On the Control of Axially Moving Material Systems

2006 ◽  
Vol 128 (4) ◽  
pp. 527-531 ◽  
Author(s):  
Haiyu Zhao ◽  
Christopher D. Rahn
Keyword(s):  
String Model ◽  
Forced Response ◽  
Control Analysis ◽  
Web Handling ◽  
Material Derivative ◽  
Stabilizing Controllers ◽  
Regular Time ◽  
Material Systems ◽  
Ill Posed ◽  
Axially Moving

Vibration control can improve the performance of many axially moving material systems (e.g., web handling machines and tape drives). Researchers have used Lyapunov analysis to develop vibration stabilizing controllers for distributed parameter models of axially moving material systems. Both the material and regular time derivatives have been used in these analyses despite the fact that they give different results. This paper proves that for a pinned axially moving string model: (i) Lyapunov stability analysis using the material derivative incorrectly predicts that a time-varying functional is constant and (ii) neglect of the coupled domain in boundary control analysis is ill posed and incorrectly predicts bounded forced response and exponentially decaying transients.

Vibration and Guiding of Moving Media With Edge Weave Imperfections

2005 ◽  
Author(s):  
V. Kartik ◽  
J. A. Wickert
Keyword(s):  
Elastic Foundation ◽  
Moving Load ◽  
Lateral Position ◽  
Harmonic Excitation ◽  
Forced Response ◽  
Moving Medium ◽  
Web Handling ◽  
Moving Strip ◽  
Axially Moving ◽  
Technical Interest

This paper examines the steady-state forced vibration of a moving medium that is guided by a partial elastic foundation, and where geometric imperfections on the medium’s edge act as an excitation source. Such a system is of technical interest in the areas of web handling and magnetic tape transport where externally-pressurized air bearing guides are sometimes used to control lateral position. The axially-moving strip is modeled here as a string that is guided by elastic foundation segments and that is subjected to traveling wave excitation as the edge imperfections interact with the foundations. The equation of motion for this “moving medium and moving load” system incorporates a skew-symmetric Coriolis acceleration component that arises from convection. The governing equation is cast in the state-space form, with one symmetric and one skew-symmetric operator, as is characteristic of gyroscopic systems. Through modal analysis, the forced response of the system is obtained to the complex harmonic excitation associated with the interaction between the edge weave and the guides. Parameter studies are presented in the transport speed, foundation stiffness, guide placement, guide width, and imperfection wavelength. Of potential technological application, for a given wavelength of the edge imperfection, it is possible to reduce the medium’s vibration at a certain location by judiciously selecting the locations and spans of the foundation segments.

Free and Forced Response of an Axially Moving String Transporting a Damped Linear Oscillator

1993 ◽  
Author(s):  
W. D. Zhu ◽  
C. D. Mote
Keyword(s):  
Initial Conditions ◽  
Numerical Algorithms ◽  
Time History ◽  
Coupled System ◽  
Interaction Force ◽  
Forced Response ◽  
Linear Oscillator ◽  
Response Analysis ◽  
History Of ◽  
Axially Moving

Abstract The transverse response of a cable transport system, which is modelled as an ideal, constant tension string travelling at constant speed between two supports with a damped linear oscillator attached to it, is predicted for arbitrary initial conditions, external forces and boundary excitations. The exact formulation of the coupled system reduces to a single integral equation of Volterra type governing the interaction force between the string and the payload oscillator. The time history of the interaction force is discontinuous for non-vanishing damping of the oscillator. These discontinuities occur at the instants when transverse waves propagating along the string interact with the oscillator. The discontinuities are treated using the theory of distributions. Numerical algorithms for computing the integrals involving generalized functions and for solution of the delay-integral-differential equation are developed. Response analysis shows a discontinuous velocity history of the payload attachment point. Special conditions leading to absence of the discontinuities above are given.

Variational Principles for Non-Material Systems Within an Arbitrary Lagrangian Eulerian Description of Motion

2020 ◽  
Author(s):  
Giuseppe Pennisi ◽  
Olivier Bauchau
Keyword(s):  
Finite Element ◽  
Dynamic Behavior ◽  
Variational Principles ◽  
Material System ◽  
Arbitrary Lagrangian Eulerian ◽  
Complex Dynamic ◽  
Eulerian Description ◽  
Material Systems ◽  
Axially Moving ◽  
Dynamic Description

Abstract Dynamics of axially moving continua, such as beams, cables and strings, can be modeled by use of an Arbitrary La-grangian Eulerian (ALE) approach. Within a Finite Element framework, an ALE element is indeed a non-material system, i.e. a mass flow occurs at its boundaries. This article presents the dynamic description of such systems and highlights the peculiarities that arise when applying standard mechanical principles to non-material systems. Starting from D’Alembert’s principle, Hamilton’s principle and Lagrange’s equations for a non-material system are derived and the significance of the additional transport terms discussed. Subsequently, the numerical example of a length-changing beam is illustrated. Energetic considerations show the complex dynamic behavior non-material systems might exhibit.

Non-contact detection of polyester filament yarn tension in the spinning process by the laser Doppler vibrometer method

2021 ◽  
pp. 004051752110342
Author(s):  
Dongjian Zhang ◽  
Qihua Ma ◽  
Yuan Tan ◽  
He Liao ◽  
Chenhui Lu ◽  
...  
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Laser Doppler Vibrometer ◽  
String Model ◽  
Contact Detection ◽  
Polyester Filament ◽  
Axially Moving String ◽  
Spinning Process ◽  
Polyester Filament Yarn ◽  
Axially Moving

The precise detection of polyester filament yarn (PFY) tension in the spinning process is critical to ensure product quality. The laser Doppler vibrometer (LDV) method is proposed to achieve non-contact detection of PFY tension in this paper. By employing the Hamilton principle, the transverse dynamics differential equations of PFY are derived, which are discretized and solved by the Galerkin method and Runge–Kutta method, respectively. In the equations, the PFY between two adjacent rollers is simplified as an axially moving string to verify the generality of calculating natural frequencies. The calculated natural frequencies from the axially moving string model are compared with solved results from the transverse dynamics differential equations. It is shown that the approximation of natural frequencies can be obtained from the axially moving string model. This study attempts to establish an approximate generic model among the PFY tension, the spinning speed and the first natural frequency based on axially moving string model, from which the PFY tension can be calculated efficiently by employing the measured natural frequencies. The LDV method is used to measure the natural frequencies. A major advantage of the proposed method is to realize non-contact detection of PFY tension. The method is more useful under high-speed spinning conditions where contact tension detectors are not available. An experimental analysis is carried out to verify the effectiveness and accuracy of the proposed method. Therefore, it is believed that the non-contact detection of PFY tension in the spinning process by the LDV method is feasible.

Forced response of translating media with variable length and tension: Application to high-speed elevators

2005 ◽  
Vol 219 (1) ◽  
pp. 35-53 ◽  
Author(s):  
W D Zhu ◽  
Y Chen
Keyword(s):  
High Speed ◽  
Initial Conditions ◽  
Control Volume ◽  
String Model ◽  
Forced Response ◽  
Rate Of Change ◽  
Variable Length ◽  
Lateral Response ◽  
Discretization Schemes ◽  
Fixed Boundaries

The lateral response of vertically translating media with variable length and tension, subjected to general initial conditions and external excitation, are determined. The translating media are modelled as a taut string and tensioned beams with pinned and fixed boundaries. In each model a rigid body is attached to the lower end of the medium and has a prescribed displacement along the horizontal direction. The rate of change in the energy of the translating medium is analysed from the control volume and system viewpoints. The models are used to predict the forced response of a moving cable in a high-speed elevator. Three spatial discretization schemes are used to calculate the response and shown to yield the same results. The convergence of the solution for each model is investigated. The approximate solution for the string model with constant tension is compared with its exact solution from the wave method.

Aeroelastic Stability of Axially Moving Webs Coupled to Incompressible Flows

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Merrill Vaughan ◽  
Arvind Raman
Keyword(s):  
High Speed ◽  
Incompressible Flows ◽  
Mixed Boundary ◽  
Free Edge ◽  
Mode Shapes ◽  
Aeroelastic Stability ◽  
Machine Direction ◽  
Web Handling ◽  
Axially Moving ◽  
The Web

The aeroelastic flutter of thin flexible webs severely limits their transport speeds and consequently the machine throughputs in a variety of paper, plastics, textiles, and sheet metal industries. The aeroelastic stability of such high-speed webs is investigated using an assumed mode discretization of an axially moving, uniaxially tensioned Kirchhoff plate coupled with cross and machine direction flows of a surrounding incompressible fluid. The corresponding aerodynamic potentials are computed using finite element solutions of certain mixed boundary value problems that arise in the fluid domain. In the absence of air coupling, the cross-span mode frequencies tightly cluster together, and the web flutters via mode coalescence at supercritical transport speed. Web coupling to an initially quiescent incompressible potential flow significantly reduces the web frequencies, substantially modifies the mode shapes, and separates the frequency clusters, while only marginally affecting the flutter speed and frequency. The inclusion of machine direction base flows significantly modifies the web stability and mode shapes. Cross machine direction flows lead to the flutter with vanishing frequency of very high cross-span nodal number modes, and the unstable vibration naturally localizes at the leading free edge. These results corroborate several previous experimental results in literature and are expected to guide ongoing experiments and the design of reduced flutter web handling systems.

Forced Response of Translating Media With Variable Length and Tension: Application to High-Speed Elevators

2004 ◽  
Author(s):  
W. D. Zhu ◽  
Y. Chen
Keyword(s):  
High Speed ◽  
Initial Conditions ◽  
Control Volume ◽  
String Model ◽  
Forced Response ◽  
Rate Of Change ◽  
Variable Length ◽  
Lateral Response ◽  
Discretization Schemes ◽  
Fixed Boundaries

The lateral response of vertically translating media with variable length and tension, subjected to general initial conditions and external excitation, are determined. The translating media are modeled as a taut string and tensioned beams with pinned and fixed boundaries. In each model a rigid body is attached to the lower end of the medium and has a prescribed displacement along the horizontal direction. The rate of change of the energy of the translating medium is analyzed from the control volume and system viewpoints. The models are used to predict the forced response of a moving cable in a high-speed elevator. Three spatial discretization schemes are used to calculate the response and shown to yield the same results. The convergence of the solution for each model is investigated. The approximate solution for the string model with constant tension is compared with its exact solution from the wave method.

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