Nonlinear Response of Cantilever Beams
The nonlinear nonplanar steady-state responses of cantilever beams to direct and parametric harmonic excitations are investigated using perturbation techniques. Modal interactions between the bending-bending and bending-bending-twisting motions are studied. Using a variational formulation, we obtained the governing equations of motion and associated boundary conditions for monoclinic composite and isotropic metallic inextensional beams. The method of multiple scales is applied either to the governing system of equations and associated boundary conditions or to the Lagrangian and virtual-work term to determine the modulation equations that govern the slow dynamics of the responses. These equations are shown to exhibit symmetry properties, reflecting the conservative nature of the beams in the absence of damping.
It is popular to first discretize the partial-differential equations of motion and then apply a perturbation technique to the resulting ordinary-differential equations to determine the modulation equations. Due to the presence of quadratic as well as cubic nonlinearities in the governing system for the bending-bending-twisting oscillations of beams, it is shown that this approach leads to erroneous results. Furthermore, the symmetries are lost in the resulting equations.
Nontrivial fixed points of the modulation equations correspond, generally, to periodic responses of the beams, whereas limit-cycle solutions of the modulation equations correspond to aperiodic responses of the beams. A pseudo-arclength scheme is used to determine the fixed points and their stability. In some cases, they are found to undergo Hopf bifurcations, which result in limit cycles. A combination of a long-time integration, a two-point boundary-value continuation scheme, and Floquet theory is used to determine in detail branches of periodic and chaotic solutions and assess their stability. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations. The chaotic attractors undergo attractor-merging and boundary crises as well as explosive bifurcations.
For certain cases, it is determined that the response of a beam to a high-frequency excitation is not necessarily a high-frequency low-amplitude oscillation. In fact, low-frequency high-amplitude components that dominate the responses may be activated by resonant and nonresonant mechanisms. In such cases, the overall oscillations of the beam may be significantly large and cannot be neglected.