Nonlinear vibration analysis of fractional viscoelastic cylindrical shells
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O R I G I NA L PA P E R
M. R. Permoon · H. Haddadpour · M. Shakouri
Nonlinear vibration analysis of fractional viscoelastic cylindrical shells
Received: 2 February 2020 / Revised: 7 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Nonlinear vibrations of viscoelastic thin cylindrical shells are studied in this paper. The viscoelastic properties are modeled using the Kelvin–Voigt fractional-order constitutive relationship. Based on the nonlinear Love thin shell theory, the structural dynamics of the cylindrical shell is modeled by using the Newton’s second law, and the Galerkin method is used to discretize the nonlinear partial differential equations into the set of nonlinear ordinary differential equations. The method of multiple scales is used to solve the nonlinear ordinary differential equations, and the amplitude–frequency and phase–frequency equations are extracted. The obtained results are verified with available investigations, and the effects of fractional parameters, excitation, and nonlinearity on the amplitude–frequency and phase–frequency responses of the viscoelastic cylindrical shells are outlined.
1 Introduction Cylindrical shells are among the most commonly used structural elements in a wide range of engineering applications such as automotive, aerospace, transportation, marine, and other industries. In most of the applications, the frequency and amplitude of the vibration should be controlled by the designers to be far enough from resonant frequencies that cause catastrophic failures. The problems of vibration and stability of viscoelastic cylindrical shells are studied by many researchers and engineers. Using Flügge’s theory and three-dimensional elasticity theory, Okazaki et al. [1] studied the effects of damping properties in bending and extensional vibrations of two-layered cylindrical shells with an unconstrained viscoelastic layer using a numerical method. Cheng and Zhang [2] studied the dynamical behavior of viscoelastic cylindrical shells under axial harmonic pressure employing Kármán–Donnell theory for a thin shell and the Boltzmann law for viscoelastic materials. They showed that the harmonic response of viscoelastic cylindrical shells has various dynamical behaviors such as chaos, limit cycle oscillation, etc. Considering the effect of initial imperfection, Cederbaum and Touati [3] investigated the post-buckling behavior of imperfect cylindrical panels made of a nonlinear viscoelastic material and showed that nonlinear viscoelastic constitutive law predicts higher deflections than the linear viscoelastic ones. Khudayarov and Bandurin [4] studied the effects of the viscoelastic parameters on the nonlinear vibrations of cylindrical panels in a gas flow and showed that the viscoelastic properties have significant effect on the vibrations of the cylindrical panel. Based on the Kirchhoff–Love hypothesis, Eshmatov et al. [5–9] investigated the linear and nonlinear vibration and dynamic stability of a viscoelastic cylinder. They considered the effect of visco
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