Baisheng Wu

Summation of perturbation solutions to nonlinear oscillations

SummaryBy coupling the Lindstedt-Poincaré perturbation technique with a rational approximation, we propose a method for summing up the perturbation solutions of the nonlinear oscillation of a conservative single-degree-of-freedom system. The equation of motion contains a parameter. This method can represent the singularities of the period at certain values of the oscillation amplitude and extend the range of validity of the perturbation solution. For constructing the summation, all is needed are the coefficients of the pertubation expansion of the periodic solution. Approximate formulas for the period and the corresponding periodic solution of the nonlinear oscillation are established. Two examples are used to illustrate the effectiveness of the method.

ANALYTICAL APPROXIMATE SOLUTIONS TO LARGE-AMPLITUDE FREE VIBRATIONS OF UNIFORM BEAMS ON PASTERNAK FOUNDATION

This paper is concerned with the large-amplitude vibration behavior of simply supported and clamped uniform beams, with axially immovable ends, on Pasternak foundation. The combination of Newtons method and harmonic balance one is used to deal with these vibrations. Explicit and brief analytical approximations to nonlinear frequency and periodic solution of the beams for various values of the two stiffness parameters of the Pasternak foundation, small as well as large amplitudes of oscillation are presented. The analytical approximate results show excellent agreement with those from numerical integration scheme. Due to brevity of expressions, the present analytical approximate solutions are convenient to investigate effects of various parameters on the large-amplitude vibration response of the beams.

Influence of shear deformation on the buckling of elastically supported beams

SummaryThe influence of shear deformation on the buckling behavior of a beam supported laterally by a Winkler elastic foundation is studied. A full investigation of the bifurcation points at which, under axial load, the beam becomes critical with respect to one or two simultaneous buckling modes is made. The configurations and stabilities of the equilibrium paths that bifurcate from the critical points are derived. From the results of theoretical analysis, it becomes evident that shear deformation has a considerable effect upon the equilibriums and stabilities of the post-buckling of the beam. The results for the Bernoulli-Euler beam can be obtained as a limiting case for those of the present beam by letting the shear stiffness tend to infinity.

Highly Accurate Analytical Approximate Solution to a Nonlinear Pseudo-Oscillator

Abstract A second-order Newton method is presented to construct analytical approximate solutions to a nonlinear pseudo-oscillator in which the restoring force is inversely proportional to the dependent variable. The nonlinear equation is first expressed in a specific form, and it is then solved in two steps, a predictor and a corrector step. In each step, the harmonic balance method is used in an appropriate manner to obtain a set of linear algebraic equations. With only one simple second-order Newton iteration step, a short, explicit, and highly accurate analytical approximate solution can be derived. The approximate solutions are valid for all amplitudes of the pseudo-oscillator. Furthermore, the method incorporates second-order Taylor expansion in a natural way, and it is of significant faster convergence rate.

Analytical Approximate Prediction of Thermal Post-Buckling Behavior of the Spring-Hinged Beam

This paper is concerned with thermal post-buckling of uniform isotropic beams with axially immovable spring-hinged ends. The ends of the beam with elastic rotational restraints represent the actual practical support conditions and the classical hinged and clamped conditions can be achieved as the limiting cases of the rotational spring stiffness. The governing differential–integral equation is solved by assuming suitable admissible function for lateral displacement and by employing the Galerkin method. A brief and explicit analytical approximate formulation is established to predict the thermal post-buckling behavior of the beam. The present analytical approximate expressions show excellent agreement with the corresponding numerical solutions based on the shooting method. This confirms the effectiveness and verifies the accuracy of the formulas established.