This article aims to facilitate the precise transmission design of modified gears. Based on the Hertz formula, it calculates the maximum contact stress in the transmission of modified involute spur cylindrical gears, analyzes the location where the maximum contact stress occurs, derives the ratio function (termed as stress ratio) between the maximum contact stress and the contact stress at the gear node engagement, and expresses the calculation formula of the maximum contact stress of the modified gear as the product of the stress ratio and the nodal stress. It also analyzes the variation rule of the stress ratio with the number of teeth of the small gear and the transmission ratio in the modified gear transmission. The results show that when the number of teeth of the small gear is greater than 17, the stress ratio decreases as increases; when
is less than 17, the stress ratio increases as increases; when is equal to 17, the stress ratio is the largest, and as the transmission ratio increases, the stress ratio will also increase. If the design criterion of the gear contact strength is that the maximum contact stress is 8% or more than the nodal contact stress, precise design should be carried out according to the maximum contact stress. The numerical ranges of the number of teeth and the transmission ratio required for precise contact strength calculation are provided. In the example analysis, the correctness of the derived stress ratio calculation formula is verified by the finite element method.
0. Introduction
The gear mechanism has superior properties such as high transmission efficiency, long service life, and stable transmission ratio, making it the most widely used mechanism in machinery. Gears are important transmission parts, and the performance of gear transmission will affect the working life and performance of mechanical equipment. Many researchers have conducted studies on gear contact issues and achieved many results. However, the calculation of the contact stress at the inner boundary point of the single-tooth meshing area of the large and small wheels when calculating the contact fatigue strength of the tooth surface in the standard does not consider the influence of gear modification on the maximum contact stress, and the calculation method of the maximum contact stress in the transmission of modified gears is not given. Therefore, the authors of this article will study the calculation problem of the maximum contact stress in the transmission of modified gears, analyze the influence of the number of teeth and the transmission ratio on the maximum contact stress of the modified gears, and determine under what conditions it is necessary to design the modified gears according to the maximum contact stress. The conclusions obtained can provide references for gear design and gear failure analysis, and meet the requirements of gear design in actual production.
1. Hertz Stress Calculation When Gears are Meshing
To analyze the difference between the contact stress at the nodal meshing and the maximum contact stress of the modified gear, a pair of modified involute spur cylindrical gears is now set up. The transmission torque is , the rotational speed is , the load correction factor is , the normal force of a single pair of teeth meshing is , the tooth width is , the tangential force is , the modulus is , the transmission ratio is , the number of teeth is and respectively, the base circle radius is and respectively, the radius of any meshing point on the tooth profile of the two gear teeth is and respectively, and the pressure angle is and respectively; the pitch circle radius is and , the pitch circle pressure angle is , the pitch circle radius is and , and the meshing angle is . Ignore the influence of the degree of coincidence on the contact line length . The meshing process of a pair of gear teeth is shown.
2. Calculation of the Maximum Contact Stress When a Single Pair of Teeth is Meshing
Let the limit meshing line of the gear be as shown in Figure 3. Point and are the meshing limit points of the small gear and the large gear respectively. Take point as the coordinate origin of the axis, and the direction from to is the positive direction of the axis. and represent the endpoints of the actual meshing line, and and are the boundary points of the double-tooth meshing area and the single-tooth meshing area respectively.
5. Stress Ratio Analysis
The calculation formula of the stress ratio can not only be applied to the design of modified gear transmission, but also to the design of standard gear transmission. When it is applied to modified gears, the modification coefficients and are substituted into equations (18), (17), (16), and (15) in sequence to calculate the stress ratio . Then, the stress ratio is substituted into equation (19) to calculate the maximum contact stress of the modified gear.
5.1 Isometric Modified Gear Transmission
It can be known from equation (19) that the main factors affecting the maximum contact stress are: the number of teeth of the small gear, the transmission ratio, and the meshing angle. In order to analyze the variation rule of the stress ratio, a pair of isometric modified gear transmission is set up. When the number of teeth of the driving gear is less than , the modification coefficient of the small gear is calculated according to the equation , and the modification coefficient of the small gear is selected as , then ; for the isometric modified gear transmission, equation (18) shows that the meshing angle is equal to the pitch circle pressure angle, and the influence of the meshing angle on the maximum contact stress does not need to be analyzed separately. Suppose the value range of the number of teeth of the driving gear is 10 to 17, and the value range of the transmission ratio is 1.7 to 6, then the variation curves of the stress ratio relative to the number of teeth of the active (or small) gear and the variation curves relative to the transmission ratio can be obtained, as shown in Figure 4.
shows that when the number of teeth of the small gear increases from 10 to 17, the modification coefficient of the small gear gradually decreases, the stress ratio increases, and the maximum contact stress increases. Figure 4(b) shows that as the transmission ratio of the isometric modified gear transmission increases, the stress ratio increases, and the maximum contact stress increases. Both curves show that if the transmission ratio is greater than or equal to 3, and the number of teeth of the small gear is greater than or equal to 13 (less than 17), the maximum contact stress is 8% or more than the nodal contact stress.
5.2 Non-Isometric Modified Gear Transmission
When the non-isometric modified gear transmission , it can be divided into positive transmission and negative transmission. If it is installed according to the standard center distance and without considering the topping, when it is positive transmission, it can be known from equation (18) that the meshing angle will increase, and when it is negative transmission, the meshing angle will decrease. The equation shows that when , the modification coefficient of the small gear is greater than 0, and the pressure angle of the top circle of the small gear increases. Suppose the number of teeth of the small gear is less than 17, the number of teeth of the large gear is greater than 17, when it is positive transmission, the modification coefficient ; when it is negative transmission, . In order to analyze the variation rule of the stress ratio, suppose that increases from to and the transmission ratio changes from to, and the variation curves of the stress ratio are shown in the figure.
