Analysis of Contact Part of Error Tooth Surface and Dynamic Performance Prediction for Involute Gear

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.


Introduction
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 [1].Due to gear manufacturing and installation errors, bearing clearance, load bearing parts' deformation, and so on, these factors will cause the gears to deviate from the ideal contact state.In order to accurately predict the dynamic performance of involute gears, the construction of error tooth surface is very important.
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. [2] assessed the large gear tooth profile deviation by using the NURBS (Nonuniform Rational B-Splines) surface.Zhang et al. [3] obtain digital tooth surface by using double three NURBS surfaces.And the tooth contact analysis of the digital tooth surface was carried according to the spatial meshing theory.
Li et al. [4] construct the digital real model of hypoid gear aiming at the wear of tooth surfaces by using non-feature block interpolation technique.Sun et al. [5] propose an algorithm for constructing G1 continuous cubic triangular Bezier Surfaces from triangular mesh surfaces.In these studies, the detection of contact parts is mainly based on the traditional roll test and manual identification method.However, there are few studies about the contact surface of the tooth surface by the gear bearing contact analysis, and also there are studies about the prediction of meshing performance of the error tooth surface by contact part.
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.

Mathematical Problems in Engineering
Based on the original research, an algorithm combining linear interpolation method with area weight interpolation method is proposed for constructing error tooth surface [6][7][8][9][10].Besides, the digital construction of error tooth surface of involute gears is realized on the basis of NURBS surface.In this paper, the traditional smoothing algorithm is improved to ensure accuracy and increase the smoothness of digital tooth surface.The feasibility of the method proposed in this paper is verified by comparing the examples.
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.

Synthesis and Reconstruction of Error Tooth Surface
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.

Synthesis of Error Tooth Surface.
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 1.
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  = 1, 2, . . .,  ( is the total number of contact lines) in the order of start contact to the end of the contact.The length of a contact line is divided into 19 equal parts, and the corresponding division points are numbered with  = 1, 2, . . ., 20 in the order of the tooth root to the tooth top.The tooth surfaces meshing with the large gears are divided by the same method.Therefore, the reference plane position on the different size of gears can be expressed by the combination of the serial number of the contact line  and the division number  of the contact line.The shape deviation of the gear can be expressed as data 20 × .
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  of a geometric contact line being the -axis and the contact line number  is the plane of the  axis, which is shown in Figure 2. The direction of the synthetic error coordinate system  is positive and the highest point is zero.The minimum of the synthetic error on the geometric contact line is defined as the ridge.The projection of the ridge to the synthetic error coordinate system - surface is called the ridge shape.

Tooth Surface Model
Based on NURBS.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 where  is the number of control vertices for ,  is the number of control vertices for V, and   is the control vertex of the surface.  is the weight factor of   . , is the th spline basis function along . , is the th spline basis function along V.Among them, the basis function recursive formula is (2)

Delaunay Triangulation of Discrete Data.
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 threedimensional discrete data points.Assuming that the scan line is ℓ  ,  = 1, 2, . . ., , the number of measurement points on the scan line is , where two adjacent scanning lines are ℓ  , ℓ +1 , 1 ≤  <  + 1 ≤ .The measurement points on each of the two scanning lines are ℓ  (),  = 1, 2, . . ., ; ℓ +1 (),  = 1, 2, . . ., .
As shown in Figure 3, the starting points for connecting two scan lines are ℓ  (1) and ℓ +1 (1); the end point are ℓ  () and ℓ +1 ().And find another scan line that is closest to the measurement point in the measurement points on ℓ  and ℓ +1 and connect, where the area between the scanning lines ℓ  and ℓ +1 is divided into  ( ≥  − 1) spatial regions.Finally, the triangular and spatial quadrilateral regions are retrieved.
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 4.

Linear Interpolation Method and Area Weight Interpolation Method.
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 5, assume that , , ,  are the measurement points and the tooth surfaces of these points have been determined, and the data of point  can be obtained.Firstly, calculate the 4 points around point for linear interpolation.The value of the point  surrounded by point , , ,  is calculated by using the area weights of the interpolation values of the four triangles , , , .If point  is close to a region, the area weighting factor is close to one.If point  is far from the area, the area weighting factor is close to zero.When the interpolation point coincides with the measurement point, the interpolation value is the measured value itself.
As shown in Figure 5, assumptions are as follows: the linear interpolation value where the triangles  pass through the interpolation point  is  1 ; the area of triangle  is   , The interpolation value of triangle  is  2 ; the area of triangle  is   .Thus the interpolation value is  1 .
where   is the area of quadrilateral .
On the other hand, the triangular  and 's interpolation values for point  are  3 ,  4 and the triangular  and 's areas are   ,   , respectively.Thus the interpolation value is  2 .
The final interpolation value  is (5)

The Basic Theory of Contact Parts.
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 6.

Analytical Method of
where The contact point path  According to Figure 7

Contact spot Contact ellipse
Thus Coordinate transformation matrix  2 →   is where [ The vector diameter of point  1 in the coordinate system   is where ( 2 ) 2 is the  2 vector in coordinate system  2 and ( 2 ) 2 is a vector of point  2 to point  in coordinate system  2 .( 1 )  is the  1 vector in coordinate system   .( 2 )  is a vector of point  2 to point  in coordinate system   .
( 1  2 )  is a vector of point  1 to point  2 in coordinate system   .The expression of vector ( 1  2 )  is where [ 2 , − 2 ] is the coordinate transformation matrix of  2 →   .
In coordinate system   , the projection for difference between  2 and  1 in the normal vector direction  1 at the point  on the tooth surface Σ 2 is the clearance in the direction of the engagement of the two tooth surfaces at the point  1 and point  2 .When the pinion turns through angle  1 , the gear turns through angle  2 : In order to ensure that the point  2 on the tooth surface Σ 2 of the large gear is still the point  1 on the tooth surface Σ 1 of the pinion in the direction of , the vectors  1 and  2 must satisfy the following relationships: The clearance between the two engaging tooth surfaces Σ 1 and Σ 2 in the direction of  at points  1 and point  2 is 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  1 = 0.01 mm.When Δ ≤ 0.01 mm, it is considered that the two tooth surfaces Σ 1 and Σ 2 are in contact at point  1 and point  2 .By dividing the tooth surface into the grid, the clearance in the direction of  the corresponding grid node is calculated, so is the tooth contact part.

Prediction of Dynamic Performance of Error Tooth Surface
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.

Gear Transmission
Error.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 8. 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 where In the above formula,   ( = 1, 2) is the torsional vibration displacement of gear and pinion.
( = 1, 2) is the inertia of the gear and pinion.
( = 1, 2) is the base radius of the gear and pinion.
( = 1, 2) is the integrated stiffness of the pair of th gears.
( = 1, 2) is the damping coefficient of the th gear.
() is the combined meshing stiffness of gear pair.
is meshing damping of gear pair.
( = 1, 2) is the error of the th gear.
( = 1, 2) is the external load moment on the driving and driven gears.
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.

Experiment Study
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 [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].This paper would not study the best trimming tooth surface; that is, select two measurements modification gears; it is short for group A and group B in the following paper, and its dynamic performance in analysis.The transmission mechanism in the gearbox is composed of three spur gears, and the structure of the gearbox is shown in Figure 9.The two gears with 35 teeth are the pinion gear, and the gears with the number of teeth are 42.The speed of the drive wheel is 1800 r/min and the rated power is 25 KW.The basic parameters of the gear are shown in Table 1.
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 10.Through the calculation of the points of the measurement area through the interpolation algorithm, tooth surface with the shape deviation on the different mesh nodes is obtained, which are shown in Figure 11.
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 12.
The theoretical calculation of the transmission error is shown in Figure 13.
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 14, The transmission error of the gear is shown in Figure 15.
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 16.
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.

Figure 1 :Figure 2 :
Figure 1: The geometrical contact line of the tooth surface during the meshing process.: the corresponding contact line number during the engagement process; : split points on the contact line.

Figure 6 :
Figure 6: Schematic diagram of contact parts.

Figure 7 :
Figure 7: Conversion diagram of tooth surface meshing coordinate system.

Figure 8 :
Figure 8: Torsional vibration model of gear pair.
A d d e n d u m A d d e n d u m Group B pinion (max.13.8 m) Group A pinion (max.18.4 m) e n d u m A d d e n d u m

Figure 10 :
Figure 10: The tooth surface with shape deviation in the measurement area.

Figure 11 :
Figure 11: The tooth surface with shape deviation in the measurement area after interpolation.
1  2 =  is the gear offset distance, and   coincides with the vertical axis of the two axes.The two gears are rotated  1 ,  2 angles, respectively.The coordinate systems are  1 ,  2 , respectively.A coordinate system   (;   ,   , ) is established at the contact point  of the tooth surface Σ 2 , where  is the unit normal vector at point  on the tooth surface Σ 2 .and   are two vertical unit vectors at the contact point  of the tooth face Σ 2 .Take any node  2 on the tooth face Σ 2 .And draw a straight line from this point  2 along a normal direction  to the tooth surface Σ.Then get the intersection  1 of the straight line and the tooth surface Σ 1 .Coordinate transformation matrix  2 →   is Contact Part of Involute Gear.As shown in Figure7,  0 ( = 1, 2) represents two coordinate systems whose origin is  1 ,  2 .  coincides with  axis of the gear coordinate system,

Table 1 :
Factors and their levels.