Unmanned aerial vehicles (UAVs) have received plenty of attention due to their high flexibility and enhanced communication ability, nonetheless, the limited onboard energy restricts UAVs' application on persistent data collection missions in large areas. In this paper, we propose a rechargeable UAV-assisted periodic data collection scheme, where a UAV is dispatched to periodically collect data from sensor nodes (SNs) in the mission area and charged by a wireless charging platform. Specifically, the periodic data collection completion time is minimized by optimizing the UAV trajectory to reach the optimal balance among the collection time, flight time, and recharging time. The formulated problem is non-convex and difficult to solve directly. To tackle this problem, we divide the main problem into two sub-problems and address them by leveraging successive convex approximation (SCA), bisection search, and heuristic methods. Then, we propose a periodic trajectory optimization algorithm to iteratively solve the two sub-problems to minimize the completion time. Furthermore, to deal with the dynamics of SNs, we propose a low-complexity trajectory adjustment strategy, where the trajectory can be maintained or adjusted locally at the SNs change, which significantly mitigates the computation cost of re-optimization. The simulation results show the superiority and robustness of the proposed scheme and the completion time is on average 39% and 33% lower than the two benchmarks, respectively.