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.     

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.     

Forced vibrations of a non-uniform thickness rectangular plate with two free sides

A forced vibration of a rectangular plate with thickness varying linearly in one direction is studied. The plate is simply supported along two opposite sides, and is free along the other sides. Approximate formulae are proposed for estimating the maximum deflection and surface stresses of a rectangular plate subject to a uniformly distributed harmonic lateral load. The accuracy of the formulae is discussed. The approximate formulae show the possibility that one can obtain solutions for the forced vibration of a plate with non-uniform thickness from those for the static bending of the same plate when the plate is subjected to a uniformly distributed load.     

Forced vibrations of a rectangular plate with non-uniform thickness

The dependence on frequency of the maximum deflection and surface stresses of a simply supported rectangular plate subjected to a uniformly distributed sinusoidal excitation is discussed and simple formulae are proposed for estimating the deflection and surface stresses. The thickness of the plate varies linearly in one direction parallel to a side of the plate.     

Transverse vibrations of rectangular plates with thickness varying in two directions and with edges elastically restrained against rotation

The title problem is solved in the case of linear variation of the thickness in the x- and y- directions, a simple polynomial co-ordinate function being used. An approximate but quite convenient frequency equation is derived, by using the Rayleigh method. Apparently the open technical literature does not contain a previous analytical treatment of this problem which is as general and simple as that presented here.     

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.     

Vibration and stability of an orthotropic plate

Using previously derived governing equations, this paper presents closed form solutions for the statics and dynamics of thick rectangular plates. A higher-order shear deformation theory is developed for the vibration and buckling of simply supported orthotropic plates. Present results based on higher-order theory are compared with the results of other theories. It can be concluded from the natural frequency results that the accuracy of the present theory is better than the other theories. In addition, the effects of various orthotropic parameters on the natural frequencies and buckling loads are studied.     

The effect of self-equilibrating stresses on the natural frequencies of a free-free rectangular plate

A rectangular plate supported with all edges free vibrates in both torsional and flexural modes. A self-equilibrating stress system, induced by running a weld on the longitudinal centre-line, reduces the natural frequencies of the plate. The effect is more marked in the torsional than in the flexural modes. It becomes less significant with increase in frequency.The paper describes an experimental and theoretical study of this effect and seeks to suggest a physical interpretation of the phenomenon.     

Note on the forced vibration of an orthotropic plate on an elastic foundation

An analysis is presented of the response of an infinite, linearly elastic, orthotropic plate, resting on an elastic foundation, to a forcing function which may vary in both time and space. Some numerical results are given for a case of concentrated blast loading.     

Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic annular plates of variable thickness

Free axisymmetric vibrations of a polar orthotropic annular plate of linearly varying thickness resting on an elastic foundation of Winkler type are studied on the basis of classical theory of plates. The fourth order linear differential equation with variable coefficients governing the motion is solved by using the quintic spline interpolation technique for three different combinations of boundary conditions. The effect of the elastic foundation together with the orthotropy on the natural frequencies of vibration is illustrated for different values of the radii ratio and the thickness variation parameter for the first three modes of vibration. Transverse displacements and moments are presented for a specified plate. The validity of the spline technique is demonstrated by presenting a comparison of present results with those available in the literature.     

Analytical solution for vibration of an incompressible isotropic linear elastic rectangular plate, and frequencies missed in previous solutions

An analytical solution is given for free vibration of a simply supported rectangular plate made of an incompressible linear elastic isotropic material. Through-the-thickness modes of vibration valid for compressible and incompressible materials and missed by previous investigators are also identified.     

孔板厚度对平衡型节流器件流场特性影响

平衡型节流器件是一种新型非标准节流装置,在流体流量测量上具有一定的优势。采用数值模拟方法,分析了孔板厚度对平衡型节流器件流场特性的影响。结果表明:节流件的孔板厚度存在优选值,随着孔板厚度的增大,流出系数逐渐增大且趋于定值,而压降损失逐渐减小且最终也趋于定值。     

Lamb modes with multiple zero-group velocity points in an orthotropic plate

In this paper the spectrum of Lamb modes in an orthotropic plate is studied. Infinitely many modes are found which undergo a change of sign in their slopes twice, thus leading to two zero-group velocity points. A few modes possessing three such points also exist.     

Transverse vibrations of a non-uniform rectangular plate on an elastic foundation

Free transverse vibrations of an isotropic rectangular plate of variable thickness resting on an elastic foundation has been studied on the basis of classical plate theory. The fourth-order differential equation governing the motion is solved by using the quintic spline interpolation technique. Characteristic equations for plates of exponentially varying thickness have been obtained for three combinations of boundary conditions at the edges. Frequencies, mode shapes and moments have been computed for different values of the taper constant and the foundation moduli for the first three modes of vibration.     

无限大板上矩形面导纳的测量

研究在实验条件下测量均布速度激励下无限大板面导纳的方法.实验中利用有限板模拟无限大板,用具有矩形接触面积的刚性棱形锥产生均匀速度的激励,并建立消除附加质量影响的测量方法。实验结果与理论预测值一致,证明了实验模型与方法的可行。     

Non-linear axisymmetric transient analysis of an orthotropic thin circular plate with an elastically restrained edge

Results are presented for the geometrically non-linear axisymmetric transient elastic stress and deflection responses of a cylindrically orthotropic thin circular plate with an elastically restrained edge, including both rotational and in-plane displacements. In the analysis the dynamic analogue of the von Kárman governing differential equations in terms of the normal displacement w and the stress function ψ are employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in the space domain and the Newmark-β scheme in the time domain are used. Four types of uniformly distributed transient loadings have been considered: step function, sinusoidal and N-shaped pulses, and exponentially decaying loads. The influence of the orthotropic parameter β and the elastic rotational and in-plane edge restraint parameters (K_b, K_i) on the large amplitude response has been investigated. The effect of a prescribed in-plane displacement on the non-linear transient response has also been studied.