Abstract A sandwich annular plate is built with two face layers of polar orthotropic material and a viscoelastic core layer. The non-axisymmetric vibration and stability problem of the rotating sandwich plate are analyzed by using the finite element method. By employing the discrete layer annular element and Hamilton’s principle, the finite element equations of motion that consider the effects of transverse shear and rotary inertia are derived. The viscoelastic material in the core layer is assumed to be incompressible, and its extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved. The effects of various properties, such as orthotropic properties of face layers, and the viscoelastic core layer stiffness and thickness are discussed. Numerical results show that the larger modulus ratio and higher rotational speed tend to increase the nature frequency of the sandwich annular plate system. In addition, the critical speeds decrease while increasing the thickness of the viscoelastic core layer.