The nonlinear response of an initially buckled beam in the neighborhood of 1 : I internal resonance is investigated analytically, numerically, and experimentally. The method of multiple time scales is applied to derive the equations in amplitudes and phase angles. Within a small range of the internal detuning parameter, the first mode; which is externally excited, is found to transfer energy to the second mode. Outside this region, the response is governed by a unimodal response of the first mode. Stability boundaries of the unimodal response are determined in terms of the excitation level, and internal and external detuning parameters. Boundaries separating unimodal from mixed mode responses are obtained in terms of the excitation and internal detuning parameters. Stationary and non-stationary solutions are found to coexist in the case of mixed mode response. For the case of non-stationary response, the modulation of the amplitude depends on the integration increment such that the motion can be periodically or chaotically modulated for a choice of different integration increments. The results obtained by multiple time scales are qualitatively compared with those obtained by numerical simulation of the original equations of motion and by experimental measurements. Both numerical integration and experimental results reveal the occurrence of multifurcation, escaping from one well to the other in an irregular manner. and chaotic motion.

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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.

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.

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