Bending problems of non-homogeneous cylindrical orthotropic circular plate

ＢＥＮＤＩＮＧＰＲＯＢＬＥＭＳＯＦＮＯＮ－ＨＯＭＯＧＥＮＥＯＵＳＣＹＬＩＮＤＲＩＣＡＬＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥＱｉｎＳｈｅｎｇ－ｌｉ（秦圣立）（ＱｕｆｕＮｏｒｍａｌＵｎｉｃｅｒｓｉｔｙ,Ｑｕｆｕ,Ｓｈａｎｄｏｎｇ）（Ｒｅｃｅｉ...     

Nonlinear flexural vibration of moderately thick orthotropic circular plates

Summary In this paper a nonlinear vibration theory which includes the effects of transverse shear deformation and rotatory inertia is formulated for rectilinearly orthotropic circular plates using Berger's method. A solution to the governing equations for rigidly clamped plates is obtained on the basis of a single-mode approach by use of Galerkin's method and numerical Runge-Kutta procedure. A good agreement is found between present results and those obtained by a more accurate theory for nonlinear static and dynamic cases. Numerical results indicate significant influence of these effects on the vibration behaviour of moderately thick plates, especially high-modulus composite plates. These effects, however, are not so significant for the radius-to-thickness ratio greater than 10.

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Nichtlineare Schwingungen orthotroper Kreisplatten

Übersicht Eine Theorie nichtlinearer Schwingungen orthotroper Kreisplatten wird mit der Methode von Berger formuliert, wobei Schubverformung und Drehträgheit berücksichtigt werden. Für eingespannte Platten wird mit dem Verfahren von Galerkin eine numerische Lösung bestimmt und mit bekannten Lösungen verglichen. Die numerischen Ergebnisse zeigen, daß der Einfluß von Schubverformung und Drehträgheit bei mäßig dicken Platten beträchtlich sein kann.

     

弹性约束边界圆环板面内自由振动的二维弹性解

基于二维线弹性理论,应用哈密顿原理导出弹性约束边界圆环板面内自由振动的控制微分方程。采用微分求积法(DQM)数值研究了弹性约束边界圆环板面内自由振动的频率特性。通过设置弹性刚度系数为0或∞,问题退化为四种典型边界圆环板的面内自由振动,与已有文献的计算数值结果进行比较,证实本文的分析求解方法行之有效。最后全面考虑了圆环板边界条件、几何系数及刚度系数对自振频率的影响。     

Multiple mode large amplitude vibration of thick orthotropic circular plates

This paper is analytically concerned with the large amplitude vibration of thick orthotropic circular plates incorporating the effects of transverse shear and rotatory inertia. Von Kármán-type field equations written in terms of the three displacement components of the plate are utilized to obtain solutions to clamped stress-free and immovable plates. By means of Galerkin's technique and a numerical Runge-Kutta procedure a multiple-mode analysis is carried out in both cases. Exact solutions are reported for two of the three governing equations. Effects of transverse shear deformation and modal interaction are found to be significant for orthotropic thick plates. The method given here could be extended to the multiple-mode analysis of circular plates with other boundary conditions.     

On the analyses of vibration and stability for orthotropic circular cylindrical shells conveying fluid

Based on flügge's thin orthotropic shell equations and a linear potential flow theory, buckling instability and flutter-type instability of orthotropic circular cylindrical shells conveying fluid are studied. By means of the expansion in beam-mode functions and Galerkin's method they can be reduced to solving a generalized complex eigenvalue problem. In calculating the generalized forces defined as Eq. (23), a direct numerical integration technique, which has proven to be more efficient than others, is used. For clamped-clamped orthotropic shells several numerical examples show that the effects of axial elastic modulus on the dynamic behaviors are quite different from those of circumferential elastic modulus.     

Unsteady vibration of a growing circular plate

Forced vibration of a thickening circular plate is studied within the framework of the small deformation theory. The plate material is assumed to be elastic and isotropic and the plate thickness to be continually increasing due to influx of material from outside. It is also assumed that the plate thickness varies with time but is independent of the space coordinates. Moreover, in the process of growth, the midsurface position does not change, which suggests that the plate grows symmetrically on both its surfaces.     

Thermoelastic free vibration of clampe circular plate

An analysis was given for the free vibration of clamped circular plate when temperature and stress fields were coupled. A nonlinear differential equation about time was obtained by using Galerkin’s method. The numerical results of vibration amplitude vs time were compared with the uncoupled case. It is found that if the given initial displacement is small, the effect of thermoelastical coupling will make the natural frequency increase; if the given initial displacement is large, it will be the opposite case. Effects of some different vibration factors are also discussed.     

Flexural vibration analysis of an eccentric annular Mindlin plate

A weak-form quadrature element method is presented to study the flexural vibrations of an eccentric annular Mindlin plate. Typical combinations of boundary conditions are considered and the natural frequencies are obtained for both thin and moderately thick plates. All results are verified using the commercial computer code ANSYS. Excellent agreement is reached in all cases. Comparison of the present predictions with other available results for thin plates is also made.     

周边嵌入弹性圆薄板非线性自由振动直接摄动解法

采用弹性圆薄板中心无量纲振幅和板厚与半径的比值为参数。将挠度、应力函数对半径的导数以及自由振动频率展开为双参数的幂级数。用直接摄动法获得各级递推线性偏微分方程。应用变分法求得各级递推方程的近似解，从而给出弹性圆薄板非线性自由振动频率的基本公式。     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULA RPLATE（I）

ＴＨＥＰＲＯＢＬＥＭＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＵＮＳＹＭＭＥＴＲＩＣＡＬＢＥＮＤＩＮＧＦＯＲＣＹＬＩＮＤＲＩＣＡＬＬＹＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥ（Ｉ）ＱｉｎＳｈｅｎｇ－Ｉｉ（秦圣立）ＨｕａｎｇＪｉａ－ｙｉｎ（黄家寅）（Ｑｕｆ...     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATE（II）

ＴＨＥＰＲＯＢＬＥＭＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＵＮＳＹＭＭＥＴＲＩＣＡＬ．ＢＥＮＤＩＮＧＦＯＲＣＹＬＩＮＤＲＩＣＡＬＬＹＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥ（ＩＩ）ＨｕａｎｇＪｉａｙｉｎ（黄家寅）;ＱｉｎＳｈｅｎｇｌｉ（秦圣立）;Ｘｉ...     

Nonlinear vibration of circular sandwich plate under the uniformed load

ＩｎｔｒｏｄｕｃｔｉｏｎＳｏｆａｒ,ｏｎｌｙａｆｅｗｐｅｏｐｌｅｈａｖｅｓｔｕｄｉｅｄｔｈｅｌａｒｇｅｄｅｆｌｅｃｔｉｏｎｐｒｏｂｌｅｍｓｏｆｓａｎｄｗｉｃｈｐｌａｔｅｓａｎｄｓｈｅｌｌｓｂｅｃａｕｓｅｏｆｔｈｅｄｉｆｆｉｃｕｌｔｙｏｆｎｏｎｌｉｎｅａｒｍａｔｈｅｍａｔｉｃｓ.ＬｉｕＲｅｎｈｕａｉｈａｓｄｏｎｅｍｕｃｈｔｏｆｉｎｄａｓｅｒｉｅｓｏｆｒｅｓｕｌｔｓｗｉｔｈｔｈｅｖａｌｕｅｏｆａｐｐｌｉｃａｔｉｏｎｉｎｅｎｇｉｎｅｅｒｉｎｇｐｒａｃｔｉｃｅ[1～5].Ａｕｔｈｏｒ[6,7]ｈａｄｔｈｅｉｎｉｔｉ…     

Stability of annular orthotropic plates

It is assumed that the orthotropy of the plate material is rectangular or polar, and a uniformly distributed, external compressive or tensile load is applied to the internal boundary of the plate. Stability is analyzed by the Ritz method with the use of Alfutov-Balabukh and Bryan energy criteria. Diagrams of the critical external load and buckling modes as a function of the plate dimensions are given. Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 166–170, March–April, 2000.     

The problem of the non-linear unsymmetrical bending for cylindrically orthotropic circular thin plate with variable thickness

I-Intr0ducti0nThebendingproblemofcircularthinplateisimriartantbothintheoryandinengineeringpractice,butbecausethebasicequationsoftheproblemsofnon-linearunsymmetricalbendingf0rcircularthinplate'arenonlineardifferentialequations,itisquitedifficulttoobtainana…     

Finite amplitude vibrations of an orthotropic circular plate with an isotropic core

The free and forced non-linear vibrations of a fixed orthotropic circular plate, with a concentric core of isotropic material, are studied. Existence of harmonic vibrations is assumed and thus the time variable is eliminated by a Ritz-Kantorovich method. Hence, the governing non-linear partial equations for the axisymmetric vibration of the composite circular plate are reduced to a set of ordinary differential equations which form a non-linear eigen-value problem. Solutions are obtained by utilizing the related initial-value problems in conjunction with Newton's integration method. The results reveal the effects of finite amplitude and anisotropy of materials upon the dynamic responses. Further, the method developed in this paper, which is used to solve the title problem, is one of some generality. It can be applied to many differential eigenvalue problems with piecewise continuous functions.     

Determination of the orthotropic plate parameters of stiffened plates and grillages in free vibration

Summary It is well known that the analysis of vibration of orthogonally stiffened rectangular plates and grillages may be simplified by replacing the actual structure by an orthotropic plate. This needs a suitable determination of the four elastic rigidity constants D _x, D _y, D _xy, D ₁ and the mass of the orthotropic plate. A method is developed here for determining these parameters in terms of the sectional properties of the original plate-stiffener combination or the system of interconnected beams. Results of experimental work conducted on aluminium plates agree well with the results of the theory developed here.Notation a, b span and width of stiffened plates, grillages and orthotropic plates - c, spacing of beams in the y and x directions - D rigidity of an unstiffened isotropic plate - D _x, D _y rigidities in the x and y directions of an orthotropic plate - D _xy torsional rigidity of an orthotropic plate - D ₁ a parameter associated with a poisson type ratio - e depth of the common neutral axis of the beam-plate combination below the middle surface of the plate - E, G modulus of elasticity and modulus of shear - H =D ₁+2D _xy - i, j integers - I, , I _i, I _j moment of inertia of beams - J, J _i, J _j polar moment of inertia of beams - m, n integers referring to mode numbers in the x and y directions - p _{m n} natural frequency of an orthotropic plate - r, s number of beams in the transverse and longitudinal direction - T _g, T _o, T _s kinetic energy of grillages, orthotropic plates and stiffened plates - u frequency parameter of grillages - V _g, V _o, V _s potential energy of grillages, orthotropic plates and stiffened plates - W=W(x, y, t) deflection of an orthotropic plate - w=w(x, y) amplitude of vibration of orthotropic plates, stiffened plates and grillages - x, y, x _i, y _i cartesian coordinates - X _m, Y _n beam eigen functions - X _mj, Y _ni X _mand Y _nat x=x _jand y=y _i - X _m, Y _n; X_m, Y_n first and second derivatives of X _mand Y _nrespectively - Y _mn plate eigen functions - _n, _n parameters occurring in the expression for the beam functions - , , _i, _j mass per unit length of beams - _mn = frequency parameter of beam and slab bridges - = - Poisson's ratio - , mass per unit area of unstiffened plates and orthotropic plates - _mn natural frequency of stiffened plates and grillages