## 1 Introduction  
Gear transmission has been widely used in aircrafts, ships, automobiles, wind turbines, excavators, and so on. With the above mechanical equipment entering the development stage of high speed, high precision, and high reliability, the dynamic performance and transmission reliability of gear transmission are facing higher demands. However, in practical engineering, uncertain factors exist widely in machine tools, cutting tools, and machining motions, so gear machining errors are inevitable and uncertain. The sensitivity of gear transmission performance to random error enlarges the gap between the expected design goal and the practical application effect and affects the feasibility of the gear design scheme and the consistency of gear transmission performance in the batch. Therefore, it is necessary to study the influence of the law of random errors on the dynamic performance of gear systems and observe the error sensitivity of the system dynamic performance when the working conditions change. The influence of machining error and its randomness has been studied in the design and analysis of gear transmission. [PERSON] and [PERSON] [1] predicted the gear rattle when considering the eccentricities and tooth indexing errors, and it was proved by experiments that such errors are likely to increase the gear rattle severity. [PERSON] et al. [2] derived the similarity relationship of a gear system when considering the machining error and proposed two modification methods for the scale model prediction to solve machining error distortion. [PERSON] et al. [3] proposed a crack initiation life model of standard and error gears in accordance with tooth stress history and a critical plane method for non-proportional loading and investigated the effects of centering error and angular misalignment on crack initiation life in herringbone gears. [PERSON] et al. [4] performed tooth contact analysis of spiral bevel gears to investigate the simulation of gear alignment errors. Some studies revealed the influence of machining accuracy on gear transmission performance. [PERSON] and [PERSON] [5] treated the profile error as a random variable and analyzed the effects of profile errors and their variances on the gear vibration. [PERSON] and [PERSON] [6] predicted the variability of the dynamic behavior of a gear pair system due to tolerances. [PERSON] et al. [7] predicted the range of the transmission error for the spur gear pair under different grades. [PERSON] and [PERSON] [8] performed a statistical analysis on the effect of random manufacturing errors on the system's dynamic performance of the gears under different accuracy grades. Some other scholars have studied the uncertainty propagation of errors and the impact on system reliability. [PERSON] et al. [9] proposed a surrogate model-based uncertainty quantification and studied the uncertainty propagations of the shaft deviations and tooth surface deviations in a helical gear pair by using the model. [PERSON] et al. [10] analyzed the sensitivities of random gear parameters to the wind turbine gear system reliability. The randomness of errors indeed leads to the uncertainty of the dynamic transmission of the system. When changing operating parameters, such as torque or speed, the uncertainty may be aggravated. Therefore, it is of great significance to predict the dynamic behaviors of gear systems with random errors efficiently and accurately to reveal the influence mechanism of random errors under multiple working conditions. In the research of dynamic uncertain behaviors of gear systems, two uncertain analysis methods of probability and interval are mainly used. [PERSON] et al. [11] analyzed the dynamic behavior of a bevel gear system with uncertainty parameters based on the projection of polynomial chaos. [PERSON] and [PERSON] [12] analyzed the nonlinear stochastic vibration of a wind turbine's planetary gear train by using the statistical linearization technique.