Parametric resonance of fractional multiple-degree-of-freedom damped beam systems

2021 ◽  
Author(s):  
Beatrice Pomaro ◽  
Carmelo E. Majorana
Keyword(s):  
Parametric Resonance ◽  
Degree Of Freedom

scholarly journals Dynamic Instability of High-Speed Rotating Shaft with Torsional Effect

2018 ◽  
Vol 15 (4) ◽  
pp. 6034-6051
Author(s):  
A. M. A. Wahab ◽  
Z. Yusof ◽  
Z. A. Rasid ◽  
A. Abu ◽  
N. F. M. N. Rudin
Keyword(s):  
Parametric Resonance ◽  
High Speed ◽  
Dynamic Instability ◽  
Parametric Instability ◽  
Degree Of Freedom ◽  
Rotating Shaft ◽  
Torsional Deformation ◽  
High Rotational Speed ◽  
The Difference ◽  
Frequency Ratios

Today’s design of machine rotor requires the rotor to operate at a high rotational speed to improve the efficiency of the machine. However, the existence of disturbances such as periodic axial load may cause parametric resonance to the rotor system in addition to the common force resonance. Previous studies on this parametric resonance of shaft typically included the element of translational and rotary inertia, gyroscopic moments and bending and shear deformation but surprisingly neglected the effect of the axial torque. This paper investigated the parametric instability behaviour of the shaft rotating at high speed while considering the torsional effect of the shaft. Based on the finite element method, a shaft model that includes torsional deformation as one of its degree of freedom was established. The Mathieu-Hill equation was derived, and then the Bolotin’s method was used to solve the equation by establishing the parametric instability chart. Two types of the rotary system were studied: a shaft with different boundary conditions and shaft with different bearing types. The results were initially validated with past findings. Following that the results were compared to the results correspond to the Timoshenko’s beam formulation that omits the torsional degree of freedom. The effect of axial torsional deformation was found to be very significant especially at high speed. The developed model in this study shows that at the shaft speed of 40000 rpm, the effect of torsional deformation has given the difference of more than 100% in the frequency ratios correspond to the 4DOF and 5DOF models for the case of fix-free boundary condition.

PARAMETRICALLY EXCITED NONLINEAR TWO-DEGREE-OF-FREEDOM SYSTEMS WITH NONSEMISIMPLE ONE-TO-ONE RESONANCE

1995 ◽  
Vol 05 (03) ◽  
pp. 725-740 ◽  
Author(s):  
C. CHIN ◽  
A.H. NAYFEH
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Degree Of Freedom ◽  
Parametrically Excited ◽  
Parametric Resonances ◽  
Quadratic Nonlinearities ◽  
One To One ◽  
Two Degree Of Freedom

The response of a parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities and a nonsemisimple one-to-one internal resonance is investigated. The method of multiple scales is used to derive four first-order differential equations governing the modulation of the amplitudes and phases of the two modes for the cases of fundamental and principal parametric resonances. Bifurcation analysis of the case of fundamental parametric resonance reveals that the quadratic nonlinearities qualitatively change the response of the system. They change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.

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