Natural vibration of cantilever porous twisted plate with variable thickness in different directions

In this paper,the blade is assumed to be a rotating variable thickness cantilever twisted plate structure,and the natural vibrations of variable thickness cantilever twisted plate made of metal porous material are studied.It is assumed that the thickness of the plate changes along spanwise direction and chordwise direction,respectively,and it changes in both directions.The classical thin shell theory,the first and second fundamental forms of surface and von Karman geometric relationship are employed to derive the total potential energy and kinetic energy of the cantilever twisted plate,in which the centrifugal force potential due to high rotational speed is included.Then,according to the Rayleigh-Ritz procedure and applying the polynomial functions which satisfy the cantilever boundary conditions,the dynamic system expressed by equations of motion is reduced to an eigenvalue problem.By numerical simulation,the frequency curves and the mode shapes of the twisted plate can be obtained to reveal the internal connection between natural vibration and the parameters.A series of comparison studies are performed to verify the accuracy of the present formulation and calculations,in which compared data come from experimental,finite element method and theoretical calculation,respectively.The influence of pre-twist angle,three different forms of thickness taper ratio and rotational speed on natural vibration,mode exchange and frequency veering phenomenon of the system is discussed in detail.In addition,the approach proposed here can efficiently extract analytical expressions of mode functions for rotating variable thickness cantilever twisted plate structures.

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness, variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.

Cubic Spline Solutions of Nonlinear Bending and Buckling of Circular Plates with Arbitrarily Variable Thickness

The cubic B-splines taken as trial function, the large deflection of a circular plate with arbitrarily variable thickness,as well as the buckling load, have been calculated by the method of point collocation. The support can be elastic. Loads imposed can be polynomial distributed loads, uniformly distributed radial forces or moments along the edge respectively or their combinations. Convergent solutions can still be obtained by this method under the load whose value is in great excess of normal one. Under the action of the uniformly distributed loads, linear solutions of circular plates with linearly or quadratically variable thickness are compared with those obtained by the parameter method. Buckling of a circular plate with identical thickness beyond critical thrust is compared with those obtained by the power series method.

Effect of variable thickness cross rolling on edge crack and microstructure gradient of AZ31 magnesium alloy

The hot rolling experiment of AZ31 magnesium alloy was carried out by laying anoverlay mold at the initial temperature of 400℃.According to the Mizushima automatic plan view pattern control system(MAS)rolling theory and the cross rolling process,different reductions in the middle and edges of the magnesium alloy were realized,and the influence of the regional controlled reduction rolling on the edge cracks and microstructure gradient of the magnesium alloy were analyzed.It is shown that this rolling approach has reduced the maximum edge crack depth of the rolled piece by 56.85%,and there is a weakening tendency in the base surface texture of the strip edge,the base surface texture density drops from 23.97 to 17.48 after ordinary flat rolling.It exhibits basal texture gradients from the edge to the middle of the sheet along the RD direction,which reflected the uneven deformation of the sheets.It is suitable for the processing of metal molds that require large edge reductions such as mobile phone shells,and provided a theoretical basis for the variable thickness rolling of the magnesium alloy strip.

Elasticity solution of clamped-simply supported beams with variable thickness

This paper studies the stress and displacement distributions of continuously varying thickness beams with one end clamped and the other end simply supported under static loads. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of the plane stress problem, the general expressions of displacements, which satisfy the governing differefitial equations and the boundary conditions attwo ends of the beam, can be deduced. The unknown coefficients in the general expressions are then determined by using Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence properties. Comparing the numerical results to those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.

Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and fiexural rigidity with the consideration of the effect of Poisson＇s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

NONLINEAR NATURAL FREQUENCY OF SHALLOW CONICAL SHELLS WITH VARIABLE THICKNESS

The nonlinear dynamical variation equation and compatible equation of the shallow conical shell with variable thickness are obtained by the theory of nonlinear dynamical variation equation and compatible equation of the circular thin plate with variable thickness. Assuming the thin film tension is composed of two items. The compatible equation is transformed into two independent equations. Selecting the maximum amplitude in the center of the shallow conical shells with variable thickness as the perturbation parameter, the variation equation and the differential equation are transformed into linear expression by theory of perturbation variation method. The nonlinear natural frequency of shallow conical shells with circular bottom and variable thickness under the fixed boundary conditions is solved. In the first approximate equation, the linear natural frequency of shallow conical shells with variable thickness is obtained. In the third approximate equation, the nonlinear natural frequency of it is obtained. The figures of the characteristic curves of the natural frequency varying with stationary loads, large amplitude, and variable thickness coefficient are plotted. A valuable reference is given for dynamic engineering.

UNSYMMETRICAL NONLINEAR BENDING PROBLEM OF CIRCULAR THIN PLATE WITH VARIABLE THICKNESS

Firstly, the cross large deflection equation of circular thin plate with variable thickness in rectangular coordinates system was transformed into unsymmetrical large deflection equation of circular thin plate with variable thickness in polar coordinates system. This cross equation in polar coordinates system is united with radical and tangential equations in polar coordinates system, and then three equilibrium equations were obtained. Physical equations and nonlinear deformation equations of strain at central plane are substituted into superior three equilibrium equations, and then three unsymmetrical nonlinear equations with three deformation displacements were obtained. Solution with expression of Fourier series is substituted into fundamental equations; correspondingly fundamental equations with expression of Fourier series were obtained. The problem was solved by modified iteration method under the boundary conditions of clamped edges. As an example, the problem of circular thin plate with variable thickness subjected to loads with cosin form was studied. Characteristic curves of the load varying with the deflection were plotted. The curves vary with the variation of the parameter of variable thickness. Its solution is accordant with physical conception.

Nonlinear Vibration of Circular Plate with Variable Thickness

This paper is concerned with the nonlinear vibration problems of circular plates with variable thickness.The nonlinear equations of plates with variable thickness are extended to the dynamic case.The resulting equations can be solved by using an iterative method,a Galerkin's approach and a perturbation method.Detailed solutions and numerical results are given for two kinds of boundary conditions,the clamped edge and the supported edge.The results show that the solutions for the case of the plates with uniform thickness can be included in the solution herin as a special case.The effect of various thickness parameters is investigated in detail.Also,a Runge Kutta method is used to solve the free and forced vibrations of plates with variable thickness,and the results are obtained firstly.It has shown that the adoption of variable thickness plate would be useful in engineering design.

BASIC EQUATIONS OF THE PROBLEM OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE WITH VARIABLE THICKNESS

Under the case of ignoring the body forces and considering components caused by diversion of membrane in vertical direction (z-direction),the constitutive equations of the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with variable thickness are given;then introducing the dimensionless variables and three small parameters,the dimensionaless governing equations of the deflection function and stress function are given.

LARGE DEFLECTION PROBLEM OF THIN ORTHOTROPIC CIRCULAR PLATE ON ELASTIC FOUNDATION WITH VARIABLE THICKNESS UNDER UNIFORM PRESSURE

Basic equations for large deflection theory of thin orthotropic circular plate on elastic foundation with variable thickness under uniform pressure are derived in this paper. The second opproximation solutions are obtained by means of the modified iteration method. The relation curves of the nondimensional loading and foe deflection, as to the differential ε and μrθ and λ are shown in Figs. 2, 3, 4. In special circumstance, the results are in accordance with those in [1], [6].

NAVIER SOLUTION FOR THE ELASTIC EQUILIBRIUM PROBLEMS OF ANISOTROPIC SKEW THIN PLATE WITE VARIABLE THICKNESS IN NONLINEAR THEORIES

This paper discusses the elastic equilibrium problems of anisotropic skew thin plate of variable thickness simply supported on all four sides in nonlinear theories, and uses the Navier method to seek an approach to the problem, and to illustrate the solution with the examples. In conclusion, the mention is made of the scope of application and the convergency of the solution.

THE PROBLEM OF THE NON-LINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR THIN PLATE WITH VARIABLE THICKNESS

To begin with, in this paper, the governing equations of the problem of the non-linear unsymmetrical bending for cylindrically orthotropic circular thin plate with variable thickness are derived. By using 'the method of two-variable' and introducing four small parameters, the problem of the non-linear unsymmetrical bending for cylindrically orthotropic circular thin plate with linear variable thickness are studied, and the uniformly valid asymptotic solution of Nth-order for epsilon(1) and Mth-order for epsilon(2) are obtained.

THE METHOD OF MULTIPLE SCALES APPLIED TO THE NONLINEAR STABILITY PROBLEM OF A TRUNCATED SHALLOW SPHERICAL SHELL OF VARIABLE THICKNESS WITH THE LARGE GEOMETRICAL PARAMETER

Using the modified method of multiple scales, the nonlinear stability of a truncated shallow spherical shell of variable thickness with a nondeformable rigid body at the center under compound loads is investigated. When the geometrical parameter k is larger, the uniformly valid asymptotic solutions of this problem are obtained and the remainder terms are estimated.

NONLINEAR STABILITY OF TRUNCATED SHALLOW CONICAL SANDWICH SHELL WITH VARIABLE THICKNESS

The theory of nonlinear stability for a truncated shallow conical shell with variable thickness under the action of uniform pressure was presented. The fundamental equations and boundary conditions were derived by means of calculus of variations. An analytic solution for the critical buckling pressure of the shell with a hyperbolically varying thickness is obtained by use of modified iteration method. The results of numerical calculations are presented in diagrams, which show the influence of geometrical and physical parameters on the buckling behavior.

Nonlinear Analysis and Optimization of Shallow Shells of Variable Thickness

This paper is concerned with the nonlillear bending, stability and optimal design of revolution shallowshells with variable thickness. The problems are investigated by means of a modified iterative method proposedearlier by the author. Solutiolls for nonlinear bedding and stability problems of revolution shallow shells withvariable thickness, such as spherical and conical shells, are presented. Deflections and critical loads for stability arecalculated and the numerical results are plotted and given in tabular forms. It is showil that the equation determiningthe maximum deflection and the load coincides with the cusp catastrophe manifold. The optimal design of plates andshells, in Which the volullle is minimized or the critical load of shells is maximized, is investigated. When the volumeoftlle shell and the arch height of the shell are given, the variable thickness parameter can be solved. In addition, thispaper also gives tile constraint optimization of nonlinear bedding of circular plates.

UNIFORMLY VALID ASYMPTOTIC SOLUTIONS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE OF FOUR CLAMPED EDGES WITH VARIABLE THICKNESS

By using “the method of modified two-variable”,“the method of mixing perturbation” and introducing four small parameters,the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with linear variable thickness is studied.And the uniformly valid asymptotic solution of Nth-order for ε_1 and Mth-order for ε_2 of the deflection functions and stress function are obtained.

NONLINEAR STABILITY OF TRUNCATED SHALLOW SPHERICALSHELL WITH VARIABLE THICKNESS UNDERUNIFORMLY DISTRIBUTED LOAD

In this paper, to begin with. the nonlinear differential equations of a truncaled shallow spherical shell with variable thickness under uniformal distributed load are linearized by step-by-step loading method. The linear differential equations can be solved by spline collocanon method. Critical loads have been obtained accordingly.

LARGE DEFLECTION PROBLEM OF CIRCULAR PLATES WITH VARIABLE THICKNESS UNDER THE ACTION OF COMBINED LOADS

By using the modified iteration method of large deflection theory of plates with variable thichness[1], we solve the problem of circular plates with variable thickness subjected to combined loads under the boundary conditions of the clamped edges and get comparatively more accurate second-order approximate analytical solution. If the results of this paper are degraded into the special cases, the results coinciding with those of papers [1,2] can be obtained. In this paper, the characteristic curves are plotted and some comparisons are made. The results of this paper are satisfactory.

AXISYMMETRIC PROBLEMS OF CYLINDRICAL SHELLS WITH VARIABLE WALL THICKNESS

The purpose of this paper is to give the general solutions for axisymmetric eylindrical shells with paraholically varying wall thickness.