This paper is focused on the diagnostics of multicopter UAV propulsion system, in which the temporary transient states occur during operation in faulty conditions (eg. not all motor phases working properly). As a diagnostic sensor, the piezo strip has been used, which is very sensitive to any vibrations of the multi-rotor frame. The paper concerns the precise location of the sensor for more effective monitoring of the propulsion system state. For this purpose, a nonlinear analysis of the vibration times series was carefully presented. The obtained non-linear time series were studied with the recurrence analysis in short time windows, which were sensitive to changes in Unmanned Aerial Vehicle motor speeds. The tests were carried out with different percentage of the pulse width modulation signal used for the operation of the brushless motor and for different locations of the piezosensor (side and top planes of the multicopter arm). In the article, it was shown that the side location of the piezosensor is more sensitive to changes in the Unmanned Aerial Vehicle propulsion system, which was studied with the Principal Component Analysis method applied for four main recurrence quantifications. The research presented proves the possibility of using nonlinear recurrence analysis for propulsion system diagnostics and helps to determine the optimal sensor location for more effective health monitoring of multicopter motor.