Nonlinear vibration of fractional Kelvin–Voigt viscoelastic beam on nonlinear elastic foundation

Abstract

The main objective of the present study is to analyze the nonlinear vibration behavior of fractional Kelvin–Voigt viscoelastic beams on a nonlinear elastic foundation under harmonic excitation. Attention is concentrated on the primary, superharmonic, and subharmonic resonances with the fractional Kelvin–Voigt constitutive model. To describe some viscoelastic material behaviors such as biomaterials, intelligent and polymeric materials, the fractional Kelvin–Voigt constitutive equation is used. Governing equations are derived using Hamilton’s principle based on the Euler–Bernoulli beam theory, with nonlinear elastic foundation and Von Karman’s nonlinearities due to stretching. These nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by the Galerkin projection technique. The method of multiple scales is utilized to obtain the response of beams under hard and soft excitations. Results are verified by the available literature. A parametric analysis is conducted to determine the influence of the fractional Kelvin–Voigt viscoelastic model on primary and secondary resonances. The numerical results illustrate that effect of the fractional Kelvin–Voigt model on the frequency-response and amplitude-response is remarkable. Therefore, the obtained results provide a useful benchmark for further nonlinear analysis of fractional Kelvin–Voigt viscoelastic beams with elastic foundations.