A Simulation Method for Free-form Milling Cylindrical Gears With Disc Cutters

： Free-form milling is a flexible gear machining method that allows using general disc cutters to machine various gear types on a 5-axis machine tool. This paper proposes a two-dimensional simulation method for free-form milling of cylindrical gears with disc cutters. A mathematical model for free-form milling of cylindrical gears with disc cutters is constructed. By analyzing the spatial positional relationships between the cutter and the workpiece, the instantaneous contact line is derived and projected onto the gear end face, and then the projected curves are intersected to obtain the final profile. This calculation method realizes the rapid simulation of the actual cutting condition on the gear end face, which is beneficial to the judgment of machining interference and the analysis of the tooth profile accuracy during the gear machining process.

instantaneous contact line of the disc cutter.In the existing research, Li and Wu [13,14] studied a solution method for calculating the contact line while the disc cutter moving along the helix of the workpiece(see Fig. 1(a)).On this basis, a method for calculating the contact line of the disc cutter with tangential offset is proposed (see Fig. 1(b)).
Fig. 1.The geometrical relations between the gear and the disc cutter Liang et al. [6] presented a numerical procedure for obtaining grinding wheel profiles using the relative motion of the grinding process.In doing so, he showed that it was possible to solve the interference and double enveloping problems with the algorithm.In addition, because the cutting process can be simulated using CAD software like AutoCAD and Pro-E [15,16], many researchers achieved such simulation using CAD approach programs [8][9][10][11][12].However, because these programs were developed based on CAD platforms, their efficiency was limited by the CAD system.
Therefore, this paper proposes a two-dimensional simulation method for free-form milling cylindrical gears on a 5-axis machine with disc cutters.The geometrical relations and the relative motion during the free-form milling process between the disc cutter and the workpiece are considered.The instantaneous contact line between the disc cutter and the gear surface at any tool position is derived.The instantaneous contact lines are projected onto the gear end face along the helix of the workpiece.The final milling profile is obtained by intersecting the projected contours at adjacent tool positions.The tooth profile accuracy can be obtained on the gear end face by comparing the final milling profile with the designed workpiece surface.

2．Simulation of the free-form milling process
At present, the simulation of 5-axis machining is usually implemented by popular 3D software such as CAD, UG, PRO-E, etc., which is realized through complex Boolean operations through solid modeling.The method of this paper breaks away from the dependence on 3D software and simplifies the calculation process.It is an efficient two-dimensional numerical simulation method.
It is assumed that the configuration of the machine tool, the axial profile of the disc cutter, and the relative helical motion relationship between the cutter and the workpiece during the free-form milling process are known.Then, the two-dimensional numerical simulation method can be implemented in the following steps: Step 1. Modeling of the geometric relationship of the disc cutter and the workpiece in the free-form milling process; Step 2. Contact analysis of the disc cutter and the workpiece at any tool position in the free-form milling process; Step 3. Spiral projection of the contact line at each tool position onto the gear end face; Step 4. Obtain the final profile of the projected curves.

Geometric model for the free-form milling with the disc cutter
The axial profile of the disc cutter discussed in this paper must be C 1 continuous, and discrete points are used in the calculation process.The cutter is a rotating body, which is formed by rotating around the cutter shaft.
Fig. 2. Free-form milling mode As shown in Fig. 2, a point on the axial profile and the vector at the point in the coordinate system   −      , can be parametrically expressed as where R is the parameter of the axial profile of the disc cutter.
The cutter surface is generated by rotating the axial profile on the xd-zd plane around the zd-axis.
It can be assumed that point M in Fig. 2 is the cutting point, and φ is the rotation angle of the axial profile through the cutting point.
Fig. 3.The geometrical relations between the gear and the disc cutter during the free-form milling During the free-form milling, the spatial relative motion between the gear and the disc cutter is shown in Fig. 3.The coordinate system  −      is rigidly attached to the gear, while the coordinate system   −      is rigidly attached to the disc cutter.The cutter setting parameters (ax, ay, az) are used to define the cutting positions along the  axis,   axis and   axis respectively.
The gear and the disc cutter are rotated about their own axes   and   , respectively, while the cutter moves linearly in a direction parallel to the gear axis   .The angle between the axes   and   is Ʃ.
The rotary surface of the disc cutter in the coordinate system   −      can be obtained: The normal vector of any point on the rotary surface of the disc cutter is The helical surface of the gear in the coordinate system   −      can be expressed as The unit normal vector of the tooth surface can be expressed as where  is the radius of base circle, 0 is the space width half-angle on the base circle,  is the screw parameter,  is the roll angle, and  is a parameter variable, indicating the angle that the end involute turns around the   axis.
When the cutter spirally moves, during the free-form milling process, the cutting point on the cutter is in tangential contact with the theoretical surface at each instant.At the contact point, the normal of the gear surface and the cutter surface are identical.In the case of a given setting angle, the machining tool position (ax, ay, az) and the rotation angle cgi from the initial position to the machining position can be obtained by the following equations: It can be observed from Eq.( 6) that, during the free-form milling process, the workpiece is rotated.In addition to the radial translation and the axial translation, the cutter also needs to perform the tangential translation.And then the final machined surface is made up of spiral surfaces at each cutting point.

Contact analysis and contact line projection
According to the method in Section 2.1, the cutting position (ax, ay, az) and the rotation angle cgi of the workpiece are shown in Fig. 3.
During the free-form milling with the disc cutter, the cutter is in point contact with the surface of the gear at a certain tool position.According to the principle of spatial surface meshing, whether a pair of conjugate surfaces is in point contact or line contact, the meshing equation must be satisfied at the contact point，  (12) . = 0 (7) where,  (12) is the relative velocity of two surfaces at the point;  is the normal vector of the surface at the point, expressed as (nx, ny, nz) in the coordinate system   −      .
Let the unit vectors of   ,   and   directions be ,  and .Let the unit vectors of   ,   and   directions be ', ' and '.It is assumed that the radial vectors of the cutting point M with respect to the two coordinate systems are rm and Rm.The angular velocity of the disc cutter is  ′ , and the angular velocity of the workpiece rotation is .
The linear velocity  () of point M moving with the helix can be given as follows,  (1) = ( ×  + ) (8) The linear velocity  () of point M moving with the disc cutter can be expressed as,  (2) =  ′ ( ′ × ) Then,  (12) =  (1) −  (2)  (10) Since the rotary surface of the disc cutter is known, the contact condition on the rotary surface is converted to the following equation, After simplification, Eq. ( 13) can be reconstructed as The contact line conditional equation is combined with the known disc cutter surface equation to obtain the contact line on the rotary surface of the disc cutter.The point (xi, yi, zi) on the contact line is spirally moved around the axis of the workpiece and projected into the gear end face.The

Final cutting profile on the gear end face
All the above calculations are in the form of discrete points.According to the general idea, based on the discrete points of the projection profiles at each tool position, the intersection points between these curves are calculated.Then the invalid points are removed, the final cutting profile can be obtained.In this paper, a more efficient interpolation scanning method of discrete points is used to obtain the final profile.The specific implementation process is as follows: Step 1: In yg >0, the projection profiles at each cutting position on the gear end face shows in Fig. 4(a).
Step 2. For the specific precision requirement, the discrete step size of the coordinate value of the yg direction is selected to be 0.5 mm, and the discrete points of the projection profile at each cutting position are linearly interpolated.Scan all projected curves along with the coordinate range of the yg-axis, take the minimum value of the corresponding xg-axis, and the corresponding coordinate point is (xxi, yyi).The interpolated points are shown in Fig. 4(b).
Step 3. Connect the discrete points with straight segments to obtain the final cutting profile as shown in Fig. 4(c).Tooth profile error is an important indicator to assess the smoothness of gear operation.It refers to the normal distance between the actual cutting profile and the theoretical tooth profile of the gear on a certain cross-section of the gear.The error analyzed in this paper is the normal distance between the simulated tooth profile and the theoretical tooth profile of the gear.It reflects the influence of the cutting on the tooth profile accuracy under theoretical conditions, which is the basis for analyzing the accuracy and compensation of the machine tool.
As shown in Fig. 5, T is the point of tangency.On the gear end face, if the error at the designed point P (xgi, ygi) to the practical point P'(xxi, yyi) is ep.The unit normal vector at P is (exgi,eygi,ezgi).Then the following equation is given, The error between the actual tooth profile and the designed tooth profile of the gear can be determined.

. Calculation of tooth profile deviation
The tooth profile error curve is obtained by plotting the error of each point on the actual tooth profile.In the standard ISO 1328-1:2013, there are three deviation indexes for evaluating the tooth profile, which includes the tooth profile deviation ffa, the tooth profile inclination deviation fhα, and the total tooth profile deviation Fα, as shown in Fig. 6.The polar radius xx is the abscissa and the tooth profile error ep is the ordinate.Within the evaluation range, the tooth profile errors are fitted to the middle dotted line by the least square method.And the error curve in the evaluation range is included by translating the dotted line to the left and right for the minimum distance.The horizontal distance between the left and right dotted lines is the tooth profile deviation ffα; the corresponding abscissa span of the dotted line is the tooth profile inclination deviation fhα; the total tooth profile deviation Fα is the difference between the maximum profile error and the minimum profile error within the evaluation range [17].

Fig. 6. Profile error curve and its evaluation 4．Numerical examples and discussion
This model can be used to analyze the influence of the position error of each position during the free-form milling process on the tooth profile error.The deviation of the tooth profile of the actual cutting profile can be found, when the uniaxial or multi-axis position is offset from the theoretical position at any tool position.An example is introduced to illustrate this condition.The machine for free-form milling has three linear axes(X, Y, and Z), and two rotary axes (A and C).The basic parameters of the gear are shown in Table1.The disc cutter with the straight edge is shown in Fig. 7.The parameters of the disc cutter are given in Table2.The given tool position are shown in Table 3. Taking the deviation of the Y-axis direction as an example, the simulation results of the tooth profile error during the cutting process of the disc cutter enveloping are obtained.Example 1 is the case of Single-axis offset; Example 2 is the case of position-dependent coordinate offset.In the figures below, the abscissa is the radius value and the ordinate is the corresponding tooth profile error.The Y-axis is shifted by ±0.02mm, respectively.The simulation results of the tooth profile error of the gear surface during the cutting process with the disc cutter are shown in Fig. 8. Table 4 shows the tooth profile errors at different offsets of the Y-axis.It can be seen that the uniaxial offset of the Y-axis has almost no influence on the tooth profile precision.The uniaxial offset of the Y-axis only affects the gear tooth thickness.Furthermore, the uniaxial offset of the Y-axis has the opposite effects on both sides of the tooth thickness, that is, the overall does not affect the tooth thickness.Assuming the offset of the Y-axis δy is linear with the Z-axis coordinate zi, and the relation can be expressed as δy=±0.002*zi+0.005.The simulation results of the tooth profile error of the gear surface during the cutting process of the disc cutter are shown in Fig. 9. Table 5 shows the tooth profile errors at different offsets in the Y-axis.It can be seen that the Y-axis is offset with the Z-axis position, which affects fha and Fa obviously, and the effects on both sides are almost the same.The slope of the relation has the same influence on the tooth thickness deviation generated on both sides of the tooth profile, and the intercept of the relation has opposite effects on the tooth thickness deviation generated on both sides of the tooth profile.

5．Conclusions
A simulation method is proposed for free-form milling.Based on the results from the numerical examples, the following conclusions are drawn: 1) From the cutting projection of the disc cutter, it can be visually seen whether there is interference and overcutting in the machining process.
2) The model shows the distribution of machining error, and can also be used to evaluate the rationality of the tool path planning strategy.
3) The model can visually show the influence of machine geometry error on gear machining accuracy.
By applying this two-dimensional simulation model, the tooth profile error caused by a tool path planning strategy is quickly obtained.The analysis of tooth profile error caused by coordinate deviation is the basis of error source analysis and error compensation strategy.It is of great significance for the improvements of the machining accuracy in the free-form milling.In summary, this method is very practical in the engineering field.

Fig. 4 .
Fig. 4. Calculation procedure of the final cutting profile of gear end face

Fig. 5
Fig. 5. Calculation of tooth profile deviationThe tooth profile error curve is obtained by plotting the error of each point on the actual tooth profile.In the standard ISO 1328-1:2013, there are three deviation indexes for evaluating the tooth profile, which includes the tooth profile deviation ffa, the tooth profile inclination deviation fhα, and the total tooth profile deviation Fα, as shown in Fig.6.The polar radius xx is the abscissa and the tooth profile error ep is the ordinate.Within the evaluation range, the tooth profile errors are fitted to the middle dotted line by the least square method.And the error curve in the evaluation range is included by translating the dotted line to the left and right for the minimum distance.The horizontal distance between the left and right dotted lines is the tooth profile deviation ffα; the corresponding abscissa span of the dotted line is the tooth profile inclination deviation fhα; the total tooth profile deviation Fα is the difference between the maximum profile error and the minimum profile error within the evaluation range[17].

Fig. 7 .
Fig. 7.The disc cutter with the straight edge