| The parametric resonances and internal resonances of beams with axial force and supporting motion have been investigated. Attention is mainly concentrated on the parametric resonances of beams with pulsating axial force, and the internal resonances of beams subjected to axial tension and with excitation of supporting motion. Theoretical analysis is carried out firstly in the process of investigation, and then the results are verified by quantitative analysis and numerical simulations.In the analysis of the parametric resonances of beams with pulsating axial force, we take four boundary conditions to determine and analyze the regions of stability of sub-harmonic parametric resonances of the second mode. The stability of zero and non-zero solutions is also analyzed. The influence of system parameters on unstable regions is discussed in the sense of quantitative analysis. The analytical solutions of amplitude-frequency and their curves are obtained by using the method of averaging, and the existence intervals of these curves are also analyzed.In the analysis of the internal resonances of beams subjected to axial tension and with excitation of supporting motion, the bifurcation equation of zero-to-one internal resonance and a near-integrable four-degree-of-freedom Hamiltonian system are obtained by the averaging method and a series of transformations. Using the energy-phase criterion, the conditions under which a Silnikov type orbit may exist near a Hamiltonian system are examined.Through numerical simulations of original governing equation, the effects of axial tension and supporting motion on the dynamic behavior of beams have been investigated, respectively, and it is found that the system gives rise to chaotic motion when the second mode natural frequency of the beam is twice the frequency of motion of supporting base. Some regions where the chaotic motions occur are determined numerically, and the dynamical behavior near these regions of chaotic motions is analyzed. |
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