###### Abstract  
This paper uses modal analysis and harmonic response analysis to study the vibration characteristics and vibration response of the frame of a volute-less centrifugal fan. The accuracy of the finite element model is verified by comparing the experimental results with the numerical simulation results. Our focus is on the resonance behavior of the fan frame under the maximum operating conditions. Therefore, the structural parameters are taken as design variables, and the maximum axial amplitude at 2600 rpm and its corresponding axial modal frequencies are taken as target parameters for research. A combination of response surface methodology and multi-objective optimization is used to optimize the structure parameters and solve for the Pareto solution set. Three feasible solutions were selected to verify the method's accuracy. The results show that the optimized fan frame structure enhances the axial modal frequencies to 52.3 Hz, reflecting a 24% improvement over the baseline. Furthermore, we reduce the maximum vibration velocity at resonance to 2.15 mm/s, effectively minimizing vibration responses and preventing excitation and resonance issues at peak speeds. volute-less centrifugal fan frame; vibration characteristics; response surface; multi-objective optimization 2 of the structure, avoid the resonance range, and achieve vibration reduction. [PERSON] et al. [4] investigated the effects of different rib forms on the vibration characteristics of cantilever plates. The results indicate that the presence of ribs increases the torsional stiffness of the structure. However, different forms of ribs have different effects on the quality and stiffness of the structure. [PERSON] [5] investigated axial vibration suppression in fan wall structures by implementing fixed beams on wall surfaces and incorporating additional fixed constraints on cantilever components. The results indicate that reinforcing the cantilevered structures, setting a wall thickness of 3 mm, and installing elongated crossbeams suppressed maximum axial structural vibrations significantly. [PERSON] et al. [6] demonstrated that relief processes on motor mounting frames significantly improved modal frequencies, with frequency enhancement correlated to the relief's height but independent of feature density. Modal analysis shows that under uniform height constraint, the modal frequencies of free-form relief configuration are 15-20% higher than that of regular geometry (trapezoid, circle, etc.). The reinforcement designs of structures have limited effectiveness in enhancing frequency and cannot be applied to some specific components. Many scholars have designed structures through topology optimization. Unlike strengthening the structure, topology optimization eliminates unnecessary parts without affecting the structure's mechanical properties, reducing the structure's mass while increasing the modal frequency. [PERSON] et al. [7] conducted topology optimization on the installation frame, industrial design, and fabrication of the optimized structure. After optimization, the first natural frequency of the structure increased by 10.2%, and the mass decreased by 37%, indicating that the optimization potential of the load-bearing components was not fully exploited. [PERSON] et al. [8] have topologically optimized the structure whilst maintaining the shape of specific regions. This optimized design slightly increased the overall deformation but effectively reduced the deformation of specific regions, significantly increasing the structure's natural frequency. This morphology preservation optimization method is useful for the design of specific engineering components. [PERSON] [9] researched the vibration of liquid storage systems and improved the system's natural frequency through structural optimization of frames and beams. The analysis demonstrated that optimizing the support beams and frames increased natural frequencies by 78% and reduced random vibration amplitude. The structure's lifespan enhanced from 242 to 2297 cycles. [PERSON] et al.