A Perturbation Solution of the Equations of Motion of a Gyroscope

1959 ◽  
Vol 26 (3) ◽  
pp. 349-352
Author(s):  
Robert Goodstein
Keyword(s):  
Perturbation Method ◽  
Nonlinear Equations ◽  
Forced Vibration ◽  
Equations Of Motion ◽  
Rate Of Change ◽  
Steady Rate ◽  
Center Position ◽  
Method Of Solution ◽  
Gimbal Gyro ◽  
Nonlinear Equations Of Motion

Abstract A perturbation method of solution is outlined for the nonlinear equations of motion of free and forced vibration of a two-gimbal gyro. Results are given for the term displayed in the solution which indicates that the outer gimbal of the gyro will not oscillate about its initial center position, but will acquire a steady rate of change of position.

Frequency Response of Fluid Lines With Nonlinear Boundary Conditions

1971 ◽  
Vol 93 (3) ◽  
pp. 365-372 ◽  
Author(s):  
R. D. Strunk
Keyword(s):  
Boundary Conditions ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Frequency Response ◽  
Nonlinear Equations ◽  
Nonlinear Boundary ◽  
Harmonic Distortion ◽  
Nonlinear Boundary Conditions ◽  
Qualitative Information ◽  
Method Of Solution

The harmonic distortion generated when a fluid line is terminated by a nonlinear orifice characteristic is analyzed by using a perturbation method of solution. The perturbation method is shown to be representative of the true phenomenon and to give very good quantitative as well as qualitative information by comparing the results to a numerical solution of the nonlinear equations. The results presented describe the distortion phenomenon as a function of several dimensionless ratios.

Direct Linearization of Continuous and Hybrid Dynamical Systems

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Julie J. Parish ◽  
John E. Hurtado ◽  
Andrew J. Sinclair
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Equations ◽  
Equilibrium Configuration ◽  
Equations Of Motion ◽  
Hybrid Dynamical Systems ◽  
Linearized Equations ◽  
Direct Linearization ◽  
Taylor Series Expansions ◽  
Nonlinear Equations Of Motion ◽  
And Control

Nonlinear equations of motion are often linearized, especially for stability analysis and control design applications. Traditionally, the full nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. This procedure is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems. The results presented require only velocity-level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of selected partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest, without first generating the entire nonlinear equations of motion. Given an equilibrium configuration of interest, the directly constructed linearized equations of motion allow one to bypass first forming the full nonlinear governing equations for the system. Examples are presented to illustrate the method for both continuous and hybrid systems.

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