Dynamic Response of a Coupled Spinning Timoshenko Shaft System

1999 ◽  
Vol 121 (1) ◽  
pp. 110-113 ◽  
Author(s):  
E. Wong ◽  
J. W. Zu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Time Varying ◽  
Chebyshev Series ◽  
Bending Vibration ◽  
Linear Ordinary Differential Equations ◽  
Simply Supported ◽  
Shaft System

The dynamic behavior of a simply-supported spinning Timoshenko shaft with coupled bending and torsion is analyzed. This is accomplished by transforming the set of nonlinear partial differential equations of motion into a set of linear ordinary differential equations. This set of ordinary differential equations is a time-varying system and the solution is obtained analytically in terms of Chebyshev series. A beating phenomoenon is observed from the numerical simulations, which is not observed for shaft systems where only bending vibration is considered.

Dynamic Response of a Spinning Timoshenko Shaft With Coupled Bending and Torsion

1997 ◽  
Author(s):  
Jean W. Zu ◽  
Edward Wong
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Full Range ◽  
Nonlinear Partial Differential Equations ◽  
Time Varying ◽  
Chebyshev Series ◽  
Linear Ordinary Differential Equations ◽  
Simply Supported

Abstract The dynamic behavior of a simply-supported spinning Timoshenko shaft with coupled bending and torsion is analyzed. This is accomplished by transforming the set of nonlinear partial differential equations of motion into a set of linear ordinary differential equations. This set of ordinary differential equations is a time-varying system and the solution is obtained analytically in terms of Chebyshev series. The analytical method is a viable alternative to numerical methods and can provide the full range of the required solutions. A beating phenomenon is observed from the numerical simulations. This phenomenon occurs when the system has two natural frequencies close to each other. It is also shown that the period of torsional vibrations is much shorter than the period of oscillations in transverse deflections and in bending angles.

Control of Vibration Amplitude, Frequency and Damping of an Electrostatically Actuated Microbeam Using Capacitive, Inductive and Resistive Elements

2010 ◽  
Author(s):  
Abdolreza Pasharavesh ◽  
Y. Alizadeh Vaghasloo ◽  
A. Fallah ◽  
M. T. Ahmadian
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vibration Amplitude ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Electrical Current ◽  
Oscillatory System ◽  
Electrostatically Actuated ◽  
In Series

In this study vibration amplitude, frequency and damping of a microbeam is controlled using a RLC block containing a capacitor, resistor and inductor in series with the microbeam. Applying this method all of the considerable characteristics of the oscillatory system can be determined and controlled with no change in the geometrical and physical characteristics of the microbeam. Euler-Bernoulli assumptions are made for the microbeam and the electrical current through the microbeam is computed by considering the microbeam deflection and its voltage. Considering the RLC block, the voltage difference between the microbeam and the substrate is calculated. Two coupled nonlinear partial differential equations are obtained for the deflection and the voltage. The one parameter Galerkin method is employed to transform the equations of motion to a set of nonlinear coupled ordinary differential equations. Differential quadrature method (DQM) is implemented to solve the governing nonlinear ordinary differential equations. The effect of the controller parameters such as capacitance, resistance and inductance on the amplitude, frequency and damping is studied. Also the internal resonance between the electrical and mechanical parts of the system is studied. Results indicate using these elements, amplitude, frequency and damping can be controlled as desired by the user.

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