Abstract
This paper focuses on the design and manufacture of face milling spiral bevel gear, establishing a mathematical model for the machining method based on free-form surface machine tools. By fixing the workpiece shaft on the free-form surface machine tool, the active design and manufacture of spiral bevel gear milling are completed. Taking meshing performance parameters as input variables, a mathematical model of gear meshing is established, and the gear tooth surface is actively designed according to the designed contact attributes. In addition to controlling the first-order and second-order parameters of the average contact point on each pinion side, this method also ensures each contact path point on the common driving side of the pinion, avoiding errors and manufacturing difficulties caused by poor workpiece shaft motion accuracy. Finally, the method is verified through tooth contact analysis (TCA) and rolling tests.
1. Introduction
Spiral bevel gear is crucial components for transmitting rotation and torque in the aerospace and automotive industries. With the development of computer numerical control (CNC) technology, free-form surface machine tools have been widely applied in the design and manufacture of spiral bevel gear. These machine tools can execute any desired tool motion on the workpiece, providing almost unlimited freedom for the tooth surface design and manufacturing of spiral bevel gear.
Extensive research has been conducted on spiral bevel gear by scholars domestically and internationally. For example, some studies have investigated the root cutting issue of involute spiral bevel gear based on gear meshing models, establishing algorithms to solve for the number of root-cut teeth, thus providing theoretical analysis methods for the design and manufacture of spiral bevel gear. Other studies have researched tooth surface modification methods based on the contact characteristics of spiral bevel gear, proposing discrete tooth surface calculation methods that effectively control contact characteristics. Additionally, numerical models have been used to study the contact characteristics of spiral bevel gear tooth surfaces, presenting solutions for contact characteristics and establishing virtual simulation models of spiral bevel gear to demonstrate the effectiveness of the proposed methods.
This study builds on previous research, focusing on the face milling of spiral bevel gear and employing a workpiece shaft fixation method on free-form surface machine tools to achieve active design and manufacture of spiral bevel gear milling. The meshing performance parameters are used as input variables to describe the active design of the microscopic geometry of the pinion tooth surface, ensuring the acquisition of machining parameters for both the target tooth surface and the target tooth depth. Finally, the proposed method is verified through simulation analysis and experimentation.
2. Gear Pair Meshing Model
2.1 Gear Pair Contact Attributes
The contact attributes of spiral bevel gear pair studied in this research include: for the driving side, the contact attributes encompass the location and shape of the contact path, the length of the major axis of the instantaneous contact ellipse at the average contact point, and the transmission error function; for the sliding side, the contact attributes include the location of the average contact point, the direction of the contact path, the length of the major axis of the instantaneous contact ellipse at the average contact point, and the angular acceleration at the average contact point.
In this method, the contact path on the driving side can be designed as a straight line or curve in the rotating projection section, and the transmission error on the driving side can be designed as a quadratic parabola or higher-order polynomial, achieved by controlling the side shape of the pinion. Specifically, the contact path is designed as a straight line in the rotating projection section, and the transmission error is designed as a quadratic parabola. The four contact attribute targets are described below, with targets 1 and 2 represented in spiral bevel gear rotating projection coordinate system.
- Coordinates of the Average Contact Point (hˉRef, vˉRef): Determines the position of the contact pattern.
- Contact Path Equation: Determined based on the position of the average contact point and the direction angle θ of the contact path, as shown in Equation (1). The direction angle θ is the angle between the contact path and the hˉ axis. For the convex surface of spiral bevel gear, the trend of the contact path is from heel tip to toe root, with θ being positive; for the concave surface of spiral bevel gear, the trend is from heel root to toe tip, with θ being negative.barh−hˉRef=tan(θ)(vˉ−vˉRef)(1)
- Length of the Major Axis of the Contact Ellipse (L) at the average contact point: Determines the width of the contact pattern.
- Transmission Error Function: The transmission error is the difference between the actual gear rotation angle φW and the theoretical gear rotation angle φ*W, denoted as TE. In this study, TE is designed as a quadratic parabola function, with the TE value at the meshing instant of the average contact point set to 0. Due to the existence of second-order convexity, the TE value is not 0 when other regions of the side engage. The transmission error at the meshing input and output points is denoted as TEmax, and the range from meshing input to meshing output is one pinion angular pitch, as shown in Figure 2. The motion function is expressed in Equations (2) to (5), with the transmission error stated as:TE=φW−φW∗=c2(πZPφP−φRefP)2(2)varphiW=φRefW+iWP∗(φP−φRefP)(3)iWP=iWP∗+2c2(πZPφP−φRefP)(4)fracdiWPdφP=dωPdωW=2c2(5)Where:
- φW: Actual gear rotation angleφ*W: Theoretical gear rotation angle calculated based on the theoretical transmission ratioφP: Actual rotation angle of the pinionφRefP: Rotation angle of the pinion under the meshing moment at the average contact point of the pinion, obtained through the meshing equation.φRefW: Gear rotation angle at the meshing moment at the average contact point of spiral bevel geari*WP: Theoretical transmission ratioi*WP = -ZP / ZW, where ZW is the number of spiral bevel gear teeth and ZP is the number of pinion teethiWP: Instantaneous transmission ratioωW: Angular velocity of spiral bevel gearc2: Quadratic coefficient of the transmission error, calculated using Equation (6)
- Where TEmax is given based on actual requirements, and the φRefP value is determined through the meshing equation, thereby establishing the transmission error function of spiral bevel gear pair.
Figure 2: Transmission Error
The implementation method for the above-listed contact attributes is to control the side shape of the pinion through pinion machining parameters. For the driving side, targets 1, 2, 3, and 4 are known and guaranteed during the process of solving for the pinion machining parameters. For the sliding side, target 1 is known and guaranteed during the process of solving for the pinion machining parameters, while targets 2, 3, and 4 are guaranteed through subsequent adjustments.
2.2 Gear Side Parameters
In this study, spiral bevel gear is cut using the form method, with the grinding wheel cutting blade being straight. Within the effective range of the tooth surface, points (xW, yW, zW) on the spatial tooth surface correspond one-to-one with points (hˉ, vˉ) in the rotating projection coordinates. Given the rotating projection coordinates of any point on the gear side, the position vector rW and unit normal vector nW of the corresponding point on the spatial tooth surface, as well as the normal curvature KW and geodesic torsion GW in any direction from that point, can be determined.
Since spiral bevel gear side shape is determined, actively designing the microscopic geometry of the pinion side can satisfy the contact attribute targets. On any point on the gear contact path, the unit tangent vector along spiral bevel gear contact path is denoted as tWline, and the unit vector perpendicular to tWline is denoted as tWvertical. The grinding wheel contact path can be expressed as a function of the gear tool parameters (hc, θc):
fWline(hc,θc)=0(7)
Where hc is the height of any point on the blade from the tool tip, and θc is the orientation angle in spiral bevel gear cutting transverse plane.
The above function can be expressed as θc = θc(hc), i.e., θc is a function of hc. The position vector rWline of a point on spiral bevel gear contact path can be expressed as Equation (8), making rWline also a function of hc.
mathbf{r}_{Wline} = \begin{bmatrix} x_W(h_c, \theta_c) \\ y_W(h_c, \theta_c) \\ z_W(h_c, \theta_c) \\ 1 end{bmatrix} \quad (8)
Since the unit tangent vector tWline should be the direction in which the instantaneous contact point moves on spiral bevel gear side, it can be expressed as:
mathbftWline=dhcdrWline/dhcdrWline(9)
Where:
fracdrWlinedhc=(∂hc∂xW+∂θc∂xW⋅dhcdθc)iW+(∂hc∂yW+∂θc∂yW⋅dhcdθc)jW+(∂hc∂zW+∂θc∂zW⋅dhcdθc)kW(10)
Where iW, jW, kW are the unit vectors along the xW-yW- and zW-axes of the moving coordinate system oW−xWyWzW.
Based on the above method, for the convex side of spiral bevel gear, the first-order parameters of each point on the contact path, the first- and second-order parameters of the average contact point, and the unit tangent vector of the average contact point along the contact path can be obtained. For the concave tooth surface, the first- and second-order parameters of the average contact point can be determined.
3. Pinion Tooth Surface Design Based on Contact Attributes
3.1 Rotation Angles of the Average Contact Point for Gear and Pinion
During meshing, the tooth surface contact points constantly change. When the average contact points of spiral bevel gear and pinion mesh, spiral bevel gear and pinion rotate by angles φrefW and φrefP around their respective axes from their initial positions. At this time, φrefW and φrefP satisfy the meshing equation for spiral bevel gear and pinion:
mathbfvWPM⋅nWM=0(11)
Where vWPM is the relative velocity of spiral bevel gear and pinion at the average contact point, and nWM is the unit normal vector of the gear at the average contact point. Equation (11) can be simplified to:
Usin(φrefW)+Vcos(φrefW)=0(12)
Where:
U=xW⋅nWxyW⋅sin(Σ)−yW⋅nWxsin(Σ)+E⋅nWzcos(Σ)
V=xW⋅nWzsin(Σ)−zW⋅nWxsin(Σ)−E⋅nWycos(Σ)
W=E⋅nWxsin(Σ)+(yW⋅nWz−zW⋅nWy)(iWP∗−cos(Σ))
And the meshing rotation angle φrefW of spiral bevel gear’s average contact point is calculated as:
varphirefW=2⋅arctan(V+W−U±U2+V2−W2)(14)
The meshing rotation angle φrefP of the pinion’s average contact point is calculated based on the theoretical transmission ratio as:
varphirefP=iWP∗φrefW(15)
Thus, the transmission error function can be determined when the TEmax value is given in advance.
3.2 First-Order Parameters of the Pinion Tooth Surface
After fully determining the transmission error function and gear side parameters, the first-order parameters of the meshing pinion can be obtained. Taking the magnitude ωP of the angular velocity vector ωPM of the pinion as 1, other variables are expressed based on this. For any point on spiral bevel gear side, when this point meshes with a point on the pinion side, Equation (16) is satisfied, which becomes an equation with only one unknown φP, solvable through iterative calculations:
mathbfvWPM(φW,φP)⋅nWM=0(16)
Where vWPM(φP, φP) = ωWPM × rWM – ωWPM × αWPM. The relative velocity between spiral bevel gear and pinion is given by the vector vWPM. nWM is the unit normal vector of the gear side; ωWPM is the relative angular velocity vector between spiral bevel gear and pinion; rWM is the position vector of spiral bevel gear side; ωPM and ωWM are the angular velocity vectors of the pinion and gear, respectively; αWPM is the vector pointing from point oP to point oW, which can be obtained from the following equations:
mathbf{\omega}_{WM} = \begin{bmatrix} i_{WP} \\ 0 \\ 0 end{bmatrix} \quad (17)
mathbf{\omega}_{PM} = \begin{bmatrix} cos(\Sigma) \\ -\sin(\Sigma) \\ 0 end{bmatrix} \quad (18)
mathbfωWPM=ωWM−ωPM(19)
mathbf{a}_{PWM} = \begin{bmatrix} E \\ 0 \\ 0 end{bmatrix} \quad (20)
mathbfrWM=MMWrW(21)
mathbfnWM=mMWnW(22)
Where rW and nW are the position vector and unit normal vector of spiral bevel gear side in the coordinate system oW−xWyWzW. Thus, spiral bevel gear rotation angle φW can be determined based on the defined motion function:
varphiW=φP⋅iWP(23)
Then, the position vector rP and unit normal vector nP of the corresponding point on the pinion side that meshes with the given gear point can be calculated using:
mathbfrP=MPPdMPdMMMWrW(24)
mathbfnP=mPPdmPdMmMWnW(25)
Where MPPd and mPPd are the transformation matrices from the coordinate system oPd−xPdyPdzPd to oP−xPyPzP; MPdM and mPdM are the transformation matrices from the coordinate system oM−xMyMzM to oPd−xPdyPdzPd; MMW and mMW are the transformation matrices from oW−xWyWzW to oM−xMyMzM; mPPd, mPd and MPPd represent the inverse transformation matrices corresponding to MPPd and mPPd, respectively; similarly, mPdM and MPdM denote the inverse transformation matrices of MPdM and mPdM; MMW_inv and mMW_inv are the inverse transformation matrices of MMW and mMW, respectively.
To summarize, we have the following sets of transformation and inverse transformation matrices:
- Transformation from oPd−xPdyPdzPd to oP−xPyPzP:
- MPPd (direct)
- mPPd (inverse)
- Transformation from oM−xMyMzM to oPd−xPdyPdzPd:
- MPdM (direct)
- mPdM (inverse)
- Transformation from oW−xWyWzW to oM−xMyMzM:
- MMW (direct)
- mMW (inverse)
Additionally, the inverses specifically named:
- MPdM_inv (same as mPdM if the matrices are properly defined inverses)
- MMW_inv (same as mMW if the matrices are properly defined inverses)
These matrices allow for transformations between different coordinate systems within a given frame of reference, facilitating the conversion of coordinates from one system to another in various applications such as robotics, computer graphics, and engineering.