Piezoelectric ring transducer is one of the most common underwater transducers, and its radial vibration, bending vibration in-plane $r - \theta $ and out-of-plane $r - \theta $ have been widely studied. However, the current research on the bending vibration in the plane $r - z$ of the ring is insufficient, although it may have a noticeable influence on the applicability of the underwater transducers. In this study, mechanical analysis and related mathematical calculations of the bending vibrations in the plane $r - z$ are carried out by using the thin-shell theory. Herein, the following three aspects are studied: (1) free vibration theory solution, (2) forced vibration: multi-order modal excitation theory, and (3) related finite element calculations and experimental verification. In this study, the bending vibration equations under electrical short and electric open condition 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. The non-homogeneous equations under forced vibration are solved. By utilizing the orthogonal completeness of the vibration mode function, an integral transformation with the vibration mode function can be defined as the basis vector, 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 excite 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. This study involves finite element simulation, experimental and theoretical comparative verification, which are in good agreement. The relevant conclusions can provide a theoretical basis for identifying the vibration modes of piezoelectric ring and the fine tuning of modal excitation.