Abstract
Complex modes and traveling waves in axially moving Timoshenko beams are studied. Due to the axially moving velocity, complex modes emerge instead of real value modes. Correspondingly, traveling waves are present for the axially moving material while standing waves dominate in the traditional static structures. The analytical results obtained in this study are verified with a numerical differential quadrature method.
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Wickert, J. A. and Mote, C. D. Classical vibration analysis of axially moving continua. ASME Journal of Applied Mechanics, 57, 738–744 (1990)
Lin, C. C. Stability and vibration characteristics of axially moving plates. International Journal of Solids and Structures, 34, 3179–3190 (1997)
Lee, U., Kim, J., and Oh, H. Spectral analysis for the transverse vibration of an axially moving Timoshenko beam. Journal of Sound and Vibration, 271, 685–703 (2004)
Wickert, J. A. Non-linear vibration of a traveling tensioned beam. International Journal of Non-Linear Mechanics, 27, 503–517 (1992)
Pellicano, F. and Vestroni, F. Nonlinear dynamics and bifurcations of an axially moving beam. ASME Journal of Vibration and Acoustics, 122, 21–30 (2000)
Chen, L. Q. and Zhao, W. J. A conserved quantity and the stability of axially moving nonlinear beams. Journal of Sound and Vibration, 286, 663–668 (2005)
Ding, H., Chen, L. Q., and Yang, S. P. Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load. Journal of Sound and Vibration, 331, 2426–2442 (2012)
Ding, H. and Zu, J. W. Steady-state responses of pulley-belt systems with a one-way clutch and belt bending stiffness. ASME Journal of Vibration and Acoustics, 136, 041006 (2014)
Yan, Q. Y., Ding, H., and Chen, L. Q. Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations. Applied Mathematics and Mechanics (English Edition), 36, 971–984 (2015) https://doi.org/10.1007/s10483-015-1966-7
Sahoo, B., Panda, L. N., and Pohit, G. Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam. Nonlinear Dynamics, 82, 1721–1742 (2015)
Michon, G., Manin, L., Remond, D., Dufour, R., and Parker, R. G. Parametric instability of an axially moving belt subjected to multifrequency excitations: experiments and analytical validation. ASEM Journal of Applied Mechanics, 75, 041004 (2008)
Öz, H. R., Pakdemirli, M., and Boyaci, H. Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. International Journal of Non-Linear Mechanics, 36, 107–115 (2001)
Öz, H. R. and Pakdemirli, M. Vibrations of an axially moving beam with time-dependent velocity. Journal of Sound and Vibration, 227, 239–257 (1999)
Ding, H., Zhang, G. C., Chen, L. Q., and Yang, S. P. Forced vibrations of supercritically transporting viscoelastic beams. ASME Journal of Vibration and Acoustics, 134, 051007 (2012)
Li, H. Y., Li, J., and Liu, Y. J. Internal resonance of an axially moving unidirectional plate partially immersed in fluid under foundation displacement excitation. Journal of Sound and Vibration, 358, 124–141 (2015)
Tang, Y. Q., Zhang, D. B., and Gao, J. M. Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Nonlinear Dynamics, 83, 401–418 (2016)
Ghayesh, M. H. and Amabili, M. Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance. Nonlinear Dynamics, 73, 39–52 (2013)
Ding, H., Huang, L. L., Mao, X. Y., and Chen, L. Q. Primary resonance of traveling viscoelastic beam under internal resonance. Applied Mathematics and Mechanics (English Edition), 38, 1–14 (2017) https://doi.org/10.1007/s10483-016-2152-6
Mao, X. Y., Ding, H., Lim, C.W., and Chen, L. Q. Super-harmonic resonance and multi-frequency responses of a super-critical translating beam. Journal of Sound and Vibration, 385, 267–283 (2016)
Mao, X. Y., Ding, H., and Chen, L. Q. Parametric resonance of a translating beam with pulsating axial speed in the super-critical regime. Mechanics Research Communications, 76, 72–77 (2016)
Zhang, G. C., Ding, H., Chen, L. Q., and Yang, S. P. Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds. Journal of Sound and Vibration, 331, 1612–1623 (2012)
Liu, Y., Zhao, Z., and He, W. Boundary control of an axially moving system with high acceleration/deceleration and disturbance observer. Journal of the Franklin Institute, 354, 2905–2923 (2017)
He, W., He, X. Y., and Sun, C. Y. Vibration control of an industrial moving strip in the presence of input deadzone. IEEE Transactions on Industrial Electronics, 64, 4680–4689 (2017)
Zhao, H. and Rahn, C. D. On the control of axially moving material systems. ASME Journal of Vibration and Acoustics, 128, 527–531 (2006)
Ghayesh, M. H. and Balar, S. Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams. Applied Mathematical Modelling, 34, 2850–2859 (2010)
Tang, Y. Q., Chen, L. Q., and Yang, X. D. Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions. International Journal of Mechanical Sciences, 50, 1448–1458 (2008)
Tang, Y. Q., Chen, L. Q., and Yang, X. D. Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations. Journal of Sound and Vibration, 320, 1078–1099 (2009)
Chen, L. Q., Tang, Y. Q., and Lim, C. W. Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams. Journal of Sound and Vibration, 329, 547–565 (2010)
Yan, Q. Y., Ding, H., and Chen, L. Q. Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam. Nonlinear Dynamics, 78, 1577–1591 (2014)
Ding, H., Tan, X., Zhang, G. C., and Chen, L. Q. Equilibrium bifurcation of high-speed axially moving Timoshenko beams. Acta Mechanica, 227, 3001–3014 (2016)
Wickert, J. A. and Mote, C. D. Classical vibration analysis of axially moving continua. ASME Journal of Applied Mechanics, 57, 738–744 (1990)
Yang, X. D., Yang, S., Qian, Y. J., Zhang, W., and Melnik, R. V. N. Modal analysis of the gyroscopic continua: comparison of continuous and discretized models. ASME Journal of Applied Mechanics, 83, 084502 (2016)
Yang, X. D., Chen, L. Q., and Zu, J. W. Vibrations and stability of an axially moving rectangular composite plate. ASME Journal of Applied Mechanics, 78, 011018 (2011)
Ghayesh, M. H. and Amabili, M. Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support. Mechanism and Machine Theory, 67, 1–16 (2013)
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Project supported by the National Natural Science Foundation of China (Nos. 11672007 and 11672186), the Training Scheme for the Youth Teachers of Higher Education of Shanghai (No.ZZyyy12035), and the “Chen Guang” Project (No. 14CG57)
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Tang, Y., Luo, E. & Yang, X. Complex modes and traveling waves in axially moving Timoshenko beams. Appl. Math. Mech.-Engl. Ed. 39, 597–608 (2018). https://doi.org/10.1007/s10483-018-2312-8
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DOI: https://doi.org/10.1007/s10483-018-2312-8