Aimed at the random vibration problem of welded metal bellows in work,based on the theory of virtual excitation method,the signals of the bellows in the axial and radial time domain under random excitation were obtained by using MATLAB programming.The frequency domain signals was obtained by transforming the time domain signals with fast Fourier transform and imported into Workbench software for random vibration analysis.The response surface optimization analysis of the dangerous structure size (wave plate thickness,welding band radius and minimum bending radius) were carried out.The results show that the frequency of the bellows under random external excitation is between the first and third orders of the natural frequency.The maximum axial response deformation of the bellows occurs at the free end of the embedded static ring,the maximum radial response deformation occurs at the larger curvature of the wave plate near the fixed end of the bellows,and the equivalent stress is concentrated near the near the welding band under random excitation.These positions are most vulnerable to damage in the working process,which is consistent with the actual damage position.Through the response surface optimization analysis and adaptive multi-objective optimization,a set of optimized key material parameters was obtained,which greatly improves the vibration performance of the bellows,reduces the damage rate of the bellows.It provides a theoretical reference for the design of the bellows.