When a load is attached to one end of an extensible beam whose ends are a fixed distance apart, the mathematical model describing the vibrations of the beam contains a nonlinearity in the partial differential equation as well as in the dynamical boundary condition. We show that uniform exponential stability can be achieved when closed loop feedback stabilization, consisting of two boundary controls, is incorporated at the point of contact between the load and the beam. Our analytical results are complemented by numerical results for the linear case.
The Local Linear Timoshenko (LLT) model for the planar motion of a rod that undergoes flexure, shear and extension, was recently derived in Van Rensburg et al. (2021). In this paper we present an algorithm developed for this model. The algorithm is based on the mixed finite element method, and projections into finite dimensional subspaces are used for dealing with nonlinear forces and moments. The algorithm is used for an investigation into elastic waves propagated in the LLT rod. Interesting properties of the LLT rod include the increased propagation speed of elastic waves when compared to the linear Timoshenko beam, and the appearance of buckled states or equilibrium solutions for compressed LLT beams. It is also shown that the LLT rod is applicable to large displacements and rotations for a wide range of slender elastic objects; from beams to highly slender flexible rods.
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