The reciprocating relative motions of friction pairs introduce nonlinearities into the host structure, while also generating frictional heat, leading to a noticeable temperature rise at the contact interface. Consequently, the contact mechanics properties are not invariant but highly dependent on temperature. To elucidate the thermo-mechanical interaction mechanism in dry friction systems, we propose a novel multi-physical nonlinear modal analysis method called Thermo-Mechanical damped Nonlinear Normal Mode (TM-dNNM). This numerical method enables the determination of nonlinear modal characteristics considering thermo-mechanical coupling and the prediction of detailed temperature distribution on contact interface. The efficiency and convergence of the TM-dNNM are ensured through three key techniques: multi-physics reduced order modeling, analytical Jacobian matrix, and scaling adjustment. We apply the proposed numerical method to a typical turbine blade with a flat underplatform damper for engineering purposes. Results indicate that frictional heat significantly affects the nonlinear modal characteristics, exhibiting complex behavior such as softening/stiffening reversal and resonance amplitude backtracking. Neglecting the thermo-mechanical coupling can lead to a deviation of more than 13 Hz in the predicted modal frequency and an overestimation of the modal damping ratio by a factor of three. Interface temperature rises considerably at higher vibration levels, with the hotspot located near the lower left corner of the contact surface, where the average temperature exceeds 1180 ℃, approaching the melting point. By employing the TM-dNNM, designers can predict critical resonance amplitudes and identify potential local hotspots, thus helping prevent excessive temperature at the interface of frictionally damped turbine blades.