Stability and Limit Cycles of Parametrically Excited, Axially Moving Strings

1996 ◽  
Vol 118 (3) ◽  
pp. 346-351 ◽  
Author(s):  
E. M. Mockensturm ◽  
N. C. Perkins ◽  
A. Galip Ulsoy
Keyword(s):  
Limit Cycles ◽  
Parametric Instability ◽  
Parametrically Excited ◽  
Weakly Nonlinear ◽  
Closed Form Solutions ◽  
Belt Drives ◽  
Existence And Stability ◽  
Instability Regions ◽  
Axially Moving ◽  
The Stability

Tension fluctuations are the dominant source of excitation in automotive belts. In particular designs, these fluctuations may parametrically excite large amplitude transverse belt vibrations and adversely impact belt life. This paper evaluates an efficient discrete model of a parametrically excited translating belt. The efficiency derives from the use of translating string eigenfunctions as a basis for a Galerkin discretization of the equations of transverse belt response. Accurate and low-order models lead to simple closed-form solutions for the existence and stability of limit cycles near parametric instability regions. In particular, simple expressions are found for the stability boundaries of the general nth-mode principal parametric instability regions and the first summation and difference parametric instability regions. Subsequent evaluation of the weakly nonlinear equation of motion leads to an analytical expression for the amplitudes (and stability) of nontrivial limit cycles that exist around the nth-mode principal parametric instability regions. Example results highlight important conclusions concerning the response of automotive belt drives.

scholarly journals Parametric Instability of a Traveling Plate Partially Supported by a Laterally Moving Elastic Foundation

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
V. Kartik ◽  
J. A. Wickert
Keyword(s):  
Data Storage ◽  
Parametric Instability ◽  
Primary Resonance ◽  
Free Edge ◽  
Moving Plate ◽  
Torsional Mode ◽  
Plane Vibration ◽  
Out Of Plane ◽  
Axially Moving ◽  
The Stability

The parametric excitation of an axially moving plate is examined in an application where a partial foundation moves in the plane of the plate and in a direction orthogonal to the plate’s transport. The stability of the plate’s out-of-plane vibration is of interest in a magnetic tape data storage application where the read/write head is substantially narrower than the tape’s width and is repositioned during track-following maneuvers. In this case, the model’s equation of motion has time-dependent coefficients, and vibration is excited both parametrically and by direct forcing. The parametric instability of out-of-plane vibration is analyzed by using the Floquet theory for finite values of the foundation’s range of motion. For a relatively soft foundation, vibration is excited preferentially at the primary resonance of the plate’s fundamental torsional mode. As the foundation’s stiffness increases, multiple primary and combination resonances occur, and they dominate the plate’s stability; small islands, however, do exist within unstable zones of the frequency-amplitude parameter space for which vibration is marginally stable. The plate’s and foundation’s geometry, the foundation’s stiffness, and the excitation’s amplitude and frequency can be selected in order to reduce undesirable vibration that occurs along the plate’s free edge.

Stability Analysis of Parametrically Excited Systems Using the Energy-Growth Exponent/Coefficient

2017 ◽  
Vol 17 (02) ◽  
pp. 1750018 ◽  
Author(s):  
Yuchun Li ◽  
Lishi Wang ◽  
Yanqing Yu
Keyword(s):  
Parametric Instability ◽  
Mathieu Equation ◽  
Energy Criterion ◽  
Growth Exponent ◽  
Parametrically Excited ◽  
Floquet Exponent ◽  
Energy Growth ◽  
Triple Pendulum ◽  
Analytical Formulae ◽  
The Stability

In this paper, the energy exponent is used to study the instability of parametrically excited systems governed by the damped Mathieu equation. Through the numerical tests, an energy-growth exponent (EGE) is adopted to evaluate the instability intensity and instability boundary of the system. The EGE can be expressed as a product of the modal frequency and a dimensionless coefficient, defined as the energy-growth coefficient (EGC). Based on the Floquet theory, the relationship between the EGE, Floquet exponent and Lyapunov exponent are derived. An energy criterion of parametric instability is proposed by using the EGE. Using a simple pendulum as an example, the geometric characteristics of the EGC are investigated, and approximate analytical formulae of the EGC/EGE for three different unstable patterns are developed. The EGC/EGE formulae are applicable to the parametrically excited systems governed by the damped Mathieu equation. The unstable behaviors and properties of parametric vibrations are analyzed and discussed in details with EGE/EGC for three systems including a triple pendulum, two-dimensional sloshing fluid, and a two-span continuous beam. The stability boundaries established by using EGE/EGC agree well with those by the conventional theory and experiment. As a practical tool, the EGE/EGC formulae can be easily applied to analyzing the unstable intensities and boundaries of the parametrically excited systems.

scholarly journals Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field

2014 ◽  
Vol 758 ◽  
pp. 180-220 ◽  
Author(s):  
Raphaël C. Assier ◽  
Xuesong Wu
Keyword(s):  
Acoustic Field ◽  
Numerical Solutions ◽  
Parametric Instability ◽  
Linear Instability ◽  
Weakly Nonlinear ◽  
Observed Behaviour ◽  
Hydrodynamic Field ◽  
The Stability ◽  
Sivashinsky Equation ◽  
Asymptotic Formulation

AbstractThe stability of premixed flames in a duct is investigated using an asymptotic formulation, which is derived from first principles and based on high-activation-energy and low-Mach-number assumptions (Wu et al., J. Fluid Mech., vol. 497, 2003, pp. 23–53). The present approach takes into account the dynamic coupling between the flame and its spontaneous acoustic field, as well as the interactions between the hydrodynamic field and the flame. The focus is on the fundamental mechanisms of combustion instability. To this end, a linear stability analysis of some steady curved flames is undertaken. These steady flames are known to be stable when the spontaneous acoustic perturbations are ignored. However, we demonstrate that they are actually unstable when the latter effect is included. In order to corroborate this result, and also to provide a relatively simple model guiding active control, we derived an extended Michelson–Sivashinsky equation, which governs the linear and weakly nonlinear evolution of a perturbed flame under the influence of its spontaneous sound. Numerical solutions to the initial-value problem confirm the linear instability result, and show how the flame evolves nonlinearly with time. They also indicate that in certain parameter regimes the spontaneous sound can induce a strong secondary subharmonic parametric instability. This behaviour is explained and justified mathematically by resorting to Floquet theory. Finally we compare our theoretical results with experimental observations, showing that our model captures some of the observed behaviour of propagating flames.

Parametric Instability of a Traveling Plate Partially Supported by a Laterally-Moving Foundation

2007 ◽  
Author(s):  
V. Kartik ◽  
J. A. Wickert
Keyword(s):  
Data Storage ◽  
Parametric Instability ◽  
Primary Resonance ◽  
Free Edge ◽  
Moving Plate ◽  
Torsional Mode ◽  
Plane Vibration ◽  
Out Of Plane ◽  
Axially Moving ◽  
The Stability

The parametric excitation of an axially-moving plate is examined in an application where a partial foundation moves in the plane of the plate and in a direction orthogonal to the plate’s transport. The stability of the plate’s out-of-plane vibration is of interest in a magnetic tape data storage application where the read/write head is substantially narrower than the tape’s width, and is repositioned during track following maneuvers. In this case, the model’s equation of motion has time-dependent coefficients, and vibration is excited both parametrically and by direct forcing. The parametric instability of out-of-plane vibration is analyzed by using the Floquet theory for finite values of the foundation’s range of motion. For a relatively soft foundation, vibration is excited preferentially at the primary resonance of the plate’s fundamental torsional mode. As the foundation’s stiffness increases, multiple primary and combination resonances occur, and they dominate the plate’s stability; small islands, however, do exist within unstable zones of the frequency-amplitude parameter space for which vibration is marginally stable. The plate’s and foundation’s geometry, the foundation’s stiffness, and the excitation’s amplitude and frequency can be selected in order to reduce undesirable vibration that occurs along the plate’s free edge.

On the Stability of Chain Drive Systems Under Periodic Sprocket Oscillations

1992 ◽  
Vol 114 (1) ◽  
pp. 119-126 ◽  
Author(s):  
K. W. Wang
Keyword(s):  
Longitudinal Vibration ◽  
Stability Boundary ◽  
Total System ◽  
Chain Drive ◽  
Longitudinal Vibrations ◽  
Transverse Vibrations ◽  
Instability Regions ◽  
Axially Moving ◽  
Drive Systems ◽  
The Stability

Experimental observations have shown that periodic torsional oscillations of engine camshafts induced by powertrain loads can cause significant tension variation in the timing chain and magnify the chain transverse vibrations and noise level. This result indicates that the sprocket dynamic characteristics and the chain vibration behavior are closely coupled. The chain drive models to-date are not able to address these phenomena. This paper presents a nonlinear model of an integrated chain drive system which couples the sprocket motion with the transverse and longitudinal vibration of the axially moving chain spans. With this model, the effects of the sprocket shaft periodic loads upon the total system are investigated. It is concluded that the sprocket oscillations will cause chain longitudinal vibrations. This could destabilize the system and induce the chains to undergo large transverse vibration. Both the subharmonic and summation types of parametric resonance are found and the instability regions are derived. The effects of various system parameters, such as the sprocket inertia, the chain speed, and the speed dependent excitation frequencies upon the instability regions have been studied. The significance of the gyroscopic terms of the stability boundary has been shown.

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.

On the Stability of Chain Drive Systems Under Periodic Sprocket Oscillations

1991 ◽  
Author(s):  
K. W. Wang
Keyword(s):  
Longitudinal Vibration ◽  
Stability Boundary ◽  
Total System ◽  
Chain Drive ◽  
Longitudinal Vibrations ◽  
Instability Regions ◽  
Axially Moving ◽  
Drive Systems ◽  
The Stability ◽  
Drive Dynamics

Abstract Experimental observations have shown that periodic torsional oscillations of engine camshafts induced by powertrain loads can cause significant tension variation in the timing chain and magnifies the chain transverse vibrations and noise level. This result indicates that the sprocket dynamic characteristics and the chain vibration behavior are closely coupled. The chain drive models to-date are not able to address these phenomena. This paper presents a nonlinear model of an integrated chain drive system which couples the sprocket motion with the transverse and longitudinal vibration of the axially-moving chain spans. With this model, the effects of the sprocket shaft periodic loads upon the total system are investigated. It is concluded that the sprocket oscillations will cause chain longitudinal vibrations. This could destabilize the system and induce the chains to undergo large transverse vibration. Both the subharmonic and summation types of parametric resonance are found and the instability regions are derived. The effects of various system parameters, such as the sprocket inertia, the chain speed, and the speed dependent excitation frequencies upon the instability regions have been studied. The significance of the gyroscopic terms on the prediction of the stability boundary has been shown. This paper presents an approach that provides new insight to the research of chain drive dynamics.

Summation Resonance of Parametrically Excited Moving Viscoelastic Belts

2003 ◽  
Author(s):  
Tao Chen ◽  
Zhichao Hou
Keyword(s):  
Order Approximation ◽  
Nonlinear Dynamic Analysis ◽  
Material Behavior ◽  
Nontrivial Solutions ◽  
Parametrically Excited ◽  
Kelvin Model ◽  
Closed Form Solutions ◽  
The Stability ◽  
First Order Approximation ◽  
Moving Speed

The nonlinear dynamic analysis is performed on parametrically excited, viscoelastic moving-belts at summation resonance. The belt material behavior is described by a Voigt-Kelvin model. Closed-form solutions are derived at the first order approximation. Focus is put on the stability of the nontrivial solutions. The explicit expressions on the stability conditions are obtained, and then simplified through numerical simulations. The influences of moving speed and tension fluctuation on the stability of the nontrivial solutions are also demonstrated.

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

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