The non-linear response of a slender cantilever beam carrying a lumped mass to a principal parametric base excitation is investigated theoretically and experimentally. The Euler-Bernoulli theory for a slender beam is used to derive the governing non-linear partial differential equation for an arbitrary position of the lumped mass. The non-linear terms arising from inertia, curvature and axial displacement caused by large transverse deflections are retained up to third order. The linear eigenvalues and eigenfunctions are determined. The governing equation is discretized by Galerkin's method, and the coefficients of the temporal equation—comprised of integral representations of the eigenfunctions and their derivatives—are computed using the linear eigenfunctions. The method of multiple scales is used to determine an approximate solution of the temporal equation for the case of a single mode. Experiments were performed on metallic beams and later on composite beams because all of the metallic beams failed prematurely due to the very large response amplitudes. The results of the experiment show very good qualitative agreement with the theory.

The Non-Linear Response of a Slender Beam Carrying a Lumped Mass to a Principal Parametric Excitation: Theory and Experiment

Application of the incremental harmonic balance method to cubic non-linearity systems

A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented.

Nonlinear response of a parametrically excited buckled beam

The Response of a Single-Degree-of-Freedom System with Quadratic and Cubic Non-Linearities to a Fundamental Parametric Resonance

An analysis is given of the dynamic of a one-degree-of-freedom oscillator with quadratic and cubic nonlinearities subjected to parametric and external excitations having incommensurate frequencies. A new method is given for constructing an asymptotic expansion of the quasi-periodic solutions. The generalized averaging method is first applied to reduce the original quasi-periodically driven system to a periodically driven one. This method can be viewed as an adaptation to quasi-periodic systems of the technique developed by Bogolioubov and Mitropolsky for periodically driven ones. To approximate the periodic solutions of the reduced periodically driven system, corresponding to the quasi-periodic solution of the original one, multiple-scale perturbation is applied in a second step. These periodic solutions are obtained by determining the steady-state response of the resulting autonomous amplitude-phase differential system. To study the onset of the chaotic dynamic of the original system, the Melnikov method is applied to the reduced periodically driven one. We also investigate the possibility of achieving a suitable system for the control of chaos by introducing a third harmonic parametric component into the cubic term of the system.

Lawrence D. Zavodney

Papers

The Response of a Single-Degree-of-Freedom System with Quadratic and Cubic Non-Linearities to a Fundamental Parametric Resonance

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance