Objectives Aiming at the complex vibration modes and excessive number of modal test points in underwater vehicle stern structures, this paper proposes an optimal sensor placement method.

Method First, a finite-element model of a typical stern structure is constructed using S3 and S4R plate elements to simulate the structure's shell and rib plates. Subsequently, the structural parameters, such as nodes, elements, and the stiffness and mass matrices are extracted from the model file. To optimize sensor placement, a composite objective function is constructed by integrating a three-dimensional redundancy elimination index and the modal assurance criterion (MAC). The three-dimensional redundancy elimination index ensures adequate spatial separation between sensors in each direction and from the center of the stern structure. The MAC helps maintain the linear independence of modal shapes. Then, the GA−BPSO algorithm is employed to optimize sensor placement using binary coding. Each particle in the particle-swarm algorithm represents a placement plan, with its dimensionality corresponding to the number of candidate degrees of freedom. Each degree of freedom can either as 1 (sensor placed) or 0 (no sensor). The algorithm updates particle velocities and positions based on an inertia weight, learning factors, and random numbers. Positions are then discretized to 0 or 1 through a threshold. Genetic algorithm operators such as replication, crossover, and mutation are introduced to improve the performance of the binary discrete particle-swarm algorithm. The parameters of the GA−BPSO algorithm are carefully set. For example, the population size is set to 600, the number of iterations to 2 000, and the inertia weight decreases from 0.9 to 0.4 over the course of the iterations.

Results The optimized sensor layout reduces the number of required sensor locations from 840 (uniform layout) to 200. The maximum off-diagonal value in the MAC matrix in the optimized layout drops to 0.0333, the frequency deviation remains below 1%, and the modal shapes show a high degree of consistency.

Conclusion The proposed method effectively achieves a balance between the linear independence and visualization of modal shapes, demonstrating its applicability for modal testing of underwater stern structures.