Real-time simulation and data assimilation with use of extended reality (XR) to visualize results of structural dynamics analysis are expected to realize more interactive applications of digital twins. This paper shows real-time forward and inverse analyses of the damped free vibration of a cantilever beam by developing sequential physics-informed neural networks (PINNs). Here, trainings and predictions of PINNs were performed at each time interval close to the dominant vibration period and the NN weights obtained for prediction in the previous time interval are carried over to the training in the next time interval. This transfer learning made possible to reduce the training time at each interval for bringing the process closer to real-time, which was shown in the numerical study. In the experiment, an real-time data acquisition system of strains and displacements on the vibrating cantilever beam specimen was connected to the sequential PINNs, and the predicted bending moment distribution at each time interval was transmitted to the augmented reality (AR) visualization on the beam. In the results, the bending momentum distribution was almost accurately predicted, and the estimation of bending stiffness could be conducted at the same time. Although there still needs consideration to issues, such as data transmission speed and performance of inverse analysis in PINNs, the results showed the possibility of realizing the real-time simulation and data assimilation for the structural dynamics by the physics-informed machine learning.