In many sensitive measurement systems such as gravitational wave detectors, multistage low-loss vacuum-compatible suspension chains are required to effectively isolate the test mass from seismic disturbances. These chains usually have high quality factor normal modes which require damping. A technique termed "self-damping" in which the motion of orthogonal modes of the same stage mass is deliberately viscously cross-coupled to each other - thereby damping both modes - was engineered into the suspension chains used in an 80 m suspended high-power optical cavity. In this report, we investigate in detail the performance of a single stage of these chains. We model the system using numerical simulation and compare this with experimental measurements with different damping parameters in order to optimize the self-damping obtained using this technique.