Nonlinear equations of motion of L-shaped beam structures

An L-shaped inextensional isotropic Euler-Bernoulli beam structure is considered in this paper and all the nonlinear equations of motion are derived up to and including second order nonlinearity. The associated nonlinear boundary conditions have also been derived up to and including second order terms. The global displacements of the secondary beam have been used in the equations of motion and the rotary inertia terms have also been considered. It has been demonstrated that the nonlinear equations couple the in-plane and the out-of-plane motions. It is also shown that the associated linear mode shapes are orthogonal to each other, which is essential for discretization of the equations of motion. Considering that the linear global displacements have been used in deriving the equations of motion then these equations can easily be discretised by projecting the dynamics onto the infinite linear mode shape basis, for reduce order modelling. Completion of this work is judged to have been essential in order to be able to perform nonlinear modal analysis on the L-shaped beam structure. It should be mentioned that this is the only nonlinear model describing the out-of-plane motions of L-Shaped beam structure which have been neglected so far in the literature. The presented process of deriving the mathematical model of the L-shaped beam structure paves the way for modelling of the nonlinear dynamics of more complicated structures comprised with several elementary substructures considering geometric nonlinearities and this analytical modelling is essential for further reduced order modelling and analysis of the nonlinear dynamics of these models.