The various modeling methods of gears are compared and analyzed, and a more accurate and feasible modeling method is proposed. The errors of various modeling methods and their influence on the accuracy of gear motion are explored and compared. An attempt is made to accurately detect virtual gears in a virtual environment.
At that time, some experts raised their accuracy problems. How to detect the accuracy of virtual gears in the virtual environment and analyze its influence on the accuracy of gear motion is a problem worth exploring. In recent years, many literatures have proposed various three-dimensional modeling methods of gears, but they all have certain shortcomings, and their modeling accuracy has not been analyzed. In this paper, we try to analyze various modeling methods and propose a kind of A more accurate and feasible modeling method, and a comparative analysis of the accuracy of various modeling methods and its impact on the accuracy of gear motion.
1 Gear 3D Modeling Method 1.1 Overview of Gear Modeling Methods In recent years, many literatures have proposed various 3D modeling methods. The key issue is the construction of gear tooth profiles, which are mainly divided into the following categories: (1) ) Replace the involute with an arc. This method is a relatively simple method, in which the involute and the root transition line are directly replaced by a circular arc between the crest and the root. For example, SolidEdge's engineering manual uses this modeling approach, and it is clear that the modeling accuracy of this approach is very low. The advantage of the SolidEdge engineering manual is its powerful design calculations.
(2) Drawing the tooth involute using the LawCurve (form curve) provided by the 3D CAD software or using the secondary development method. The essence of this method is to calculate several points on the involute according to the involute equation, and then connect them with splines or straight segments. Most of the literatures currently use this method. From the introduction of this kind of literature, it can be seen that the crest and the root are directly connected by a spline curve of approximately involute, but no treatment is performed in the root portion. When the base circle is larger than the root circle (number of teeth zâ‰¤41), the tooth profile is theoretically composed of an involute and a transition line; when the base circle is smaller than the root circle (number of teeth z>41), the gear teeth The profile is theoretically an involute, but in reality, due to the existence of the arc of the tool tip, there is still a transition curve at the root portion. Although the tooth root transition curve theoretically does not affect the movement of the gear, it will affect the structural and mechanical analysis of the gear model. Therefore, the root transition curve should be considered in the three-dimensional modeling of the gear, and most of the literature The processing of the gear profile curve is not described.
(3) Consider the root transition curve. For example, the "4-point method" proposed in the literature draws a tooth profile. According to the number of teeth, the method uses a B-spline curve to form a tooth profile curve. When the base circle is larger than the root circle (number of teeth z â‰¤ 41), the four points indicated are taken from the point A on the root circle, the point B on the base circle, the point C on the index circle, and the point D on the addendum circle. When the base circle is smaller than the root circle (the number of teeth z>41), since the transition curve is small, the four points pointed at this time are taken from the point A on the root circle, the point C on the index circle, and the addendum circle. The upper point D and the point E between the root circle and the index circle, the advantage of this method is that the modeling speed of the gear is fast, but the modeling accuracy is affected to some extent. Modeled in a similar way is the US CNC Software Inc.
MasterCAM gear styling module, the root of the tooth is replaced by a circular arc, and the involute part generally adopts 6 points with high precision.
(4) The simulation model is used to generate a gear model. In order to obtain accurate tooth contours, a method for drawing precise tooth contours by simulated cutting method has been proposed in the literature. It is based on the principle of cutting gears in AutoCAD, and is simulated on a computer with a standard rack tool. Cutting out the precise tooth profile of a gear. Since the method completely simulates the cutting of the gear, the generated tooth profile curve is exactly the same as the real gear regardless of the gear number, modulus and displacement coefficient of the gear. Obviously, the method is very accurate, but it can only be implemented in a two-dimensional environment, and then the generated two-dimensional profile is input into the three-dimensional CAD software to establish a three-dimensional model, which is cumbersome to operate. In the 3D CAD software SolidEdge, the method of building a three-dimensional model of gears by using the simulation method is proposed. However, in the three-dimensional environment, the gears formed have a large number of characteristic elements, occupying a large disk storage space and generating gears. It takes a long time to be used in 3D assembly and 2D drawings of gear transmission systems.
To this end, the original method has been improved on the basis of the "4-point method", and the involute and transition curves of the gear teeth are calculated by the analytical method, and the three-dimensional mode 1.2 analysis method of the gear is generated to generate the teeth of the gear tooth profile gear. The profile includes an engaging portion and a non-engaging portion, wherein the non-engaging portion refers to a root circle and a transition curve. Although the tooth root transition curve theoretically does not affect the meshing motion of the gear, it will affect the structural and mechanical analysis of the gear model. Therefore, the root transition curve should be considered in the three-dimensional modeling of the gear. The original "4-point method" was improved, and the tooth profile involute curve and transition curve were calculated by analytical method, and then connected by a B-spline curve.
The involute portion of the tooth profile can be easily obtained by the involute parameter equation, and will not be described again. In the tooth profile transition curve part, according to the introduction of the literature, the gear is processed by the rack type tool, which is equivalent to the meshing of the rack gear. The involute portion of the tooth profile of the machined tool is cut by the cutting edge, and the transition curve portion is cut out by the rounded portion of the tool. During the machining process, the machining section line of the tool is tangent to the machining pitch circle of the gear, and the rounded corner of the tool will describe the extended involute, so that the transition line of the gear is an equidistant curve extending the involute.
Within the range, for different Î±' angles, substituting the parameters of the transition curve tool, the coordinates of different points on the transition curve can be obtained.
2 Error Analysis of 3D Gear Model 2.1 Error Analysis According to the above gear modeling method, no matter which method is adopted, the tooth profile curve of the gear is fitted by spline curve, straight line segment or arc segment, therefore, both There is a certain tooth shape error. Research shows that the base section error and the tooth profile error are the two errors that have the greatest influence on the dynamic performance of the gear in the gear error project. For the gears processed by the Fan method, there are two kinds of base section errors, one is the constant base section deviation caused by the base circle radius error, or the pressure angle error; the other is the change caused by the shape error of the tooth shape itself. Value base section deviation. Different from the actual machining of the gears by Fan Chengfa, by accurately calculating the virtual gears that are shaped, there will be no constant base section error due to machining factors, and only the tooth shape error will cause the variable base section deviation. Therefore, for the virtual gear, the main influence of the accuracy is the tooth shape error. The main sources are: (1) using the arc or straight line segment to approximate the tooth profile error caused by the tooth profile curve; (2) using the spline curve When the tooth profile curve is combined, due to the limited number of data points, a certain tooth shape error is caused; (3) When the tooth profile curve is fitted by the spline curve, a certain tooth shape error is caused due to the uneven distribution of the data points.
2.2 Error Analysis Model In order to analyze the influence of gear tooth profile error on the accuracy of gear motion, it is common practice to use a standard gear or rack to mesh with the gear to be measured, and to reflect the angle error or angular velocity error of the measured gear to reflect the tooth. The magnitude of the shape error and its effect on the accuracy of the gear movement. In the virtual environment, due to the error of the tooth profile curve, the standard gear is difficult to accurately realize, but the rack tooth profile is composed of straight lines and arc segments, and there is no tooth shape error during the modeling. For this reason, a standard rack is used. Engage with the measured gear and establish the UG-based kinematics model shown in Figure 2.
In order to avoid installation errors, in the UG assembly environment, by the distance h between the given rack and the center of the gear, the midpoint A of the top circle of the gear is vertically aligned with the midpoint B of the rack groove to ensure the accuracy of the installation, as shown in Fig. 2. Shown. In the dynamic model of the rack and pinion, a rotation pair constraint is applied to the gear so that it can only rotate around the center; a sliding pair is applied to the rack so that the rack can only move linearly in the horizontal direction and give the rack a uniform speed The speed of the motion (v = 10 mm / s), in order to allow the rack to drive the gear movement, a three-dimensional contact is applied between the rack and pinion, as shown in Figure 2. The so-called three-dimensional contact in UG is to generate a set of baseline distance relationship between the active contact body (rack) and the contact surface (gear profile). In each step of analysis and calculation, the system will check these distance relationships to determine Whether there is contact. Once a contact is detected, the ADAMS solver calculates the contact force and subsequent contact motion response. According to the principle equation of contact force, the following equations can be used to calculate the contact force. It can be seen from the above model analysis that the analysis model of the rack and pinion can be simplified to the dynamic model, which is consistent with the classical single-degree-of-freedom gear transmission dynamics model. Through the calculation and simulation of the model, the fluctuation result of the gear rotation speed is obtained, which reflects the tooth profile error of the measured gear.
UG-based kinematics model dynamics model 2.3 analysis results on the computer, using the SolidEdge engineering manual, "4-point method", "analytical method" and "Fan Cheng method" for gear modeling (modulo m = 2, number of teeth z = 18 , pressure angle Î± = 20 Â°), and then with the same standard rack, according to the above analysis model in the UG assembly, modeling and motion simulation, the results obtained 35 March 2005 Xu Wubin, et al: (b) "analysis Method" gear angular velocity curve (c) "4-point method" gear angular velocity curve (d) "Fan Chengfa" gear angular velocity curve analysis result Since the given speed of the rack is Ï… = 10mm / s, the theoretical angular velocity of the gear is 0.555556 / s, Therefore, the standard deviation s of the angular velocity error of various methods can be calculated, and the calculation results are: 0.45026, 0.016556, 0.011326, 0.002098. It can be seen that the "Fan Cheng method" has the highest modeling accuracy, followed by the "analytical method" and "4 points." The law, because the SolidEdge engineering manual directly replaces the tooth profile curve with an arc, it is obviously the biggest error.
3 Conclusions The various three-dimensional modeling methods of gears are compared. The errors of various modeling methods and their influence on the accuracy of gear motion are explored and compared, and how to implement virtual gears in virtual environment. The accuracy check presented an attempt.
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