## 1 Introduction  
With the rapid development of 3D printing technology, the application of 3D printing technology is increasingly extensive [1; 2; 3]. Due to the demand for individuation, color 3D printing technology has developed rapidly recently [4; 5]. For the FDM 3D printer, the mode of multi-nozzle cooperative work is mainly used for color printing, which will lead to a series of problems [6; 7]. On the one hand, the accuracy of the 3D printer has always been its problem and limits its development to a certain extent due to its unique printing mode [8; 9; 10], especially when color products are printed, accuracy deviations are easily detected due to the existence of color boundaries, so the FDM color 3D printer requires higher accuracy [11]. On the other hand, with the generation and iterative improvement of the color multi-nozzle device, the weight of whole nozzle is increased compared with that of the single nozzle [12]. The vibration characteristic of the nozzle is one of key factors that affect the printing accuracy. In the printing process, if the vibration frequency is close to the natural frequency of the 3D printer, the 3D printer will resonate, which will seriously affect the printing accuracy, and even lead to printing failure [13; 14]. [PERSON] et al. [15] analyzed the influence of the layer thickness, extrusion speed, and printing speed on printing surface roughness, and obtained the most effective method to improve surface quality of the printing parts. [PERSON] et al. [16] analyzed the technological parameters of the FDM 3D printer by the orthogonal experiment method and found that the layer thickness was the main factor affecting the accuracy of the FDM 3D printer. [PERSON] et al. [17] identified the influencing factors and estimated their individual and interaction effects on warpage, and obtained the maximum deformation at the middle value of the height of the part, which mainly related to the thermal conduction and bending stress of the layer. [PERSON] et al. [18] proposed an I-optimality criterion for the optimization of the FDM process parameters in order to address the limitations of the commonly used traditional designs. While some research has been carried out on improving the printing accuracy by the method of parameter optimization, no studies have been found to prove the effect of mechanical vibration on the accuracy. Thus, it is crucial to study the vibration characteristics of the 3D printer to improve the printing accuracy. Finite element analysis (FEA) is a simulation of real operating conditions by mathematical approximation. With simple and interacting elements, a finite number of unknowns could be used to approximate an infinitely unknown real system. [PERSON] [19] simulated the parallel 3D printer and performed a series of dynamics and performance analysis. [PERSON] et al. [20] proposed a technique to monitor the state of the FDM 3D printer nozzle by using a vibration sensor, the nozzle clogging in the 3D printer was detected by measuring the vibration of the extruder. [PERSON] et al. [21] analyzed the effect of lattice structure bars on the vibration characteristics of a 3D printer, the vibration testing on models was performed in order to find its natural frequencies. Such expositions are unsatisfactory because researchers mainly used experimental analysis or FEA to analyze mechanical structures of the 3D printer. To a certain extent, the data obtained by the above researchers could reflect the inherent characteristics of the structure, but it is not comprehensive enough. The methodological approach taken in this study is a mixed methodology based on the combination of experimental analysis and FEA. Firstly, the modal analysis of the color 3D printer is carried out, and the low-order natural vibration frequency and its corresponding mode shape displacement are calculated. Then the vibration signal is obtained by experiments, and the power spectrum is obtained by analyzing the vibration signal.