Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion |
| |
Authors: | Feng Zhihua Hu Haiyan |
| |
Institution: | (1) Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China |
| |
Abstract: | A set of nonlinear differential equations is established by using Kane's method for the planar oscillation of flexible beams
undergoing a large linear motion. In the case of a simply supported slender beam under certain average acceleration of base,
the second natural frequency of the beam may approximate the tripled first one so that the condition of 3∶1 internal resonance
of the beam holds true. The method of multiple scales is used to solve directly the nonlinear differential equations and to
derive a set of nonlinear modulation equations forthe principal parametric resonance of the first mode combined with 3∶1 internal
resonance between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response
and the stability condition of the beam. The abundant nonlinear dynamic behaviors, such as various types of local bifurcations
and chaos that do not appear for linear models, can be observed in the case studies. For a Hopf bifurcation, the 4-dimensional
modulation equations are reduced onto the central manifold and the type of Hopf bifurcation is determined. As usual, a limit
cycle may undergo a series of period-doubling bifurcations and become a chaotic oscillation at last.
The project supported by the Fund of Doctorial Program, Ministry of Education of China (20020287011) |
| |
Keywords: | flexible beam nonlinear dynamics method of multiple scales principal parametric resonance internal resonance |
本文献已被 维普 SpringerLink 等数据库收录! |
|