Aiming at the problem of constructing digital model of involute gear with error, the method of linear interpolation combined with area weight interpolation is proposed. Based on the non-feature discrete data block technique, the true tooth surface discrete data obtained by the coordinate measuring instrument is divided into blocks, and then the interpolation method is used to interpolate the nonmeasurement area to construct the real tooth surface with errors. The contact part and dynamic performance of the gear are predicted by using the constructed error tooth surface. The contact error of the tooth surface and the transmission error of the gear are verified by the test, and the reliability of the judgment result is judged by measuring the vibration in the direction of the gear meshing line. Compared with the example, this method not only reduces the computational complexity of the interpolation algorithm, but also improves the accuracy of the tooth interpolation data points and the smoothness of the error tooth surface.
It is well known that the contact situation of the gears in the gearbox directly affects the gearbox’s carrying capacity, vibration, and noise. The gear transmission error, the distribution area, and shape of tooth contact part have become an important index to measure the meshing performance of gear [
The reconstruction of complex surfaces is one of the most important research techniques in digital construction of tooth surface, and scholars at home and abroad have also done a lot of research about it. Lin et al. [
When the coordinate measurement is performed on the tooth surface, these methods are mainly applied to the smooth tooth surface because of the limited measurement data points of tooth surface. As for irregular tooth surfaces such as error tooth surfaces, the reason why the structural tooth surface can not reflect the true tooth surface is that most of the measurement data is separated from the structural tooth surface. In the above discrete data preprocessing technology, data segmentation is the bottleneck problem of constructing digital real tooth surface.
Based on the original research, an algorithm combining linear interpolation method with area weight interpolation method is proposed for constructing error tooth surface [
In contrast to constructing the digital model of single tooth surface, NURBS surface is used to construct the error tooth surface model of modified involute gears in this paper. Combined with the principle of involute gear meshing, the solution of the contact part of involute gear is obtained.
The sum of the deviation values of the normal direction at the corresponding contact point on the tooth surface is called the synthesis error. In this paper, a grinding method with high grinding precision is used to grind the gear. This method is through the grinding wheel to grind the gear, and then get the relevant tooth shape. When the standard gear is ideally engaged, the contact line will cover the entire tooth surface, which is called the geometric contact line. This line also serves as the reference for the contact position. The tooth surface error is defined as the size of the angle in the normal direction when the actual tooth surface corresponds to the reference plane.
In this paper, a base section of the gear is divided into 20 equal parts for gear contact analysis. The ideal contact part for the different size of gears is shown in Figure
The geometrical contact line of the tooth surface during the meshing process.
The ideal contact part for spur gears
The ideal contact part for helical gears
The interval between the contact lines is one twentieth of the rotation angle of a base section. The contact lines on the tooth surface of the large gear are numbered with
The sum of the deviation values of the normal direction at the corresponding contact point on the tooth surface is called the synthesis error. The curve of the 20 synthetic error values on a geometric contact line is called the synthetic error curve. In this paper, the synthetic error surface is expressed in a coordinate system with the split number
The creation of synthetic error surfaces.
NURBS surface is the only representation standard of product data model defined in STEP standard (Product data exchange standard), and it has been applied in real tooth surface simulation. However, in the process of real tooth surface reconstruction by using NURBS surface, the precision of the reconstructed tooth surface in the nongrid node is difficult to control due to the limited number of measurement data points. In order to ensure the smoothness of reconstructed tooth surface, this paper proposes a linear interpolation method combined with area weight interpolation method. This method is used to estimate the tooth surface data points which are outside the measurement area to realize the construction of the error tooth surface and to ensure the accuracy of the reconstructed tooth surface to the utmost extent.
NURBS surfaces are tensor product forms based on NURBS curves, and the double three NURBS surface can be expressed as
In this paper, the scanning method is used to obtain the tooth surface discrete point data. In order to describe the real tooth surface situation more intuitively, this paper deals with the discrete data points by means of Delaunay triangulation principle because the topological relations between the discrete data points are complex. The traditional Delaunay triangulation method often projects the three-dimensional discrete data points onto the two-dimensional plane and then triangles the projection points. However, the triangulation of this method cannot truly reflect its spatial angle, which will affect the quality of the section. On the basis of these methods, this paper presents a method of direct triangulation of three-dimensional discrete data points.
Assuming that the scan line is
As shown in Figure
Preliminary Delaunay triangulation of adjacent tooth surface scan lines.
For the spatial quadrilateral region, the minimum internal angle maximum criterion is used to complete the Delaunay triangulation of the discrete data points in this paper, respectively, as shown in Figure
Polygon area triangulation.
In order to describe the error of the tooth surface quickly and accurately, it is necessary to block the discrete data points of the tooth surface in the construction of the digital real tooth surface. And use the interpolation method to accurately describe the error surface to reduce the amount of tooth interpolation calculation. The traditional data block method is often based on the geometric characteristics of the part, which does not apply to free surfaces that describe the error tooth surface. In this paper, an interpolation method based on linear interpolation combined with area weight interpolation is proposed to accurately identify tooth surface errors.
The error tooth surface shape is calculated by using the area weight method according to the following interpolation method. In Figure
Four-point interpolation algorithm.
As shown in Figure
On the other hand, the triangular
When the tooth surface is transferring load, the contact tooth surface will be elastically deformed so that the area around the contact part will make contact. As the teeth continue meshing, the area left of the tooth surface is the actual tooth contact area, as shown in Figure
Schematic diagram of contact parts.
As shown in Figure
Conversion diagram of tooth surface meshing coordinate system.
Coordinate transformation matrix
According to Figure
The vector diameter of point
In coordinate system
In order to ensure that the point
The clearance between the two engaging tooth surfaces
According to GB/Z18620.4-2008, the thickness of the detection coating for contact part is set to 0.01 mm. The light load contact part exists in the area of contact point separation amount
The main causes of gear vibration are gear manufacturing error, installation error, gear time varying meshing stiffness and load gear, shaft, gearbox, and other deformation caused by the gear position error. The gear contact part and transmission error is the visualization of gear meshing performance. In the example, the reliability of the prediction results is judged by measuring the vibration of the gears in the direction of the meshing line.
The factors that affect the dynamic excitation of the gear are not only the structural form of the gear itself, the geometric characteristics, and the error, but also the other components in the system. The cylindrical gear system can be simplified as a torsional vibration system for the gear pair when the stiffness of the support member such as the drive shaft, bearing, and housing is large. The dynamics model is shown in Figure
Torsional vibration model of gear pair.
Suppose the gear pair of coincidence between one and two. According to the mechanical dynamics theory and the mechanical vibration theory, the torsional vibration equation of a pair of gear pair is
When the dynamic term such as inertia force and damping force term is ignored, the static transmission error is
In the quasi-static transmission error test of the gear pair, it is necessary to know exactly the rotation angle of the driving and driven gears in the transmission process.
This paper chooses the gears of a certain type of solar generator gearbox to carry on the research. According to the support stiffness of the gear and other factors, refer to optimizing the gear modification [
The transmission mechanism in the gearbox is composed of three spur gears, and the structure of the gearbox is shown in Figure
Factors and their levels.
Parameters | Letter symbols | |
---|---|---|
Gear modulus |
|
2.5 |
Pressure angle |
|
20 |
Addendum |
|
2.5 |
Dedendum |
|
3.125 |
Teeth |
|
35/42 |
Gear distribution diagram.
Using the measurement method of ZEISS coordinate measuring instrument, the topological measurement of tooth width and tooth height is taken. According to the actual measurement data, the surface of the tooth surface deviation is directly constructed, and the shape deviation surface in the measurement area of the two sets of gear driving wheels is shown in Figure
The tooth surface with shape deviation in the measurement area.
The tooth surface with shape deviation in the measurement area after interpolation.
On the basis of the analysis of the contact part of the error tooth surface after interpolation, the contact surface of the tooth surface is shown in Figure
Tooth surface contact parts (theoretical calculation).
The theoretical calculation of the transmission error is shown in Figure
Gear drive error (theoretical calculation).
A gear loading test is performed for gears whose tooth surfaces are coated with steel red after steel red is dried, and the gear contact parts are shown in Figure
Contact part of test gear.
Transmission error of test gear.
In order to test the accuracy of the structural error surface, the vibration measuring instrument is used to measure the time-domain diagram of the vibration in the direction of the gear meshing line, and its FFT is shown in Figure
Gearbox vibration spectrum.
It can be seen from the figure that the predictive gear contact part and gear transmission error is basically the same as test results. The vibration measurement of the gearbox along the gear meshing line shows that the vibration condition of the gearbox is basically the same as the transmission error of the gear. This shows that the model of error tooth surface is accurate.
An algorithm combining linear interpolation with area weight interpolation is proposed to realize the construction of error tooth surface. This method not only reduces the computational complexity of the tooth surface interpolation algorithm, but also improves the efficiency of the digital tooth surface construction. The contact part and the transmission error of the gear are analyzed by using the constructed error tooth surface, which provides a new idea for the prediction of the dynamic performance of the gear. Through the gear loading test, the contact part and gear transmission error of the gear are obtained. The analysis result and the test result are basically the same, which shows the accuracy of the constructed error tooth surface. The construction of the error tooth surface in this paper lays the foundation for further improving the grinding accuracy of the tooth surface and the grinding of the high performance gear.
Control vertex of the surface
Weight factor of
External load moment on the driving and driven gears
Combined meshing stiffness of gear pair
Meshing damping of gear pair
the error of the
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors would like to thank Shanghai Science and Technology Commission (Science and Technology Support Program for the People’s Livelihood) for the process adaptability test and evaluation and application on NC equipment manufacturing for aerospace complex structural parts, Grant no. 15110502300.