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

Yıl 2019, Cilt: 2 Sayı: 1, 104 - 112, 01.07.2019

Ruşen Sınır , Berra Gultekin Sınır

Öz

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.

Anahtar Kelimeler

Viscoelastic, Multiple time scale, Stability, Pipe

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

Öz

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.

Anahtar Kelimeler

viscoelastic, multiple time scale, pipe, stability

Kaynakça

• 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.