Determination of Undercutting Avoidance for Designing the Production Technology of Worm Gear Drives with a Curved Proﬁle

: One of the most difﬁcult production geometry tasks arising in the machining process of the elements of a drive pair is to avoid undercuts. It is a serious technological challenge to determine the production of the elements of worm gear drives avoiding the phenomenon undercut, especially in the case of a pair consisting of a curved proﬁle worm and its mating wheel. The technology of forming the tooth surface requires a separate examination in each case, running the simulation procedure of the tool geometry and the movement conditions when forming different teeth. This article proposes a new concept for determining and then avoiding the positions of undercutting by examining the patented worm with a circular arc proﬁle in axial section, due to its extremely advantageous aspect in terms of production technology. The cutting edge of the hob, formed from the substitutional worm, moves on the tooth surface of the worm, and produces the tooth surface of the conjugate wheel. The gear tooth surface has been determined based on the main law of gearing with the lines consisting of the contact points of the conjugated surfaces. The conditions for the disappearance of the common normal or the relative velocity ﬁtting to the common tangent plane of the contacting points are deﬁned in this paper.


Introduction
Gear theory belongs to the scientific fields of constructive geometry, manufacturing, design, measurement technology and computer methods.All these disciplines are necessary for modern developments [1][2][3], some of which have affected our research work at the Worm Gear Science School [4,5], which was founded at the University of Miskolc.Gear tooth theory has evolved into an independent discipline following much theoretical and practical research [6][7][8].Works about examining the necessary and sufficient conditions for the existence of an envelope have also influenced our research (e.g., [9,10]).With the development of gear technology and the use of computers both in gearing theory and gear manufacturing, researchers have modified it to a modern theory of gearing and extended its methodology [10] and industrial applications [11][12][13].Many researchers dealing with this topic have had a significant impact on our work [13,14].Our studies have also been influenced by writings focused on the study of undercuts in involute shafts and bevel gears [15,16].For the present work, tool profile distortion analysis in the case of the machining of worm gear drive pairs with a circle arc profile in the axial section has been completed using the methods of the constructive geometry [17][18][19].Studies supported by valuable simulation procedures have been performed for the contact analysis of the drive pair elements, which were also useful in the research leading to the present paper [20,21].Particularly noteworthy is the research on the industrial implementation of the production of cylindrical worms, supported by theory, for the purpose of this paper [22,23].
In this article, an undercutting analysis in relative motion is presented, which can occur even if there is no singular point on the generating surface.At singular points the surface normal vectors become indeterminate, so undercutting can occur.The analysis Machines 2023, 11, 56 2 of 13 has been performed in a constructive geometric model created for the development of the production geometry of the elements of a conical and cylindrical worm gear pair (see Figure 1).The rotating coordinate system K 1F (x 1F , y 1F , z 1F ) has been fixed to the worm or hob, while the rotating coordinate system K 2F (x 2F , y 2F , z 2F ) has been fixed to the gear or grinding wheel, the coordinate system K 1 (x 1 , y 1 , z 1 ) has been connected to the linear moving table, the stationary coordinate system K 2 (x 2 , y 2 , z 2 ) has been connected to the grinding wheel or gear, and the stationary coordinate system K 0 (x 0 , y 0 , z 0 ) has been fixed to the frame.
In this article, an undercutting analysis in relative motion is presented, which can occur even if there is no singular point on the generating surface.At singular points the surface normal vectors become indeterminate, so undercutting can occur.The analysis has been performed in a constructive geometric model created for the development of the production geometry of the elements of a conical and cylindrical worm gear pair (see Figure 1).The rotating coordinate system K1F(x1F, y1F, z1F) has been fixed to the worm or hob, while the rotating coordinate system K2F(x2F, y2F, z2F) has been fixed to the gear or grinding wheel, the coordinate system K1(x1, y1, z1) has been connected to the linear moving table, the stationary coordinate system K2(x2, y2, z2) has been connected to the grinding wheel or gear, and the stationary coordinate system K0(x0, y0, z0) has been fixed to the frame.

Figure 1.
The frames defined for the analysis of the production geometry of the surface Σ1 of the worm and the Σ2 of the wheel, based on [18].
The geometrical parameters used, such as a for the distances of axes, c for the tool offset, α for the tilting angle of the tool to the helical surface in a characteristic section, γ 12  for the angle between worm and wheel or tool axes, which is equal to helix lead angle γ on the worm's reference surface in case of manufacturing with a grinding wheel, pa for the axial screw parameter, pr for the radial spiral parameter, and zax for axial displacement of the helicoid surface to the manufacturing position, are indicated in Figure 1.The motion geometrical parameters used, such as φ1 for the rotation angle of the helical surface, φ2 for the rotation angle of the gear or the tool surface, ω1 for the angular velocity of the helical surfaces and ω2 for the angular velocity of the gear or the tool, have been specified as shown in Figure 1.In the case of the reported constructive geometric model, the geometric parameters must be set according to the task.In the case of worm gear drive meshing analysis, the shaft angle γ 12 is −90°, taking into account the orientation of the axes.Our further analyses relate to the examination of rigid bodies.In this discussion, the worm The frames defined for the analysis of the production geometry of the surface Σ 1 of the worm and the Σ 2 of the wheel, based on [18].
The geometrical parameters used, such as a for the distances of axes, c for the tool offset, α for the tilting angle of the tool to the helical surface in a characteristic section, γ 12  for the angle between worm and wheel or tool axes, which is equal to helix lead angle γ on the worm's reference surface in case of manufacturing with a grinding wheel, p a for the axial screw parameter, p r for the radial spiral parameter, and z ax for axial displacement of the helicoid surface to the manufacturing position, are indicated in Figure 1.The motion geometrical parameters used, such as ϕ 1 for the rotation angle of the helical surface, ϕ 2 for the rotation angle of the gear or the tool surface, ω 1 for the angular velocity of the helical surfaces and ω 2 for the angular velocity of the gear or the tool, have been specified as shown in Figure 1.In the case of the reported constructive geometric model, the geometric parameters must be set according to the task.In the case of worm gear drive meshing analysis, the shaft angle γ 12 is −90 • , taking into account the orientation of the axes.Our further analyses relate to the examination of rigid bodies.In this discussion, the worm gear hob created from the worm is labeled Σ 1 and the derived gear tooth surface Σ 2 to differentiate between generator and generated surfaces.
For the analysis, the vector parametric form of the regulator helical surface Σ 1 in the coordinate system K 1F will be suitable: where η is the internal distance parameter and ϑ is the internal angle parameter.The regulator surface Σ 1 is free from singularities if the normal vectors exist, so the following condition is fulfilled: The transformation matrix between the coordinate system K 1F and the coordinate system K 2F can be determined based on Figure 1 The transformation matrices M 1,1F and M 1F,1 based on Figure 1 are as follows The transformation matrices M 0,1 and M 1,0 based on Figure 1 are as follows The transformation matrices M K,0 and M 0,K based on Figure 1 are as follows The transformation matrices M 2,K and M K,2 based on Figure 1 are as follows The transformation matrices M 2F,2 and M 2,2F based on Figure 1 are as follows The matrix of the transformation from the frame K 2F to the frame K 1F is as follows The matrix of the transformation from the frame K 1F to the frame K 2F is as follows The v 2F (12) relative velocity vector between surface Σ 1 and surface Σ 2 can be determined using the transformation matrix M 2F,1F from the frame K 1F (x 1F , y 1F , z 1F ) of the worm to the frame K 2F (x 2F , y 2F , z 2F ) of the mating gear, in the form Using the transformation matrix M 1F,2F from the frame K 2F (x 2F , y 2F , z 2F ) of the mating gear to the frame K 1F (x 1F , y 1F , z 1F ) of the worm, the relative velocity vector v 1F (12) can be calculated according to the following formula Machines 2023, 11, 56 5 of 13 where the "kinematic transformer" matrix is based on [18]: where the P 1a is as follows The equation of meshing can be written in the following form The tooth surface Σ 2 can be produced as the enveloping surface of the instantaneous contact lines in such a way that any contact point C of the contact lines l described in K 1F can be converted into the frame K 2F using the transformation matrix M 2F,1F between them, which can be written The geometric location of those points of the l ϕ1 contact curves occurring for any ϕ 1 parameter should be determined on the generator surface Σ 1 , which results in singular points on the generated gear tooth surface Σ 2 .

Singularity Avoidance Method
Different points of the tooth surfaces created by the tool surfaces can be distinguished from the perspective of differential geometry.Definition 1.By elementary surface we mean a shape that can be produced as the endpoints of the position vectors of a two-parameter vector function r = r(η, ϑ) interpreted on a simply connected region of the plane (η, ϑ), where (a) the mapping defined by r = r(η, ϑ) is topological (b) r = r(η, ϑ) is continuously differentiable (c) vectors ∂r/∂η and ∂r/∂ϑ are not parallel at any point.
Those surface productions that fulfil the conditions (a)-(c) are called regular productions.Definition 2. A point that does not meet the definition of a regular point is called a singular point.
To avoid undercutting it is necessary to determine the geometrical location of the points on the regulator surface Σ 1 that create the singular points on the regulated surface Σ 2 , where the velocity vector or the normal vector of the surface become indeterminate, resulting in undercutting.
Undercutting during relative motion can also occur even if there is no singular point on the regulator surface Σ 1 , but the generated surface Σ 2 may contain not only regular points but also singular points.
In order to carry out the matrix algebraic analysis, it is necessary to make some definitions regarding the projections falling on the coordinate planes.

Definition 3. Let the value of the determinant of matrix
be ∆ xy in the mathematical kinematical model.

Definition 4.
Let the value of the determinant of matrix be ∆ yz in the mathematical kinematical model.

Definition 5. Let the value of the determinant of matrix
be ∆ zx in the mathematical kinematical model.Definition 6.Let the value of the determinant of matrix be ∆ ηϑ in the mathematical kinematical model.
The contact point C is located on both the generator surface Σ 1 and the generated surface Σ 2 at the same time.To determine the relative velocity, the contact point C should be examined simultaneously as point C (1) fitted to the generator surface Σ 1 and as point C (2)  fitted as to the generated surface Σ 2 .The relative velocity v (12) of the contact point C can be represented as the velocity of point C (1) with respect to point C (2) by the following equation v (12) = v (1) − v (2)  (22 where v (i) are the velocity vectors of the coincident points C (i) of surface Σ i , and i is the index of the frame in which the velocity is written.If a regular point of Σ 1 generates a singular point on the meshed surface Σ 2 , then using the equality of the absolute velocity vectors of the contact points C i the following equation must be fulfilled: v (2) = v (1) + v (12) = 0 (23) where v (i) (i = 1, 2) are the velocity vectors of the contacting points on the portant surfaces in the common tangent plane.The differentiation of function f (η, ϑ, ϕ 1 ) = 0 according to the time parameter t from Equation (6) helps to filter the points on Σ 1 , which generates singular points on Σ 2 as follows: Theorem 1.In order for the surface Σ 1 with regular points to create singular points on the surface Σ 2 enveloped by it, the sufficient condition is the fulfilment of the next equation Proof of Theorem 1. Equations ( 23) and (24) in the frame K 1F of the mathematical kinematical model result in the forms contributing to the elimination of points on the surface Σ 1 , that generate singular points.
From Equations ( 26) and ( 27) prescribing dϕ 1 /dt = 1rad/1 sec, an overdetermined system of four linear equations arises with two unknowns, which are dη/dt and dϑ/dt.This system leads to the matrix G 4×3 with rank r = 2 and a certain solution for the unknowns The G 4×3 yields on the coordinate planes in our kinematical model to the determinants of the matrices M xy , M yz , M zx and M ηϑ , which respectively take the value ∆ 1 = 0, ∆ 2 = 0, ∆ 3 = 0 and ∆ 4 = 0.
The ∆ 4 = 0 yields to the equation of meshing [9], and it is fulfilled for the contact points of surfaces Σ 1 and Σ 2 , which are taken into account during the tests.Thus only the equalities ∆ 1 = 0, ∆ 2 = 0 and ∆ 3 = 0 need to apply to determine the singularity conditions of the surface, which procedure yields to the sufficient condition F(η, ϑ, ϕ 1 ) = 0.
The simultaneous fulfilment of equations determines the points on the surface Σ 1 that form singular points on the surface Σ 2 .
The presented procedure is a suggested way to determine the singularities on the created surface and thus to avoid its undercutting during production.
Through G 4×3 it is possible to study different types of helicoid drives.

Application
In the following, the analysis will have been applied to a cylindrical worm with circular arc profile in the axial section and its connected gear.
The gear is machined with a hob derived from the worm, designed using a complicated mathematical process [10,18].The solid model of the simultaneous coupling wheel, worm and hob is shown in Figure 2. The gear is machined with a hob derived from the worm, designed using a complicated mathematical process [10,18].The solid model of the simultaneous coupling wheel, worm and hob is shown in Figure 2. The geometrical parameters of the worm profile, such as the radius of the circle arc ρax, the distance between the worm axis and the center of the profile circle K are shown in Figure 3.
Based on Figure 1, the manufacturing geometry of cylindrical worm drives with parameters α = 0, c = 0, pr = 0 and γ = −90° can be examined using the kinematic model.The coordinates of the helical surface with circular arc profile curve in axial section can be written in the next form The geometrical parameters of the worm profile, such as the radius of the circle arc ρ ax , the distance between the worm axis and the center of the profile circle K are shown in Figure 3.The gear is machined with a hob derived from the worm, designed using a complicated mathematical process [10,18].The solid model of the simultaneous coupling wheel, worm and hob is shown in Figure 2. The geometrical parameters of the worm profile, such as the radius of the circle arc ρax, the distance between the worm axis and the center of the profile circle K are shown in Figure 3.
Based on Figure 1, the manufacturing geometry of cylindrical worm drives with parameters α = 0, c = 0, pr = 0 and γ = −90° can be examined using the kinematic model.The coordinates of the helical surface with circular arc profile curve in axial section can be written in the next form The cylindrical worm with a circle arc profile in axial section with the profile parameters, as the distance K between the arc center and the worm axis, and the arc radius ρax at standard marks, which has been patented [11,[17][18][19]24].
The normal vectors of this worm surface can be described as follows Figure 3.The cylindrical worm with a circle arc profile in axial section with the profile parameters, as the distance K between the arc center and the worm axis, and the arc radius ρ ax at standard marks, which has been patented [11,[17][18][19]24].
Based on Figure 1, the manufacturing geometry of cylindrical worm drives with parameters α = 0, c = 0, p r = 0 and γ = −90 • can be examined using the kinematic model.The coordinates of the helical surface with circular arc profile curve in axial section can be written in the next form The normal vectors of this worm surface can be described as follows Based on (4) the coordinates of the relative velocity vector v (12) 1F are calculated in the form With the method outlined, the elements of G 4×3 can be calculated with a clear mathematical process [25].The elements of the determinants of Equation (26) derived from r 1F can be found in the following formulas The element ∂ f /∂η of the matrix G 4×3 is presented in the equation And the element ∂ f /∂ϑ of the matrix G 4×3 is presented in the following equation Similarly to the previous calculations, the element ∂ f /∂ϕ 1 of the matrix G 4×3 is as follows The parameters indicated in the input part of the table are the geometrical parameters of the worm gear drive, and the information on the movement and surface parameters.The For the cylindrical arched worm, LA and LB curves can be found having points to cause the singularity on the gear tooth surface.To avoid undercutting on the gear surface Σ2, it is sufficient to delimit the worm surface Σ1 during design by eliminating the LA and LB curves.
The patented worm and the connected gear have been manufactured with the given parameters, as it can be seen in Figure 6.

Discussion
The different drives, which were mostly investigated by simulation methods in excellent papers [12,14,15,20,21], can now be determined using a targeted calculation procedure.Until now, the matrix algebra solution was performed separately for each type of gear pair in its own frame, but this paper presents a step towards generalization.In the model developed for production geometry developments used here, an experiment in the direction of generalization using the methods of matrix algebra was presented, as opposed to the tests carried out separately for each type of gear pair until now.
The worm with a circular profile in the axial section was patented due to its outstanding advantages from the point of view of production technology.Undercutting has not For the cylindrical arched worm, L A and L B curves can be found having points to cause the singularity on the gear tooth surface.To avoid undercutting on the gear surface Σ 2 , it is sufficient to delimit the worm surface Σ 1 during design by eliminating the L A and L B curves.
The patented worm and the connected gear have been manufactured with the given parameters, as it can be seen in Figure 6.For the cylindrical arched worm, LA and LB curves can be found having points to cause the singularity on the gear tooth surface.To avoid undercutting on the gear surface Σ2, it is sufficient to delimit the worm surface Σ1 during design by eliminating the LA and LB curves.
The patented worm and the connected gear have been manufactured with the given parameters, as it can be seen in Figure 6.

Discussion
The different drives, which were mostly investigated by simulation methods in excellent papers [12,14,15,20,21], can now be determined using a targeted calculation procedure.Until now, the matrix algebra solution was performed separately for each type of gear pair in its own frame, but this paper presents a step towards generalization.In the model developed for production geometry developments used here, an experiment in the direction of generalization using the methods of matrix algebra was presented, as opposed to the tests carried out separately for each type of gear pair until now.
The worm with a circular profile in the axial section was patented due to its outstanding advantages from the point of view of production technology.Undercutting has not

Discussion
The different drives, which were mostly investigated by simulation methods in excellent papers [12,14,15,20,21], can now be determined using a targeted calculation procedure.Until now, the matrix algebra solution was performed separately for each type of gear pair in its own frame, but this paper presents a step towards generalization.In the model developed for production geometry developments used here, an experiment in the direction of generalization using the methods of matrix algebra was presented, as opposed to the tests carried out separately for each type of gear pair until now.
The worm with a circular profile in the axial section was patented due to its outstanding advantages from the point of view of production technology.Undercutting has not yet appeared in the literature dealing with the manufacturing geometrical problems of a cylindrical worm with a circular profile in the axial section and the related gear [4,11,[17][18][19], whose deficiencies in this field has been fulfilled by this study.The processing of the worm wheel connected to this worm was examined in this paper with the aim of determining the possible locations of undercutting occurring during production.The cutting edge of the worm gear hob made from a worm with strict manufacturing geometrical conditions forms the tooth surface of the gear.According to the tests carried out by means of matrix algebra procedures, as a result of a computer program run with specified data, the points of the characteristic curves of the worm that make up the nodes can create singular points on the tooth surface of the gear.In order to avoid undercutting, it is advisable to exclude knot lines when determining the limits of the worm during designing.

Conclusions
This paper dealt with the problematics of avoiding undercutting in the general mathematical kinematic model.The undercut can be mapped in the case of a helical drive with a circular profile in the axial section by taking into account the mutual influence of the relationships of the interacting mathematical parameters.If these interacting parameters are performed in one model with one procedure, it is desirable to perform them in the case of other worm drives as well.

1 δFigure 1 .
Figure 1.The frames defined for the analysis of the production geometry of the surface Σ 1 of the worm and the Σ 2 of the wheel, based on[18].

Figure 2 .
Figure 2. The worm gear drive and the hob derived from the worm [17].

Figure 2 .
Figure 2. The worm gear drive and the hob derived from the worm [17].

Figure 2 .
Figure 2. The worm gear drive and the hob derived from the worm [17].

Figure 5 .
Figure 5. Contact lines with the LA and LB curves on the worm surface Σ1.

Figure 6 .
Figure 6.The manufactured worm wheel drive pair.

Figure 5 .
Figure 5. Contact lines with the L A and L B curves on the worm surface Σ 1 .

Machines 2023 , 14 Figure 5 .
Figure 5. Contact lines with the LA and LB curves on the worm surface Σ1.

Figure 6 .
Figure 6.The manufactured worm wheel drive pair.

Figure 6 .
Figure 6.The manufactured worm wheel drive pair.