scholarly journals Theory of Frequency Captured of Eccentric Rotor by Vibration Environment with Same Direction

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Su Ming ◽  
Li Rong ◽  
Xie Zhiping ◽  
Zheng Jiming
Keyword(s):  
Differential Equation ◽  
Stability Criterion ◽  
Periodic Coefficient ◽  
Second Order ◽  
Frequency Synchronization ◽  
Synchronization Problem ◽  
Eccentric Rotor ◽  
Small Parameters ◽  
The Stability ◽  
Rotating Bodies

Aiming at the frequency synchronization phenomena of oscillating or rotating bodies, this paper proposes a novel solution to address the self-synchronization problem of vibration systems. An integral mean method with small parameters and periodic coefficient (IMM-SPPC) is proposed, which converts the relative motion of the electrically driven eccentric rotor and the vibration environment into a second-order periodic coefficient differential equation. Through the calculation of the equilibrium point of the second-order periodic coefficient differential equation and the study of its stability, the synchronization criterion and the stability criterion of the eccentric rotor and the vibration environment are deduced. The simulation results show the validity of the deduced synchronization criterion and stability criterion. The proposed IMM-SPPC provides a new way for studying vibration synchronization.

scholarly journals Theoretical Study on Synchronization of the Counter-Rotating Exciter in the Compound Vibrating Field

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yisen Sun ◽  
Ming Su ◽  
Xu Huang ◽  
Rongchen Zhao ◽  
Rong Li ◽  
...  
Keyword(s):  
Differential Equation ◽  
Theoretical Study ◽  
Order Differential Equation ◽  
Periodic Coefficient ◽  
Vibrating System ◽  
Synchronization Problem ◽  
Small Parameters ◽  
Coupling Characteristics ◽  
The Impact ◽  
Theoretical Results

Aiming at the impact of the complex vibration environment generated by the integrated vibration equipment on the vibration testing equipment, this paper proposes a new method to solve the vibratory synchronization problem in the compound vibration environment. A new concept of the compound vibrating field is proposed, and a new simple vibrating system with a single counter-rotating exciter in a compound vibrating field is established. The motion differential equation of the system is established by the integral mean method with small parameters, and then the periodic coefficient differential equation is obtained through linearization. Based on the relevant theory of the second-order differential equation with periodic coefficient, the synchronization criterion and stability criterion of the vibrating system are derived. According to the theoretical criteria, the coupling characteristics of the exciter and the vibrating field are numerically simulated and analyzed, which supports the theoretical results. The proposed compound vibrating field provides a new way for studying vibratory synchronization.

scholarly journals On the Existence of Eigenmodes of Linear Quasi-Periodic Differential Equations and their Relation to the MHD Continuum

1982 ◽  
Vol 37 (8) ◽  
pp. 830-839 ◽  
Author(s):  
A. Salat
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Infinite Sequence ◽  
Periodic Coefficient ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Magnetic Surfaces ◽  
Toroidal Geometry

The existence of quasi-periodic eigensolutions of a linear second order ordinary differential equation with quasi-periodic coefficient f{ω1t, ω2t) is investigated numerically and graphically. For sufficiently incommensurate frequencies ω1, ω2, a doubly indexed infinite sequence of eigenvalues and eigenmodes is obtained.The equation considered is a model for the magneto-hydrodynamic “continuum” in general toroidal geometry. The result suggests that continuum modes exist at least on sufficiently ir-rational magnetic surfaces

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

scholarly journals Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part I: Theoretical Analysis

2009 ◽  
Vol 16 (5) ◽  
pp. 505-515 ◽  
Author(s):  
Chunyu Zhao ◽  
Hongtao Zhu ◽  
Ruizi Wang ◽  
Bangchun Wen
Keyword(s):  
Zero Solution ◽  
Moment Of Inertia ◽  
Moments Of Inertia ◽  
Vibrating System ◽  
Frequency Capture ◽  
Synchronization Problem ◽  
Small Parameters ◽  
Existence And Stability ◽  
Synchronous Operation ◽  
The Stability

In this paper an analytical approach is proposed to study the feature of frequency capture of two non-identical coupled exciters in a non-resonant vibrating system. The electromagnetic torque of an induction motor in the quasi-steady-state operation is derived. With the introduction of two perturbation small parameters to average angular velocity of two exciters and their phase difference, we deduce the Equation of Frequency Capture by averaging two motion equations of two exciters over their average period. It converts the synchronization problem of two exciters into that of existence and stability of zero solution for the Equation of Frequency Capture. The conditions of implementing frequency capture and that of stabilizing synchronous operation of two motors have been derived. The concept of torque of frequency capture is proposed to physically explain the peculiarity of self-synchronization of the two exciters. An interesting conclusion is reached that the moments of inertia of the two exciters in the Equation of Frequency Capture reduce and there is a coupling moment of inertia between the two exciters. The reduction of moments of inertia and the coupling moment of inertia have an effect on the stability of synchronous operation.

A Comprehensive Stability Criterion for Forced Vibrations in Nonlinear Systems

1953 ◽  
Vol 20 (1) ◽  
pp. 9-12
Author(s):  
K. Klotter ◽  
E. Pinney
Keyword(s):  
Differential Equation ◽  
Nonlinear Systems ◽  
Stability Criterion ◽  
Practical Importance ◽  
Nonlinear Function ◽  
Forced Vibrations ◽  
Maximum Value ◽  
The Stability

Abstract This paper deals with the forced vibrations described by the differential equation a q .. + c q + c Φ ( q , q . ) = P cos Ω t wherein Φ denotes a nonlinear function of q and/or q̇. It presents a criterion for determining their stability. It is shown that under very weak restrictions, which equivalently means, for a large variety of cases (including all of practical importance) the stability depends on the sign of ∂q*/∂P (q* denoting the maximum value of q(t) within a period). The motion is stable if this derivative is positive; it is unstable if it is negative.

scholarly journals The stability of collocation methods for VIDEs of second order

2005 ◽  
Vol 2005 (7) ◽  
pp. 1049-1066 ◽  
Author(s):  
Edris Rawashdeh ◽  
Dave McDowell ◽  
Leela Rakesh
Keyword(s):  
Differential Equation ◽  
Jordan Block ◽  
Solution Procedure ◽  
Polynomial Spline ◽  
Second Order ◽  
Collocation Methods ◽  
Spline Functions ◽  
Stability Criteria ◽  
Spline Collocation ◽  
The Stability

Simplest results presented here are the stability criteria of collocation methods for the second-order Volterra integro differential equation (VIDE) by polynomial spline functions. The polynomial spline collocation method is stable if all eigenvalues of a matrix are in the unit disk and all eigenvalues with|λ|=1belong to a1×1Jordan block. Also many other conditions are derived depending upon the choice of collocation parameters used in the solution procedure.

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