This work investigates the hyper-viscoelastic behavior of a thermoplastic polyurethane IROGRAN A 92 E 5670 FCM used in the gear rim of a jaw coupling. The aim is to provide the material parameters for the modeling of jaw couplings in engineering tasks. Uniaxial compression tests were conducted at various temperatures and loading speeds to fit a hyper-viscoelastic material model. The material model uses the Yeoh free energy and the power law model with one Maxwell element. The parametrized material model showed very good results in comparison to experimental data. In addition, the influence of friction on the experiments was modeled, investigated, and discussed. Further component tests for a jaw coupling of size 28 were conducted with two different specimen types. The original gear rim and a modified gear rim with central boreholes for integrated sensors were used in the test. The influence of such a modification was investigated with four different loading cases, focusing on the torsion-torque characteristic and the relative damping. These component tests enable the validation of numerical models of jaw couplings.