Dynamic Stability of Axially Moving Viscoelastic Plates under Non-Uniform In-Plane Edge Excitations
This paper investigates the parametric stability of an
axially moving web subjected to non-uniform in-plane edge
excitations on two opposite, simply-supported edges. The web is
modeled as a viscoelastic plate whose constitutive relation obeys the
Kelvin-Voigt model, and the in-plane edge excitations are expressed
as the sum of a static tension and a periodical perturbation. Due to the
in-plane edge excitations, the moving plate may bring about
parametric instability under certain situations. First, the in-plane
stresses of the plate due to the non-uniform edge excitations are
determined by solving the in-plane forced vibration problem. Then,
the dependence on the spatial coordinates in the equation of transverse
motion is eliminated by the generalized Galerkin method, which
results in a set of discretized system equations in time. Finally, the
method of multiple scales is utilized to solve the set of system
equations analytically if the periodical perturbation of the in-plane
edge excitations is much smaller as compared with the static tension of
the plate, from which the stability boundaries of the moving plate are
obtained. Numerical results reveal that only combination resonances
of the summed-type appear under the in-plane edge excitations
considered in this work.
[1] R. F. Fung, J. S. Huang, and Y. C. Chen, “The transient amplitude of the
viscoelastic travelling string: An integral constitutive law,” Journal of
Sound and Vibration, vol. 201, pp. 153-167, 1997.
[2] L. Q. Chen, W. J. Zhao, and J. W. Zu, “Transient response of an axially
accelerating viscoelastic string constituted by a functional differential
law,” Journal of Sound and Vibration, vol. 278, pp. 861-871, 2004.
[3] L. Q. Chen, Y. Q, Tang, and C. W. Lim, “Dynamic stability in parametric
resonance of axially accelerating viscoelastic Timoshenko beams,”
Journal of Sound and Vibration, vol. 329, pp. 547-565, 2010.
[4] M. Marynowski, “Two-Dimensional rheological element in modeling of
axially moving viscoelastic web”. European Journal of Mechanics –
A/Solids, vol. 25, pp. 729-744, 2006.
[5] C. Shin, J. Chung, and H. Heeyoo, “Dynamic responses of the in-plane
and out-of-plane vibrations for an axially moving membrane,” Journal of
Sound and Vibration, vol. 297, pp. 794-809, 2006.
[6] C. C. Lin, “Stability and vibration characteristics of axially moving
plates,” International Journal of Solids and Structures, vol. 34, pp.
3179-3190, 1997.
[7] N. Banichuk, J. Jeronen, P. Neittaanmaki, and T. Tuovinen, “On the
instability of an axially moving elastic plate,” International Journal of
Solids and Structures, vol. 47, pp. 91-99, 2010.
[8] L. 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.
[9] M. Marynowski, “Non-Linear vibrations of an axially moving
viscoelastic web with time-dependent tension,” Chaos, Solitons &
Fractals, vol. 21, pp. 481-490, 2010.
[10] Y. Q. Tang, and L. Q. Chen, “Stability analysis and numerical
confirmation in parametric resonance of axially moving viscoelastic
plates with time-dependent speed,” European Journal of Mechanics
A/Solids, vol. 37, pp. 106-121, 2013.
[11] A. C. Liu, “Dynamic stability of axially moving viscoelastic plates with
pulsating speeds and nonuniform edge tensions,” Master Thesis, National
Taiwan University of Science and Technology, 2011. (in Chinese)
[12] L. Lengoc, and H. Mccallion, “Wide bandsaw blade under cutting
conditions, Part II: Stability of a plate moving in its plane while subjected
to parametric excitation,” Journal of Sound and Vibration, vol.186, pp.
143-162, 1995.
[13] C. Shin, W. Kim, and J. Chung, “Free in-plane vibration of an axially
moving membrane,” Journal of Sound and Vibration, vol. 272, pp.
137-154, 2004.
[14] A. H. Nayfeh, and D. T. Mook, Nonlinear Oscillations. New York: John
Wiley and Sons, 1979.
[1] R. F. Fung, J. S. Huang, and Y. C. Chen, “The transient amplitude of the
viscoelastic travelling string: An integral constitutive law,” Journal of
Sound and Vibration, vol. 201, pp. 153-167, 1997.
[2] L. Q. Chen, W. J. Zhao, and J. W. Zu, “Transient response of an axially
accelerating viscoelastic string constituted by a functional differential
law,” Journal of Sound and Vibration, vol. 278, pp. 861-871, 2004.
[3] L. Q. Chen, Y. Q, Tang, and C. W. Lim, “Dynamic stability in parametric
resonance of axially accelerating viscoelastic Timoshenko beams,”
Journal of Sound and Vibration, vol. 329, pp. 547-565, 2010.
[4] M. Marynowski, “Two-Dimensional rheological element in modeling of
axially moving viscoelastic web”. European Journal of Mechanics –
A/Solids, vol. 25, pp. 729-744, 2006.
[5] C. Shin, J. Chung, and H. Heeyoo, “Dynamic responses of the in-plane
and out-of-plane vibrations for an axially moving membrane,” Journal of
Sound and Vibration, vol. 297, pp. 794-809, 2006.
[6] C. C. Lin, “Stability and vibration characteristics of axially moving
plates,” International Journal of Solids and Structures, vol. 34, pp.
3179-3190, 1997.
[7] N. Banichuk, J. Jeronen, P. Neittaanmaki, and T. Tuovinen, “On the
instability of an axially moving elastic plate,” International Journal of
Solids and Structures, vol. 47, pp. 91-99, 2010.
[8] L. 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.
[9] M. Marynowski, “Non-Linear vibrations of an axially moving
viscoelastic web with time-dependent tension,” Chaos, Solitons &
Fractals, vol. 21, pp. 481-490, 2010.
[10] Y. Q. Tang, and L. Q. Chen, “Stability analysis and numerical
confirmation in parametric resonance of axially moving viscoelastic
plates with time-dependent speed,” European Journal of Mechanics
A/Solids, vol. 37, pp. 106-121, 2013.
[11] A. C. Liu, “Dynamic stability of axially moving viscoelastic plates with
pulsating speeds and nonuniform edge tensions,” Master Thesis, National
Taiwan University of Science and Technology, 2011. (in Chinese)
[12] L. Lengoc, and H. Mccallion, “Wide bandsaw blade under cutting
conditions, Part II: Stability of a plate moving in its plane while subjected
to parametric excitation,” Journal of Sound and Vibration, vol.186, pp.
143-162, 1995.
[13] C. Shin, W. Kim, and J. Chung, “Free in-plane vibration of an axially
moving membrane,” Journal of Sound and Vibration, vol. 272, pp.
137-154, 2004.
[14] A. H. Nayfeh, and D. T. Mook, Nonlinear Oscillations. New York: John
Wiley and Sons, 1979.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:70537", author = "T. H. Young and S. J. Huang and Y. S. Chiu", title = "Dynamic Stability of Axially Moving Viscoelastic Plates under Non-Uniform In-Plane Edge Excitations", abstract = "This paper investigates the parametric stability of an
axially moving web subjected to non-uniform in-plane edge
excitations on two opposite, simply-supported edges. The web is
modeled as a viscoelastic plate whose constitutive relation obeys the
Kelvin-Voigt model, and the in-plane edge excitations are expressed
as the sum of a static tension and a periodical perturbation. Due to the
in-plane edge excitations, the moving plate may bring about
parametric instability under certain situations. First, the in-plane
stresses of the plate due to the non-uniform edge excitations are
determined by solving the in-plane forced vibration problem. Then,
the dependence on the spatial coordinates in the equation of transverse
motion is eliminated by the generalized Galerkin method, which
results in a set of discretized system equations in time. Finally, the
method of multiple scales is utilized to solve the set of system
equations analytically if the periodical perturbation of the in-plane
edge excitations is much smaller as compared with the static tension of
the plate, from which the stability boundaries of the moving plate are
obtained. Numerical results reveal that only combination resonances
of the summed-type appear under the in-plane edge excitations
considered in this work.", keywords = "Axially moving viscoelastic plate, in-plane periodic
excitation, non-uniformly distributed edge tension, dynamic stability.", volume = "9", number = "7", pages = "1290-8", }