Abstract

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 LShaped
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.
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