Parametric Instability of Axially Moving Media Subjected to Multifrequency Tension and Speed Fluctuations

2000 ◽  
Vol 68 (1) ◽  
pp. 49-57 ◽  
Author(s):  
R. G. Parker ◽  
Y. Lin
Keyword(s):  
Multiple Scales ◽  
Parametric Instability ◽  
Translation Speed ◽  
Excitation Frequencies ◽  
Axially Moving ◽  
Frequency Changes ◽  
Limit Cycle Amplitude ◽  
Moving Media ◽  
Secondary Instabilities ◽  
Primary Instability

This work investigates the stability of axially moving media subjected to parametric excitation resulting from tension and translation speed oscillations. Each of these excitation sources has spectral content with multiple frequencies and arbitrary phases. Stability boundaries for primary parametric instabilities, secondary instabilities, and combination instabilities are determined analytically through second-order perturbation. The classical result that primary instability occurs when one of the excitation frequencies is close to twice a natural frequency changes as a result of multiple excitation frequencies. Unusual interactions occur for the practically important case of simultaneous primary and secondary instabilities. While sum type combination instabilities occur, no difference type instabilities are detected. The nonlinear limit cycle amplitude that occurs under primary instability is derived using the method of multiple scales.

Dynamic Stability of Axially Moving Plates With Periodically Varying Speeds Under Partially Distributed Edge Tensions

2012 ◽  
Author(s):  
T. H. Young ◽  
S. J. Huang ◽  
A. C. Liu
Keyword(s):  
Dynamic Stability ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Plane Elasticity ◽  
Moving Plate ◽  
System Equations ◽  
Axially Moving ◽  
Axially Moving Web ◽  
Moving Speed

This paper investigates the dynamic stability of an axially moving web which translates with periodically varying speeds and is subjected to partially distributed tensions on two opposite edges. The web is modeled as a rectangular plate simply supported at two opposite edges where the tension is applied, and free at the other two edges. The plate is assumed to possess internal damping, which obeys the Kelvin-Voigt model. The moving speed of the plate is expressed as the sum of a constant speed and a periodical perturbation with a zero mean. Due to the periodically varying speed of the moving plate, terms with time-dependent coefficients appear in the equations of motion, which may bring about parametric instability under certain situations. First, the in-plane stresses of the plate due to the partially distributed edge tensions is determined exactly by the theory of plane elasticity. Then, the dependence on the spatial coordinates in the equations of motion is eliminated by the Galerkin method, which results in a set of discretized system equations in time. Finally, the method of multiple scales is utilized to solve this set of system equations analytically if the periodical perturbation of the moving speed is much smaller as compared with the average speed of the plate, from which the stability boundaries of the moving plate are obtained. Numerical results reveal that only combination resonances of the sum-type appear between modes having the same symmetry class in the transverse direction. Unstable regions of main resonances are generally larger than those of sum-type resonances.

Nonlinear Vibration of Parametrically Excited, Viscoelastic, Axially Moving Strings

2005 ◽  
Vol 72 (3) ◽  
pp. 374-380 ◽  
Author(s):  
Eric M. Mockensturm ◽  
Jianping Guo
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Time Derivative ◽  
Excitation Frequency ◽  
Viscoelastic Behavior ◽  
Solution Procedure ◽  
Governing Equations ◽  
Translation Speed ◽  
Parametrically Excited ◽  
Axially Moving

The dynamic response of parametrically excited, axially moving viscoelastic belts is investigated in this paper. Results are compared to previous work in which the partial, not material, time derivative was used in the viscoelastic constitutive relation. It is found that this added “steady state” dissipation greatly affects both the existence and amplitudes of nontrivial limit cycles. The discrepancy increases with increasing translation speed. To limit the comparison to the additional physics included in the model, the solution procedure of Zhang and Zu [1,2], who applied the method of multiple scales to the governing equations prior to discretization, is retained. The excitation here is provided by physically stretching the belt. In this case, viscoelastic behavior and excitation frequency also affects the amplitude of the tension fluctuations.

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.

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.

Vibration Localization in Band/Wheel Systems: Theory and Experiment

1993 ◽  
Author(s):  
A. A. N. Al-jawi ◽  
A. G. Ulsoy ◽  
Christophe Pierre
Keyword(s):  
Simple Model ◽  
Coupling Model ◽  
Complex Model ◽  
Translation Speed ◽  
Mode Localization ◽  
Vibration Localization ◽  
Natural Modes ◽  
Theoretical Predictions ◽  
Axially Moving Beams ◽  
Axially Moving

Abstract An investigation of the localization phenomenon in band/wheel systems is presented. The effects of tension disorder, interspan coupling, and translation speed on the confinement of the natural modes of free vibration are investigated both theoretically and experimentally. Two models of the band/wheel system dynamics are discussed; a simple model proposed by the authors [1] and a more complete model originally proposed by Wang and Mote [9]. The results obtained using the simple interspan coupling model reveal phenomena (i.e., eigenvalue crossings and veerings and associated mode localization) that are qualitatively similar to those featured by the more complex model of interspan coupling, thereby confirming the usefulness of the simple coupling model. The analytical predictions of the two models are validated by an experiment. A very good agreement between the experimental results and the theoretical ones for the simple model is observed. While both the experimental observations and the theoretical predictions show that a beating phenomenon takes place for ordered stationary and axially moving beams, beating is destroyed (indicating the occurrence of localization) when any small tension disorder is introduced especially for small interspan coupling (i.e., when localization is strongest).

Investigation of Power Harvesting via Parametric Excitations

2008 ◽  
Vol 20 (5) ◽  
pp. 545-557 ◽  
Author(s):  
Mohammed F. Daqaq ◽  
Christopher Stabler ◽  
Yousef Qaroush ◽  
Thiago Seuaciuc-Osório
Keyword(s):  
Output Power ◽  
Coupling Coefficient ◽  
Electromechanical Coupling ◽  
Multiple Scales ◽  
Broad Band ◽  
Parametric Instability ◽  
Load Resistance ◽  
Resistive Load ◽  
Parametrically Excited ◽  
Critical Excitation

This article presents an analytical and experimental investigation of energy harvesting via parametrically excited cantilever beams. To that end, we consider a lumped-parameter non-linear model that describes the first-mode dynamics of a parametrically excited cantilever-type harvester. The model accounts for the beam's geometric and inertia non-linearities as well as non-linearities representing air drag. Using the method of multiple scales, we obtain approximate analytical expressions describing the beam response, voltage drop across a purely resistive load, and output power in the vicinity of the first principle parametric resonance. Using these expressions, we study the effect of the electromechanical coupling and load resistance on the output power. We show that these parameters play an imperative role in determining the magnitude of the output power and characterizing the broad-band properties of the harvester. Specifically, we show that the region of parametric instability wherein energy can be harvested shrinks as the coupling coefficient increases. Furthermore, we show that there exists a coupling coefficient beyond which the peak power decreases. We also demonstrate that there is a critical excitation level below which no energy can be harvested. The amplitude of this critical excitation increases with the coupling coefficient and is maximized for a given load resistance. Theoretical findings that were compared to experimental results show good agreement and reflect the general trends.

Friction-Induced Dynamics of Axially-Moving Media in Contact With an Actively-Positioned Surface

2008 ◽  
Author(s):  
V. Kartik ◽  
Evangelos Eleftheriou
Keyword(s):  
Data Storage ◽  
Surface Friction ◽  
Design Parameters ◽  
Moving Medium ◽  
Critical Design ◽  
Flexible Material ◽  
Potential Benefits ◽  
Axially Moving ◽  
The Stability ◽  
Moving Media

The dynamics of an axially-moving flexible medium are examined in the context of an application where the medium is partially supported by a frictional surface, that actively-orients itself relative to the direction of transport. The stability and motion of the medium are of interest in a magnetic tape data storage application where the orientation of a sensing surface is continuously altered in order to ‘follow’ the medium’s motion. Moving media that are in contact with such guiding surfaces experience friction excitations induced by the relative motion in addition to what is observed with a stationary guiding surface. Friction-induced bending moments, as well as tension fluctuation beyond the permissible limits for the flexible material can erode the potential benefits of such active positioning. This paper describes some of these dynamic phenomena using the simplified example of a planar guiding surface whose orientation is dynamically altered relative to the moving medium. A physical model for the friction-induced excitation of the moving medium is developed, and the dynamics are analyzed for their effect on critical design parameters such as the achievable bandwidth of the active control algorithm, as well as with respect to constraints on the geometry and positioning of the guiding surface.

scholarly journals Parametric Instability of an Axially Moving Belt Subjected to Multifrequency Excitations: Experiments and Analytical Validation

2008 ◽  
Vol 75 (4) ◽  
Author(s):  
Guilhem Michon ◽  
Lionel Manin ◽  
Didier Remond ◽  
Regis Dufour ◽  
Robert G. Parker
Keyword(s):  
Equations Of Motion ◽  
Parametric Instability ◽  
Transmission Error ◽  
Angle Measurements ◽  
Longitudinal Stiffness ◽  
Axially Moving ◽  
Belt Transmission ◽  
Multifrequency Excitations ◽  
First Time ◽  
Acquisition Technique

This paper experimentally investigates the parametric instability of an industrial axially moving belt subjected to multifrequency excitation. Based on the equations of motion, an analytical perturbation analysis is achieved to identify instabilities. The second part deals with an experimental setup that subjects a moving belt to multifrequency parametric excitation. A data acquisition technique using optical encoders and based on the angular sampling method is used with success for the first time on a nonsynchronous belt transmission. Transmission error between pulleys, pulley/belt slip, and tension fluctuation are deduced from pulley rotation angle measurements. Experimental results validate the theoretical analysis. Of particular note is that the instability regions are shifted to lower frequencies than the classical ones due to the multifrequency excitation. This experiment also demonstrates nonuniform belt characteristics (longitudinal stiffness and friction coefficient) along the belt length that are unexpected sources of excitation. These variations are shown to be sources of parametric instability.

Transverse Vibration Instabilities in Multiribbed Belt Transmission Subjected to Multi-Frequency Excitations: Modelling and Experiments

2007 ◽  
Author(s):  
Guilhem Michon ◽  
Lionel Manin ◽  
Didier Remond ◽  
Regis Dufour ◽  
Robert G. Parker
Keyword(s):  
Equations Of Motion ◽  
Parametric Instability ◽  
Transmission Error ◽  
Angle Measurements ◽  
Frequency Excitation ◽  
Axially Moving ◽  
Set Up ◽  
Belt Transmission ◽  
First Time ◽  
Acquisition Technique

This paper experimentally investigates the parametric instability of an industrial axially moving belt subjected to multifrequency excitation. Based on the equations of motion, an analytical perturbation analysis is achieved to identify instabilities. The second part deals with an experimental set-up that subjects a moving belt to multi-frequency parametric excitation. A data acquisition technique using optical encoders and based on the angular sampling method is used with success for the first time on a non-synchronous belt transmission. Transmission error between pulleys, pulley/belt slip and tension fluctuation are deduced from pulley rotation angle measurements. Experimental results validate the theoretical analysis. Of particular note is that the instability regions are shifted to lower frequencies than the classical ones due to the multi-frequency excitation.

Stability and Vibration of a Rotating Circular Plate Subjected to Stationary In-Plane Edge Loads

1996 ◽  
Vol 63 (1) ◽  
pp. 121-127 ◽  
Author(s):  
I. Y. Shen ◽  
Y. Song
Keyword(s):  
Circular Plate ◽  
Rotational Speed ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Parametric Instability ◽  
Single Mode ◽  
Equation Of Motion ◽  
Asymmetric Membrane ◽  
Parametric Resonances ◽  
Hill’S Equations

This paper predicts transverse vibration and stability of a rotating circular plate subjected to stationary, in-plane, concentrated edge loads. First of all, the equation of motion is discretized in a plate-based coordinate system resulting in a set of coupled Hill’s equations. Through use of the method of multiple scales, stability of the rotating plate is predicted in closed form in terms of the rotational speed and the in-plane edge loads. The asymmetric membrane stresses resulting from the stationary in-plane edge loads will transversely excite the rotating plates to single-mode parametric resonances as well as combination resonances at supercritical speed. In addition, introduction of plate damping will suppress the parametric instability when normalized edge loads are small. Moreover, the radial in-plane edge load dominates the rotational speed at which the instability occurs, and the tangential in-plane edge load dominates the width of the instability zones.

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