Motion limiting nonlinear dynamics of initially curved beams: Fid-Bau Portal

Motion limiting nonlinear dynamics of initially curved beams

Farokhi, Hamed / Ghayesh, Mergen H.

Abstract An initially curved beam is considered and its motion is constrained using two elastic constraints; the corresponding non-smooth nonlinear transverse dynamics is investigated for the first time. A clamped-clamped beam with one axially movable end is modelled via Bernoulli-Euler beam theory together with the inextensibility condition, giving rise to nonlinear inertial terms along with nonlinear geometric terms. Furthermore, the damping is modelled via Kelvin-Voigt internal damping model. The proposed model is verified for linear and nonlinear behaviours via comparison to a finite element model. The impact between beam and constraints is incorporated via calculating its work contribution. The nonlinear equation of motion is derived while incorporating geometric, damping, inertial, and constraints nonlinearities. A series of spatial basis functions together with corresponding vibration modes are used as the proposed solution of the transverse displacement. A modal discretisation is performed via the weighted-residual method of Galerkin and the corresponding non-smooth terms are kept intact while conducting numerical integration. A numerical continuation technique is utilised to solve the resultant equations. The non-smooth response is obtained for various cases and the effects of several parameters are studied thoroughly.

Highlights • The nonlinear dynamics of unconstrained system showed that the presence of initial curvature gives rise strong modal interactions as well as quasiperiodic motion. • For unconstrained system, increasing the initial curvature amplitude results in decreased natural frequency. • Presence of elastic constraints renders the motion much more complex and gives rise to many bifurcation points and many stable and unstable attractors. • For a constrained system, at larger base excitation amplitudes, period-doubling bifurcation occurs resulting in a new bifurcation solution branch with both quasiperiodic and periodic attractors. • Decreasing the gap width reduces the vibration amplitude and causes the slope-change to occur at smaller oscillation amplitudes.