In the present work, the integral equation approach is employed to investigate the nonlinear dynamic characteristics of two elastically coupled nanomechanical resonators. It is found that the mode localization phenomenon can occur in the vibrations of two nearly identical coupled Duffing oscillators. Our numerical experiments help us determine a group of specific parameters in the system consisting of two elastically coupled Duffing nanomechanical resonators. For this system, a tiny added mass on beam 1 can trigger a sudden jump of beam 2’s response from one periodic solution to another. Therefore, this nonlinear system can be tuned to a mass sensing device which can detect the tiny added mass. It sheds light on the wide application of the nonlinearity of the nonlinear dynamic system in the ultrasensitive mass sensing technology.