Abstract
Nonlinear forced vibrations of a fractional viscoelastic pipe conveying fluid exposed to the time-dependent excitations is investigated in the present work. Attention is focused in particular on the primary and secondary resonances with the Kelvin–Voigt fractional order constitutive relationship model. The nonlinear geometric partial differential equations due to stretching effect have been expressed by assumptions with Von Karman’s strain-displacement relation and Euler–Bernoulli beam theory. Viscoelastic fractional model for dam** and stiffness, and also plug flow model for fluid flow are considered to derive the equation of motion. Based on the Galerkin truncation, the coupled Fluid-Solid interaction nonlinear equation transferred to ordinary differential equations. The method of multiple scales is adopted to analyze steady-state solutions for the primary, superharmonic, and subharmonic resonances. Finally, the detailed parametric studies on the nonlinear dynamic behavior are discussed. Results delineate that the fractional derivative order and the retardation time have significant effects on the oscillation exhibited for different values of flow velocity.
Similar content being viewed by others
References
Zhang M, Shen Y, **ao L, Wenzhong Q (2017) Application of subharmonic resonance for the detection of bolted joint looseness. Nonlinear Dyn 88(3):1643–1653
Andreaus U, Baragatti P (2012) Experimental damage detection of cracked beams by using nonlinear characteristics of forced response. Mech Syst Signal Process 31:382–404
Mohammadi Ghazi R, Büyüköztürk O (2016) Damage detection with small data set using energy-based nonlinear features. Struct Control Health Monit 23(2):333–348
Andreaus U, Baragatti P (2011) Cracked beam identification by numerically analysing the nonlinear behaviour of the harmonically forced response. J Sound Vib 330(4):721–742
Peng ZK, Lang ZQ, Billings SA (2007) Crack detection using nonlinear output frequency response functions. J Sound Vib 301(3–5):777–788
Tsyfansky SL, Beresnevich VI (2000) Non-linear vibration method for detection of fatigue cracks in aircraft wings. J Sound Vib 236(1):49–60
Paidoussis MP (2014) Fluid-structure interactions : slender structures and axial flow, vol 1, 2nd edn. Elsevier, Amsterdam
Lin Y-H, Tsai Y-K (1997) Nonlinear vibrations of timoshenko pipes conveying fluid. Int J Solids Struct 34(23):2945–2956
Semler C, Paıdoussis MP (1996) Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe. J Fluids Struct 10(7):787–825
Semler C, Li GX, Paıdoussis MP (1994) The non-linear equations of motion of pipes conveying fluid. J Sound Vib 169(5):577–599
Lee S-I, Chung J (2002) New non-linear modelling for vibration analysis of a straight pipe conveying fluid. J Sound Vib 254(2):313–325
Szabó Z (2003) Nonlinear analysis of a cantilever pipe containing pulsatile flow. Meccanica 38(1):163–174
Ni Q, Tang M, Luo Y, Wang Y, Wang L (2014) Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation. Nonlinear Dyn 76(1):867–886
Mao X-Y, Ding H, Chen L-Q (2016) Steady-state response of a fluid-conveying pipe with 3: 1 internal resonance in supercritical regime. Nonlinear Dyn 86(2):795–809
Liu Z-Y, Wang L, Sun X-P (2018) Nonlinear forced vibration of cantilevered pipes conveying fluid. Acta Mech Solida Sin 31(1):32–50
Rong Bao L, **ao-Ting KR, Ni X-J, Tao L, Wang G-P (2018) Nonlinear dynamics analysis of pipe conveying fluid by riccati absolute nodal coordinate transfer matrix method. Nonlinear Dyn 92(2):699–708
Tang Y, Yang T (2018) Post-buckling behavior and nonlinear vibration analysis of a fluid-conveying pipe composed of functionally graded material. Compos Struct 185:393–400
Taylor G, Ceballes S, Abdelkefi A (2018) Insights on the point of contact analysis and characterization of constrained pipelines conveying fluid. Nonlinear Dyn 93(3):1261–1275
Bagley Ronald L, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27(3):201–210
Bagley Ronald L, Torvik J (1983) Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA J 21(5):741–748
Caputo M, Mainardi F (1971) A new dissipation model based on memory mechanism. Pure Appl Geophys 91(1):134–147
Caputo M, Mainardi F (1971) Linear models of dissipation in anelastic solids. La Rivista del Nuovo Cimento (1971–1977) 1(2):161–198
Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol 111. Elsevier, Amsterdam
Rossikhin Y, Shitikova MV (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 45:22–27
Yang T-Z, Fang B (2012) Stability in parametric resonance of an axially moving beam constituted by fractional order material. Arch Appl Mech 82(12):1763–1770
Di Paola M, Heuer R, Pirrotta A (2013) Fractional visco-elastic Euler–Bernoulli beam. Int J Solids Struct 50(22–23):3505–3510
Yang Tianzhi, Fang B (2013) Asymptotic analysis of an axially viscoelastic string constituted by a fractional differentiation law. Int J Non-Linear Mech 49:170–174
Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2016) Fractional calculus. World Scientific Publishing Company, Singapore
Colinas-Armijo N, Cutrona S, Di Paola M, Pirrotta A (2017) Fractional viscoelastic beam under torsion. Commun Nonlinear Sci Numer Simul 48:278–287
Permoon MR, Haddadpour H, Javadi M (2018) Nonlinear vibration of fractional viscoelastic plate: primary, subharmonic, and superharmonic response. Int J Non-Linear Mech 99:154–164
Asgari M, Permoon MR, Haddadpour H (2017) Stability analysis of a fractional viscoelastic plate strip in supersonic flow under axial loading. Meccanica 52(7):1495–1502
Agrawal OP (2004) Analytical solution for stochastic response of a fractionally damped beam. J Vib Acoust 126(4):561–566
Giuseppe F, Adolfo S, Massimiliano Z (2013) A non-local two-dimensional foundation model. Arch Appl Mech 83(2):253–272
Di Lorenzo S, Di Paola M, Pinnola FP, Pirrotta A (2014) Stochastic response of fractionally damped beams. Probab Eng Mech 35:37–43
Spanos PD, Malara G (2014) Nonlinear random vibrations of beams with fractional derivative elements. J Eng Mech 140(9):04014069
Wojciech S, Tomasz B, Christian L (2015) Fractional Euler–Bernoulli beams: theory, numerical study and experimental validation. Eur J Mech-A/Solids 54:243–251
Di Paola M, Scimemi GF (2016) Finite element method on fractional visco-elastic frames. Comput Struct 164:15–22
Jan Kazimierz Freundlich (2016) Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load. J Theor Appl Mech 54(4):1433–1445
Gioacchino A, Di Mario P, Giuseppe F, Pinnola FP (2018) On the dynamics of non-local fractional viscoelastic beams under stochastic agencies. Compos Part B Eng 137:102–110
Liaskos KB, Pantelous AA, Kougioumtzoglou IA, Meimaris AT (2018) Implicit analytic solutions for the linear stochastic partial differential beam equation with fractional derivative terms. Syst Control Lett 121:38–49
Jan F (2019) Transient vibrations of a fractional Kelvin–Voigt viscoelastic cantilever beam with a tip mass and subjected to a base excitation. J Sound Vib 438:99–115
Sinir BG, Donmez DD (2015) The analysis of nonlinear vibrations of a pipe conveying an ideal fluid. Eur J Mech-B/Fluids 52:38–44
Tang Y, Yang T, Fang B (2018) Fractional dynamics of fluid-conveying pipes made of polymer-like materials. Acta Mech Solida Sin 31(2):243–258
Tang Y, Zhen Y, Fang B (2018) Nonlinear vibration analysis of a fractional dynamic model for the viscoelastic pipe conveying fluid. Appl Math Model 56:123–136
Javadi M, Noorian MA, Irani S (2019) Stability analysis of pipes conveying fluid with fractional viscoelastic model. Meccanica 54(3):399–410
Amabili Marco (2008) Nonlinear vibrations and stability of shells and plates. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Javadi, M., Noorian, M.A. & Irani, S. Primary and secondary resonances in pipes conveying fluid with the fractional viscoelastic model. Meccanica 54, 2081–2098 (2019). https://doi.org/10.1007/s11012-019-01068-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-019-01068-2