Nonlinear dynamics for transverse motion of axially moving strings
Introduction
Axially moving strings can represent many engineering devices, such as power transmission belts, magnetic tapes and fiber winding. Despite many advantages of these devices, transverse vibrations associated with the devices have limited their applications. The investigations on transverse vibrations of axially moving strings have theoretical significance as well, because an axially moving string is a simplest representative of distributed gyroscopic systems. Also, the investigation on nonlinear transverse vibrations of axially viscoelastic moving strings is a challenging subject. The approaches developed in analysis of transverse vibrations of an axially moving string can be applied to other more complicated distributed gyroscopic systems. As the development of the theories of nonlinear dynamics and chaotic dynamics, predictions and understanding become possible for more complicated nonlinear phenomena in viscoelastic moving strings, such as the global bifurcations and multi-pulse Shilnikov type chaotic dynamics.
The relevant researches on transverse vibrations of axially moving strings can be dated back to more than 100 years ago. Analysis of such vibrations are challenging subjects that have been investigated for many years and are still of interest nowadays. Although there are two review papers on the topic [1], [2], the present paper is still necessary based on the following reasons. In paper [1], Chen and Zu summarized the related progresses dated to 2000, and did not cover new results afterwards. Literature [2] is a comprehensive survey paper with a complete and detailed representation of current researches. It does not highlight the development in the field in China, even if they are mentioned there. Several nonlinear models of transverse vibrations are accounted in [2], but the author failed to put them into a general framework. Besides, some new results or views were worked out after finishing [2].
The paper is organized as follows: Section 2 proposes dynamical models of an axially accelerating string. The governing equation for planar motion is derived from the Eulerian equation of motion of a continuum, and the equation reduces to the governing equation for transverse motions. Section 3 summarizes perturbation analysis on nonlinear parametric vibration due to the tension variation and the axial acceleration. Section 5 documents the existence of the heteroclinic bifurcations and the Silnikov type multi-pulse homoclinic orbits in the averaged equations of axially moving viscoelastic strings. Section 5 is devoted to chaotic behaviors revealed numerically via the Galerkin truncation. Section 6 presents Section 6 covers some numerical algorithms including the Galerkin method and the finite difference. Section 7 discusses energetics, conserved quantity and the applications. Section 8 recommends future research directions.
Section snippets
Governing equations
The governing equation is the base of all analytical or numerical investigations. Generally, an axially moving string undergoes both the longitudinal vibration and the transverse vibration, and they are coupled. Thurman and Mote actually obtained the governing equation of coupled longitudinal and transverse vibrations of an axially moving string [3]. Koivurova and Salonen revisited the same modeling problem and clarified its kinematic aspects [4]. Their nonlinear formulation for the moving
Nonlinear parametric vibrations
Moving string-like devices in engineering applications may undergo large transverse vibrations due to periodical variation of their parameters. Such vibration is termed as parametric vibration. Parametric vibration of an axially moving string results from two major factors: the tension variation and the axial transport acceleration.
Zhang and Zu first applied the method of multiple scales directly to analyze nonlinear parametric vibration of an axially moving viscoelastic string excited by the
Global bifurcations
The global bifurcations and chaotic dynamics of high-dimensional nonlinear systems have been at the forefront of nonlinear dynamics for the last two decades. The global bifurcations and chaotic dynamics for high-dimensional nonlinear systems are an important theoretical problem in science and engineering applications as they can reveal the instabilities of motion and complicated dynamical behaviors in high dimensional nonlinear systems. Some new phenomena on the global bifurcations and chaotic
Chaotic behaviors
An axially moving string may undergo periodic transverse motion. More complicated types of motion, such as chaotic motion, may also occur in nonlinear system. Chaos implies motions with the continuous frequency spectrum, which may be significant in applications.
If longtime nonlinear dynamical behaviors (especially the chaotic behavior) of elastic or viscoelastic mechanisms and structures are concerned, only the low order Galerkin truncation is feasible. The suitability of the Galerkin
Numerical algorithms
Numerical methods form an important approach to study nonlinear transverse vibrations of axially moving strings.
Some numerical simulation research is based on the Galerkin method to discretize the governing equation into a set of ordinary differential equations. Fung and his collaboraters [43], [44] applied the 4-term Galerkin truncation based on stationary string eigenfunctions to study the transient motion of an axially moving viscoelastic stings constituted by Eq. (10). Zhang et al. [45]
Energetics and conserved quantity
The total mechanical energy associated with an axially moving string that travels between two supports is not constant [55], [56]. It is a fundamental feature of the free transverse vibration of axially moving elastic strings, while the total energy is constant for an undamped non-translating elastic string. Suweken and van Horssen used the energy to prove the boundedness of vibration of an axially moving string [57]. Although the total mechanical energy of an axially moving string is generally
Conclusions
In recent years, many research activities in the area have been witnessed because the axially moving string is a mechanical model that can be used in diverse engineering fields. Therefore, transverse vibration analysis of axially moving strings will remain to be an active research field. There are many promising topics for future researches, including development of analytical and numerical approaches to analyze moving strings with material and geometric nonlinearities under complex constraint
Acknowledgements
This work is supported by the National Natural Science Foundation of China through grants Nos. 10172056, 10372008, 10328204 and 100672092, the Natural Science Foundation of Beijing through Grant No. 3032006, the Natural Science Foundation of Shanghai Municipality through Grant No. 04ZR14058, Shanghai Municipal Education Commission Scientific Research Project through Grant No. 07ZZ07, and Shanghai Leading Academic Discipline Project through Grant No.Y0103.
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