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The parametric instability of simply-supported sandwich cylindrical shell with a functionally graded (FG) core under static and time dependent periodic axial compressive loads is investigated in this study. The governing equations of sandwich cylindrical shell with an FG core are derived based on Donnell׳s shell theory using first order shear deformation theory (FSDT). Considering the time-dependent periodic axial compressive load, the governing equations are reduced the second order differential equation with the time-dependent periodic coefficient or Mathieu-type equation by using the Galerkin׳s method. This equation is solved by Bolotin׳s method and the closed form solutions for dimensionless excitation frequencies of parametric vibration of sandwich cylindrical shells with an FG core based on the FSDT are obtained. Numerical simulations are conducted to verify the analytical results. Finally, the effects of variations of volume fraction index of the FG core, static and dynamic load factors and shell characteristics on the dimensionless excitation frequencies of parametric vibration are studied numerically. [ABSTRACT FROM AUTHOR]