Free vibration of a closed thin spherical shell on a fractional-order viscoelastic foundation

This paper investigates the free vibration of a closed thin spherical shell on a fractional-order viscoelastic foundation. First, the mechanical model of foundation is established, from which the expressions of the complex modulus and the foundation reaction force are derived. By introducing the concept of complex frequency, the attenuation coefficient is then determined from the governing equations of motion that include the foundation’s reaction. Subsequently, the method of separation of variables is employed to decouple the equations, leading to an ordinary differential equation for the normal displacement. This equation is solved analytically using associated Legendre polynomials, providing an explicit expression for the natural frequency. Finally, numerical examples are presented to examine the effects of the foundation’s viscous coefficient and fractional-order coefficient on the natural frequencies and mode shapes of the shell. Notably, the analysis reveals the distinct roles played by the fractional-order coefficient and the viscous coefficient.