Nonlinear Response of Cantilever Beams
Abstract
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.
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- Doctoral Dissertations [13614]