Jointed precast concrete systems typically have low inherent damping and are thus particularly suitable for applying supplemental damping systems. Analytical modelling is utilised to characterise jointed beam-to-column rocking connections, using a rate-dependent tri-linear compound version of the well-known Menegotto-Pinto rule. The analytical model is verified against near full-scale experimental results. The beam-column connections are constructed utilising Damage Avoidance Design (DAD) principles with unbonded post-tensioned tendons. High force-to-volume extrusion-based energy dissipaters are externally fitted to provide supplemental energy dissipation and modify joint hysteretic performance. Multiple joint configurations are analysed, with supplemental damping systems modified to investigate the effect of damping forces on joint hysteresis. Particular attention is given to the re-centring limit. Good agreement between the analytical models and experimental results is demonstrated, with discussion of possible improvements. Overall, system damping behaviour is significantly improved by adding the extrusion based damping system.