The energy dissipation efficiencies of ball-in-tube dampers oriented with their sensitive axes orthogonal to (nutation) and parallel to (precession) the nominal spin axis of an asymmetric spacecraft are studied using the energy-sink method and numerical integration. Energy-sink equations for each of the two types of dampers are obtained in the forms of damped, periodically forced Hill-type equations. The stability of solutions to Mathieu equation approximations to these equations is considered. An approximate analytical expression for the ratio of the average energy dissipation rates of the two types of dampers is obtained. Results for ideally tuned, identical (except for their spring constants and orientations) dampers are presented and compared with those of a previous study. For a particular spacecraft configuration, analytical results for the energy dissipation ratio are compared with numerical results, and excellent agreement is achieved.