In cylindrical grinding, it is known that using balls in place of conventional cone machine centers (CMCs), makes it possible to grind a workpiece without the influences of the misalignments and the geometric deviations of the workpiece center holes and machine centers. This study determines the characteristics of these ball machine centers (BMCs) and CMCs. In this paper, the stiffness and the damping capacity of CMC and BMC systems are measured to investigate the potential of chatter in stability of BMCs compared to conventional 60° CMCs. The main results obtained are as follows: 1) The stiffness difference between BMCs and CMCs was small or negligible, because the stiffness of the taper connection of our machine center shank was remarkably low; 2) the damping ratio of the BMCs was slightly larger than that of the CMCs; 3) the misalignments of the workpiece and CMC reduced the stiffness, while the stiffness of the BMCs was inherently independent of any misalignment; 4) using BMCs, a lubricant with a center hole diameter appropriate to the workpiece size can be used because the balls run-in to the center holes rapidly; 5) using CMCs, the stiffness depends on the center hole diameter; and 6) using BMCs, the waviness of the socket center hole containing the steel ball influences the stiffness. The ball diameter, moreover, affects the damping. Furthermore, it is unnecessary to worry about a more frequent occurrence of chatter when using BMCs, because the dynamic characteristics of BMCs are almost the same as those of CMCs.