Structural Modal Multifurcation With International Resonance: Part 1—Deterministic Approach

Abstract

The bifurcation and multifurcation in multimode interaction of nonlinear continuous structural systems is investigated. Under harmonic excitation the nonstationary response of multimode interaction is considered in the neighborhood of fourth-order internal resonance condition. The response dynamic characteristics are examined via three different approaches. These are the multiple scales method, numerical simulation, and experimental testing. The model considered is a clamped-clamped beam with initial static axial load. Under certain values of the static load the first three normal modes are nonlinearly coupled and this coupling results in a fourth-order internal resonance. The method of multiple time scales yields nonstationary response in the neighborhood of internal resonance. Within a small range of internal detuning parameter the third mode, which is externally excited, is found to transfer energy to the first two modes. Outside this region, the response is governed by a unimodal response of the third mode which follows the Duffing oscillator characteristics. The bifurcation diagram which represents the boundaries that separate unimodal and mixed mode responses is obtained in terms of the excitation level, damping ratios, and internal resonance detuning parameter. The domains of attraction of the two response regimes are also obtained. The numerical simulation of the original equations of motion suggested the occurrence of complex response characteristics for certain values of damping ratios and excitation amplitude. Both numerical integration and experimental results reveal the occurrence of multifurcation as reflected by multi-maxima of the response probability density curves.

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

“…A deterministic beam test specimen [8] was installed on the shaker platform such that torsional and out-of-plane motions were minimized. Adjusting the initial axial load P = 2.76 lb for internal resonance was difficult because the clamped ends affected the setting.…”

“…Using Galerkin's method the PDE governing equation is reduced to three ordinary differential equations of motion by substituting the assumed mode solutions for U(z, t) and requiring the equation to be orthogonal to a selected mode shape [8]. The modal expansion is given as…”

“…Assuming a lumped parameter approach and multiplying by a finite interval, Ax, the beam's partial differential equation of motion takes the form [8] m--P 02U(t)j 02U(t)J Ax q-EI 04U(t)j Ax q-/kx 0t; 2 OX 4 OZ 2…”

“…The results, referenced to the initial equilibrium position, were obtained in terms of initial axial displacements but the effects of damping, asymmetric modes and internal resonance were not studied. Ibrahim et al [8], examined a harmonically excited clamped-clamped beam analytically, numerically and experimentally. The axial static load was adjusted to give the internal resonance relation wl + 2co2 --co3.…”

Nonlinear response of a beam under distributed moving contact load

“…Imposing appropriate assumptions, it is shown that the problem is fundamentally reduced to a nonlinear Leipholz beam subjected to two frequency excitations of the moving contact loads. There have been a number of theoretical and experimental studies on the chaotic vibration of a forced beam element, e.g., [24][25][26][27][28][29]. In these reports, various techniques such as multiple scales and direct numerical integration have been employed to simulate the chaotic motions.…”