A variational formulation is developed for calculating liquid sloshing effects on the dynamic response of spherical containers under external dynamic excitation. The velocity potential is expressed in a series form, where each term is the product of a time function and the associated spatial function. Because of the configuration of the containers, the associated spatial functions are non-orthogonal, and the problem is not separable and results in a system of coupled non-homogeneous ordinary linear differential equations, which is solved numerically. The solution can be obtained through either direct integration or modal analysis. Sloshing frequencies and masses are calculated rigorously for arbitrary liquid height, and convergence of the solution is thoroughly examined. Particular emphasis is given on the cases of half-full spheres, where explicit expressions for the coefficients of the governing equations are derived. Furthermore, the behavior of nearly-full and nearly-empty vessels is briefly discussed.

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