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Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales

Year 2019, Volume: 2 Issue: 1, 104 - 112, 01.07.2019

Abstract

This study investigates the transverse vibrations taking place tensioned viscoelastic pipes conveying fluid with time-dependent velocity taking into account simple supports condition. The governing equation is derived from Newton’s second law, Boltzmann’s superposition principle, and the stress-strain relation given for Maxwell viscoelastic model. The time-dependent velocity is assumed to vary harmonically about mean velocity. This system experiences a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is solved using the multiple time scale method. Principal parametric resonance is investigated. Stability boundaries are determined analytically. It is demonstrated that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the system with constant velocity or when the frequency is close to the sum of any two natural frequencies.

References

  • H. R. Öz, M. Pakdemirli, and E. Özkaya, “Transition Behaviour From String To Beam for an Axially Accelerating Material,” J. Sound Vib., vol. 215, no. 3, pp. 571–576, 1998.
  • J. A. Wickert and J. Mote C. D., “Classical Vibration Analysis of Axially Moving Continua,” J. Appl. Mech., vol. 57, no. 3, pp. 738–744, Sep. 1990.
  • R. F. Fung, J. S. Huang, Y. C. Chen, and C. M. Yao, “Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed,” Comput. Struct., vol. 66, no. 6, pp. 777–784, 1998.
  • K. Marynowski and T. Kapitaniak, “Kelvin–Voigt versus Burgers internal damping in modeling of axially moving viscoelastic web,” Int. J. Non. Linear. Mech., vol. 37, pp. 1147–1161, 2002.
  • K. Marynowski and T. Kapitaniak, “Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension,” Int. J. Non. Linear. Mech., vol. 42, pp. 118–131, 2007.
  • L. Zhang and J. W. Zu, “Non-linear vibrations of viscoelastic moving belts, part I and II: Forced vibration analysis,” J. Sound Vib., vol. 216, no. 1, pp. 75–105, 1998.
  • Z. Hou and J. W. Zu, “Non-linear free oscillations of moving viscoelastic belts,” Mech. Mach. Theory, vol. 37, pp. 925–940, 2002.
  • R. F. Fung, J. S. Huang, and Y. C. Chen, “The transient amplitude of the viscoelastic travelling string: An integral constitutive law,” J. Sound Vib., vol. 201, no. 2, pp. 153–167, 1997.
  • L. Q. Chen, J. Wu, and J. W. Zu, “The chaotic response of the viscoelastic traveling string: an integral constitutive law,” Chaos, Solitons &Fractals, vol. 22, pp. 349–357, 2004.
  • X.-D. Yang and L.-Q. Chen, “Stability in parametric resonance of axially accelerating beams constituted by Boltzmann’s superposition principle,” J. Sound Vib., vol. 289, pp. 54–65, 2006.
  • C. Jo, J. Fu, and H. E. Naguib, “Constitutive modeling for mechanical behavior of PMMA microcellular foams,” Polymer (Guildf)., vol. 46, pp. 11896–11903, 2005.
  • M. P. Kruijer, L. L. Warnet, and R. Akkerman, “Modelling of the viscoelastic behaviour of steel reinforced thermoplastic pipes,” Compos. Part A, vol. 37, pp. 356–367, 2006.
  • M. P. Paı̈doussis and G. X. Li, “Pipes Conveying Fluid: A Model Dynamical Problem,” J. Fluids Struct., vol. 7, no. 2, pp. 137–204, 1993.
  • M. P. Païdoussis, Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 1. London: Academic Press, 1998.
  • M. P. Païdoussis, Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 2. London: Academic Press, 2003.
  • R. F. Gibson, Principles of Composite Material Mechanics. Singapore: McGraw-Hill Book Co, 1994.
  • H. R. Öz and M. Pakdemirli, “Vibrations of an axially moving beam with time-dependent velocity,” J. Sound Vib., vol. 227, no. 2, pp. 239–257, 1999.

Stability analysis of principal parametric resonance of viscoelastic pipes by using multiple time scales

Year 2019, Volume: 2 Issue: 1, 104 - 112, 01.07.2019

Abstract

This study
investigates the transverse vibrations taking place tensioned viscoelastic
pipes conveying fluid with time-dependent velocity taking into account simple
supports condition. The governing equation is derived from Newton’s second law,
Boltzmann’s superposition principle, and the stress-strain relation given for Maxwell
viscoelastic model. The time-dependent velocity is assumed to vary harmonically
about mean velocity. This system experiences a Coriolis acceleration component
which renders such systems gyroscopic. The equation of motion is solved using
the multiple time scale method. Principal parametric resonances is
investigated. Stability boundaries are determined analytically. It is
demonstrated that instabilities occur when the frequency of velocity
fluctuations is close to two times the natural frequency of the system with
constant velocity or when the frequency is close to the sum of any two natural
frequencies.

References

  • H. R. Öz, M. Pakdemirli, and E. Özkaya, “Transition Behaviour From String To Beam for an Axially Accelerating Material,” J. Sound Vib., vol. 215, no. 3, pp. 571–576, 1998.
  • J. A. Wickert and J. Mote C. D., “Classical Vibration Analysis of Axially Moving Continua,” J. Appl. Mech., vol. 57, no. 3, pp. 738–744, Sep. 1990.
  • R. F. Fung, J. S. Huang, Y. C. Chen, and C. M. Yao, “Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed,” Comput. Struct., vol. 66, no. 6, pp. 777–784, 1998.
  • K. Marynowski and T. Kapitaniak, “Kelvin–Voigt versus Burgers internal damping in modeling of axially moving viscoelastic web,” Int. J. Non. Linear. Mech., vol. 37, pp. 1147–1161, 2002.
  • K. Marynowski and T. Kapitaniak, “Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension,” Int. J. Non. Linear. Mech., vol. 42, pp. 118–131, 2007.
  • L. Zhang and J. W. Zu, “Non-linear vibrations of viscoelastic moving belts, part I and II: Forced vibration analysis,” J. Sound Vib., vol. 216, no. 1, pp. 75–105, 1998.
  • Z. Hou and J. W. Zu, “Non-linear free oscillations of moving viscoelastic belts,” Mech. Mach. Theory, vol. 37, pp. 925–940, 2002.
  • R. F. Fung, J. S. Huang, and Y. C. Chen, “The transient amplitude of the viscoelastic travelling string: An integral constitutive law,” J. Sound Vib., vol. 201, no. 2, pp. 153–167, 1997.
  • L. Q. Chen, J. Wu, and J. W. Zu, “The chaotic response of the viscoelastic traveling string: an integral constitutive law,” Chaos, Solitons &Fractals, vol. 22, pp. 349–357, 2004.
  • X.-D. Yang and L.-Q. Chen, “Stability in parametric resonance of axially accelerating beams constituted by Boltzmann’s superposition principle,” J. Sound Vib., vol. 289, pp. 54–65, 2006.
  • C. Jo, J. Fu, and H. E. Naguib, “Constitutive modeling for mechanical behavior of PMMA microcellular foams,” Polymer (Guildf)., vol. 46, pp. 11896–11903, 2005.
  • M. P. Kruijer, L. L. Warnet, and R. Akkerman, “Modelling of the viscoelastic behaviour of steel reinforced thermoplastic pipes,” Compos. Part A, vol. 37, pp. 356–367, 2006.
  • M. P. Paı̈doussis and G. X. Li, “Pipes Conveying Fluid: A Model Dynamical Problem,” J. Fluids Struct., vol. 7, no. 2, pp. 137–204, 1993.
  • M. P. Païdoussis, Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 1. London: Academic Press, 1998.
  • M. P. Païdoussis, Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 2. London: Academic Press, 2003.
  • R. F. Gibson, Principles of Composite Material Mechanics. Singapore: McGraw-Hill Book Co, 1994.
  • H. R. Öz and M. Pakdemirli, “Vibrations of an axially moving beam with time-dependent velocity,” J. Sound Vib., vol. 227, no. 2, pp. 239–257, 1999.
There are 17 citations in total.

Details

Primary Language English
Subjects Civil Engineering
Journal Section Research Articles
Authors

Ruşen Sınır 0000-0003-0700-1041

Berra Gultekin Sınır 0000-0002-9478-1666

Publication Date July 1, 2019
Published in Issue Year 2019 Volume: 2 Issue: 1

Cite

APA Sınır, R., & Sınır, B. G. (2019). Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales. Bayburt Üniversitesi Fen Bilimleri Dergisi, 2(1), 104-112.
AMA Sınır R, Sınır BG. Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales. Bayburt Üniversitesi Fen Bilimleri Dergisi. July 2019;2(1):104-112.
Chicago Sınır, Ruşen, and Berra Gultekin Sınır. “Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales”. Bayburt Üniversitesi Fen Bilimleri Dergisi 2, no. 1 (July 2019): 104-12.
EndNote Sınır R, Sınır BG (July 1, 2019) Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales. Bayburt Üniversitesi Fen Bilimleri Dergisi 2 1 104–112.
IEEE R. Sınır and B. G. Sınır, “Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales”, Bayburt Üniversitesi Fen Bilimleri Dergisi, vol. 2, no. 1, pp. 104–112, 2019.
ISNAD Sınır, Ruşen - Sınır, Berra Gultekin. “Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales”. Bayburt Üniversitesi Fen Bilimleri Dergisi 2/1 (July 2019), 104-112.
JAMA Sınır R, Sınır BG. Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales. Bayburt Üniversitesi Fen Bilimleri Dergisi. 2019;2:104–112.
MLA Sınır, Ruşen and Berra Gultekin Sınır. “Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales”. Bayburt Üniversitesi Fen Bilimleri Dergisi, vol. 2, no. 1, 2019, pp. 104-12.
Vancouver Sınır R, Sınır BG. Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales. Bayburt Üniversitesi Fen Bilimleri Dergisi. 2019;2(1):104-12.

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