Symmetry group classification and optimal reduction of a class of damped   Timoshenko beam system with a nonlinear rotational moment
  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. 
S. M. Al-Omari, F. D. Zaman, A. Y. Al-Dweik and Ryad A Ghanam
math.AP