Symmetry group classification and optimal reduction of a class of damped Timoshenko beam system with a nonlinear rotational moment

TL;DR

This paper classifies symmetries of a nonlinear damped Timoshenko beam system with arbitrary rotation moment dependence, deriving optimal reductions to simpler ODEs for various nonlinear cases.

Contribution

It provides a comprehensive Lie symmetry classification and optimal reduction framework for a class of nonlinear damped Timoshenko systems with arbitrary rotation moment functions.

Findings

Symmetry classification for all nonlinear cases

Derivation of optimal one-dimensional subalgebras

Reduced ODE systems for invariant variables

Abstract

We consider a nonlinear Timoshenko system of partial differential equations (PDEs) with a frictional damping term in rotation angle. The nonlinearity is due to the arbitrary dependence on the rotation moment. A Lie symmetry group classification of the arbitrary function of rotation moment is presented. An optimal system of one-dimensional subalgebras of the nonlinear damped Timoshenko system is derived for all the non-linear cases. All possible invariant variables of the optimal systems for the three non-linear cases are presented. The corresponding reduced systems of ordinary differential equations (ODEs) are also provided.