On the Parametric Excitation of a Dynamic System Having Multiple Degrees of Freedom

1963 ◽  
Vol 30 (3) ◽  
pp. 367-372 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Dynamic System ◽  
Ordinary Differential Equations ◽  
Parametric Excitation ◽  
Degrees Of Freedom ◽  
Periodic Coefficients ◽  
Multiple Degrees Of Freedom ◽  
First Approximation ◽  
Approximation Analysis

A dynamical system having multiple degrees of freedom and under parametric excitation is governed by a system of ordinary differential equations with periodic coefficients. In this paper a first-approximation analysis is carried out and criteria for instability are derived explicitly.

Further Results on Parametric Excitation of a Dynamic System

1965 ◽  
Vol 32 (2) ◽  
pp. 373-377 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamic System ◽  
Parametric Excitation ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Constant Matrix ◽  
System Parameters ◽  
Multiple Degrees Of Freedom ◽  
The Stability ◽  
The Given

A dynamic system having multiple degrees of freedom and being under parametric excitation has been studied in an earlier paper [2]. However, the analysis given there necessitates certain restrictions on the distribution of the natural frequencies of the system. In this paper those restrictions are removed. The analysis presented here shows how to obtain a constant matrix whose eigenvalues determine the stability or instability of a system of ordinary differential equations with periodic coefficients at a given excitation frequency. The constant matrix is expressed entirely in terms of the given system parameters and the excitation frequency.

scholarly journals ON THE SELF-DISPLACEMENT OF DEFORMABLE BODIES IN A POTENTIAL FLUID FLOW

2008 ◽  
Vol 18 (11) ◽  
pp. 1945-1981 ◽  
Author(s):  
ALEXANDRE MUNNIER
Keyword(s):  
Fluid Flow ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Order System ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Deformable Bodies ◽  
Potential Fluid

Understanding fish-like locomotion as a result of internal shape changes may result in improved underwater propulsion mechanism. In this paper, we study a coupled system of partial differential equations and ordinary differential equations which models the motion of self-propelled deformable bodies (called swimmers) in a potential fluid flow. The deformations being prescribed, we apply the least action principle of Lagrangian mechanics to determine the equations of the inferred motion. We prove that the swimmers' degrees of freedom solve a second-order system of nonlinear ordinary differential equations. Under suitable smoothness assumptions on the boundary of the fluid's domain and on the given deformations, we prove the existence and regularity of the bodies rigid motions, up to a collision between two swimmers or between a swimmer with the boundary of the fluid. Then we compute explicitly the Euler–Lagrange equations in terms of the geometric data of the bodies and of the value of the fluid's harmonic potential on the boundary of the fluid.

scholarly journals The Vibrations of a Particle about a Position of Equilibrium—Part 2 :The Relation between the Elliptic Function and Series Solutions

1921 ◽  
Vol 40 ◽  
pp. 34-49 ◽  
Author(s):  
Bevan B. Baker
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Elliptic Function ◽  
Degrees Of Freedom ◽  
Elliptic Functions ◽  
Series Solutions ◽  
Two Degrees Of Freedom ◽  
Function Solution

In a previous paper, entitled the “Vibrations of a Particle about a Position of Equilibrium,” by the author in collaboration with Professor E. B. Ross (Proc. Edin. Math. Soc., XXXIX, 1921, pp. 34–57), a particular dynamical system having two degrees of freedom was chosen and solutions of the corresponding differential equations were obtained in terms of periodic series and also in terms of elliptic functions. It was shown that for certain values of the frequencies of the principal vibrations, the periodic series become divergent, whereas the elliptic function solution continues to give finite results.

A Simplified and Consistent Nonlinear Transient Analysis Method for Gas Bearing: Extension to Flexible Rotors

2014 ◽  
Author(s):  
Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Rational Functions ◽  
Special Treatment ◽  
Air Bearings ◽  
Fluid Forces ◽  
Stability Diagrams ◽  
Nonlinear Transient

A simplified, new method for evaluating the nonlinear fluid forces in air bearings was recently proposed in [1]. The method is based on approximating the frequency dependent linearized dynamic coefficients at several eccentricities, by second order rational functions. A set of ordinary differential equations is then obtained using the inverse of Laplace Transform linking the fluid forces components to the rotor displacements. Coupling these equations with the equations of motion of the rotor lead to a system of ordinary differential equations where displacements and velocities of the rotor and the fluid forces come as unknowns. The numerical results stemming from the proposed approach showed good agreement with the results obtained by solving the full nonlinear transient Reynolds equation coupled to the equation of motion of a point mass rotor. However the method [1] requires a special treatment to ensure continuity of the values of the fluid forces and their first derivatives. More recently, the same authors [2] showed the benefits of imposing the same set of stable poles to the rational functions approximating the impedances. These constrains simplified the expressions of the fluid forces and avoided the introduction of false poles. The method in [2] was applied in the frame of the small perturbation analysis for calculating Campbell and stability diagrams. This approach enhances also the consistency of the fluid forces approximated with the same set of poles because they become naturally continuous over the whole bearing clearance while their increments were not. The present paper shows how easily the new formulation may be applied to compute the nonlinear response of systems with multiple degrees of freedom such as a flexible rotor supported by two air bearings.

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