Uniform Convergence Series to Solve Nonlinear Partial Differential Equations: Application to Beam Dynamics

Extended trigonometric series of uniform convergence are proposed as a method to solve the nonlinear dynamic problemsgoverned by partial differential equations. In particular, the method isapplied to the solution of a uniform beam supported at its ends withnonlinear rotational springs and subjected to dynamic loads. The beam isassumed to be both material and geometrically linear and the end springs are of the Duffing type. The action may be a continuous load q = q(x, t) within a certain range and/or concentrated dynamic moments at the boundaries. The adopted solution satisfies the differential equation, the initial conditions, andthe nonlinear boundary conditions. It has been previously demonstrated that, due to the uniform convergence of the series, the method yieldsarbitrary precision results. An illustration example shows theefficiency of the method.     

Three different guises for the dynamics of a rotating beam

The dynamics of a flexible beam forced by a prescribed rotation around an axis perpendicular to its plane is addressed. Three approaches are considered, two of them related with simplified theories, within Strength of Materials, and the third one using Finite Elasticity. In the Strength of Materials approaches, the governing equations of motion are derived by superposing the deformations and the rigid motion in the first model, and in the second by stating the stationarity of the Lagrangian (including first- and second-order effects in order to capture the stiffening due to the centrifugal forces) through Hamilton's principle. Two actions are considered: gravity forces (pendulum) and prescribed rotation. Comparison of the two Strength of Materials models with the model derived from Finite Elasticity is carried out. Predictions for the same problems, interpreted in the context of the specific model, are compared and it was found that sometimes they give rather different results, both in the results and in the computational cost. Energy analyses are performed in order to obtain information about the quality of the numerical solutions. The paper ends with an example of a pendulum with a finite pivot including friction and flexibility. When the structural elements are sufficiently slender and the rotational speeds are low, so that the resulting deformations are small, the Strength of Material model that includes the load stiffening and the Finite Elasticity approach, lead to similar results. It can be concluded that the stiffening phenomenon is appropriately considered in the first model. On the contrary, when the Strength of Material hypothesis are not fulfilled, the problem should be addressed via the Finite Elasticity model. Additionally, cases with complexities such as friction at a finite pivot can only be addressed by Finite Elasticity.     

A note on the vibrations of rectangular plates of variable thickness with two opposite simply supported edges and very general boundary conditions on the other two

An approximate solution for the title problem is obtained by making use of the Galerkin method. The plate displacement function is approximated by means of a sinusoid multiplied by a polynomial. Translational and rotational flexibilities are taken into account at x = ±a/2. It is shown that the free edge situation (Kirchoff's boundary condition) can be treated as a special case by means of the approach developed herein. A simple algorithm which allows evaluation of the fundamental frequency of vibration is derived and rough estimates of amplitudes and stress resultants are also given when the plate is subjected to a p₀cosωt-type excitation.     

Static and dynamic behavior of circular plates of variable thickness elastically restrained along the edges

Simple polynomial approximations and a variational approach are used to solve a rather complex elasto-mechanics problem. It is assumed that the plate is elastically restrained against rotation and translation along the edge. The approach developed in the present paper allows for a unified solution of both free and forced vibration problems, the static situation being a special situation of the dynamic state.