Yeh Kai-yuan

An analytical formula of the exact solution to von Kármán's equations of a circular plate under a concentrated load

Abstract It is extremely difficult to obtain an exact solution of the non-linear coupled von Karmans equations in large deflection problems of a circular plate. So far only a few exact solutions have been investigated but no strict proof of convergence has been presented. In this paper, we first reduce von Karmans equations to be equivalent to integral equations which are non-linear, coupled and singular. The sequences of continuous functions with general form are then constructed using an iterative technique. Based on the sequences being uniformly convergent, an exact solution of von Karmans equations related to the large deflection problem of circular plates subjected to a concentrated load at the center is obtained.

General analytic solution of dynamic response of beams with nonhomogeneity and variable cross-section

In this paper, a new method, the step-reduction method, is proposed to investigate the dynamic response of the Bernoulli-Euler beams with arbitrary nonhomogeneity and arbitrary variable cross-section under arbitrary loads. Both free vibration and forced vibration of such beams are studied. The new method requires to discretize the space domain into a number of elements. Each element can be treated as a homogeneous one with uniform thickness. Therefore, the general analytical solution of homogeneous beams with uniform cross-section can be used in each element. Then, the general analytic solution of the whole beam in terms of initial parameters can be obtained by satisfying the physical and geometric continuity conditions at the adjacent elements. In the case of free vibration, the frequency equation in analytic form can be obtained, and in the case of forced vibration, a final solution in analytical form can also be obtained which is involved in solving a set of simultaneous algebraic equations with only

Exact analytic method for solving variable coefficient differential equation

Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation format is obtained. Its convergence is proved. We can get analytic expressions which converge to exact solution and its higher order derivatives unifornuy. Four numerical examples are given, which indicate that satisfactory results can be obtained by this method.

Nonlinear stability of thin elastic circular shallow spherical shell under the action of uniform edge moment

In this paper, nonlinear stability of thin elastic circular shallow spherical shell under the action of uniform edge moment is considered by the modified iteration method to obtain second and third approximations to decide the upper and lower critical loads. Results are plotted in curves for the engineering use and are compared with results of Hu Hai-changs. We also investigate the neighbour situation of the critical point, i.e. the double points of the upper and lower critical loads and denote the range of validity of the second approximation. In the end, we obtain the special case, the design formulas of rigidity and stress as well as the corresponding curves as ν=0.3 of large deflection of circular plate under the same load. These results are also compared with Huang Tse-yens.

On solving high-order solutions of Chien's perturbation method to study convergence by computer

In this paper, we have obtained the analytical formula of arbitrary n-th order perturbation solution which perturbation parameter is central deflection for circular thin plate under a concentrated load by means of its integral equation. All unkown constants of every perturbation solution can be determined by computer. Hence higher order perturbation solution is obtained. Based on the solution of higher order perturbation asymptotical property and suitable interval of Chiens perturbation method are also discussed.

A Study of Belleville Spring and Diaphragm Spring in Engineering

This paper deals with the static response of a Belleville spring and a diaphragm spring by using the finite rotation and large deflection theories of a beam and conical shell, and an experimental method as well

Deformations and stability of spherical caps under centrally distributed pressures

In this paper the deformations and stability in large axisymmetric deflection of spherical caps under centrally distributed pressures are investigated. The Newton- spline method for solving the nonlinear equations governing large axisymmetric deflection of spherical caps is presented. The buckling behavior is studied for a cap with fixed geometry when the size of the loaded radius is allowed to vary, and for a fixed loaded radius when the shell geometry is allowed to vary. The influence of the buckling modes on the critical loads is analysed. Numerical results are given for ν=0.3.

Bifurcation and Chaos of the Circular Plates on the Nonlinear Elastic Foundation

According to the large amplitude equation of the circular plate on nonlinear elastic foundation, elastic resisting force has linear item, cubic nonlinear item and resisting bend elastic item. A nonlinear vibration equation is obtained with the method of Galerkin under the condition of fixed boundary. Floquet exponent at equilibrium point is obtained without external excitation. Its stability and condition of possible bifurcation is analysed. Possible chaotic vibration is analysed and studied with the method of Melnikov with external excitation. The critical curves of the chaotic region and phase figure under some foundation parameters are obtained with the method of digital artificial.

On some properties and calculation of the exact solution to von Kármán's equations of circular plates under a concentrated load

Abstract In this paper, we prove some properties of the exact solution obtained in our earlier paper for large deflection of a circular plate under a concentrated load. Using Newtons method, we also solve for the unknown constants Aij and Bij in the formula of an exact solution. Finally, we use the exact solution to check the accuracy of Chiens perturbation solution.

Nonlinear Dynamical Behavior of Shallow Cylindrical Reticulated Shells

By using the method of quasi-shells, the nonlinear dynamic equations of three-dimensional single-layer shallow cylindrical reticulated shells with equilateral triangle cell are founded. By using the method of the separating variable function, the transverse displacement of the shallow cylindrical reticulated shells is given under the conditions of two edges simple support. The tensile force is solved out from the compatible equations, a nonlinear dynamic differential equation containing second and third order is derived by using the method of Galerkin. The stability near the equilibrium point is discussed by solving the Floquet exponent and the critical condition is obtained by using Melnikov function. The existence of the chaotic motion of the single-layer shallow cylindrical reticulated shell is approved by using the digital simulation method and Poincare mapping.By using the method of quasi-shells, the nonlinear dynamic equations of three-dimensional single-layer shallow cylindrical reticulated shells with equilateral triangle cell are founded. By using the method of the separating variable function, the transverse displacement of the shallow cylindrical reticulated shells is given under the conditions of two edges simple support. The tensile force is solved out from the compatible equations, a nonlinear dynamic differential equation containing second and third order is derived by using the method of Galerkin. The stability near the equilibrium point is discussed by solving the Floquet exponent and the critical condition is obtained by using Melnikov function. The existence of the chaotic motion of the single-layer shallow cylindrical reticulated shell is approved by using the digital simulation method and Poincaré mapping.