Based on the Donnell theory of thin shells，considering the level offset of the cone vertex in all layers, the governing equations of host shell and constraining shell can be obtained . For the viscoelastic layer, only shear deformations are considered. Employing the linear viscoelastic theory, combined with the governing equations of host shell and constraining shell, as well as the normal balance equation of viscoelastic layer, the integrated first order constant differential matrix equation of a sandwich conical shell with circular passive constrained layer damping is derived. The coefficient matrix of the equation is not constant, and the expression for each element is more complex than PCLD cylindrical shell. After that, a semi-analytical, semi-numerical method is developed to analyze the vibration and damping characteristics of a sandwich conical shell by means of the transfer matrix method which is based on precise integration technology. Comparisons with literature results show that the proposed approach is effective. The effects of parameters on natural frequency and dissipative factors are also discussed.