Vibration Characteristics of Functionally Graded Fractional Derivative Viscoelastic Fluid–Conveying Pipe with Initial Geometric Defects
Publication: Journal of Engineering Mechanics
Volume 150, Issue 5
Abstract
A viscoelastic constitutive model constructed with fractional derivatives can accurately describe the mechanical phenomena caused by materials with time-dependent and memory properties. Using this model, the vibration characteristics and stability of a functionally graded fractional derivative viscoelastic fluid–conveying pipe with initial geometric defects are investigated in this study. First, the differential equation of motion for a functionally graded fractional derivative viscoelastic fluid–conveying pipe with initial geometric defects is derived by applying the Hamilton principle. Dimensionless differential equations and boundary conditions are discretized using a differential quadrature method, and numerical solutions of the equations under three boundary conditions are obtained. Further, the effects of geometric defect types, fractional order, power law exponent, and other factors on the complex frequency of this fluid-conveying pipe are discussed. The findings revealed that the geometric defects of the fluid-conveying pipe can significantly affect the complex frequency of the system. Moreover, the pipe’s damping characteristics vary with the fractional order of the viscoelastic material. In other words, the higher the vibration mode of the pipe, the more susceptible it is to the fractional order.
Practical Applications
Because the fractional derivative relates to values near the current moment and overall history, it can effectively describe the temporal effect of materials, such as polymers, synthetic rubber, and coatings. Many naturally occurring materials (including geological materials such as soil, oil, and minerals and biological materials such as muscle, bone, and blood) exhibit time-dependent properties. The deformation of such materials relates to the current stress level and their overall history. Because the fractional derivative constitutive relation can describe the dynamic properties of a structure in a wide frequency range with only a few parameters and terms, it represents a potential candidate for addressing the mechanical problems of viscoelastic materials and structures. In this study, functionally graded fractional derivative viscoelastic materials were applied to a fluid-conveying pipe with initial geometric curvature, as common in practical engineering. The effects of the initial geometric defect type, dimensionless delay time, power law exponent, and other parameters on the stability of the fluid-conveying pipe were discussed, providing some theoretical reference for the design of such fluid-conveying pipes.
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Data Availability Statement
All data and models that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No.: 11972286).
References
Aldraihem, O. J. 2007. “Analysis of the dynamic stability of collar-stiffened pipes conveying fluid.” J. Sound Vib. 300 (3–5): 453–465. https://doi.org/10.1016/j.jsv.2006.09.007.
Alshorbagy, A. E., M. A. Eltaher, and F. Mahmoud. 2011. “Free vibration characteristics of a functionally graded beam by finite element method.” Appl. Math. Modell. 35 (1): 412–425. https://doi.org/10.1016/j.apm.2010.07.006.
Amirinezhad, H., A. Tarkashvand, and R. Talebitooti. 2020. “Acoustic wave transmission through a polymeric foam plate using the mathematical model of functionally graded viscoelastic (FGV) material.” Thin Walled Struct. 148 (Mar): 106466. https://doi.org/10.1016/j.tws.2019.106466.
Caputo, M. 1966. “Linear models of dissipation whose Q is almost frequency independent.” Ann. Geophys. 19 (4): 383–393.
Caputo, M., and F. Mainardi. 1971. “A new dissipation model based on memory mechanism.” Pure Appl. Geophys. 91 (1): 134–147. https://doi.org/10.1007/BF00879562.
Chang, S. 2000. Differential quadrature and its application in engineering. London: Springer.
Chen, L.-Q., and X.-D. Yang. 2005. “Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed.” J. Sound Vib. 284 (3–5): 879–891. https://doi.org/10.1016/j.jsv.2004.07.024.
Chen, L.-Q., W.-J. Zhao, and J. W. Zu. 2004. “Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law.” J. Sound Vib. 278 (4–5): 861–871. https://doi.org/10.1016/j.jsv.2003.10.012.
Culshaw, R. V., and S. Ruan. 2000. “A delay-differential equation model of HIV infection of CD4+ T-cells.” Math. Biosci. 165 (1): 27–39. https://doi.org/10.1016/S0025-5564(00)00006-7.
Dal, F. 2011. “Multiple time scales solution of an equation with quadratic and cubic nonlinearities having fractional order derivative.” Math. Comput. Appl. 16 (1): 301–308. https://doi.org/10.3390/mca16010301.
Deng, J., Y. Liu, Z. Zhang, and W. Liu. 2017a. “Size-dependent vibration and stability of multi-span viscoelastic functionally graded material nanopipes conveying fluid using a hybrid method.” Compos. Struct. 179 (Nov): 590–600. https://doi.org/10.1016/j.compstruct.2017.07.084.
Deng, J., Y. Liu, Z. Zhang, and W. Liu. 2017b. “Stability analysis of multi-span viscoelastic functionally graded material pipes conveying fluid using a hybrid method.” Eur. J. Mech. A. Solids 65 (Sep–Oct): 257–270. https://doi.org/10.1016/j.euromechsol.2017.04.003.
Fallah, A., and M. M. Aghdam. 2011. “Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation.” Eur. J. Mech. A. Solids 30 (4): 571–583. https://doi.org/10.1016/j.euromechsol.2011.01.005.
Farshidianfar, A., and P. A. Y. Soltani. 2012. “Nonlinear flow-induced vibration of a SWCNT with a geometrical imperfection.” Comput. Mater. Sci. 53 (1): 105–116. https://doi.org/10.1016/j.commatsci.2011.08.014.
Gu, F., C. K. Huang, J. Zhou, and L. P. Li. 2009. “Mechanical response of steel wire wound reinforced rubber flexible pipe under internal pressure.” J. Shanghai Jiaotong Univ. 14 (6): 747–756. https://doi.org/10.1007/s12204-009-0747-2.
Hosseini-Hashemi, S., A. R. Abaei, and M. R. Ilkhani. 2015. “Free vibrations of functionally graded viscoelastic cylindrical panel under various boundary conditions.” Compos. Struct. 126 (Aug): 1–15. https://doi.org/10.1016/j.compstruct.2015.02.031.
Ibrahim, H. H., M. Tawfik, and H. M. Negm. 2011. “Thermal buckling and nonlinear flutter behavior of shape memory alloy hybrid composite plates.” J. Vib. Control 17 (3): 321–333. https://doi.org/10.1177/1077546309353368.
Ibrahim, R. A. 2010. “Overview of mechanics of pipes conveying fluids—Part I: Fundamental studies.” J. Pressure Vessel Technol. 132 (3): 034001. https://doi.org/10.1115/1.4001271.
Kiani, Y., A. H. Akbarzadeh, Z. T. Chen, and M. R. Eslami. 2012. “Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation.” Compos. Struct. 94 (8): 2474–2484. https://doi.org/10.1016/j.compstruct.2012.02.028.
Kilbas, A. A., H. M. Srivastava, and J. J. Trujillo. 2006. Theory and applications of fractional differential equations. Vol. 204 of North-Holland mathematics studies. Amsterdam, Netherlands: Elsevier.
Komijani, M., J. N. Reddy, and M. R. Eslami. 2014. “Nonlinear analysis of microstructure-dependent functionally graded piezoelectric material actuators.” J. Mech. Phys. Solids 63 (Feb): 214–227. https://doi.org/10.1016/j.jmps.2013.09.008.
Li, Y., Y. Chen, and I. Podlubny. 2010. “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability.” Comput. Math. Appl. 59 (5): 1810–1821. https://doi.org/10.1016/j.camwa.2009.08.019.
Li, Z. M., and P. Qiao. 2015. “Buckling and postbuckling behavior of shear deformable anisotropic laminated beams with initial geometric imperfections subjected to axial compression.” Eng. Struct. 85 (Feb): 277–292. https://doi.org/10.1016/j.engstruct.2014.12.028.
Loghman, E., A. Kamali, F. Bakhtiari-Nejad, and M. Abbaszadeh. 2021. “Nonlinear free and forced vibrations of fractional modeled viscoelastic FGM micro-beam.” Appl. Math. Modell. 92 (Apr): 297–314. https://doi.org/10.1016/j.apm.2020.11.011.
McTavish, D. J., and P. C. Hughes. 1993. “Modeling of linear viscoelastic space structures.” J. Vib. Acoust. 115 (1): 103. https://doi.org/10.1115/1.2930302.
Misra, A. K., M. P. Païdoussis, and K. Van. 1988. “On the dynamics of curved pipes transporting fluid—Part I: Inextensible theory.” J. Fluids Struct. 2 (3): 221–244. https://doi.org/10.1016/S0889-9746(88)80009-4.
Oyelade, A. O., and A. A. Oyediran. 2020. “The effect of various boundary conditions on the nonlinear dynamics of slightly curved pipes under thermal loading.” Appl. Math. Modell. 87 (Nov): 332–350. https://doi.org/10.1016/j.apm.2020.06.019.
Païdoussis, M. P. 1987. “Flow-induced instabilities of cylindrical structures.” Appl. Mech. Rev. 40 (2): 163–175. https://doi.org/10.1115/1.3149530.
Permoon, M. R., H. Haddadpour, and M. Javadi. 2018. “Nonlinear vibration of fractional viscoelastic plate primary subharmonic and superharmonic response.” Int. J. Non-Linear Mech. 99 (Mar): 154–164. https://doi.org/10.1016/j.ijnonlinmec.2017.11.010.
Rajidi, S. R., A. Gupta, and S. Panda. 2020. “Vibration characteristics of viscoelastic sandwich tube conveying fluid.” Mater. Today Proc. 28 (Jan): 2440–2446. https://doi.org/10.1016/j.matpr.2020.04.788.
Rossikhin, Y. A., and M. V. Shitikova. 2012. “On fallacies in the decision between the Caputo and Riemann–Liouville fractional derivatives for the analysis of the dynamic response of a nonlinear viscoelastic oscillator.” Mech. Res. Commun. 45 (Oct): 22–27. https://doi.org/10.1016/j.mechrescom.2012.07.001.
Selmi, A., and H. Hassis. 2021. “Vibration analysis of post-buckled fluid-conveying functionally graded pipe.” Composites, Part C: Open Access 4 (Mar): 100117. https://doi.org/10.1016/j.jcomc.2021.100117.
Shen, H., M. P. Païdoussis, J. Wen, D. Yu, and X. Wen. 2014. “The beam-mode stability of periodic functionally graded material shells conveying fluid.” J. Sound Vib. 333 (10): 2735–2749. https://doi.org/10.1016/j.jsv.2014.01.002.
Sinir, B. G. 2010. “Bifurcation and chaos of slightly curved pipes.” Math. Comput. Appl. 15 (3): 490–502. https://doi.org/10.3390/mca15030490.
Sumelka, W., B. Łuczak, T. Gajewski, and G. Z. Voyiadjis. 2020. “Modelling of AAA in the framework of time-fractional damage hyperelasticity.” Int. J. Solids Struct. 206 (1): 30–42. https://doi.org/10.1016/j.ijsolstr.2020.08.015.
Tang, Y., and T. Yang. 2018. “Post-buckling behavior and nonlinear vibration analysis of a fluid-conveying pipe composed of functionally graded material.” Compos. Struct. 185 (Feb): 393–400. https://doi.org/10.1016/j.compstruct.2017.11.032.
Tang, Y., Y. Zhen, and B. Fang. 2018. “Nonlinear vibration analysis of a fractional dynamic model for the viscoelastic pipe conveying fluid.” Appl. Math. Modell. 56 (Apr): 123–136. https://doi.org/10.1016/j.apm.2017.11.022.
Wang, L., H. L. Dai, and Q. Qian. 2012. “Dynamics of simply supported fluid-conveying pipes with geometric imperfections.” J. Fluids Struct. 29 (Feb): 97–106. https://doi.org/10.1016/j.jfluidstructs.2011.12.013.
Wang, Z.-M., and Y.-Z. Liu. 2016. “Transverse vibration of pipe conveying fluid made of functionally graded materials using a symplectic method.” Nucl. Eng. Des. 298 (3): 149–159. https://doi.org/10.1016/j.nucengdes.2015.12.007.
Wang, Z.-M., Z. Zhan-Wu, and Z. Feng-Qun. 2005. “Stability analysis of viscoelastic curved pipes conveying fluid.” Appl. Math. Mech. 26 (Jun): 807–813. https://doi.org/10.1007/BF02465432.
Yang, J., L.-L. Ke, and S. Kitipornchai. 2010. “Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory.” Physica E 42 (5): 1727–1735. https://doi.org/10.1016/j.physe.2010.01.035.
Yin, Y., and K.-Q. Zhu. 2006. “Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model.” Appl. Math. Comput. 173 (1): 231–242. https://doi.org/10.1016/j.amc.2005.04.001.
Zhen, Y., Y. Gong, and Y. Tang. 2021. “Nonlinear vibration analysis of a supercritical fluid-conveying pipe made of functionally graded material with initial curvature.” Compos. Struct. 268 (Jul): 113980. https://doi.org/10.1016/j.compstruct.2021.113980.
Zhu, B., X.-C. Chen, Y. Guo, and Y.-H. Li. 2020. “Static and dynamic characteristics of the post-buckling of fluid-conveying porous functionally graded pipes with geometric imperfections.” Int. J. Mech. Sci. 189 (Jan): 105947. https://doi.org/10.1016/j.ijmecsci.2020.105947.
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Published In
Journal of Engineering Mechanics
Volume 150 • Issue 5 • May 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Jun 7, 2023
Accepted: Oct 19, 2023
Published online: Feb 22, 2024
Published in print: May 1, 2024
Discussion open until: Jul 22, 2024
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