Abstract
This paper investigates the transverse 3:1 internal resonance of an axially transporting nonlinear viscoelastic Euler-Bernoulli beam with a two-frequency parametric excitation caused by a speed perturbation. The Kelvin-Voigt model is introduced to describe the viscoelastic characteristics of the axially transporting beam. The governing equation and the associated boundary conditions are obtained by Newton’s second law. The method of multiple scales is utilized to obtain the steady-state responses. The Routh-Hurwitz criterion is used to determine the stabilities and bifurcations of the steady-state responses. The effects of the material viscoelastic coefficient on the dynamics of the transporting beam are studied in detail by a series of numerical demonstrations. Interesting phenomena of the steady-state responses are revealed in the 3:1 internal resonance and two-frequency parametric excitation. The approximate analytical method is validated via a differential quadrature method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
ÖZ, H. R. On the vibrations of an axially travelling beam on fixed supports with variable velocity. Journal of Sound and Vibration, 239(3), 556–564 (2001)
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)
JACQUES, N., DAYA, E. M., and POTIER-FERRY, M. Nonlinear vibration of viscoelastic sandwich beams by the harmonic balance and finite element methods. Journal of Sound and Vibration, 329, 4251–4265 (2010)
ÖZKAYA, E. and PAKDEMIRLI, M. Vibrations of an axially accelerating beam with small flexural stiffness. Journal of Sound and Vibration, 234, 521–535 (2000)
KONG, L. and PARKER, R. G. Approximate eigensolutions of axially moving beams with small flexural stiffness. Journal of Sound and Vibration, 276(1–2), 459–469 (2004)
DING, H. and CHEN, L. Q. Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations. European Journal of Mechanics A-Solids, 27, 1108–1120 (2008)
YANG, X. D., LIM, C. W., and LIEW, K. M. Vibration and stability of an axially moving beam on elastic foundation. Advances in Structural Engineering, 13(2), 241–248 (2010)
SUWEKEN G. and VAN HORSSEN, W. T. On the weakly nonlinear, transversal vibrations of a conveyor belt with a low and time-varying velocity. Nonlinear Dynamics, 31(2), 197–223 (2003)
SUWEKEN G. and VAN HORSSEN, W. T. On the transversal vibrations of a conveyor belt with a low and time-varying velocity, part I: the string-like case. Journal of Sound and Vibration, 264(1), 117–133 (2003)
SUWEKEN, G. and VAN HORSSEN, W. T. On the transversal vibrations of a conveyor belt with a low and time-varying velocity, part II: the beam-like case. Journal of Sound and Vibration, 267(5), 1007–1027 (2003)
PONOMAREVA, S. V. and VAN HORSSEN, W. T. On transversal vibrations of an axially moving string with a time-varying velocity. Nonlinear Dynamics, 50, 315–323 (2007)
GHAYESH, M. H. and AMABILI, M. Nonlinear stability and bifurcations of an axially accelerating beam with an intermediate spring-support. Nonlinear Dynamics, 69, 193–210 (2012)
FAROKHI, H., GHAYESH, M. H., and HUSSAIN, S. Three-dimensional nonlinear global dynamics of axially moving viscoelastic beams. Journal of Vibration and Acoustics-Transactions of the ASME, 138(1), 011007 (2016)
WANG, Y. B., DING, H., and CHEN, L. Q. Kinematic aspects in modeling large-amplitude vibration of axially moving beams. International Journal of Applied Mechanics, 11, 1950021 (2019)
DING, H., YAN, Q. Y., and ZU, J. W. Chaotic dynamics of an axially accelerating viscoelastic beam in the supercritical regime. International Journal of Bifurcation and Chaos, 24, 1450062 (2014)
WANG, Y. B., DING, H., and CHEN, L. Q. Vibration of axially moving hyperelastic beam with finite deformation. Applied Mathematical Modelling, 71, 269–285 (2019)
ABDELHAFEZ, H. M. Resonance of a nonlinear forced system with two-frequency parametric and self-excitations. Mathematics and Computers in Simulation, 66(1), 69–83 (2004)
El-BASSIOUNY, A. F. Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations. Mechanics Research Communications, 32, 337–350 (2005)
MICHON, G., MANIN, L., PARKER, R. G., and DUFOUR, R. Duffing oscillator with parametric excitation: analytical and experimental investigation on a belt-pulley system. Journal of Computational and Nonlinear Dynamics, 3, 031001 (2008)
MICHON, G., MANIN, L., REMOND, D., DUFOUR, R., and PARKER, R. G. Parametric instability of an axially moving belt subjected to multi-frequency excitations: experiments and analytical validation. International Journal of Applied Mechanics, 75, 041004 (2008)
NAYFEH, A. H. Response of two degree of freedom systems to multi-frequency parametric excitations. Journal of Sound and Vibration, 88, 1–10 (1983)
NAYFEH, A. H. The response of non-linear single degree of freedom systems to multi-frequency excitations. Journal of Sound and Vibration, 102(3), 403–414 (1985)
PARKER, R. G. and LIN, Y. Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations. Journal of Applied Mechanics-Transactions of the ASME, 68, 49–57 (2001)
YANG, T. Z., FANG, B., CHEN, Y., and ZHEN, Y. Z. Approximate solutions of axially moving viscoelastic beams subject to multi-frequency excitations. International Journal of Non-Linear Mechanics, 44, 230–238 (2009)
SAHEBKAR, S. M., GHAZAVI, M. R., KHADEM, S. E., and GHAYESH, M. H. Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well. Mechanism and Machine Theory, 46, 743–760 (2011)
ÖZHAN, B. B. Vibration and stability analysis of axially moving beams with variable speed and axial force. International Journal of Structural Stability and Dynamics, 14(6), 1450015 (2014)
LV, H. W., LI, Y. H., LI, L., and LIU, Q. K. Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity. Applied Mathematical Modelling, 38, 2558–2585 (2014)
CHEN, L. Q. and TANG, Y. Q. Combination and principal parametric resonances of axially accelerating viscoelastic beams: recognition of longitudinally varying tensions. Journal of Sound and Vibration, 330, 5598–5614 (2011)
CHEN, L. Q. and TANG, Y. Q. Parametric stability of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Journal of Vibration and Acoustics-Transactions of the ASME, 134, 011008 (2012)
DING, H. and ZU, J. W. Periodic and chaotic responses of an axially accelerating viscoelastic beam under two-frequency excitations. International Journal of Applied Mechanics, 5(2), 1350019 (2013)
WANG, Y. Q., HUANG, X. B., and LI, J. Hydroelastic dynamic analysis of axially moving plates in continuous hot-dip galvanizing process. International Journal of Mechanical Sciences, 110, 201–216 (2016)
YE, C. and WANG, Y. Q. Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells: internal resonances. Nonlinear Dynamics, 104(3), 2051–2069 (2021)
MAO, X. Y., DING, H., and CHEN, L. Q. Internal resonance of a supercritically axially moving beam subjected to the pulsating speed. Nonlinear Dynamics, 95, 631–651 (2019)
SAHOO, B., PANDA, L. N., and POHIT, G. Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam. Nonlinear Dynamics, 82(4), 1721–1742 (2015)
SAHOO, B., PANDA, L. N., and POHIT, G. Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation. International Journal of Non-Linear Mechanics, 78(2016), 35–44 (2015)
ZHU, B., DONG, Y., and LI, Y. Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance. Nonlinear Dynamics, 94(4), 2575–2612 (2018)
ZHANG, D. B., TANG, Y. Q., LIANG, R. Q., YANG, L., and CHEN, L. Q. Dynamic stability of an axially transporting beam with two-frequency parametric excitation and internal resonance. European Journal of Mechanics A-Solids, 85, 104084 (2021)
MOTE, C. D. JR. A study of band saw vibrations. Journal of the Franklin Institute, 276, 430–444 (1965)
CHEN, L. Q. and ZU, J. W. Solvability condition in multi-scale analysis of gyroscopic continua. Journal of Sound and Vibration, 309, 338–342 (2008)
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(1–2), 401–418 (2016)
Acknowledgements
This work is supported by the Supporting Funds for Scientific Research of Linyi University (No. 2021PTXM001).
Author information
Authors and Affiliations
Corresponding author
Additional information
Citation: ZHANG, D. B., TANG, Y. Q., LIANG, R. Q., SONG, Y. M., and CHEN, L. Q. Internal resonance of an axially transporting beam with a two-frequency parametric excitation. Applied Mathematics and Mechanics (English Edition), 43(12), 1805–1820 (2022) https://doi.org/10.1007/s10483-022-2930-9
Project supported by the National Natural Science Foundation of China (Nos. 12002142, 11872159, and 51976087), the National Natural Science Foundation of Shanghai of China (No. 21ZR1462500), and the Natural Science Foundation of Shandong Province of China (No. ZR2021QB137)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhang, D., Tang, Y., Liang, R. et al. Internal resonance of an axially transporting beam with a two-frequency parametric excitation. Appl. Math. Mech.-Engl. Ed. 43, 1805–1820 (2022). https://doi.org/10.1007/s10483-022-2930-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-022-2930-9
Key words
- axially transporting viscoelastic beam
- internal resonance
- two-frequency excitation
- nonlinear vibration
- perturbation technique