0
Research Papers

On Parametric Stability of a Nonconstant Axially Moving String Near Resonances

[+] Author and Article Information
Rajab A. Malookani

Department of Mathematical Physics,
Delft Institute of Applied Mathematics,
Delft University of Technology,
Delft 2628 CD, The Netherlands
e-mail: R.Ali@tudelft.nl

Wim T. van Horssen

Department of Mathematical Physics,
Delft Institute of Applied Mathematics,
Delft University of Technology,
Delft 2628 CD, The Netherlands
e-mail: w.t.vanhorssen@tudelft.nl

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 2, 2015; final manuscript received August 15, 2016; published online October 25, 2016. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 139(1), 011005 (Oct 25, 2016) (12 pages) Paper No: VIB-15-1505; doi: 10.1115/1.4034628 History: Received December 02, 2015; Revised August 15, 2016

The stability of an axially moving string system subjected to parametric excitation resulting from speed fluctuations has been examined in this paper. The time-dependent velocity is assumed to be a harmonically varying function around a (low) constant mean speed. The method of characteristic coordinates in combination with the two timescales perturbation method is used to compute the first-order approximation of the solutions of the equations of motion that governs the transverse vibrations of an axially moving string. It turns out that the system can give rise to resonances when the velocity fluctuation frequency is equal (or close) to an odd multiple of the natural frequency of the system. The stability conditions are investigated analytically in terms of the displacement-response and the energy of the system near the resonances. The effects of the detuning parameter on the amplitudes of vibrations and on the energy of the system are also presented through numerical simulations.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Mahalingam, S. , 1957, “ Transverse Vibrations of Power Transmission Chains,” Br. J. Appl. Phys., 8(4), pp. 145–148. [CrossRef]
Mote, C. D. , 1965, “ A Study of Bandsaw Vibrations,” J. Franklin Inst., 279(6), pp. 430–444. [CrossRef]
Mote, C. D. , 1968, “ Parametric Excitation of an Axially Moving String,” ASME J. Appl. Mech., 35(1), pp. 171–172. [CrossRef]
Pakdemirli, M. , and Ulsoy, A. G. , 1997, “ Stability Analysis of an Axially Accelerating String,” J. Sound Vib., 203(5), pp. 815–832. [CrossRef]
van Horssen, W. T. , and Ponomareva, S. V. , 2005, “ On the Construction of the Solution of an Equation Describing an Axially Moving String,” J. Sound Vib., 287(1–2), pp. 359–366. [CrossRef]
Suweken, G. , and van Horssen, W. T. , 2003, “ On the Weakly Nonlinear, Transversal Vibrations of a Conveyor Belt With a Low and Time-Varying Velocity,” Nonlinear Dyn., 31(2), pp. 197–223. [CrossRef]
Sandilo, S. H. , and van Horssen, W. T. , 2012, “ On Boundary Damping for an Axially Moving Tensioned Beam,” ASME J. Vib. Acoust., 134(1), p. 0110051. [CrossRef]
Suweken, G. , and van Horssen, W. T. , 2003, “ On the Transversal Vibrations of a Conveyor Belt With a Low and Time-Varying Velocity. Part I: The String-Like Case,” J. Sound Vib., 264(1), pp. 117–133. [CrossRef]
Ponomareva, S. V. , and van Horssen, W. T. , 2007, “ On Transversal Vibrations of an Axially Moving String With a Time-Varying Velocity,” Nonlinear Dyn., 50(▪), pp. 315–323. [CrossRef]
Öz, H. R. , and Boyaci, H. , 2000, “ Transverse Vibrations of Tensioned Pipes Conveying Fluid With Time-Dependent Velocity,” J. Sound Vib., 236(2), pp. 259–276. [CrossRef]
Naguleswaran, S. , and Williams, C. J. H. , 1968, “ Lateral Vibration of Band-Saw Blades, Pulley Belts and the Like,” Int. J. Mech. Sci., 10(11), pp. 239–250. [CrossRef]
Ariartnam, S. , and Asokanthan, S. , 1987, “ Dynamic Stability of Chain Drives,” ASME J. Appl. Mech., 109(3), pp. 412–418.
Mockensturm, E. M. , Perkins, N. C. , and Ulsoy, A. G. , 1996, “ Stability and Limit Cycles of Parametrically Excited, Axially Moving Strings,” ASME J. Vib. Acoust., 118(1), pp. 346–351. [CrossRef]
Miranker, W. L. , 1960, “ The Wave Equation in a Medium in Motion,” IBM J. Res. Dev., 4(1), pp. 36–42. [CrossRef]
Mote, C. D. , 1975, “ Stability of Systems Transporting Accelerating Axially Moving Materials,” ASME J. Dyn. Syst., Meas., Control, 97(1), pp. 96–98. [CrossRef]
Zhu, W. D. , Song, X. K. , and Zheng, N. A. , 2011, “ Dynamic Stability of a Translating String With a Sinusoidally Varying Velocity,” ASME J. Appl. Mech., 78(1), p. 0610211.
Pakdemirli, M. , Ulsoy, A. G. , and Ceranoglu, A. , 1994, “ Transverse Vibration of an Axially Accelerating String,” J. Sound Vib., 169(2), pp. 179–196. [CrossRef]
Chen, L. Q. , Zhang, N. H. , and Zu, J. W. , ▪, “ Bifurcation and Chaos of an Axially Moving Visco-Elastic Strings,” Chaos, Solitons Fractals, 29(2), pp. ▪–▪.
Ponomareva, S. V. , and van Horssen, W. T. , 2009, “ On the Transversal Vibrations of an Axially Moving Continuum With a Time-Varying Velocity: Transient From String to Beam Behavior,” J. Sound Vib., 325(4–5), pp. 959–973. [CrossRef]
Sandilo, S. H. , and van Horssen, W. T. , 2014, “ On Variable Length Induced Vibrations of a Vertical String,” J. Sound Vib., 333(11), pp. 2432–2449. [CrossRef]
Malookani, R. A. , and van Horssen, W. T. , 2015, “ On Resonances and the Applicability of Galerkin's Truncation Method for an Axially Moving String With Time-Varying Velocity,” J. Sound Vib., 344(▪), pp. 1–17. [CrossRef]
Malookani, R. A. , and van Horssen, W. T. , 2016, “ On the Asymptotic Approximation of the Solution of an Equation for a Non-Constant Axially Moving String,” J. Sound Vib., 367(▪), pp. 203–218. [CrossRef]
Gaiko, N. V. , and van Horssen, W. T. , 2016, “ On Transversal Oscillations of a Vertically Translating String With Small Time-Harmonic Length Variations,” J. Sound Vib., 383(▪), pp. 339–348. [CrossRef]
Zhu, W. D. , and Guo, B. Z. , 1998, “ Free and Forced Vibration of an Axially Moving String With an Arbitrary Velocity Profile,” ASME J. Appl. Mech., 65(3), pp. 901–907. [CrossRef]
Suweken, G. , ▪, “ A Mathematical Analysis of a Belt System With a Low and Time-Varying Velocity,” Ph.D. thesis, TU Delft, Delft, The Netherlands.
van Horssen, W. T. , 1992, “ Asymptotics for a Class of Weakly Nonlinear Wave Equations With Applications to Some Problems,” First World Congress of Nonlinear Analysis, Vol. 48, pp. 19–26.
Darmawijoyo , van Horssen, W. T. , and Celément, P. , 2003, “ On a Rayleigh Wave Equation With Boundary Damping,” Nonlinear Dyn., 33(4), pp. 399–429. [CrossRef]
Strauss, W. A. , 1992, Partial Differential Equations, An Introduction, Wiley, New York, NY.

Figures

Grahic Jump Location
Fig. 1

A schematic model of a moving belt system between two fixed supports

Grahic Jump Location
Fig. 2

Integration in the characteristic ξ (or σ) direction

Grahic Jump Location
Fig. 6

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, and δ = 2α. (a) Approximated energy and (b) energy of the system.

Grahic Jump Location
Fig. 7

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, δ = − 2α, and x = 0.05. (a) The unstable first-order asymptotic approximation (v0) and (b) the unstable numerical solution (u).

Grahic Jump Location
Fig. 8

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, and δ = −2α. (a) Approximated energy and (b) energy of the system.

Grahic Jump Location
Fig. 9

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, δ = 15, and x = 0.05. (a) The stable first-order asymptotic approximation (v0) and (b) the stable numerical solution (u).

Grahic Jump Location
Fig. 10

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, and δ = 15. (a) Approximated energy and (b) energy of the system.

Grahic Jump Location
Fig. 11

Approximate instability region in the (α, Ω)-plane for ε = 0.1: the amplitude α of the velocity fluctuation of the belt versus the frequency Ω of the velocity fluctuation of the belt. The boundaries of the instability region are given by Ω=(2m−1)π+εδ with |δ|=2α and m = 1, 2,….

Grahic Jump Location
Fig. 3

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, δ = 0.01, and x = 0.05. (a) The unstable first-order asymptotic approximation (v0) and (b) the unstable numerical solution (u).

Grahic Jump Location
Fig. 4

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, and δ = 0.01. (a) Approximated energy and (b) energy of the system.

Grahic Jump Location
Fig. 5

m = 1, ε = 0.01, V0 = 0.8, α = 0.5, δ = 2α, and x = 0.05. (a) The unstable first-order asymptotic approximation (v0) and (b) the unstable numerical solution (u).

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In