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.

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.

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.

Multimode Dynamics and Out-of-Plane Drift in Suspended Cable Using the Kinematically Condensed Model

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
Lianhua Wang ◽  
Yueyu Zhao ◽  
Giuseppe Rega
Keyword(s):  
Primary Resonance ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Modulation Equations ◽  
Plane Vibration ◽  
Suspended Cable ◽  
Out Of Plane ◽  
Large Amplitude Vibration ◽  
Suspended Cables ◽  
Plane Mode

The large amplitude vibration and modal interactions of shallow suspended cable with three-to-three-to-one internal resonances are investigated. The quasistatic assumption and direct approach are used to obtain the condensed suspended cable model and the corresponding modulation equations for the case of primary resonance of the third symmetric in-plane or out-of-plane mode. The equilibrium, periodic, and chaotic solutions of the modulation equations are studied. Moreover, the nonplanar motion and symmetric character of out-of-plane vibration of the shallow suspended cables are investigated by means of numerical simulations. Finally, the role played by the quasistatic assumption, internal resonance, and static configuration in disrupting the symmetry of the out-of-plane vibration is discussed.

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.

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.

Nonlinear dynamics of an axially moving beam with fractional viscoelastic damping

2019 ◽  
Author(s):  
Ping Zhu
Keyword(s):  
Nonlinear Dynamics ◽  
Primary Resonance ◽  
Viscosity Coefficient ◽  
Harmonic Excitation ◽  
Axially Moving Beam ◽  
Response Curves ◽  
System P ◽  
Steady State Responses ◽  
Axially Moving ◽  
The Stability

Nonlinear dynamics of an axially moving viscoelastic beam subjected to transverse harmonic excitation is studied. The governing equation of motion of this system is discretized by employing Galerkin’s technique which yields a single-degree-of-freedom Duffing system having nonlinear fractional derivative. The viscoelastic properties of the material are described by the fractional Kelvin–Voigt model based on the Caputo definition. The primary resonance is analytically investigated by the averaging method. With the aid of response curves, a parametric study is conducted to display the influences of the fractional order and the viscosity coefficient on steady-state responses. The validations of this study are given through comparisons between the analytical solutions and numerical ones, where the stability of the solutions is determined by the Routh-Hurwitz criterion. It is found that suppression of undesirable responses can be achieved via changing the viscosity of the system.

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