Title page for ETD etd-042399-153536

Type of Document Dissertation
Author Arafat, Haider Nabhan
Author's Email Address harafat@vt.edu
URN etd-042399-153536
Title Nonlinear Response of Cantilever Beams
Degree PhD
Department Engineering Mechanics
Advisory Committee
Advisor Name Title
Nayfeh, Ali H. Committee Chair
Ahmadian, Mehdi Committee Member
Hajj, Muhammad R. Committee Member
Hendricks, Scott L. Committee Member
Kraige, Luther Glenn Committee Member
Mook, Dean T. Committee Member
  • Resonances
  • Beams
  • Vibrations
  • Modal Interactions
  • Nonlinear Responses
Date of Defense 1999-04-09
Availability unrestricted
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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