Nonlinear Vibrations of Axially Moving Beams

Axially moving beams can represent many engineering devices, such as band saws, power transmission belts, aerial cable tramways, crane hoist cables, flexible robotic manipulators, and spacecraft deploying appendages.Despite usefulness and advantages of these devices, vibrations associated with the devices have limited their applications.Therefore, understanding transverse vibrations of axially moving beams is important for the design of the devices.The investigations on vibrations of axially moving beams have theoretical importance as well, because an axially moving beam is a typical representative of distributed gyroscopic systems.The term "gyroscopic" arose in recognition of an early problem in gyrodynamics.Actually, the Coriolis acceleration component experienced by axially moving materials imparts a skew-symmetric or gyroscopic term to their governing equations.Due to particular characteristics of the gyroscopic term, the approaches developed in analyzing vibrations of an axially moving string can be applied to other more complicated distributed gyroscopic systems.Because of the practical and theoretical significance, the investigation on nonlinear vibrations of axially moving beams is a challenging subject which has been studied for many years and is still of interest today.The relevant researches on transverse vibrations of axially moving strings can be dated back to (Aiken, 1878).There are several excellent and comprehensive survey papers, notably (Mote, 1972), (Ulsoy and Mote, 1978), (Mote et al., 1982), (D'Angelo et al., 1985), (Wickert and Mote, 1988), (Wang and Liu, 1991), (Abrate, 1992), (Zhu, 2000), reviewing the state-ofthe-art in different time phases of investigations related to vibrations of axially moving beams.The present chapter emphasizes on the recent achievements, although some early results are mentioned for the sake of completeness.Besides, the chapter focuses the nonlinear problem only.If the vibration amplitude is large, the nonlinearity should be taken into account.Hence the chapter, unlike (Chen, 2005a) for axially moving strings, is not a comprehensive survey with a complete and detailed representation of current researches.Instead, the chapter is a counterpart of (Chen et al., 2008) for axially moving beams.The author tries to put the some available results into a general framework, as well as to highlight the work of the author and his collaborators.It is hoped that the chapter serves as a collection of ideas, approaches, and main results in investigations on nonlinear vibration of axially moving beams.The chapter is organized as follows: Section 2 focuses on the mathematical models of nonlinear transverse vibration.The special attentions are paid to the comparison of two different nonlinear models and the introduction of the material time derivative into the