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Chin. Phys. B, 2011, Vol. 20(4): 040501    DOI: 10.1088/1674-1056/20/4/040501
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Pseudo analytical solution to time periodic stiffness systems

Wang Yan-Zhong(王延忠) and Zhou Yuan-Zi(周元子)
School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China
Abstract  An analytical form of state transition matrix for a system of equations with time periodic stiffness is derived in order to solve the free response and also allow for the determination of system stability and bifurcation. A pseudo-closed form complete solution for parametrically excited systems subjected to inhomogeneous generalized forcing is developed, based on the Fourier expansion of periodic matrices and the substitution of matrix exponential terms via Lagrange-Sylvester theorem. A Mathieu type of equation with large amplitude is presented to demonstrate the method of formulating state transition matrix and Floquet multipliers. A two-degree-of-freedom system with irregular time periodic stiffness characterized by spiral bevel gear mesh vibration is presented to find forced response in stability and instability. The obtained results are presented and discussed.
Keywords:  parametric excitation      time periodic stiffness      stability      response  
Received:  12 June 2010      Revised:  30 December 2010      Accepted manuscript online: 
PACS:  05.10.-a (Computational methods in statistical physics and nonlinear dynamics)  
  45.20.dc (Rotational dynamics)  
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 50875009), the Defense Industrial Technology Development Program of China (Grant No. B0620060424) and the Aviation Science Foundation of China (Grant No. 20090451009).

Cite this article: 

Wang Yan-Zhong(王延忠) and Zhou Yuan-Zi(周元子) Pseudo analytical solution to time periodic stiffness systems 2011 Chin. Phys. B 20 040501

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