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Investigation on the planar nonlinear vibration of cantilever beams with various slenderness ratios

English Abstract

Investigation on the Planar Nonlinear Vibration of Cantilever Beams with Various Slenderness Ratios by Haien Du Thesis Supervisor: Prof. Guo-Kang Er Department of Civil and Environmental Engineering In this thesis, the uniform cantilever beam under the action of the lateral harmonic load on the tip and axial self-weight has been investigated. in order to find out the effects of high order terms, such as the third and fifth order nonlinearities, and the effects of slenderness of the beam, a variational approach based on the extended Hamilton principle is employed to derive the equations of motion and boundary conditions governing the planar nonlinear vibrations of the isotropic and inextensible Euler-Bernoulli cantilever beams with different slender ratios. The nonlinear second-order ordinary differential equations (ODEs) containing up to third and fifth order nonlinear terms are obtained with the Galerkin method. Then the Runge-Kutta method is applied to analyze the response and stability of the system. A comparison between linear cantilever beam, the nonlinear cantilever beam with up to cubic nonlinear terms, and the nonlinear cantilever beam with up to quintic nonlinear terms has been carried out. Furthermore, the nonlinear behaviors and the effects of the slenderness of the cantilever beams have been examined numerically. A relative value of comparison between linear stiffness term and nonlinear stiffness term is introduced to illustrate the nonlinear behaviors of the cantilever beam vibration. The convergence of the solutions is analyzed by increasing the number of mode functions of the linear cantilever beam in the Galerkin approximation. The nonlinearity of the cantilever beam under the lateral harmonic load on the tip and axial self-weight is studied with different frequencies of harmonic excitations. Some hardening and softening behaviors have been observed and investigated through the numerical analysis. Key Words: nonlinear vibration, cantilever beam, slenderness ratio, geometrical nonlinearity, inertial nonlinearity, steady state response, Galerkin method, Runge-Kutta method, multiple mode response, jump phenomenon.

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Du, Hai En


Faculty of Science and Technology


Department of Civil and Environmental Engineering





Structural dynamics -- Mathematics


Er, Guo Kang

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