###### Abstract
[10] achieved a 17.9% reduction in mass and a 2% reduction in vibration response by the lightweight design of the structure based on stress distribution, modal, and vibration response studies of the structure. The frame structure studied in this paper makes it difficult to make shape modifications due to the limitations of rotating components. These two methods are not applicable. Some structures are unsuitable for modification and can only be analyzed in terms of their structural and damping materials. Structures made of different materials will show different modal frequencies. Therefore, without changing the shape of the structure, the modal frequencies of the structure will be significantly different from the excitation frequency to avoid resonance. Damping materials achieve a reduction in vibration response by absorbing vibration energy. [PERSON] et al. [11] studied the impact of the structure and material of compressor mounting frames on vibration. Their results indicated that the modal frequency of the mounting frame is greatly influenced by its structure. Additionally, overall vibration performance has improved under the combined effects of the structure and material. [PERSON] [12], [PERSON] [13;14], and their colleagues emphasized that substituting materials in specific structural components effectively modulates modal frequencies. A comprehensive analysis of the elastic damping mechanism has shown that viscoelastic damping materials can enhance the rate of amplitude attenuation and improve vibration response characteristics. However, these materials have a minimal impact on the modal frequency of the structure [15; 16; 17]. Although changing materials and damping has a good vibration reduction effect, it increases production costs. The above scholars have researched vibration reduction through reinforcement design, topology optimization, and using different materials and damping. Some scholars have also studied the vibration characteristics of a single structure through parameter optimization methods [18; 19; 20; 21]. However, the research difficulty has increased due to the complexity of the connection of structural components in the coupled system and the coupled vibration of multiple excitation sources. These challenges have resulted in a scarcity of studies within the current literature that explore the key factors affecting the structural vibration response and the interactions within the coupled system. Therefore, there is relatively little research on the vibration of the solute-less centrifugal fan frame. This paper focuses on the significant problem of axial resonance in fan frames, studies the fan frame coupling system's vibration characteristics, and improves the fan frame's vibration response through size optimization. A hybrid approach integrating finite element analysis (FEA) simulations and experimental measurements is implemented to characterize structural vibration behavior. Parametric modeling combined with response surface methodology (RSM) systematically analyzed the influence of structural parameters on vibrational performance. Subsequently, the structural search for excellent vibration characteristics is carried out using the multi-objective optimization method. The optimization results improved the vibration performance of the frame structure, reduced the vibration response, and avoided resonance under high-speed operating conditions. The optimization improves operational reliability and provides a theoretical basis for the structural design of a solute-less centrifugal fan frame.