Combination resonance analysis of imperfect functionally graded conical shell resting on nonlinear viscoelastic foundation in thermal environment under multi-excitation

In this work, nonlinear forced vibrations of truncated conical shells are presented using a semi-analytical method. The material properties are varied along the thickness direction as a power law distribution. The functionally graded truncated conical shells are exposed to external harmonic load and placed in the thermal environment and have an initial imperfection. Furthermore, the functionally graded truncated conical shells rests on generalized nonlinear viscoelastic foundations which consisted of a Winkler and Pasternak foundation parameters augmented by a Kelvin–Voigt viscoelastic model and a nonlinear cubic stiffness. The fundamental equations are extracted using first-order shear deformation theory in conjunction with nonlinear von Kármán relationships. The partial differential equations of truncated conical shells are reduced through Galerkin’s method, and the result is extracted using the multiple scales method. To analyze the resonance analyses, a two-term external excitation is considered. In this regard, various secondary resonances are investigated, and finally, the analyses about combination resonances are represented. To investigate the presented approach, a comparison study is performed with those addressed by other researchers. To analyze the nonlinear combination resonance behavior of truncated conical shells, the effect of geometrical characteristics, material properties, power law index, thermal effects, external load amplitude, and initial imperfection are examined. Finally, the steady-state responses of the nonlinear system are analyzed. As one of the most interesting results, the softening behavior of truncated conical shells with inverse quadratic distribution is the most, and for the quadratic distribution is the least.