Transmission Design of Bevel Gear with Arbitrary Crossed Axes and Its Application in the Knotter

: The bevel gear transmission with crossed axes is widely used, but there are the difficulty of tooth profile calculation and modeling in the design and manufacturing process. This paper analyzes firstly the parameter solving equation of the bevel gear pair with arbitrary crossed axes. Based on the meshing principle of the bevel gear pair and numerical analysis method, the meshing equations of the bevel gear pair with crossed axes are derived, which provide the calculation models for the design of spatial bevel gear tooth surface. Taking the bevel gear pair with crossed axes in two kinds of the knotter as two design examples, the geometric parameters of the bevel gear pair in the knotter are solved, and the mathematical models of the tooth surface are programmed and calculated by using MATLAB. Through the graphic display of the tooth surface point set under MATLAB software and the 3D modeling function of Pro/Engineering software, the accurate 3D models of the bevel gear tooth surface are established. The meshing transmission simulations of the established bevel gear pair are respectively carried out by using ADAMS, and their physical prototype and transmission tests are also implemented. The test results showed that the transmission of the designed bevel gear pair is accurate and stable, which proves the correctness of the derived calculation model of the tooth surface of the bevel gear with crossed axes.


Introduction
It is generally known that gear transmission has an extremely important role in the equipment manufacturing of industry and agriculture. With the continuous progress of gear technology research, the requirements of higher performance are also directly put forward, such as higher transmission efficiency, transmission accuracy and reliability [1]. As an important form of gear transmission, the bevel gear with crossed axes is widely used in industrial and agricultural fields. Its design theory and modeling method have always been a hot topic in the field of gear transmission [2]. The research on the parameter calculation model and accurate three-dimensional modeling of the bevel gear with arbitrary crossed axes has great significance to improve of the bevel gear transmission.
At present, the researches of gear transmission mainly focus on gear life assessment techniques [3][4][5][6], theoretical research on gear reinforcement [7][8][9][10], computer-aided design method of gear [11][12][13] and gear fault detection research [14][15][16][17][18]. There are few theoretical studies on gear mathematical models based on numerical analysis methods. The relationship among tooth pressure, helix angle and tool system parameter was established according to the accurate geometry of tool system in hyperbolic gear forming, and a new parameterization method of gear tooth surface was introduced [19]. Based on the mathematical modeling method of helical bevel gear, the deviation between the actual tooth surface shape and the theoretical tooth surface shape was expressed by polynomial expression, and a method of *Correspondence: yinjianjun@ujs.edu.cn School of Agricultural Engineering, Jiangsu University, Zhenjiang 212013, China Full list of author information is available at the end of the article remanufacturing the existing helical bevel gear pinion by using NC machining center was proposed [20]. The concept of auxiliary surface of Camus is extended to the design case of involute gear with crossed axes [21]. The geometric characteristics of tooth profile of crossed shaft gear set are studied by using pressure angle function, meshing equation and continuity condition constraints [22]. By studying the surface modification method of helical bevel gear, an accurate computer design method of spur bevel gear and helical bevel gear is proposed [23]. According to the requirements of helical gear installation position, a pre size method of helical gear transmission data based on application requirements is proposed [24]. It can be seen that research on gear transmission design is mostly concentrated in the field of cylindrical gear, and there is relatively little knowledge about bevel gear transmission design. The derivation of meshing equation of bevel gear tooth surface based on bevel gear meshing principle and accurate modeling of bevel gear pair are a challenge faced by the machinery industry [25].
This study derives the mathematical model of tooth profile of bevel gear with arbitrary crossed axes and proposes an accurate 3D modeling method of tooth profile of bevel gear. The organization of this paper is as follows: Section 2 presents a brief summary on parameter calculation model of bevel gear pair with crossed axes; The establishment of mathematical model for tooth surface of bevel gear pair with crossed axes is introduced in Section 3; Section 4 gives the two calculation and design examples of bevel gear pair in the knotter; and Section 5 has concluding remarks.

Parameter Calculation Model of Bevel Gear Pair with Crossed Axes
The layout form of the bevel gear pair studied in this paper is shown in Figure 1 [26,27]. The axis of bevel gear 2 is located at the rear and fixed, the big ends of the two bevel gears are on the opposite side, and the vertex of the cone of bevel gear 1 points to the right side of the axis of gear 2.

Figure1 Layout form of the bevel gear pair
The installation parameter method is used to solve the geometric parameters of the bevel gear pair [28]. The known parameters include gear normal modulus mn, normal pressure angle αn, helical angle β1, number of teeth z1 and z2, pitch cone angle δ1, crossed axis angle Σ, wheel base a, and installation distance of gear 1 l1. Firstly, the tooth profile angle coefficient of the bevel gear 1 t  is preset, and the pitch radius ' 1 r of the bevel gear is calculated by Eq. (1).
The intersection angle δ ' 1 between the imaginary common gear rack indexing plane and the axis of bevel gear 1 can be calculated by Eq. (4).
To the imaginary common gear rack, the normal pressure angle α ' n, the inclination angle β ' 1 of the tooth trace of the indexing plane and the normal modulus m ' n can be calculated by Eq.
) cos tan cos ( arctan Then, wheel base a, and installation distance of gear 1 l1 can be expressed as follows: The sharp angle ε ' between two conical buses of bevel gears 1 and 2 is ) cos cos sin sin cos arccos( So, the included angle δ ' 2 between the axis of bevel gear 2 and the imaginary common gear rack indexing plane may be calculated by simultaneous Eq. (8) -(10), and ' 1  is substituted by a negative value.
Next, the inclination angle β ' 2 of the imaginary common gear rack tooth trace, the pitch cone angle δ2 of gear 2, the tooth profile angle coefficient ξt2 and helix angle β2 are obtained by Eq. (11) - (13).
If a proper preset tooth profile angle coefficient ξt1 of bevel gear 1 is selected and r ' 2 =r " 2 is satisfied finally, the above all parameters of the designed bevel gear pair will be determined.

Derivation of Mathematical Model of Imaginary Gear Rack Tooth Surface
The normal section of imaginary gear rack is shown as Figure 2. where 1 ( ) is the coordinate value of the point on the skew straight tooth profile of the gear rack in the coordinate system Sn shown in Figure 3. t is a parametric variable, and represents the distance from any point on the oblique line to point C or point F, and satisfies this inequality constraint 0 ≤ t ≤ 2h * an mn /cos αn. p is gear pitch, and p=πm.
Similarly, the arc segments BC and FG envelop the transition curve between the tooth root and the involute tooth surface, the geometric parameter equation of the arc segments can be expressedas follows: . As shown in Figure 3, the imaginary common gear rack indexing plane is tangent to the pitch cone of bevel gear, and the pitch plane is tangent to the indexing cylinder of bevel gear. Sn is the coordinate system of the imaginary common gear rack normal section. Sp is the coordinate system of the imaginary common gear rack indexing plane. β is the inclination angle of tooth trace on the indexing plane when the gear meshes with the imaginary common gear rack. Sc is the coordinate system of the pitch plane of the imaginary common gear rack, which is tangent to the indexing cylinder of bevel gear. S0 is the coordinate system of gear, the coordinate system S0 changes to the coordinate system S1 when the gear rotates at an angle of φ1. δ represents the angle between the imaginary common gear rack indexing plane and the axis of the gear.  By substituting Eq. (15) into Eq. (17), Eq. (18) can be obtained as follows: represents the equation of the tooth surface formed by the normal section straight segment of the imaginary common gear rack in the coordinate system Sc; k represents the tooth width, and t represents a parameter of the tooth depth; and superscript 1 denotes a surface generated by a straight segment of the imaginary common gear rack.
By substituting Eq. (16) into Eq. (17), Eq. (19) can be obtained as follows: where f 2 c (k, θ) represents the equation of the tooth surface formed by the transition curve section equation of the imaginary common gear rack normal section in the coordinate system Sc; k represents tooth width, and θ represents a parameter of transition radian; and superscript 2 represents the surface generated by the transition curve segment.
The coordinate transformation relationship from the coordinate system Sc to the gear coordinate system S1 is given in Eq. (20):  (20), the expression of the plane equation formed by the straight segment of the imaginary common gear rack in the coordinate system S1 may be derived and shown in Eq. (21). Accordingly, by substituting Eq. (19) into Eq. (20), the expression of the surface equation formed by the transition curve segment of the imaginary common gear rack in the coordinate system S1 may be derived and shown in Eq. (22).

Derivation of Bevel Gear Meshing Equation and Tooth
Surface Equation According to the basic theorem of tooth profile meshing [29], assuming that the meshing points of gear and gear rack are represented by ( , , ) in the coordinate system Sc, then they have a common tangent and normal at meshing point D, as shown in Figure 4.
Supposed that the gear rack is fixed and the gear rolls on the gear rack, and the unit vectors of axes xc, yc, zc, x1, y1, z1, xp, yp and zp are expressed by → , → , → , 1 → , 1 → , 1 → , → , → , → respectively. At this time, the direction of gear angular velocity vector is the negative direction of the Zc axis, and the relative motion velocity of the gear tooth surface and gear rack tooth surface at the meshing point can be expressed as follows: where 1  is angular velocity of gear in the coordinate system Sc, and MD represents the distance from node M to meshing point D.
The partial derivatives of Eq. (18) and Eq. (19) with respect to k, t and θ may be solved respectively to obtain the normal vector of any point on the gear rack straight tooth surface and circular arc surface. The solution results are expressed as follows: where, for the symbols of ∓ and ±, the upper symbols should be used when the normal vectors corresponded to arc segment BC and line segment CD are calculated, and the lower symbols should be used when the normal vectors corresponded to line segment EF and arc segment FG. By substituting the relative velocity obtained by Eq. (24) and the normal vector obtained by Eq. (25) Eq. (29) may be deduced by the expression of angle 1 , and expressed as follows:

Solution of Tooth Surface Point Set of Bevel Gear Pair
Based on the parameters described in section 2 and the mathematical models for tooth surface of bevel gear pair described in section 3.2, the flow chart of solution of tooth surface point set is shown in Figure 5.

Introduction of the Bevel Gear Pair in the Knotter
The bevel gear pair of driving the rope-gripping plate in the D-type knotter is a crossed axes bevel gear pair with shaft intersection angle of 98 degrees, as shown in Figure 6(a). In the knotter driven by double fluted discs, the bevel gear pair of driving the knotter jaw to rotate is an intersecting shaft bevel gear pair with shaft intersection angle of 98 degrees, as shown in Figure 6

Calculation of Geometric Parameters of Bevel Gear Pair in the Knotter
The known parameters of the bevel gear pair in two kinds of knotter are given in Table 1 and Table 2 respectively, and the solution process of their unknown parameters is shown in Figure 7. The solution results of the unknown parameters are shown in Table 3 and Table 4 respectively.

Establishment of Accurate 3D Model of Bevel Gear
Pair in the Knotter According to the flow chart shown in Figure 5, the tooth surface point sets of the bevel gear pair may be programmed to solve and visual display by using powerful calculation and graphic display function of MATLAB software [30,31]. During the solution, the parameter t is set to take value at 0.2 intervals within the range from 0 to 0.8. The tooth width k is set to take value at 0.2 intervals within the range from -10 to 0. The parameters θ is set to take value at 0.1° intervals within the range from 110°to 180°. The point set obtained under MATLAB software is imported into Pro/Engineering software, and the accurate 3D model of bevel gear may be established according to the rule of "point -line -surfacebody". To the bevel gear pair in the knotter driven by double fluted discs, the process of its accurate modeling is shown in Figure 8. According to the motion requirements of the knotter, the bevel gear pair is designed as an incomplete bevel gear pair. Similarly, the point set of tooth profile of the bevel gear pair in D-type knotter can be obtained, and their 3D assembly model is shown in Figure 9. The 3D assembly model shown in Figure 10(a) is firstly import into ADAMS software, the material properties of parts are defined and the constraints among parts are added, as shown in Figure 10  The rotary drive with 540° percent second is added to the rotating pair of the driving bevel gear on the large fluted disc. The simulation time is set to 1 second, and the number of steps is 500. After finishing the simulation under ADAMS, the meshing process of single tooth of the bevel gear pair is shown in Figure 11. It can be seen from the above simulation results that the meshing point moves smoothy from starting contact point to end contact point, which indicates that the calculation of the bevel gear pair based on the derived gear tooth surface equation is correct. The designed bevel gear pair may have stable meshing transmission, and complete the given transmission action.

Meshing transmission simulation of bevel gear pair with 98°crossed shaft in D-type knotter
The same method as section 4.4.1 is used to simulate the transmission of the bevel gear pair with 98°crossed shaft. The 3D assembly model shown in Figure 12(a) is firstly import into ADAMS software, the material properties of parts are defined and the constraints among parts are added, as shown in Figure 12(b). After finishing the simulation under ADAMS, the meshing process of single tooth of the bevel gear pair is shown in Figure 13, and the simulation results are similar to the simulation in section 4.4.1.

Prototype manufacturing and transmission test of bevel gear pair with 98°intersecting shaft in the knotter driven by double fluted discs
According to the accurate 3D model generated by section 4.3, the prototype of bevel gear pair was manufactured by precision casting method. The actual prototypes are shown in Figure 14. The driving incomplete bevel gear and the large gear disc are integrally cast, and the driven incomplete bevel gear is individually cast. To carry out the transmission test, the above two bevel gears are assembled with other parts of the knotter. During the transmission test, the large gear disc is firstly driven to rotate, the driven incomplete bevel gear will mesh with the incomplete bevel gear on the large gear disc, and the knotting jaw coaxially fixed with the driven incomplete bevel gear realizes the trope-winding action. The process is shown in Figure 15 by taking a set of photos. The prototype motion of the bevel gear pair is very consistent with the simulation results under ADAMS, and can meet the motion requirements of the knotter.

Prototype manufacturing and transmission test of bevel gear pair with 98°crossed shaft in D-type knotter
As mentioned in section 4.5.1, the actual prototypes are shown in Figure 16. The two bevel gears are assembled with other parts of the D-type knotter. The transmission process is shown in Figure 17 by taking another set of photos. The prototype motion of the bevel gear pair is also very consistent with the simulation results under ADAMS, and can meet the motion requirements of the D-type knotter. It can be seen from the above two transmission tests that the tooth profile of the bevel gear pair generated by the tooth surface meshing equation in section 3 may guarantee the correctness of bevel gear design and the accuracy of bevel gear transmission. The gear contour generation method based on the rule of "point -line -surface -body" is feasible and reliable.

Conclusion
(1) On the basis of analyzing the equations for solving the parameters of bevel gear pair with arbitrary crossed axes, the meshing equations of the bevel gears are derived, based on the installation parameter method and the meshing principle of bevel gears, which can provide a calculation model for the design of the tooth profile of space bevel gear.
(2) Taking the bevel gear pair in the knotter as design examples, the geometric parameters of the bevel gear pair of the knotter are solved, and the accurate 3D models of the bevel gear pair are established. Through the meshing transmission simulation of the bevel gear pair and the transmission test of the knotter, the results show that the designed bevel gear pair is accurate and stable, which proves the correctness of the mathematical model of bevel gear tooth surface.