Rayleigh-Ritz methodology and a finite element algorithmic procedure.Numerical results are obtained in the case of simply supported and clamped plates with free, square holes located at the middle of a plate edge. Experimental values are obtained in the case of clamped plates.It is concluded that the agreement is very good from an engineering viewpoint as long as the hole size is moderate. Since this constitutes a valid restriction in practice when ducts, pipes, etc., traverse plates or slabs, one can say that the Rayleigh-Ritz procedure yields fundamental frequency coefficients which possess sufficient accuracy from a designer's viewpoint.     

Adaptive non-linear control of a clamped rectangular plate with PZT patches

The dynamic behavior and control of a clamped rectangular plate with bonded piezoelectric ceramic patches are investigated in this paper. The dynamic behavior is studied experimentally, showing that the plate exhibits dense modes, varying residual stress and non-linearity. An adaptive non-linear control scheme is then presented, which introduces a non-linear function into the normal adaptive feedforward control to non-linearize a reference signal. Vibration reductions using both the normal adaptive feedforward control and the adaptive non-linear control are compared in the cases of sinusoidal excitations at the first nine modal frequencies of the plate and a swept-frequency harmonic excitation below 100 Hz, indicating that the adaptive non-linear control can suppress not only the fundamental frequency vibration that the normal adaptive feedforward control can only attenuate, but also its higher harmonic components. Significant vibration reduction achieved by the adaptive non-linear control demonstrates its validity and reliability.     

Vibration and buckling of rectangular plates under in-plane hydrostatic loading

Numerical solutions are presented for the fundamental natural frequency and mode shape of a rectangular plate loaded by in-plane hydrostatic forces for a wide variety of aspect ratios, boundary conditions, and load magnitudes. All six possible combinations of simply supported and clamped edges are considered. The limiting conditions of unloaded vibration and buckling are discussed in detail, with emphasis on the preferred mode shape. Design curves and approximate formulae are presented which provide a simple means of determining the fundamental frequency parameter.     

Large amplitude vibrations and damage detection of rectangular plates

In this work, geometrically nonlinear vibrations of fully clamped rectangular plates are used to study the sensitivity of some nonlinear vibration response parameters to the presence of damage. The geometrically nonlinear version of the Mindlin plate theory is used to model the plate behaviour. Damage is represented as a stiffness reduction in a small area of the plate. The plate is subjected to harmonic loading with a frequency of excitation close to the first natural frequency leading to large amplitude vibrations. The plate vibration response is obtained by a pseudo-load mode superposition method. The main results are focussed on establishing the influence of damage on the vibration response of the plate and the change in the time-history diagrams and the Poincaré maps caused by the damage. Finally, a criterion and a damage index for detecting the presence and the location of the damage is proposed. The criterion is based on analysing the points in the Poincaré sections of the damaged and healthy plate. Numerical results for large amplitude vibrations of damaged and healthy rectangular and square plates are presented and the proposed damage index for the considered cases is calculated. The criterion demonstrates quite good abilities to detect and localize damage.     

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.     

Geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass

Nonlinear forced vibrations of rectangular plates carrying a central concentrated mass are studied. The plate is assumed to have immovable edges and rotational springs; numerical results are presented for clamped plates. The Von Kármán nonlinear plate theory is used, but in-plane inertia in both the plate and the mass is retrained. The problem is discretized into a multi-degree-of-freedom (dof) system by using an energy approach and Lagrange equations taking damping into account. A pseudo-arclength continuation method is used in order to obtain numerical solutions. Results are presented as both (i) frequency-amplitude curves and (ii) time domain responses. The effect of gravity and the effect of the consequent initial plate deflection are also investigated.     

Vibrating rectangular plate with a free edge: The effect of a concentrated mass and an internal support

It is shown that use of a polynomial coordinate function and the Rayleigh-Schmidt method constitute a convenient approach for determining the fundamental frequency of vibration of the title system. Experimental results are presented for the case where three edges are rigidly clamped; and it is shown that the values are in good engineering agreement with the analytical predictions.     

Vibration analysis of a rectangular thin isotropic plate with a part-through surface crack of arbitrary orientation and position

In this paper, vibration analysis of a rectangular thin isotropic plate with a part-through surface crack of arbitrary orientation and position is performed by using the Kirchhoff plate theory. Simply supported (SSSS), clamped (CCCC) and simply supported–clamped (SCSC) boundary conditions are considered for the analysis. First, the governing differential equation of a cracked plate is formulated. A modified line spring model is then used to formulate the crack terms in the governing equation. Next, by the application of Burger's formulation, the differential equation is transformed into the well-known Duffing equation with cubic and quadratic nonlinearities. The Duffing equation is then solved by the method of multiple scales (MMS) to extract the frequency response curve. Natural frequencies are evaluated for different values of length, angle and position of a part-through surface crack. Some results are compared with the published literature. Amplitude variation with different values of length, angle and position of a part-through surface crack are presented, for all three types of the plate boundary conditions.     

Natural frequencies of rectangular orthotropic plates with a pair of parallel edges simply supported

Solutions of the exact characteristic equations for the title problem derived earlier by an extension of Bolotin's asymptotic method are considered. These solutions, which correspond to flexural modes with frequency factor, R, greater than unity, are expressed in convenient forms for all combinations of clamped, simply supported and free conditions at the remaining pair of parallel edges. As in the case of uniform beams, the eigenvalues in the CC case are found to be equal to those of elastic modes in the FF case provided that the Kirchoff's shear condition at a free edge is replaced by the condition $\text{?M}{}_{\mathrm{n}}\text{?n = 0}$. The flexural modes with frequency factor less than unity are also investigated in detail by introducing a suitable modification in the procedure. When Poisson's ratios are not zero, it is shown that the frequency factor corresponding to the first symmetric mode in the free-free case is less than unity for all values of side ratio and rigidity ratios. In the case of one edge clamped and the other free it is found that modes with frequency factor less than unity exist for certain dimensions of the plate—a fact hitherto unrecognized in the literature.     

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.     

Free vibrations of a rectangular plate of variable thickness elastically restrained against rotation along three edges and free on the fourth edge

A literature search has shown that the title problem has received no treatment.In this paper solutions are presented as obtained (I) by use of the Ritz method with deflection functions which are simple polynomials, and (II) by use of the extended Kantorovich Method. The natural boundary conditions along the free edge are not satisfied in the first case, while they are complied with approximately when using the second approach. The fundamental frequency coefficient is determined and good agreement is shown to exist between the results obtained by the two methods.     

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.     

The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method

A simple approximate formula for the natural frequencies of flexural vibration of isotropic plates, originally developed by Warburton using characteristic beam functions in Rayleigh's method, is modified to apply to specially orthotropic plates and extended to include the effect of uniform, direct inplane forces. The initial buckling problem is treated simply by equating the frequency expression to zero. The approach permits the ready determination of reasonably accurate natural frequencies and/or buckling loads for a given plate involving any combination of free, simply supported or clamped edges, without requiring the aid of a sophisticated calculating device or a knowledge of plate, vibration or buckling theory. To illustrate the applicability and accuracy of the approach, numerical results for a number of specific plate problems are presented.     

A note on vibrating polar orthotropic circular plates carrying concentrated masses

The fundamental frequency of vibration of a circular plate of polar orthotropy carrying concentrated masses is determined by using (a) an extension of the Rayleigh-Schmidt technique and (b) a Ritz-Lagrange multipliers method. Numerical results are presented for clamped and simply supported plates for several combinations of orthotropic parameters and values of the concentrated mass to plate mass ratio.     

NON-LINEAR VIBRATION AND THERMAL BUCKLING OF AN ORTHOTROPIC ANNULAR PLATE WITH A CENTRIC RIGID MASS

A computational analysis of the non-linear vibration and thermal post-buckling of a heated orthotropic annular plate with a central rigid mass is examined for the cases of immovably hinged as well as clamped constraint conditions of the outer edge. First, based on von Karman's plate theory and Hamilton's principles, the governing equations, in terms of the displacements of the middle plane, of the problem are derived. Then, upon assuming that harmonic responses of the system exist, the non-linear partial differential equations are converted into the corresponding non-linear ordinary differential equations through elimination of the time variable by using the Kantorovich time-averaging method. Finally, by applying a shooting method, the fundamental responses of the non-linear vibration and thermal post-buckling of the plate are numerically obtained. For some prescribed values of the parameters, such as the material rigidity ratio, temperature rise and so on, the curves of the fundamental frequency versus specified amplitude and the thermal post-buckled equilibrium paths of the plate are numerically presented.     

Vibrations of an orthotropic elliptic plate of non-uniform thickness and temperature

An analysis is presented of the free vibrations of an orthotropic elliptic plate whose temperature and thickness spatial variations both are parabolic along a line through the plate centre. An approximate but quite convenient frequency equation is derived by using Galerkin's method with a two-term deflection function. The upper bounds of frequencies corresponding to the first two modes of vibrations of a clamped orthotropic elliptic plate for various values of aspect ratio, taper constant and temperature constant are obtained.     

弯曲振动矩形板辐射阻抗的计算

本文推导了四边简支矩形板的辐射阻抗表达式, 利用高斯数值积分方法, 计算了其相对辐射阻抗的数值解. 由不同模态下相对辐射阻抗与频率以及不同长宽比对应的相对辐射阻抗与频率的关系可知, 在中低频段, 模态越低, 辐射阻抗越大, 也就意味着辐射声功率和同振质量越大; 对于一定面积和模态的矩形板, r (r=a/b, 长与宽之比)值越接近1, 即越接近正方形, 辐射阻和辐射抗越大. 本文的方法能对其他复杂边界条件下的、无振动解析解的矩形板的辐射阻抗数值量级大小提供一个参考, 也可由计算弯曲振动的阻抗自然地过渡到活塞振动阻抗的计算.     

A note on forced vibrations of a clamped rectangular plate

Simple polynomial approximations and Galerkin's method are used to determine the response of a thin, elastic, rectangular plate clamped along the boundary and subjected to sinusoidal excitation. It also is shown that a one-term polynomial solution yields good accuracy in the case of a rectangular plate with simply supported edges.     

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.