Applied Mathematics and Mechanics >
Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions
Received date: 2016-03-07
Revised date: 2016-05-01
Online published: 2016-12-01
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 11672186, 11502147, and 11602146), the Chen Guang Project supported by the Shanghai Municipal Education Commission and the Shanghai Education Development Foundation (No. 14CG57), the Training Scheme for the Youth Teachers of Higher Education of Shanghai (No. ZZyyy12035), and the Alliance Program (No.LM201663)
The dynamic stability of axially accelerating plates is investigated. Longitudinally varying tensions due to the acceleration and nonhomogeneous boundary conditions are highlighted. A model of the plate combined with viscoelasticity is applied. In the viscoelastic constitutive relationship, the material derivative is used to take the place of the partial time derivative. Analytical and numerical methods are used to investigate summation and principal parametric resonances, respectively. By use of linear models for the transverse behavior in the small displacement regime, the plate is confined by a viscous damping force. The generalized Hamilton principle is used to derive the governing equations, the initial conditions, and the boundary conditions of the coupled planar vibration. The solvability conditions are established by directly using the method of multiple scales. The Routh-Hurwitz criterion is used to obtain the necessary and sufficient condition of the stability. Numerical examples are given to show the effects of related parameters on the stability boundaries. The validity of longitudinally varying tensions and nonhomogeneous boundary conditions is highlighted by comparing the results of the method of multiple scales with those of a differential quadrature scheme.
Youqi TANG, Dengbo ZHANG, Mohan RUI, Xin WANG, Dicheng ZHU . Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions[J]. Applied Mathematics and Mechanics, 2016 , 37(12) : 1647 -1668 . DOI: 10.1007/s10483-016-2146-8
[1] Skutch, R. Uber die bewegung eines gespannten fadens, weicher gezwungen ist durch zwei feste punkte, mit einer constanten geschwindigkeit zu gehen, und zwischen denselben in transversal-schwingungen von gerlinger amplitude versetzt wird. Annalen der Physik und Chemie, 61, 190-195(1897)
[2] Sack, R. A. Transverse oscillations in traveling strings. British Journal of Applied Physics, 5, 224-226(1954)
[3] Mote, C. D., Jr. Dynamic stability of axially moving materials. The Shock and Vibration Digest, 4, 2-11(1972)
[4] Miranker, W. L. The wave equation in a medium in motion. IBM Journal of Research and Development, 4, 36-42(1960)
[5] Ulsoy, A.G. and Mote, C. D., Jr. Vibration of wide band saw blades. Journal of Engineering for Industry, 104, 71-78(1982)
[6] Lin, C. C. and Mote, C. D., Jr. Equilibrium displacement and stress distribution in a two-dimensional axially moving web under transverse loading. Journal of Applied Mechanics, 62, 772-779(1995)
[7] Lengoc, L. and McCallion, H.Wide band saw blade under cutting conditions I:vibration of a plate moving in its plane while subjected to tangential edge loading. Journal of Sound and Vibration, 186, 125-142(1995)
[8] Lengoc, L. and McCallion, H. Wide bandsaw blade under cutting conditions Ⅱ:stability of a plate moving in its plane while subjected to parametric excitation. Journal of Sound and Vibration, 186, 143-162(1995)
[9] Lengoc, L. and McCallion, H. Wide bandsaw blade under cutting conditions Ⅲ:stability of a plate moving in its plane while subjected to non-conservative cutting forces. Journal of Sound and Vibration, 186, 163-179(1995)
[10] Lin, C. C. Stability and vibration characteristics of axially moving plates. International Journal of Solids and Structures, 34, 3179-3190(1997)
[11] Lin, C. C. Finite width effects on the critical speed of axially moving materials. Journal of Vibration and Acoustics, 120, 633-634(1998)
[12] Wang, X. Numerical analysis of moving orthotropic thin plates. Computers and Structures, 70, 467-486(1999)
[13] Luo, Z. and Hutton, S. G. Formulation of a three-node traveling triangular plate element subjected to gyroscopic and in-plane forces. Computers and Structures, 80, 1935-1944(2002)
[14] Kim, J., Cho, J., Lee, U., and Park, S. Modal spectral element formulation for axially moving plates subjected to in-plane axial tension. Composite Structures, 81, 2011-2020(2003)
[15] Luo, A. C. J. and Hamidzadeh, H. R. Equilibrium and buckling stability for axially traveling plates. Communications in Nonlinear Science and Numerical Simulation, 9, 343-360(2004)
[16] Hatami, S., Azhari, M., and Saadatpour, M. M. Stability and vibration of elastically supported, axially moving orthotropic plates. Iranian Journal of Science and Technology, 30, 427-446(2006)
[17] Hatami, S., Azhari, M., and Saadatpour, M. M. Exact and semi-analytical finite strip for vibration and dynamic stability of traveling plates with intermediate supports. Advances in Structural Engineering, 9, 639-651(2006)
[18] Hatami, S., Azhari, M., and Saadatpour, M. M. Free vibration of moving laminated composite plates. Composite Structures, 80, 609-620(2007)
[19] Tang, Y. Q. and Chen, L. Q. Nonlinear free transverse vibrations of in-plane moving plates:without and with internal resonances. Journal of Sound and Vibration, 330, 110-126(2011)
[20] Tang, Y. Q. and Chen, L. Q. Natural frequencies, modes and critical speeds of in-plane moving plates. Advances in Vibration Engineering, 11, 229-244(2012)
[21] Hu, Y. D. and Zhang, J. Z. Principal parametric resonance of axially accelerating rectangular thin plate in magnetic field. Applied Mathematics and Mechanics (English Edition), 34, 1405-1420(2013) DOI 10.1007/s10483-013-1755-8
[22] Tang, Y. Q., Zhang, D. B., and Gao, J. M. Vibration characteristic analysis and numerical confirmation of an axially moving plate with viscous damping. Journal of Vibration and Control (2015) DOI 10.1177/1077546315586311
[23] Marynowski, K. Two-dimensional rheological element in modelling of axially moving viscoelastic web. European Journal of Mechanics-A/Solids, 25, 729-744(2006)
[24] Zhou, Y. F. and Wang, Z. M. Vibration of axially moving viscoelastic plate with parabolically varying thickness. Journal of Sound and Vibration, 316, 198-210(2008)
[25] Hatami, S., Ronagh, H. R., and Azhari, M. Exact free vibration analysis of axially moving viscoelastic plates. Computers and Structures, 86, 1736-1746(2008)
[26] Zhou, Y. F. and Wang, Z. M. Dynamic behaviors of axially moving viscoelastic plate with varying thickness. Journal of Sound and Vibration, 22, 276-281(2009)
[27] Marynowski, K. Free vibration analysis of the axially moving Levy-type viscoelastic plate. European Journal of Mechanics-A/Solids, 29, 879-886(2010)
[28] Tang, Y. Q. and Chen, L. Q. Primary resonance in forced vibrations of in-plane translating viscoelastic plates with 3:1 internal resonance. Nonlinear Dynamics, 69, 159-172(2012)
[29] Tang, Y. Q. and Chen, L. Q. Parametric and internal resonances of in-plane accelerating viscoelastic plates. Acta Mechanica, 223, 415-431(2012)
[30] Lee, H. P. and Ng, T. Y. Dynamic stability of a moving rectangular plate subject to in-plane acceleration and force perturbations. Applied Acoustics, 45, 47-59(1995)
[31] Yang, X. D., Chen, L. Q., and Zu, J. W. Vibrations and stability of an axially moving rectangular composite plate. Journal of Applied Mechanics, 78, 011018(2011)
[32] Tang, Y. Q. and Chen, L. Q. Stability analysis and numerical confirmation in parametric resonance of axially moving viscoelastic plates with time-dependent speed. European Journal of Mechanics-A/Solids, 37, 106-121(2013)
[33] 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, 134, 11008(2012)
[34] 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)
[35] Tang, Y. Q., Chen, L. Q., Zhang, H. J., and Yang, S. P. Stability of axially accelerating viscoelastic Timoshenko beams:recognition of longitudinally varying tensions. Mechanism and Machine Theory, 62, 31-50(2013)
[36] Chen, L. Q., Tang, Y. Q., and Zu, J. W. Nonlinear transverse vibration of axially accelerating strings with exact internal resonances and longitudinally varying tensions. Nonlinear Dynamics, 76, 1443-1468(2014)
[37] 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 Acoustic, 136, 63-69(2014)
[38] Ding, H. Periodic responses of a pulley-belt system with one-way clutch under inertia excitation. Journal of Sound and Vibration, 353, 308-326(2015)
[39] Yan, Q. Y., Ding, H., and Chen, L. Q. Nonlinear dynamics of an axially moving viscoelastic Timoshenko beam under parametric and external excitations. Applied Mathematics and Mechanics (English Edition), 36, 971-984(2015) DOI 10.1007/s10483-015-1966-7
[40] Mote, C. D., Jr. A study of band saw vibrations. Journal of the Franklin Institute, 276, 430-444(1965)
[41] 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(2015)
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