Piezoelectric ring transducer is one of the most common underwater transducers, its radial vibration, bending vibration in-plane r- θand out-of-plane r- θhave been widely studied. However, the current research on the bending vibration in-plane r- zof the ring is insufficient, although it may have a noticeable impact on the applicability of the underwater transducers. In this study, mechanical analysis and related mathematical calculations of the bending vibrations in-plane r- zhave been carried out from the thin-shell theory. The work consists of three aspects: (1) free vibration theory solution, (2) forced vibration: multi-order modal excitation theory, and (3) related finite element calculations and experimental verification. In the study, the bending vibration equations under electrical short and open are derived, and the multi-order resonance frequency prediction formulas and shape functions for both conditions are obtained by analytical solution and function fitting. Using the finite element method, the influence of piezoelectric effect and the range of applicability of these two electrical conditions are analyzed. In the paper, the non-homogeneous equations under forced vibration are solved. Benefits from the orthogonal completeness property of the vibration mode function, an integral transform with the vibration mode function as the basis vector can be defined, so that the equation is solved in a simple positive space, and the results reveal the relationship between the coefficients of the modes of different orders and the voltage distribution. By modal theory, the effects of electrical excitation conditions on the multistep bending vibration modes are investigated, and effective methods such as unimodal excitation, partial excitation and single-ended excitation acting on several different target modes are obtained. The proposed piezoelectric ring unimodal excitation and single-ended excitation methods successfully excited the target modes in the experiments: the unimodal excited ring excites only one of its corresponding bending modes, while the single-ended excitation method excites all the bending modes of the first five orders, and its modal strength characteristics are in accordance with the theoretical predictions. The paper involves finite element simulation (FEM), experimental and theoretical comparative verification, which are in good agreement. The relevant conclusions can provide a theoretical basis for the identification of vibration modes of piezoelectric ring and the fine tuning of modal excitation.