In the process of gear operation, due to the different number of teeth involved in meshing at the same time, the meshing stiffness of gear teeth changes periodically, which leads to the periodic change in the elastic deformation of gear teeth. In addition, the influence of machining error during manufacture and center distance deviation during installation may cause meshing interference and the partial load of gear teeth, resulting in vibration and noise. As an effective way to solve the above problems, gear tooth modification has been widely used in high speed and heavy load gear transmission. Therefore, it is necessary to study the parametric modeling method of a gear pair with tooth modification in order to accurately analyze the influence of gear tooth modification on the dynamic meshing performance of gear.

In this section, based on the space meshing theory, the tooth surface equation of the modified gear is deduced. On this basis, the parametric modeling program of the modified helical gear pair is compiled to automatically generate the accurate tooth surface of the helical gear with tooth profile modification, and then the solid model of the double-helical gear pair with tooth profile modification is established.

First, the two sides of the rack cutter are modified into a parabola shape, and the shape of the tooth profile modification section is obtained according to the coordinate transformation of the gear meshing principle, and then the development process of the helical gear is slightly modified. The tooth modification parameter is introduced to make the tooth surface feed the parabola in the direction of tooth height in the process of the spiral development. It can be imagined that the helical gear is machined by a grinding wheel with tooth profile modification. In the process of the spiral feed, the feed in the direction of tooth depth is distributed according to the parabola.

The gear is divided into two tooth surfaces, namely g tooth surface and h tooth surface. As shown in Fig. 1, Sg(ogxgyg) is the modification coordinate system of the left tooth surface, and Sh(ohxhyh) is the modification coordinate system of the right tooth surface. Where α is the pressure angle of indexing circle, S0 is one-quarter of the base segment of the normal plane, A is the quadratic coefficient of the modified parabola, u is the independent variable, and u0 is the axis of symmetry of the modified parabola.

The point on the g-modified tooth surface is represented in the g coordinate system as follows: 1Rg=Au-u02u01. The point on the h-modified tooth surface is represented in the h coordinate system as follows: 2Rh=-Au-u02;u;0;1.

The coordinate transformation matrices from the tooth surface to rack normal surface includes Mc_g and Mc_h. Mc_g is the transformation matrix of the g tooth surface to rack normal surface, and Mc_h is the transformation matrix of the h tooth surface to rack normal surface. 3Mc_g=cos⁡αsin⁡α0-s0-sin⁡αcos⁡α0000000001.4Mc_h=cos⁡α-sin⁡α0s0sin⁡αcos⁡α0000100001. Sc(oxcyczc) is the normal section coordinate system of a rack, and Sm(oxmymzm) is the rack end coordinate system. The coordinate conversion between Sc and Sm is shown in Fig. 2, where β is the helix angle.

The coordinate transformation matrix between Sc and Sm is as follows: 5Mm_c=cos⁡β0sin⁡βs00100-sin⁡β0cos⁡β00001. S1(ox1y1) is a dynamic coordinate system fixed to the end face of the helical gear, and Sn(oxnyn) is a fixed coordinate system connected to the end face of the helical gear. rp is the radius of the gear pitch circle, and φ is the angle of rotation of the coordinate system S1 about the Zn axis.

The coordinate transformation matrix between Sm and Sn is as follows: 6Mn_m=100-rp×φ010rp00100001. The coordinate transformation matrix between Sn and S1 is as follows: 7M1_n=cos⁡φsin⁡φ00-sin⁡φcos⁡φ0000100001. S2(ox2y2z2) is the coordinate system moving along the spiral. The coordinate conversion between S1 and S2 is shown in Fig. 4, where E=B×ε2 and k=rb/tan⁡β.

The coordinate transformation matrix between S1 and S2 is as follows: 8M2_1=1000010-B×ε2001-k×ε0001.

The coordinate transformation matrix between S2 and S3 is as follows: 9M3_2=cos⁡ε-sin⁡ε00sin⁡εcos⁡ε0000100001.

Finally, the equation of a two-way modified tooth surface is obtained as follows: 10R3g=M3_2∗M2_1∗M1_n∗Mn_m∗Mm_c∗Mc_g∗Rg,11R3h=M3_2∗M2_1∗M1_n∗Mn_m∗Mm_c∗Mc_h∗Rh. To solve the point coordinates of the working tooth surface, there are three unknowns, ε, φ, and u, which need three equations, respectively, as follows: 12(1)Radius direction:x32+y32=R2.13(2)Teeth width direction:z3=z.14(3)Meshing equation:f=(a(1:3)×b(1:3)⋅c(1:3)).

The derivation of R3 to u is a, the derivation of R3 to ε is b, the derivation of R3 to φ is c, and the meshing equation is a mixed product of the three vectors.

The modified tooth surface point cloud is obtained by programming the calculation, as shown in Fig. 6. The red arrow indicates the direction of tooth width, the white arrow indicates the direction of tooth thickness, and the blue arrow indicates the direction of tooth height.

The point set on the surface is imported into SolidWorks software, and the helical gear tooth surface with tooth modification is created. Then, the tooth surface is stitched to generate the solid, and the accurate three-dimensional solid model of the modified helical gear pair is obtained, as shown in Fig. 7.

In the process of gear transmission, with the constant change in the meshing position, the stiffness and bearing position of gear teeth are constantly changing along the tooth direction, and the distribution of the load between teeth is also changing. Accurate solution of load distribution between teeth and load distribution on tooth surface is the basis of gear tooth modification. Most of the existing calculation methods are based on some assumed contact area shape, according to Hertz contact theory, which deviates from the actual contact situation. The finite element method can solve the contact nonlinear problem well, and the instantaneous contact area shape and the pressure distribution of cylindrical gear are very typical contact nonlinear problems, so the finite element software can be used to analyze and calculate the contact state of gear.

In the process of finite element calculation, the quality and quantity of meshing have a great influence on the calculation results and calculation cost. In order to improve the quality of the contact surface mesh and reduce the overall mesh quantity in contact analysis, this paper will cut the gear model. As shown in Fig. 8, the herringbone gear is cut into two helical gears, and each tooth of the helical gear is cut into four parts to facilitate subsequent calculations.

In this paper, the three-dimensional model of the herringbone gear system is imported into the finite element software ABAQUS. The solid segmentation technology is used to divide the model by a hexahedral mesh; the mesh of the contact area of the encrypted tooth surface and the sparse mesh are used in the non-gear contact area. The mesh size of the contact tooth surface is kept at about 50 % of the contact half bandwidth, and the transition mesh size is used to divide the mesh. The mesh along the tooth width direction is 1 mm, and the rest is divided by automatic mesh division. The rationality of this meshing method has been proved in other research (Zhu, 2009; Gonzalez, 2012).

The finite element model of the herringbone gear system established in this paper is shown in Fig. 9, and the design data of the herringbone gear drive represented in Fig. 9 are shown in Table 1. The modeling process is shown in Fig. 10.

Based on the difference between the power input point and the power output point in the herringbone gear transmission system, the power flow direction of the herringbone gear transmission system is defined as follows: the power input point and the power output point in the herringbone gear transmission system are located on the same side of the gear, and it is called the U-shaped power flow direction; in the herringbone gear transmission system, the power input point and the power output point are located on both sides of the gear and are called the Z-shaped power flow direction. According to the above definition of the power flow direction of herringbone gear transmission, two schematic diagrams of power flow direction are drawn. As shown in Fig. 12, point A is the point of power input, point D is the point of power output, and the path of the power transfer is A-B-C-D.

Through the calculation, the tooth surface contact stress cloud diagram of the herringbone gear system with two different power flow directions is as shown in Fig. 13.

It can be seen from Fig. 13 that the maximum contact stress of the U-shaped power flow is 1365 MPa, and that of the Z-shaped power flow is 1307 MPa. The maximum contact stress of the U-shaped power flow is obviously higher than that of the Z-type power flow.

In order to better study the distribution of contact stress on tooth surface, the contact stress distribution map of the herringbone gear system is drawn on the basis of stress cloud diagram, as shown in Fig. 14.

In Fig. 14, the contact stress at the tooth vertex of the follower is the largest, followed by the stress at the tooth root, and there is an obvious stress concentration at the tooth root and tooth vertex of the follower. The maximum contact stress of the two different power flows to the left and right tooth surfaces is quite different, and the right tooth, which is the tooth surface close to the power input surface, is larger than the left tooth surface.

In order to explore the variation law of contact stress value of different power flow direction in one meshing cycle, this paper calculates the maximum contact stress of two tooth surfaces of each power flow direction and the difference in left and right tooth surfaces. The contact stresses of different power flow directions on the tooth surface are enumerated, as shown in Table 2.

In order to study the change in the maximum contact stress of the gear in one meshing cycle, the difference curves of the maximum contact stress on the left and right tooth surfaces with two different power flows are drawn, as shown in Fig. 15.

It can be clearly seen from Table 2 and Fig. 15 that there is a partial load on the tooth surface of the herringbone gear system with the U-shaped and the Z-shaped power flow, and the partial load of the U-shaped power flow is more serious than that of the Z-shaped power flow.

As an important supporting component of the gear transmission system, the stiffness change in the gear-bearing shaft has a great influence on the bearing contact state of the tooth surface of the whole system. In order to study the specific influence of the stiffness of the supporting shaft segment on the contact state of the tooth surface, the bearing contact state of the herringbone gear system with five different stiffness is calculated by changing the elastic modulus of the shaft section to change the stiffness of the shaft section. The parameters of the gear model are shown in Table 1; the power flow direction is the Z-shaped power flow direction, and the elastic moduli of five different shaft segments are shown in Table 3.

In order to clearly see the contact stress distribution on each contact line, this paper draws the contact stress distribution map of the left and right tooth surfaces with different shaft stiffness according to the contact stress value.

Through a comprehensive comparison of the contact stress distribution maps of the tooth surfaces with five different shaft stiffness, it can be concluded that, with the increase in the shaft stiffness, the maximum contact stress of the tooth surface decreases, and the difference between the left and right tooth surfaces gradually decreases. When the stiffness of the shaft section is large, the phenomenon of partial load on the left and right tooth surface disappears. When the stiffness of the shaft section is small, the supporting deformation of the shaft section is larger, and the stress concentration mainly appears at the tooth root of the driven wheel.

In this section, the influence of different tooth profile modifications on tooth surface bearing contact is calculated in the finite element analysis software in order to explore the best modification amount of herringbone gear system.

The gear teeth will have certain elastic deformation due to the action of the load, including the contact deformation, bending deformation, shear deformation and tooth root deformation. The deformation is related to the load on the gear teeth and the meshing stiffness of the gear teeth, which can be approximated by the following Eq. (15): 15δa=Wt/cγ. δa is the elastic deformation of tooth profile (micrometer; hereafter µm), Wt is the load per tooth width (Newtons per millimeter; hereafter N/mm), and cγ is the meshing stiffness per tooth width (Newtons per millimeter micrometer; hereafter N/mm µm).

According to the above equation, the tooth profile deformation of the model in this chapter is calculated to be 28.5 µm, which is the theoretical tooth profile modification.

According to the coordinate comparison of tooth surface before and after modification, the tooth profile modification curve is drawn. On the basis of the amount of theoretical practice, two cases which are larger than the amount of theoretical modification and smaller than the amount of theoretical modification are calculated respectively, and the curves of three different tooth profile modifications are shown in the Fig. 21.

The maximum amount of modification is designed according to the calculated tooth profile deformation, and the modified model is calculated; then, the contact stress cloud diagram of the modified tooth surface is obtained, as shown in Fig. 22.

It can be seen from Figs. 13 and 22 that the contact stress distribution of the left and right tooth surfaces after tooth profile modification is still symmetrical, and the maximum contact stress is obviously lower than that without profile modification.

The different power flow direction will make the partial load degree of the tooth surface load contact of herringbone gear system different. Considering this difference, the influence of different tooth profile modifications on the tooth surface load contact with the different power flow direction is studied. According to the calculation results, the contact stress distribution of tooth surface under the action of different modification values of two different power directions are drawn in Figs. 23 and 24.

As can be seen from Fig. 23, when the maximum amount of modification is more than 30 % of the theoretical amount of modification, the stress concentration at the root and top of the tooth can be completely eliminated, and the contact stress distribution on the tooth surface is parabolic; when the maximum amount of modification is less than 35 % of the theoretical amount, the stress concentration phenomenon still exists, but the maximum contact stress of the whole tooth surface decreases, and the contact stress distribution of the tooth surface tends to be rectangular. Comparing Figs. 23 and 24, we can see that, when the maximum tooth profile modification is greater than or equal to the theoretical modification, the tooth surface contact stress distribution of the two different power flow directions is the same, but when the maximum modification amount is less than the theoretical modification amount, then the tooth surface stress distribution of the U-type power flow direction is more uniform than that of the Z-type power flow direction.

Through the research, this paper finds that the stiffness of different shaft segments has a great influence on the load-bearing contact state of the tooth surface, so the influence of different tooth profile modification on the tooth surface contact state, according to the stiffness of different shaft segments, will be further analyzed and then determine the appropriate amount of tooth profile modification for different shaft stiffness. In order to complete the above research, aiming at the stiffness of shaft nos. 2 and 4, which are shown in Table 3, the tooth surface contact stress distributions corresponding to three different tooth profile modifications are calculated, and the tooth surface contact stress distributions of different shaft stiffness under different tooth profile modifications are drawn, as shown in Figs. 25 and 26.

It can be seen from Fig. 25 that when the maximum amount of modification is equal to or less than the theoretical amount of modification, the stress concentration at the root and top of the tooth still exists, and the smaller the amount of modifications, the more obvious the stress concentration at the root and top of the tooth is; when the amount of modification is larger, the contact stress of the tooth surface is also relatively small. In Fig. 26, three different tooth profile modifications have little effect on the contact stress distribution of the tooth surface, and the stress concentration of the tooth root and tooth top disappears.