Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 6143054, 11 pages
https://doi.org/10.1155/2017/6143054

Research Article

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

Xinrong Liu and Zhonghou Wang

School of Mechanical Engineering, University of Shanghai for Science and Technology, No. 516, Jungong Road, Yangpu, Shanghai 200093, China

Correspondence should be addressed to Zhonghou Wang; wang_zhonghou18@163.com

Received 11 August 2017; Revised 12 October 2017; Accepted 24 October 2017; Published 12 December 2017

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2017 Xinrong Liu and Zhonghou Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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.

Based on the original research, an algorithm combining linear interpolation method with area weight interpolation method is proposed for constructing error tooth surface [6–10]. 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.

2. 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.

2.1. 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.

(a) The ideal contact part for spur gears

(b) The ideal contact part for helical gears

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.

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 ( 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 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 .

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.

Figure 2: The creation of synthetic error surfaces.

2.2. 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 , 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 . Among them, the basis function recursive formula is

2.3. 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 three-dimensional discrete data points.

Assuming that the scan line is , , the number of measurement points on the scan line is , where two adjacent scanning lines are . The measurement points on each of the two scanning lines are , .

As shown in Figure 3, the starting points for connecting two scan lines are and ; the end point are and . And find another scan line that is closest to the measurement point in the measurement points on and and connect, where the area between the scanning lines and is divided into spatial regions. Finally, the triangular and spatial quadrilateral regions are retrieved.

Figure 3: 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 4.

Figure 4: Polygon area triangulation.

2.4. 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.

Figure 5: Four-point interpolation algorithm.

As shown in Figure 5, assumptions are as follows: the linear interpolation value where the triangles pass through the interpolation point is ; the area of triangle is , The interpolation value of triangle is ; the area of triangle is . Thus the interpolation value is .

where is the area of quadrilateral .

On the other hand, the triangular and ’s interpolation values for point are and the triangular and ’s areas are , respectively. Thus the interpolation value is .

The final interpolation value is

3. The Simulation of Contact Pattern

3.1. 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.

Figure 6: Schematic diagram of contact parts.

3.2. Analytical Method of Contact Part of Involute Gear

As shown in Figure 7, represents two coordinate systems whose origin is . coincides with axis of the gear coordinate system, is the gear offset distance, and coincides with the vertical axis of the two axes. The two gears are rotated angles, respectively. The coordinate systems are , respectively. A coordinate system is established at the contact point of the tooth surface , where is the unit normal vector at point on the tooth surface . and are two vertical unit vectors at the contact point of the tooth face . Take any node on the tooth face . And draw a straight line from this point along a normal direction to the tooth surface . Then get the intersection of the straight line and the tooth surface .

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

Coordinate transformation matrix is

where , , , , , , , , .

According to Figure 7

Thus Coordinate transformation matrix is

where is the coordinate transformation matrix of ; is the coordinate transformation matrix of .

is the coordinate transformation matrix of .

The vector diameter of point in the coordinate system is

The vector diameter of point in the coordinate system is

where is the vector in coordinate system and is a vector of point to point in coordinate system . is the vector in coordinate system . is a vector of point to point in coordinate system . is a vector of point to point in coordinate system . The expression of vector is

where is the coordinate transformation matrix of .

In coordinate system , the projection for difference between and in the normal vector direction at the point on the tooth surface is the clearance in the direction of the engagement of the two tooth surfaces at the point and point . When the pinion turns through angle , the gear turns through angle :

In order to ensure that the point on the tooth surface of the large gear is still the point on the tooth surface of the pinion in the direction of , the vectors and must satisfy the following relationships:

The clearance between the two engaging tooth surfaces and in the direction of at points and point 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  mm. When  mm, it is considered that the two tooth surfaces and are in contact at point and point . 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.

4. 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.

4.1. 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.

Figure 8: 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

where

In the above formula,

  is the torsional vibration displacement of gear and pinion.  is the inertia of the gear and pinion.  is the base radius of the gear and pinion.  is the integrated stiffness of the pair of th gears.  is the damping coefficient of the th gear.  is the combined meshing stiffness of gear pair.  is meshing damping of gear pair.  is the error of the th gear.  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.

5. 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 [11–27]. 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.

Table 1: Factors and their levels.

Parameters
Letter symbols

Gear modulus

2.5

Pressure angle

20

Addendum

2.5

Dedendum

3.125

Teeth

35/42

Figure 9: 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 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.

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

Figure 11: 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 12.

Figure 12: Tooth surface contact parts (theoretical calculation).

The theoretical calculation of the transmission error is shown in Figure 13.

Figure 13: 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 14, The transmission error of the gear is shown in Figure 15.

Figure 14: Contact part of test gear.

Figure 15: 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 16.

Figure 16: 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.

6. Conclusions

(1)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.(2)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.(3)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.(4)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.

Nomenclature

: | Control vertex of the surface
: | Weight factor of
: | th spline basis function along
: | th spline basis function along
: | External load moment on the driving and driven gears
: | Combined meshing stiffness of gear pair
: | Meshing damping of gear pair
: | the error of the th gear.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

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.

References

S. Zhou, G. Song, Z. Ren, and B. Wen, “Nonlinear dynamic analysis of coupled gear-rotor-bearing system with the effect of internal and external excitations,” Chinese Journal of Mechanical Engineering, vol. 29, no. 2, pp. 281–292, 2016.

J. Lin, Z. Shi, C. Pan, and B. Zhang, “Profile error evaluation of large gears based on NURBS surface fitting,” Yi Qi Yi Biao Xue Bao/Chinese Journal of Scientific Instrument, vol. 37, no. 3, pp. 533–539, 2016.

J.-H. Zhang, Z.-D. Fang, and C. Wang, “Digital simulation of spiral bevel gears' real tooth surfaces based on non-uniform rational B-spline,” Hangkong Dongli Xuebao/Journal of Aerospace Power, vol. 24, no. 7, pp. 1672–1676, 2009.

G. Li, Z. Wang, Z. Geng, and W. Zhu, “Modeling approach of digital real tooth surfaces of hypoid gears based on non-geometric-feature segmentation and interpolation algorithm,” Jixie Gongcheng Xuebao/Journal of Mechanical Engineering, vol. 51, no. 7, pp. 77–84, 2015.

D. Sun, X. Li, Y. Li, and Z. Tian, “G1-continuous algorithms of triangular mesh surface,” Jixie Gongcheng Xuebao/Journal of Mechanical Engineering, vol. 46, no. 15, pp. 125–129, 2010.

Z. H. Wang, Simulation of hypoid gears dynamic performance, 2013.

Z. Wang, A. Kubo, S. Asano, S. Kato, and T. Nonaka, “Simulation of deterioration of hypoid gear performance due to tooth flank wear (2nd report. model for tooth flank wear and the change of gear performance resulted),” Transactions of the Japan Society of Mechanical Engineers, vol. 66, no. 647, pp. 2363–2370, 2000.

G. Li, Z. Wang, and A. Kubo, “The modeling approach of digital real tooth surfaces of hypoid gears based on non-geometric-feature segmentation and interpolation algorithm,” International Journal of Precision Engineering and Manufacturing, vol. 17, no. 3, pp. 281–292, 2016.

Z. Wang, A. Kubo, and T. Nonaka, “Prediction of performance of hypoid gears by observation of tooth contact pattern (1st report, algorithm for predicting composite error surface),” TRansactions of the Japan Society of Mechanical Engineers, vol. 64, no. 624, pp. 3103–3111, 1998.

Z. Wang, A. Kubo, S. Asano, S. Kato, and T. Nonaka, “Simulation of deterioration of hypoid gear performance due to tooth flank wear. 1st report, data processing of measured tooth flank form deviation for performance simulation,” Transactions of the Japan Society of Mechanical Engineers, vol. 66, no. 647, pp. 2354–2362, 2000.

S. Chen, J. Tang, Z. Wang, and Z. Hu, “Effect of modification on dynamic characteristics of gear transmissions system,” Jixie Gongcheng Xuebao/Journal of Mechanical Engineering, vol. 50, no. 13, pp. 59–65, 2014.

Y.-Z. Sun, R.-P. Zhu, and H.-Y. Bao, “Calculation of mean friction coefficient in scuffing strength for gear drive with meshing beyond pitch point,” Hangkong Dongli Xuebao/Journal of Aerospace Power, vol. 28, no. 9, pp. 2155–2160, 2013.

L. Chang, Y.-R. Jeng, and P.-Y. Huang, “Modeling and analysis of the meshing losses of involute spur gears in high-speed and high-load conditions,” Journal of Tribology, vol. 135, no. 1, Article ID 011504, 2013.

M. Vaishya and R. Singh, “Strategies for modeling friction in gear dynamics,” Journal of Mechanical Design, vol. 125, no. 2, pp. 383–393, 2003.

S. Li and A. Kahraman, “A tribo-dynamic model of a spur gear pair,” Journal of Sound and Vibration, vol. 332, no. 20, pp. 4963–4978, 2013.

S. Li, “Effects of misalignment error, tooth modifications and transmitted torque on tooth engagements of a pair of spur gears,” Mechanism and Machine Theory, vol. 83, pp. 125–136, 2015.

T. J. Lisle, B. A. Shaw, and R. C. Frazer, “External spur gear root bending stress: a comparison of ISO 6336:2006, AGMA 2101-D04, ANSYS finite element analysis and strain gauge techniques,” Mechanism and Machine Theory, vol. 111, pp. 1–9, 2017.

J. Tang, H. Chen, and Q. Wang, “Dynamics model of gear transmission with combined friction and clearance by using bond graph method,” Jixie Gongcheng Xuebao/Journal of Mechanical Engineering, vol. 47, no. 9, pp. 53–59, 2011.

J. I. Pedrero, M. Pleguezuelos, and M. Muñoz, “Critical stress and load conditions for pitting calculations of involute spur and helical gear teeth,” Mechanism and Machine Theory, vol. 46, no. 4, pp. 425–437, 2011.

M. Miladi Chaabane, R. Plateaux, J.-Y. Choley, C. Karra, A. Riviere, and M. Haddar, “Topological modeling of a one stage spur gear transmission,” Chinese Journal of Mechanical Engineering, vol. 27, no. 5, pp. 900–908, 2014.

M. H. Jia, C. L. Wang, and B. Ren, “Manufacturing error effects on mechanical properties and dynamic characteristics of rotor parts under high acceleration,” Chinese Journal of Mechanical Engineering, vol. 30, no. 4, pp. 920–932, 2017.

V. V. Simon, “Micro tooth surface topography of face-milled hypoid gears,” Mechanism and Machine Theory, vol. 104, pp. 370–381, 2016.

V. V. Simon, “Optimal machine-tool settings for the manufacture of face-hobbed spiral bevel gears,” Journal of Mechanical Design, vol. 136, no. 8, Article ID 081004, 2014.

H. Ding, J. Tang, Y. Zhou, J. Zhong, and G. Wan, “A multi-objective correction of machine settings considering loaded tooth contact performance in spiral bevel gears by nonlinear interval number optimization,” Mechanism and Machine Theory, vol. 113, pp. 85–108, 2017.

H. Ding, J. Tang, and J. Zhong, “An accurate model of high-performance manufacturing spiral bevel and hypoid gears based on machine setting modification,” Journal of Manufacturing Systems, vol. 41, pp. 111–119, 2016.

A. Artoni, A. Bracci, M. Gabiccini, and M. Guiggiani, “Optimization of the loaded contact pattern in hypoid gears by automatic topography modification,” Journal of Mechanical Design, vol. 131, no. 1, pp. 0110081–0110089, 2009.

A. Artoni and M. Guiggiani, “Revisiting generation and meshing properties of beveloid gears,” Journal of Mechanical Design, vol. 139, no. 9, pp. 1–9, 2017.