Primary and secondary resonances in pipes conveying fluid with the fractional viscoelastic model M. Javadi . M. A. Noorian . S. Irani Received: 24 May 2019 / Accepted: 10 October 2019 Ó Springer Nature B.V. 2019 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 assump- tions with Von Karman’s strain-displacement relation and Euler–Bernoulli beam theory. Viscoelastic frac- tional model for damping 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. Keywords Pipes conveying fluid  Fractional viscoelastic model  Method of multiple scales  Superharmonic resonance  Subharmonic resonance 1 Introduction Pipes conveying fluid are one of the fundamental elements which are extensively encountered in many practical applications such as the oil extraction and transmission, hydraulic oil tubes, lubrication pipes, and military industries. There are numerous particular conditions which pipes conveying fluid will be excited by external excitations. In some cases, these external excitations may lead to dangerous effects and anoma- lous behaviors. The high amplitude excitations may result in nonlinear behaviors. Specifically, nonlinear resonances may be used in the field of damage detection. In the literature several techniques have been proposed to detect the damage and health monitoring, using nonlinear techniques [1–6]. More specifically, structural health monitoring based on the analyzing sub- and superharmonic responses describes and gives a physical interpretation to define parame- ters suitable for damage detection. There are excellent literature surveys that discuss comprehensively the problem of pipes conveying fluid. Paı ¨doussis [7] M. Javadi  M. A. Noorian (&)  S. Irani Faculty of Aerospace Engineering, K.N. Toosi University of Technology, Tehran, Iran e-mail: m.javadi@email.kntu.ac.ir M. A. Noorian e-mail: noorian@kntu.ac.ir S. Irani e-mail: irani@kntu.ac.ir 123 Meccanica https://doi.org/10.1007/s11012-019-01068-2