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.

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.

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.

Simple parametric resonance in an electrostatically actuated microelectromechanical gyroscope: Theory and experiment

2008 ◽  
Vol 222 (1) ◽  
pp. 43-52 ◽  
Author(s):  
K M Harish ◽  
B J Gallacher ◽  
J S Burdess ◽  
J A Neasham
Keyword(s):  
Parametric Resonance ◽  
Parametric Excitation ◽  
Microelectromechanical Systems ◽  
Stability Boundary ◽  
Equation Of Motion ◽  
Excitation Scheme ◽  
Primary Mode ◽  
Electrostatically Actuated ◽  
Stability And Instability ◽  
The Stability

One of the major issues facing electrostatically actuated and sensed microelectromechanical systems (MEMS) sensors is electrical feed-through between the drive and the sense electrodes due to parasitic capacitances. This feed-through, in the case of a ‘tuned’ MEMS gyroscope, limits the sensor sensitivity. In the current paper, the first practical step towards demonstrating reduced feed-through using a combined harmonic forcing and parametric excitation scheme is demonstrated. The equation of motion for the primary mode of vibration of the electrostatically actuated MEMS ring gyroscope is shown to contain a stiffness modulating term which, when modulated at a frequency near twice the natural frequency of the mode, results in parametric resonance. A solution for the equation of motion is assumed, based on Floquet theory, and the method of harmonic balance is employed for analysis. Regions of stability and instability and the stability boundary demarcating the stable and unstable regions are determined. Frequency sweeps, centred on twice the measured resonant frequency of the primary mode, were performed at various values of voltage amplitudes of the parametric excitation and the parametric resonance was observed electrically at half the excitation frequency. This data were used to map the stability boundary of the parametric resonance. The theoretical and experimental stability boundaries are shown to demonstrate significant similarity.

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.

On Instability Pockets and Influence of Damping in Parametrically Excited Systems

2018 ◽  
Vol 140 (5) ◽  
Author(s):  
Ashu Sharma ◽  
S. C. Sinha
Keyword(s):  
Degrees Of Freedom ◽  
State Transition ◽  
Degree Of Freedom ◽  
Transition Matrices ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
Parametric Space ◽  
Stability Diagrams ◽  
Wide Range ◽  
The Stability

In most parametrically excited systems, stability boundaries cross each other at several points to form closed unstable subregions commonly known as “instability pockets.” The first aspect of this study explores some general characteristics of these instability pockets and their structural modifications in the parametric space as damping is induced in the system. Second, the possible destabilization of undamped systems due to addition of damping in parametrically excited systems has been investigated. The study is restricted to single degree-of-freedom systems that can be modeled by Hill and quasi-periodic (QP) Hill equations. Three typical cases of Hill equation, e.g., Mathieu, Meissner, and three-frequency Hill equations, are analyzed. State transition matrices of these equations are computed symbolically/analytically over a wide range of system parameters and instability pockets are observed in the stability diagrams of Meissner, three-frequency Hill, and QP Hill equations. Locations of the intersections of stability boundaries (commonly known as coexistence points) are determined using the property that two linearly independent solutions coexist at these intersections. For Meissner equation, with a square wave coefficient, analytical expressions are constructed to compute the number and locations of the instability pockets. In the second part of the study, the symbolic/analytic forms of state transition matrices are used to compute the minimum values of damping coefficients required for instability pockets to vanish from the parametric space. The phenomenon of destabilization due to damping, previously observed in systems with two degrees-of-freedom or higher, is also demonstrated in systems with one degree-of-freedom.

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.

Principal Parametric Resonance of Axially Accelerating Viscoelastic Strings

2005 ◽  
Author(s):  
Li-Qun Chen
Keyword(s):  
Steady State ◽  
Parametric Resonance ◽  
Multiple Scales ◽  
Transverse Motion ◽  
Viscoelastic Parameters ◽  
Principal Parametric Resonance ◽  
Linearized Stability ◽  
Speed Fluctuation ◽  
The Stability ◽  
Axial Speed

The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) nontrivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial speed fluctuation amplitude, and the viscoelastic parameters.

A time-varying stiffness rotor-active magnetic bearings system under parametric excitation

2008 ◽  
Vol 222 (3) ◽  
pp. 447-458 ◽  
Author(s):  
U H Hegazy ◽  
Y A Amer
Keyword(s):  
Parametric Resonance ◽  
Parametric Excitation ◽  
Multiple Scales ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Principal Parametric Resonance ◽  
Non Linear ◽  
Response Function Method ◽  
Non Linear System

The method of multiple scales is applied to investigate the non-linear oscillations and dynamic behaviour of a rotor-active magnetic bearings (AMBs) system, with time-varying stiffness. The rotor-AMB model is a two-degree-of-freedom non-linear system with quadratic and cubic non-linearities and parametric excitation in the horizontal and vertical directions. The case of principal parametric resonance is considered and examined. The steady-state response and the stability of the system at the principal parametric resonance case for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical non-linear behaviours including multiple-valued solutions, jump phenomenon, hardening and softening non-linearity. Different effects of the system parameters on the non-linear response of the rotor are also reported. Results are compared with available published work.

Parametric Excitation of a Pendulum With Bilinear Hysteresis

1970 ◽  
Vol 37 (4) ◽  
pp. 1061-1068 ◽  
Author(s):  
W. K. Tso ◽  
K. G. Asmis
Keyword(s):  
Steady State ◽  
Parametric Resonance ◽  
Parametric Excitation ◽  
Viscous Damping ◽  
Sinusoidal Oscillation ◽  
Effective Mechanism ◽  
Steady State Responses ◽  
The Stability ◽  
State Responses ◽  
Steady State Solutions

The steady-state responses of a simple pendulum with a hinge exhibiting bilinear hysteretic moment-rotation characteristics and parametrically excited by a sinusoidal oscillation at the base is given. The stability of the steady-state solutions is discussed. It is shown that in contrast with viscous damping, the bilinear hysteresis is an effective mechanism to limit the growth of the response during parametric resonance.

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