Gear error is another important excitation source of gear vibration and noise. The error is the difference between the actual tooth profile and the theoretical tooth profile, including manufacturing error, installation error and tooth modification. Spur gear errors greatly affect the contact characteristics and dynamic characteristics of gear teeth. The influence of spur gear profile error on dynamic transmission error is noted. The mathematical model for calculating transmission error is established based on theoretical analysis, finite element method and experimental model.
Considering the actual tooth profile contact, a dynamic model is established to solve the transmission error, which makes the transmission error more and more complex, which brings great difficulties to the solution. Especially for helical gears, three-dimensional contact analysis needs to be considered, which undoubtedly brings difficulties to the calculation. The transmission error is usually expressed in a simplified form. Each conjugate tooth profile is in line contact in the base plane, so the deformation from the action surface is ignored. Static transmission error is considered as displacement excitation and can be analyzed by many methods. With the wide application of high-performance spur gears, more and more analysis methods are focused on analysis tools, hoping to improve the analysis performance of dynamics. Conry and seireg have made important contributions in this respect. In their research, they used optimization technology and simplex algorithm to calculate the minimum static transmission error of spur gears, and used this algorithm to study the influence of transmission error on dynamic characteristics. It is particularly important to predict the vibration and noise of spur gears in the design stage. With the enhancement of calculation ability, more complex situations can be considered to solve more complex dynamic models. In this analysis of dynamic characteristics, the analysis of vibration and noise is always in an important position.
Giorgio bonori et al. Proposed a new method to simulate the error of spur gear based on the classical single degree of freedom model, in which the manufacturing error is treated as random error and freely distributed to the tooth profile of spur gear, so as to analyze the nonlinear dynamic behavior. As shown in Figure 1:
Because the actual tooth profile deviates from the ideal tooth profile during spur gear operation, the transmission error is also an irregular transmission. The ideal involute tooth profile is consistent with the transmission of motion and torque because of the perfection of the tooth profile, but in fact this perfect spur gear does not exist, so the modification becomes more important. The transmission error comes from the modification of spur gear, manufacturing error, misalignment of transmission shaft and elastic deformation of conjugate tooth profile. Figure 2 is the schematic diagram of single degree of freedom spur gear transmission error:
The expression of longitudinal transmission error is given as:
Where:
θ 1（ θ 2) – angular displacement of driving (driven) wheel;
N1 (N2) – number of teeth of driving (driven) wheel;
RB – base circle radius of driven wheel.
Based on the tooth contact analysis theory and Monte Carlo method, the influence of spur gear deviation on spur gear tooth contact characteristics is analyzed. The calculation speed of this method is faster than that of finite element method. The transmission error depends on the instantaneous contact spur gear tooth profile surface morphology and the elastic deformation of the gear tooth. More studies focus on the optimal design of the tooth profile in order to reduce the transmission error of the spur gear.
P. Velex and V. cahouet have made an analysis on the influence of the tooth friction on the transmission error through experiments and numerical simulation. The analysis results show that the surface tooth friction has a great influence on the transmission error, as shown in Figure 3:
In the past decades, gear lumped parameter models have been widely used. Stiffness excitation and error excitation are still important considerations in the study of spur gear transmission dynamic excitation models, and have always occupied an important position in the study of spur gear dynamic characteristics. The transmission error of spur gear is an important part of the research on the dynamic characteristics of spur gear in recent years. The transmission error of spur gear is greatly affected by the processing and installation of spur gear, which has important guiding significance for the subsequent tooth modification.
As the early experimental means are not available and the theory is not perfect, the influence of tooth friction and other factors on the dynamic characteristics analysis is ignored. With the improvement of the theory and the further improvement of the dynamic solution method, more comprehensive factors should be considered to further study the dynamic characteristics of spur gears. In recent years, the influence of tooth surface friction on the dynamic characteristics of spur gear has been studied in the literature published abroad, and some conclusions that have a great impact on the dynamic characteristics of spur gear have been obtained. However, the mechanism of the influence of tooth surface friction on the dynamic characteristics of spur gear transmission is not completely clear, so it is necessary to consider the influence of tooth surface friction on the dynamic characteristics of spur gear.