Adaptive stabilization of Kirchhoff's non‐linear strings by boundary displacement feedback

Kobayashi, Toshihiro; Sakamoto, Tetsuzo

doi:10.1002/mma.839

Cited by 3 publications

(1 citation statement)

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“…In all these references, cables were considered to be massless and thus there was no vibration suppression problem involved. Kobayashi and Sakamoto studied the displacement feedback control of a Kirchhoff 's nonlinear cable and designed an adaptive boundary control to stabilize it [11]. Li and Hou designed an exponential stabilization controller for an axially moving cable using linear boundary feedback [12].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Analysis and Vibration Attenuation of Cable-Driven Parallel Manipulators for Large Workspace Applications

Qiu

2013

Advances in Mechanical Engineering

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Cable-driven parallel manipulators are one of the best solutions to achieving large workspace since flexible cables can be easily stored on reels. However, due to the negligible flexural stiffness of cables, long cables will unavoidably vibrate during operation for large workspace applications. In this paper a finite element model for cable-driven parallel manipulators is proposed to mimic small amplitude vibration of cables around their desired position. Output feedback of the cable tension variation at the end of the end-effector is utilized to design the vibration attenuation controller which aims at attenuating the vibration of cables by slightly varying the cable length, thus decreasing its effect on the end-effector. When cable vibration is attenuated, motion controller could be designed for implementing precise large motion to track given trajectories. A numerical example is presented to demonstrate the dynamic model and the control algorithm.

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Section: Introductionmentioning

confidence: 99%

Dynamic Analysis and Vibration Attenuation of Cable-Driven Parallel Manipulators for Large Workspace Applications

Qiu

2013

Advances in Mechanical Engineering

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Boundary Feedback Stabilization of Kirchhoff-Type Timoshenko System

Xue

2014

J Dyn Control Syst

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Boundary for Complete Set of Attractors for Forced–Damped Essentially Nonlinear Systems

Grinberg

Gendelman

2015

Journal of Applied Mechanics

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Forced-damped essentially nonlinear oscillators can have a multitude of dynamic attrac tors. Genetically, no analytic procedure is available to reveal all such attractors. For many practical and engineering applications, however, it might be not necessary to know all the attractors in detail. Knowledge of the zone in the state space (or the space of ini tial conditions), in which all the attractors are situated might be sufficient. We demon strate that this goal can be achieved by relatively simple means-even for systems with multiple and unknown attractors. More specifically, this paper suggests an analytic pro cedure to determine the zone in the space of initial conditions, which contains all attrac tors of the essentially nonlinear forced-damped system for a given set of parameters. The suggested procedure is an extension of well-known Lyapunov functions approach; here we use it for analysis of stability of nonautonomous systems with external forcing. Conse quently, instead of the complete state space of the problem, we consider a space of initial conditions and define a bounded trapping region in this space, so that for every initial condition outside this region, the dynamic flow will eventually enter it and will never leave it. This approach is used to find a special closed curve on the plane of initial condi tions for a forced-damped strongly nonlinear oscillator with single-degree-of-freedom (single-DOF). Solving the equations of motion is not required. The approach is illus trated by the important benchmark example of X2" potential, including the celebrated Ueda oscillator for n = 2. Another example is the well-known mode! of forced-damped oscillator with double-well potential. We also demonstrate that the boundary curve, obtained by analytic tools, can be efficiently "tightened" numerically, yielding even stricter estimation for the zone of the existing attractors. In tro d u c tio nDamped-forced nonlinear dynamical systems can have multi ple attractors of different natures-periodic, quasi-periodic, or chaotic [1][2][3][4][5][6][7]. Besides linear systems, generically, no analytical procedure is able to establish the complete structure of these attractors or even to reveal their number (if it is finite). Approxi mate analytic approaches like averaging or multiple scales meth ods [1-7] can provide some approximations for these attractors, but of limited accuracy. Besides, in order to use any approximate method, one has to make first some assumptions on the nature of attractor. Numerical treatment is also widely applied; however, by its very nature it can deliver only the solutions for finite set of ini tial conditions. Famous and vast work done by Ueda [8,9] and his followers on the special case of forced purely cubic oscillators reveals complicated co-existence patterns of the attractors in a wide range of the system parameters. In the same time, by using any of the aforementioned approaches one can never guarantee that all available attractors were revealed even for some fixed pa rameter values. Recent...

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Adaptive stabilization of Kirchhoff's non‐linear strings by boundary displacement feedback

Cited by 3 publications

References 30 publications

Dynamic Analysis and Vibration Attenuation of Cable-Driven Parallel Manipulators for Large Workspace Applications

Dynamic Analysis and Vibration Attenuation of Cable-Driven Parallel Manipulators for Large Workspace Applications

Boundary Feedback Stabilization of Kirchhoff-Type Timoshenko System

Boundary for Complete Set of Attractors for Forced–Damped Essentially Nonlinear Systems

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