Super Harmonic Nonlinear Lateral Vibration of an Axially Moving Beam with Rotating Prismatic Joint

The motion of an axially moving beam with rotating prismatic joint with a tip mass on the end is analyzed to investigate the nonlinear vibration and dynamic stability of the beam. The beam is moving with a harmonic axially and rotating velocity about a constant mean velocity. A time-dependent partial differential equation and boundary conditions with the aid of the Hamilton principle are derived to describe the beam lateral deflection. After the partial differential equation is discretized by the Galerkin method, the method of multiple scales is applied to obtain analytical solutions. Frequency response curves are plotted for the super harmonic resonances of the first and the second modes. The effects of non-linear term and mean velocity are investigated on the steady state response of the axially moving beam. The results are validated with numerical simulations.
{"references":["J. Yuh and T. Young, \"Dynamic Modeling of an Axially Moving Beam in Rotation: Simulation and Experiment,\" Journal of Dynamic Systems, Measurement, and Control, vol. 113, pp. 34-40, 1991.","H. B. S.K.Tadikonda, \"Dynamic and control of a translating flexible beam with a prismatic joint,\" Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME 114 vol. 114, p. 5, 1992.","B. O. Al-Bedoor and Y. A. Khulief, \"Finite element dynamic modeling of a translating and rotating flexible link,\" Computer Methods in Applied Mechanics and Engineering, vol. 131, pp. 173-189, 1996.","M. Kalyoncu, \"Mathematical modelling and dynamic response of a multi-straight-line path tracing flexible robot manipulator with rotating-prismatic joint,\" Applied Mathematical Modelling, vol. 32, pp. 1087-1098, 2008.","S. E. Khadem and A. A. Pirmohammadi, \"Analytical development of dynamic equations of motion for a three-dimensional flexible link manipulator with revolute and prismatic joints,\" IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 33, pp. 237-249, 2003.","J. Chung, et al., \"Stability analysis for the flapwise motion of a cantilever beam with rotary oscillation,\" Journal of Sound and Vibration, vol. 273, pp. 1047-1062, 2004.","P. K. C. Wang and J.-D. Wei, \"Vibrations in a moving flexible robot arm,\" Journal of Sound and Vibration, vol. 116, pp. 149-160, 1987.","H. Karimi and M. Yazdanpanah, \"A new modeling approach to single-link flexible manipulator using singular perturbation method,\" Electrical Engineering (Archiv fur Elektrotechnik), vol. 88, pp. 375-382, 2006.","M. H. Ghayesh and S. E. Khadem, \"Rotary inertia and temperature effects on non-linear vibration, steady-state response and stability of an axially moving beam with time-dependent velocity,\" International Journal of Mechanical Sciences, vol. 50, pp. 389-404, 2008.\n[10]\tY.-Q. Tang, et al., \"Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations,\" Journal of Sound and Vibration, vol. 320, pp. 1078-1099, 2009.\n[11]\tL.-Q. Chen and W.-J. Zhao, \"A conserved quantity and the stability of axially moving nonlinear beams,\" Journal of Sound and Vibration, vol. 286, pp. 663-668, 2005.\n[12]\tL.-Q. Chen and X.-D. Yang, \"Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models,\" International Journal of Solids and Structures, vol. 42, pp. 37-50, 2005.\n[13]\tV. Kartik and J. A. Wickert, \"Vibration and guiding of moving media with edge weave imperfections,\" Journal of Sound and Vibration, vol. 291, pp. 419-436, 2006.\n[14]\tF Rahimi Dehgolan, SE Khadem, S Bab, M Najafee, \"Linear Dynamic Stability Analysis of a Continuous Rotor-Disk-Blades System\", World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 12 (4) (2016) 349–357.\n[15]\tA. H. Nayfeh, Problems in perturbation. New York: Wiley, 1993.\n[16]\tA. Nayfeh, Introduction to perturbation techniques. New York: Wiley, 1981.\n[17]\tJ. Kevorkian and J. D. Cole, Perturbation methods in applied mathematics. New York: Springer-Verlag, 1981."]}