Structural Modal Multifurcation With Internal Resonance: Part 2—Stochastic Approach

Abstract

Stochastic bifurcation in moments of a clamped-clamped beam response to a wide band random excitation is investigated analytically, numerically, and experimentally. The nonlinear response is represented by the first three normal modes. The response statistics are examined in the neighborhood of a critical static axial load where the normal mode frequencies are commensurable. The analytical treatment includes Gaussian and non-Gaussian closures. The Gaussian closure fails to predict bifurcation of asymmetric modes. Both non-Gaussian closure and numerical simulation yield bifurcation boundaries in terms of the axial load, excitation spectral density level, and damping ratios. The results of both methods are in good agreement only for symmetric response characteristics. In the neighborhood of the critical bifurcation parameter the Monte Carlo simulation yields strong nonstationary mean square response for the asymmetric mode which is not directly excited. Experimental and Monte Carlo simulation exhibit nonlinear features including a shift of the resonance peak in the response spectra as the excitation level increases. The observed shift is associated with a widening effect in the response bandwidth.

Modal Interactions of a Randomly Excited Hinged–clamped Beam

“…When the load does not exceed the Euler buckling load [11], the Gaussian closure failed to predict bifurcation of unexcited second mode under all possible conditions of axial load and excitation level. But both the non-Gaussian closure and Monte Carlo simulation predicted second mode bifurcation.…”

“…On the other hand, Ibrahim and his colleagues [11,12] have studied the stochastic bifurcation of the unexcited mode of a clamped}clamped beam under wide band random excitation when initial static axial load is applied to the beam. When the load does not exceed the Euler buckling load [11], the Gaussian closure failed to predict bifurcation of unexcited second mode under all possible conditions of axial load and excitation level.…”

Nonlinear response of a beam under distributed moving contact load

“…6(a) shows the harmonic components included in the assumed solution in terms of x s and x d . However, since the solutions in (29) are assumed in terms ofmx 1 and x 2 , the corresponding harmonic components included in the solutions are as described Fig. 6(b).…”

“…However, when the Newton-Raphson iteration method is applied to determine the unknown coefficients a i and b i in Eq. (29), this technique requires repetitive evaluations ofĉ i ,d i , c i and d i in Eq. (30) in each iteration step to obtain the Jacobian matrix.…”

“…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.…”