AbstractTheoretical analyses are presented for the dynamic stability of a clamped‐free cylindrical shell partially filled with liquid, under vertical excitation. In the analyses, the dynamic version of the Donnell equations and the velocity potential theory were used for the motions of the shell and the liquid, respectively. The problem was solved by using the modified Galerkin method so as to satisfy the boundary conditions. The equations of motion coupling the shell and the liquid were derived from a type of coupled Mathieu's equation. It is found that the parametric principal resonance could occur, as well as the parametric combination resonance of the sum type, involving two natural vibrations with the same circumferential wave number but with different axial mode number. The latter type of parametric resonance apparently has not been previously studied. The instability regions where parametric resonance occurs were determined by using Hsu's method.17To compare with the experimental results which had been stated in a companion paper,15detailed numerical calculations were carried out for the two test cylinders partially filled with water. Excellent agreement between theory and experiment was demonstrated for the instability region