Excessive plastic deformation may occur at the beam-column joints under seismic action and lead to connection failure, increasing the possibility of collapse of the entire frame structure. In order to improve the seismic performance of assembled steel structure joints, a right-angle viscoelastic joint damper with polyurethane as the core energy dissipation material is proposed in this paper. First, the temperature scanning tests of polyurethane materials were carried out based on the dynamic mechanical analysis method. Second, the dynamic mechanical test and numerical simulation analysis of the designed right-angle viscoelastic damper were performed to reveal the dynamic energy dissipation characteristics of the right-angle damper. Finally, the dynamic time-history damping analysis was performed on the steel frame structure equipped with RVDs. The results show that the TA value of polyurethane material reached its peak at 2.0 Hz, which is the ideal frequency for the material’s damping ability. The right-angle damper made of polyurethane material has a softening nonlinear characteristic, and the peak value of the loss factor was obtained at 2.0 Hz, which is consistent with the results of the dynamic performance of the polyurethane material. The numerical simulation results demonstrate that stress on the steel plates and viscoelastic layers is reasonable. When the excitation level did not exceed 9 mm displacement amplitude, the energy input to the damper was dissipated by polyurethane, and the steel plates never showed plastic deformation. The time-history analysis of the steel structure shows that the dampers designed in this paper have a good control effect on the interstory displacement, acceleration, and interstory shear force of the structure. The research results lay the necessary foundation for the engineering application of polyurethane materials in the field of beam-column joints vibration damping.