On the Stability of a Differential Equation With Application to Parametrically Excited Systems

1970 ◽  
Vol 37 (1) ◽  
pp. 218-220
Author(s):  
R. H. Rand ◽  
H. Simon
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Computer Program ◽  
Parametric Excitation ◽  
Digital Computer ◽  
Floquet Theory ◽  
Parametrically Excited ◽  
The Stability

The stability of the equation z¨ + (Δ + ε cos t)−mz = 0, where m is a positive integer, is studied by using Floquet theory and perturbations. The results are confirmed by a digital computer program based on Floquet theory. Physical examples involving parametric excitation for m = 1, 3 are cited from the literature.

A Study of the Parametrically Excited Behavior of Passenger and Freight Railway Vehicles Using Linear Models

1991 ◽  
Vol 113 (2) ◽  
pp. 336-338 ◽  
Author(s):  
J. Lieh ◽  
I. Haque
Keyword(s):  
Parametric Resonance ◽  
Parametric Excitation ◽  
Floquet Theory ◽  
Linear Models ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
Critical Speeds ◽  
Railway Vehicles ◽  
Principal Parametric Resonance ◽  
The Stability

This paper presents a study of the parametrically excited behavior of passenger and freight vehicles on tangent track due to harmonic variations in conicity using linear models. The effect of primary and secondary stiffnesses on parametric excitation is also studied. Floquet theory is used to find the stability boundaries. The results show that wavelengths associated with conicity variation that are in the vicinity of half the kinematic wavelengths of the vehicles can lead to significant reductions in critical speeds. Results also show that the primary and warp stiffnesses can affect the severity of principal parametric resonance depending on the vehicle models and magnitude of stiffnesses chosen.

An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Ashu Sharma ◽  
S. C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Picard Iteration ◽  
Periodic Systems ◽  
Transition Matrices ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Stability Diagrams ◽  
The Stability

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

A Preliminary Investigation of the Dynamic Stability of Flexible Manipulators Performing Repetitive Tasks

1986 ◽  
Vol 108 (3) ◽  
pp. 206-214 ◽  
Author(s):  
D. A. Streit ◽  
C. M. Krousgrill ◽  
A. K. Bajaj
Keyword(s):  
Parametric Excitation ◽  
Floquet Theory ◽  
Equations Of Motion ◽  
Preliminary Investigation ◽  
Zero Solution ◽  
Flexible Manipulator ◽  
Governing Equations ◽  
Repetitive Tasks ◽  
The Stability ◽  
Two Degree Of Freedom

The governing equations of motion for the compliant coordinates describing a flexible manipulator performing repetitive tasks contain parametric excitation terms. The stability of the zero solution to these equations is investigated using Floquet theory. Analytical and numerical results are presented for a two-degree-of-freedom model of a manipulator with one prismatic joint and one revolute joint.

An Approximate Analysis of Quasi-Periodic Systems via Floquét Theory

2017 ◽  
Author(s):  
Ashu Sharma ◽  
Subhash C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Periodic Systems ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Approximate Approach ◽  
Varying Coefficients ◽  
P Transformation ◽  
The Stability

Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. Although Floquét theory is applicable only to periodic systems, it is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.

scholarly journals Dark breather using symmetric Morse, solvent and external potentials for DNA breathing

2019 ◽  
Vol 43 (4) ◽  
pp. 44
Author(s):  
Hernan Oscar Cortez Gutierrez ◽  
Milton Milciades Cortez Gutierrez ◽  
Girady Iara Cortez Fuentes Rivera ◽  
Liv Jois Cortez Fuentes Rivera ◽  
Deolinda Fuentes Rivera Vallejo
Keyword(s):  
Differential Equation ◽  
Ground State ◽  
Floquet Theory ◽  
Finite Difference Methods ◽  
Potential Effect ◽  
Quantum Thermodynamics ◽  
Bond Stretching ◽  
Difference Methods ◽  
The Stability ◽  
Quantum Thermodynamic

We analyze the dynamics and the quantum thermodynamics of DNA in Symmetric-Peyrard-Bishop-Dauxois model (S-PBD) with solvent and external potentials and describe the transient conformational fluctuations using dark breather and the ground state wave function of the associate Schrodinger differential equation.  We used the S-PBD, the Floquet theory, quantum thermodynamic and finite difference methods. We show that for lower coupling dark breather is present.  We estimate the fluctuations or breathing of DNA. For the S-PBD model we have the stability of dark breather for k<0.004 and mobile breathers with coupling k=0.004. The fluctuations of the dark breather in the S-PBD model is approximately zero with the quantum thermodynamics. The viscous and external potential effect is direct proportional to hydrogen bond stretching.

scholarly journals Dark breather using symmetric Morse, solvent and external potentials for DNA breathing

2018 ◽  
Vol 43 (4) ◽  
pp. 44
Author(s):  
Hernan Oscar Cortez Gutierrez ◽  
Milton Milciades Cortez Gutierrez ◽  
Girady Iara Cortez Fuentes Rivera ◽  
Liv Jois Cortez Fuentes Rivera ◽  
Deolinda Fuentes Rivera Vallejo
Keyword(s):  
Differential Equation ◽  
Ground State ◽  
Floquet Theory ◽  
Finite Difference Methods ◽  
Potential Effect ◽  
Quantum Thermodynamics ◽  
Bond Stretching ◽  
Difference Methods ◽  
The Stability ◽  
Quantum Thermodynamic

We analyze the dynamics and the quantum thermodynamics of DNA in Symmetric-Peyrard-Bishop-Dauxois model (S-PBD) with solvent and external potentials and describe the transient conformational fluctuations using dark breather and the ground state wave function of the associate Schrodinger differential equation.  We used the S-PBD, the Floquet theory, quantum thermodynamic and finite difference methods. We show that for lower coupling dark breather is present.  We estimate the fluctuations or breathing of DNA. For the S-PBD model we have the stability of dark breather for k<0.004 and mobile breathers with coupling k=0.004. The fluctuations of the dark breather in the S-PBD model is approximately zero with the quantum thermodynamics. The viscous and external potential effect is direct proportional to hydrogen bond stretching.

Parametrically Excited Behavior of a Railway Wheelset

1988 ◽  
Vol 110 (1) ◽  
pp. 8-17 ◽  
Author(s):  
J. Lieh ◽  
I. Haque
Keyword(s):  
Floquet Theory ◽  
Initial Conditions ◽  
Excitation Frequency ◽  
Time History ◽  
Parametrically Excited ◽  
Load Variations ◽  
Rail Vehicles ◽  
Stability And Instability ◽  
Instability Regions ◽  
The Stability

The dynamic response of rail vehicles is affected by parameters such as wheel-rail geometry, track gage, and axle load. Variations in these parameters, as a rail vehicle moves down the track, can cause instabilities that are related to parametrically excited behavior. This paper reports on the use of Floquet Theory to predict the stability and instability regions for a single wheelset subjected to harmonic variations in wheel-rail geometry, track gage and axle load. Time studies showing the response of a wheelset to various initial conditions are also included. The results show that harmonic variations in the wheel-rail geometry can influence the behavior of a wheelset significantly. The system is especially susceptible to variations in conicity. Time history studies show that the response is dependent on initial conditions, the amount of variations and the magnitude of the excitation frequency.

Parameter–Excited Instabilities of a 2UPU-RUR-RPS Spherical Parallel Manipulator With a Driven Universal Joint

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Guanglei Wu
Keyword(s):  
Parallel Manipulator ◽  
Floquet Theory ◽  
Monodromy Matrix ◽  
Critical Parameters ◽  
Parametric Stability ◽  
Parametrically Excited ◽  
Lateral Vibrations ◽  
Stability Chart ◽  
The Stability ◽  
Spherical Parallel Manipulator

This paper presents the parametrically excited lateral instabilities of an asymmetrical spherical parallel manipulator (SPM) by means of monodromy matrix method. The linearized equation of motion for the lateral vibrations is developed to analyze the stability problem, resorting to the Floquet theory, which is numerically illustrated. To this end, the parametrically excited unstable regions of the manipulator are visualized to reveal the effect of the system parameters on the stability. Critical parameters, such as rotating speeds of the driving shaft, are identified from the constructed parametric stability chart for the manipulator.

On the Stability of a Differential Equation With Application to the Vibrations of a Particle in the Plane

1969 ◽  
Vol 36 (2) ◽  
pp. 311-313 ◽  
Author(s):  
Richard H. Rand ◽  
Shoei-Fu Tseng
Keyword(s):  
Differential Equation ◽  
Fourier Analysis ◽  
Initial Stress ◽  
Floquet Theory ◽  
The Stability

The stability of the equation Z¨ + f(t)Z = 0, where f(t) = (δ − ε cos2t)/(1 − ε cos2t), is studied by using Floquet theory, Fourier analysis and perturbations. The results are used to study the stability of the vibrations of a particle constrained to a plane and restrained by two identical linear springs with initial stress.

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