A structural dynamics model was established to address the issue of thermal deformation and structural vibration of large space deployable antenna support structures under cyclic dynamic temperature loads when entering and exiting the Earth’s shadow. Equivalent thermodynamic material parameters of composite material members were provided, and thermal vibration response analysis was conducted. The nonlinear fluid damping model of Passive Nonlinear Viscous Damper (PNVD) is presented, and a spatially deployable antenna support truss structure thermal vibration suppression method based on nonlinear fluid damping is proposed. The experimental results of modal damping of large deployable antenna support structures show that before and after installing dampers, the fundamental frequency of the structure changes by no more than 1%, and the modal damping ratio increases by more than 2%.