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In this paper, we present the design of a biomimetic robotic fish with a modular caudal fin and analyze its performance. The robot's propulsion system is experimentally characterized for different caudal fin geometries by using an ad hoc thrust-measurement system. The static thrust produced by the vibrating tail is expressed in terms of the oscillatory Reynolds number and compared with similar findings in the literature. Nonlinear vibrations of the propulsive tail are modeled using modal analysis and classical results from the study of large vibrations of slender cylinders in fluids. This analysis allows for computing the oscillatory Reynolds number in terms of the input parameters to the tail vibration. Free-swimming experiments are performed to investigate the performance and maneuverability of the robot and correlate static thrust with terminal speed. This robotic platform is currently being used in ethorobotics research for investigating collective behavior of gregarious fish species and in educational fun-science activities for K-12 students.