Torque measurements can significantly enhance control and monitoring loops for many mechatronic and aerospace applications. A typical challenge is to justify incorporating a torque sensor in terms of cost, system complexity, and reliability. Recently, sensorless torque estimation methods have been developed for robotic joints that include harmonic drive transmissions (HDTs). The principle is based on their relatively low torsional stiffness, which allows for estimating the transmitted torque by measuring the torsional angles (via existing joint encoders) and a compliance model. However, these methods are based on nonlinear models that are difficult to identify and tune. In this study, a simplified torque estimation method is introduced based on the structural damping friction of the HDTs. The structural damping can be correlated to the HDT torque using a simplified linear dynamic model and torsional rate measurements. Experimental results have validated the proposed method, using a robotic joint setup with an external torque sensor that has been previously utilized for testing several torque estimation methods.