Approximate Analytical Solution for the Dynamic Model of Large Amplitude Non-Linear Oscillations Arising in Structural Engineering

TL;DR

This paper develops an approximate analytical method using Laplace transforms to solve a strongly nonlinear differential equation modeling large amplitude vibrations in a cantilever beam, providing a practical alternative to numerical solutions.

Contribution

It introduces an iterative Laplace transform-based approach to obtain approximate solutions for complex nonlinear vibration equations in structural engineering.

Findings

Approximate solutions closely match numerical results.

Method applicable to any order of approximation.

Provides a new analytical tool for nonlinear vibration analysis.

Abstract

In this work we obtain an approximate solution of the strongly nonlinear second order differential equation $\frac{d^{2}u}{dt^{2}}+\omega ^{2}u+\alpha u^{2}\frac{d^{2}u}{dt^{2}}+\alpha u\left( \frac{du}{dt}\right)^{2}+\beta \omega ^{2}u^{3}=0$, describing the large amplitude free vibrations of a uniform cantilever beam, by using a method based on the Laplace transform, and the convolution theorem. By reformulating the initial differential equation as an integral equation, with the use of an iterative procedure, an approximate solution of the nonlinear vibration equation can be obtained in any order of approximation. The iterative approximate solutions are compared with the exact numerical solution of the vibration equation.