Keywords: moving beam, free vibration, Timoshenko beam, dynamic stiffness theory.

There are many engineering structures that are modelled as axially moving beams. Some typical examples include power transmission belt and chain drives, high-speed magnetic tapes, aerial cable tramways, band saws, pipe conveying fluids and many other technological devices. The free vibration analysis of axially moving beams has generally been carried out using the Bernoulli-Euler beam theory [1]. There are, however, one or two exceptions where relatively more advanced Timoshenko beam theory which includes the effects of shear deformation and rotatory inertia has been used [2,3]. A recent literature review suggests that the free vibration behaviour of a moving Timoshenko beam has neither thoroughly nor widely been investigated. No one appears to have used the dynamic stiffness theory. Of course, the effects of shear deformation and rotatory inertia on the free vibration behaviour of a non-moving beam using the Timoshenko beam theory are well known and even included in text books. As for moving beams, these effects do not appear to be sufficiently well known. Clearly, the subject matter warrants a thorough and in-depth investigation. The effects of shear deformation and rotatory inertia may possibly be more significant for moving beams as opposed to the stationary ones, and the analysis is particularly relevant when establishing the critical speed at which the beam experiences the divergence phenomenon when it ultimately becomes unstable. The purpose of this paper is to address this issue by developing the dynamic stiffness matrix of a moving Timoshenko beam and then applying it to study its free vibration characteristics.

For harmonic oscillation the equations obtained using Hamilton's principle [2,3] are solved and the expressions for the amplitudes of flexural displacement, section rotation, shear force and bending moment are obtained in terms of the arbitrary constants. Next by applying the boundary conditions, the constants are eliminated to form the frequency dependent dynamic stiffness matrix of the moving Timoshenko beam relating the amplitudes of loads to those of responses. Finally the dynamic stiffness matrix is used to compute the natural frequencies and mode shapes of a number of carefully chosen moving Timoshenko beam examples. This is achieved by applying the well known algorithm of Wittrick and Williams [4]. The effects of shear deformation and rotatory inertia and their subsequent influence on the critical moving speed are investigated and discussed for both simply supported and fixed-fixed boundary conditions of the beam.

1

J.R. Banerjee and W.D. Gunawardana, "Dynamic Stiffness Development and Free Vibration Analysis of a Moving Beam", Journal of Sound and Vibration, 303, 135-143, 2007. doi:10.1016/j.jsv.2006.12.020

2

S. Chonan, "Steady State Response of an Axially Moving Strip Subjected to a Stationary Lateral Load", Journal of Sound and Vibration, 107, 155-165, 1986. doi:10.1016/0022-460X(86)90290-7

3

U. Lee, J. Kim and H. Oh, "Spectral Analysis for the Transverse Vibration of an Axially Moving Timoshenko Beam", Journal of Sound and Vibration, 271, 685-703, 2004. doi:10.1016/S0022-460X(03)00300-6

4

W.H. Wittrick and F.W. Williams, "A General Algorithm for Computing Natural Frequencies of Elastic Structures", Quarterly Journal of Mechanics and Applied Mathematics, 24, 263-284, 1971. doi:10.1093/qjmam/24.3.263

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