Dynamics of two impacting beams with clearance nonlinearity

Analytical solutions describing the transient dynamics of two Euler-Bernoulli beams with tips separated by clearance, are obtained. The tips of the beams impact when one of the beams is harmonically excited. Expressions of transient dynamics are presented as a superposition of particular solutions that satisfy to inhomogeneous boundary conditions, and eigenfunctions series with time dependent coefficients and homogeneous boundary conditions. The transition from impact phase to out-of-contact phase and vice versa is implemented using conditions that switch, involving construction of expressions for shear forces and relative position of beam tips. After each transition from one phase to another, the functions describing the time dependent coefficients in the eigenfunctions series are updated. This update involves the solution of ordinary differential equations with initial conditions corresponding to the end of the previous phase. The system of impacting beams reveals complex dynamics, including chaotic behaviour. Transient dynamics surfaces, time histories of beams deflections, impact forces, coefficients of restitution and phase planes are presented.