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

Abstract

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.

1. 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 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 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.

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, γ¹² 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 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 γ¹² 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 Σ₁ and the derived gear tooth surface Σ₂ to differentiate between generator and generated surfaces.

For the analysis, the vector parametric form of the regulator helical surface Σ₁ in the coordinate system K_1F will be suitable:

$$r_{1F}=r_{1F}\left(\eta ,\vartheta \right)$$

(1)

where η is the internal distance parameter and ϑ is the internal angle parameter. The regulator surface Σ₁ is free from singularities if the normal vectors exist, so the following condition is fulfilled:

$$\frac{\partial r_{1F}}{\partial \eta }\times \frac{\partial r_{1F}}{\partial \vartheta }=n_{1F}\ne 0$$

(2)

The transformation matrix between the coordinate system K_1F and the coordinate system K_2F can be determined based on Figure 1

$$M_{2\mathrm{F},1\mathrm{F}}=M_{2\mathrm{F},2}\cdot M_{2,\mathrm{K}}\cdot M_{\mathrm{K},0}\cdot M_{0,1}\cdot M_{1,1\mathrm{F}}$$

(3)

$$M_{1\mathrm{F},2\mathrm{F}}=M_{1\mathrm{F},1}\cdot M_{1,0}\cdot M_{0,\mathrm{K}}\cdot M_{\mathrm{K},2}\cdot M_{2,2\mathrm{F}}$$

(4)

The transformation matrices M_1,1F and M_1F,1 based on Figure 1 are as follows

$$M_{1,1\mathrm{F}}=\left[\begin{array}{cccc}\cos {\phi }_1& -\sin {\phi }_1& 0& 0\\ \sin {\phi }_1& \cos {\phi }_1& 0& 0\\ 0& 0& 1& -{\mathrm{z}}_{\mathrm{ax}}\\ 0& 0& 0& 1\end{array}\right]\hspace{1em}\hspace{1em}{M}_{1\mathrm{F},1}=\left[\begin{array}{cccc}\cos {\phi }_1& \sin {\phi }_1& 0& 0\\ -\sin {\phi }_1& \cos {\phi }_1& 0& 0\\ 0& 0& 1& {\mathrm{z}}_{\mathrm{ax}}\\ 0& 0& 0& 1\end{array}\right]$$

(5)

The transformation matrices M_0,1 and M_1,0 based on Figure 1 are as follows

$$M_{0,1}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& {\phi }_1\cdot {\mathrm{p}}_{\mathrm{a}}\\ 0& 0& 0& 1\end{array}\right] M_{1,0}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& -{\phi }_1\cdot {\mathrm{p}}_{\mathrm{a}}\\ 0& 0& 0& 1\end{array}\right]$$

(6)

The transformation matrices M_K,0 and M_0,K based on Figure 1 are as follows

$$M_{\mathrm{K},0}=\left[\begin{array}{cccc}{\cos \mathsf{\gamma }}^{12}& 0& {\sin \mathsf{\gamma }}^{12}& -\mathrm{c}\cdot {\cos \mathsf{\gamma }}^{12}\\ 0& 1& 0& \mathrm{a}-{\phi }_1\cdot {\mathrm{p}}_{\mathrm{r}}\\ -{\sin \mathsf{\gamma }}^{12}& 0& {\cos \mathsf{\gamma }}^{12}& \mathrm{c}\cdot {\sin \mathsf{\gamma }}^{12}\\ 0& 0& 0& 1\end{array}\right] M_{0,\mathrm{K}}=\left[\begin{array}{cccc}{\cos \mathsf{\gamma }}^{12}& 0& -{\sin \mathsf{\gamma }}^{12}& \mathrm{c}\\ 0& 1& 0& -\mathrm{a}+{\phi }_1\cdot {\mathrm{p}}_{\mathrm{r}}\\ {\sin \mathsf{\gamma }}^{12}& 0& {\cos \mathsf{\gamma }}^{12}& 0\\ 0& 0& 0& 1\end{array}\right]$$

(7)

The transformation matrices M_2,K and M_K,2 based on Figure 1 are as follows

$$M_{2,\mathrm{K}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \mathsf{\alpha }& \sin \mathsf{\alpha }& 0\\ 0& -\sin \mathsf{\alpha }& \cos \mathsf{\alpha }& 0\\ 0& 0& 0& 1\end{array}\right] M_{\mathrm{K},2}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \mathsf{\alpha }& -\sin \mathsf{\alpha }& 0\\ 0& \sin \mathsf{\alpha }& \cos \mathsf{\alpha }& 0\\ 0& 0& 0& 1\end{array}\right]$$

(8)

The transformation matrices M_2F,2 and M_2,2F based on Figure 1 are as follows

$$M_{2\mathrm{F},2}=\left[\begin{array}{cccc}\cos {\phi }_2& -\sin {\phi }_2& 0& 0\\ \sin {\phi }_2& \cos {\phi }_2& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right] M_{2,2\mathrm{F}}=\left[\begin{array}{cccc}\cos {\phi }_2& \sin {\phi }_2& 0& 0\\ -\sin {\phi }_2& \cos {\phi }_2& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(9)

The matrix of the transformation from the frame K_2F to the frame K_1F is as follows

$$\begin{array}{l}{M}_{1\mathrm{F},2\mathrm{F}}=\\ \left[\begin{array}{cccc}\begin{array}{l}-\cos \mathsf{\alpha }\cdot \sin {\phi }_1\cdot \sin {\phi }_2\\ +\sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\cdot \cos {\phi }_1\cdot \sin {\phi }_2\\ +{\cos \mathsf{\gamma }}^{12}\cdot \cos {\phi }_1\cdot \cos {\phi }_2\\ \end{array}& \begin{array}{l}+{\cos \mathsf{\gamma }}^{12}\cdot \cos {\phi }_1\cdot \sin {\phi }_2\\ +\cos \mathsf{\alpha }\cdot \sin {\phi }_1\cdot \cos {\phi }_2\\ -\sin \mathsf{\alpha }\cdot \sin {\phi }_1\cdot \cos {\phi }_2\\ \end{array}& \begin{array}{l}-{\sin \mathsf{\alpha }}_0\cdot \sin {\phi }_1\\ -\cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \cos {\phi }_1\end{array}& \begin{array}{l}-\mathrm{a}\cdot \sin {\phi }_1\\ +\mathrm{c}\cdot \cos {\phi }_1\\ +{\mathrm{p}}_{\mathrm{r}}\cdot {\phi }_1\cdot \sin {\phi }_1\\ \end{array}\\ \begin{array}{l}-\sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\cdot \sin {\phi }_1\cdot \sin {\phi }_2\\ -\cos \mathsf{\alpha }\cdot \cos {\phi }_1\cdot \sin {\phi }_2\\ -{\cos \mathsf{\gamma }}^{12}\cdot \sin {\phi }_1\cdot \cos {\phi }_2\\ \\ \end{array}& \begin{array}{l}-{\cos \mathsf{\gamma }}^{12}\cdot \sin {\phi }_1\cdot \sin {\phi }_2\\ +\sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\cdot \sin {\phi }_1\cdot \cos {\phi }_2\\ +\cos \mathsf{\alpha }\cdot \cos {\phi }_1\cdot \cos {\phi }_2\\ \\ \end{array}& \begin{array}{l}+\cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \sin {\phi }_1\\ -\sin \mathsf{\alpha }\cdot \cos {\phi }_1\\ \end{array}& \begin{array}{l}-\mathrm{a}\cdot \cos {\phi }_1\\ -\mathrm{c}\cdot \sin {\phi }_1\\ +{\mathrm{p}}_{\mathrm{r}}\cdot {\phi }_1\cdot \sin {\phi }_1\\ \end{array}\\ \begin{array}{l}-\sin \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\cdot \sin {\phi }_2\\ +{\sin \mathsf{\gamma }}^{{}_{12}}\cdot \cos {\phi }_2\\ \end{array}& \begin{array}{l}+{\sin \mathsf{\gamma }}^{12}\cdot \sin {\phi }_2\\ +{\sin \mathsf{\gamma }}^{12}\cdot \cos {\phi }_2\\ \end{array}& \begin{array}{l}\cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ \end{array}& \begin{array}{l}{\mathrm{p}}_{\mathrm{a}}\cdot {\phi }_1+{\mathrm{z}}_{\mathrm{ax}}\\ \end{array}\\ 0& 0& 0& 1\end{array}\right]\end{array}$$

(10)

The matrix of the transformation from the frame K_1F to the frame K_2F is as follows

$$\begin{array}{l}{M}_{2\mathrm{F},1\mathrm{F}}=\\ \left[\begin{array}{cccc}\begin{array}{l}-\cos \mathsf{\alpha }\cdot \sin {\phi }_1\\ \cdot \sin {\phi }_2\\ +\sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \cos {\phi }_1\cdot \sin {\phi }_2\\ +{\cos \mathsf{\gamma }}^{12}\cdot \cos {\phi }_1\\ \cdot \cos {\phi }_2\\ \\ \end{array}& \begin{array}{l}-\sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \sin {\phi }_1\cdot \sin {\phi }_2\\ -\cos \mathsf{\alpha }\cdot \cos {\phi }_1\\ \cdot \sin {\phi }_2\\ -{\cos \mathsf{\gamma }}^{12}\cdot \sin {\phi }_1\\ \cdot \cos {\phi }_2\\ \\ \end{array}& \begin{array}{l}-{\sin \mathsf{\alpha }}_0\cdot {\cos \mathsf{\gamma }}^{12}\\ \cdot \sin {\phi }_2\\ +{\sin \mathsf{\gamma }}^{12}\cdot \cos {\phi }_2\\ \\ \end{array}& \begin{array}{l}-\left(\begin{array}{l}\mathrm{a}\cdot \cos \mathsf{\alpha }\\ +\mathrm{c}\cdot \sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ -{\mathrm{z}}_{\mathrm{ax}}\cdot \sin \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\end{array}\right)\cdot \sin {\phi }_2\\ -\left(\begin{array}{l}{\mathrm{p}}_{\mathrm{a}}\cdot \sin \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ -{\mathrm{p}}_{\mathrm{r}}\cdot \cos \mathsf{\alpha }\end{array}\right)\cdot {\phi }_1\cdot \sin {\phi }_1\\ -\left(\begin{array}{l}\mathrm{c}\cdot {\cos \mathsf{\gamma }}^{12}\\ +{\mathrm{z}}_{\mathrm{ax}}\cdot {\sin \mathsf{\gamma }}^{12}\end{array}\right)\cdot \cos {\phi }_2\\ +{\mathrm{p}}_{\mathrm{a}}\cdot {\sin \mathsf{\gamma }}^{12}\cdot {\phi }_1\cdot \cos {\phi }_2\\ \end{array}\\ \begin{array}{l}+{\cos \mathsf{\gamma }}^{12}\cdot \cos {\phi }_1\\ \cdot \sin {\phi }_2\\ +\cos \mathsf{\alpha }\cdot \sin {\phi }_1\\ \cdot \cos {\phi }_2\\ -\sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \cos {\phi }_1\cdot \cos {\phi }_2\\ \\ \end{array}& \begin{array}{l}-{\cos \mathsf{\gamma }}^{12}\cdot \sin {\phi }_1\\ \cdot \sin {\phi }_2\\ +\sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \sin {\phi }_1\cdot \cos {\phi }_2\\ +\cos \mathsf{\alpha }\cdot \cos {\phi }_1\\ \cdot \cos {\phi }_2\\ \\ \end{array}& \begin{array}{l}+{\sin \mathsf{\gamma }}^{12}\cdot \sin {\phi }_2\\ -\sin \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ \cdot \cos {\phi }_2\\ \\ \end{array}& \begin{array}{l}-\left(\begin{array}{l}\mathrm{c}\cdot {\cos \mathsf{\gamma }}^{12}\\ +{\mathrm{z}}_{\mathrm{ax}}\cdot {\sin \mathsf{\gamma }}^{12}\end{array}\right)\cdot \sin {\phi }_2\\ +{\mathrm{p}}_{\mathrm{a}}\cdot {\sin \mathsf{\gamma }}^{12}\cdot {\phi }_1\cdot \sin {\phi }_2\\ +\left(\begin{array}{l}\mathrm{a}\cdot \mathrm{sos}\mathsf{\alpha }\\ +\mathrm{c}\cdot \sin \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ -{\mathrm{z}}_{\mathrm{ax}}\cdot \sin \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\end{array}\right)\cdot \cos {\phi }_2\\ +\left(\begin{array}{l}{\mathrm{p}}_{\mathrm{a}}\cdot \sin \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ -{\mathrm{p}}_{\mathrm{r}}\cdot \cos \mathsf{\alpha }\end{array}\right)\cdot {\phi }_1\cdot \cos {\phi }_2\\ \end{array}\\ \begin{array}{l}-\sin \mathsf{\alpha }\cdot \sin {\phi }_1\\ -\cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{{}_{12}}\\ \cdot \cos {\phi }_1\\ \end{array}& \begin{array}{l}+\cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \sin {\phi }_1\\ -\sin \mathsf{\alpha }\cdot \cos {\phi }_1\\ \end{array}& \begin{array}{l}+\cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ \end{array}& \begin{array}{l}+\left(\begin{array}{l}{\mathrm{p}}_{\mathrm{a}}\cdot \cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ +{\mathrm{p}}_{\mathrm{r}}\cdot \sin \mathsf{\alpha }\end{array}\right)\cdot {\phi }_1\\ \mathrm{a}\cdot \sin \mathsf{\alpha }+\mathrm{c}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ -{\mathrm{z}}_{\mathrm{ax}}\cdot \cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ \end{array}\\ 0& 0& 0& 1\end{array}\right]\end{array}$$

(11)

The v_2F⁽¹²⁾ relative velocity vector between surface Σ₁ and surface Σ₂ 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

$$v_{2\mathrm{F}}^{\left(12\right)}=\frac{d{M}_{2F,1F}}{dt}\cdot r_{1\mathrm{F}}$$

(12)

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

$$v_{1\mathrm{F}}^{\left(12\right)}=M_{1F,2F}\cdot \frac{d{M}_{2F,1F}}{dt}\cdot r_{1\mathrm{F}}$$

(13)

where the “kinematic transformer” matrix is based on [18]:

$$P_{1a}=M_{1F,2F}\cdot \frac{d{M}_{2F,1F}}{dt}$$

(14)

where the P_1a is as follows

$$\begin{array}{l}{P}_{1\mathrm{a}}=\\ \left[\begin{array}{cccc}\begin{array}{l}0\\ \\ \end{array}& \begin{array}{l}-1-\mathrm{i}\cdot \cos \mathsf{\alpha }\\ \cdot {\cos \mathsf{\gamma }}^{12}\\ \\ \end{array}& \begin{array}{l}\mathrm{i}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \sin {\phi }_1\\ -\mathrm{i}\cdot \sin \mathsf{\alpha }\cdot \cos {\phi }_1\\ \end{array}& \begin{array}{l}-\left(\begin{array}{l}\mathrm{i}\cdot \mathrm{c}\cdot \cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ +{\mathrm{a}}_0+{\mathrm{p}}_{\mathrm{r}}\\ +\mathrm{i}\cdot {\mathrm{z}}_{\mathrm{ax}}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\end{array}\right)\cdot \sin {\phi }_1\\ +\left(\begin{array}{l}\mathrm{i}\cdot {\mathrm{p}}_{\mathrm{a}}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ +{\mathrm{p}}_{\mathrm{r}}\end{array}\right)\cdot {\phi }_1\cdot \sin {\phi }_1\\ -\left(\begin{array}{l}\mathrm{i}\cdot {\mathrm{a}}_0\cdot \cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ -\mathrm{c}-\mathrm{i}\cdot {\mathrm{z}}_{\mathrm{ax}}\cdot \sin \mathsf{\alpha }\end{array}\right)\cdot \cos {\phi }_1\\ -\mathrm{i}\cdot \left(\begin{array}{l}{\mathrm{p}}_{\mathrm{a}}\cdot \sin \mathsf{\alpha }\\ -{\mathrm{p}}_{\mathrm{r}}\cdot \cos \mathsf{\alpha }\\ \cdot {\cos \mathsf{\gamma }}^{12}\end{array}\right)\cdot {\phi }_1\cdot \cos {\phi }_1\\ \begin{array}{c}\\ \end{array}\end{array}\\ \begin{array}{l}1+\mathrm{i}\cdot \cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ \\ \end{array}& \begin{array}{l}0\\ \\ \end{array}& \begin{array}{l}\mathrm{i}\cdot \sin \mathsf{\alpha }\cdot \sin {\phi }_1\\ +\mathrm{i}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \cos {\phi }_1\\ \end{array}& \begin{array}{l}\left(\begin{array}{l}\mathrm{i}\cdot {\mathrm{a}}_0\cdot \cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ -\mathrm{c}-\mathrm{i}\cdot {\mathrm{z}}_{\mathrm{ax}}\cdot \sin \mathsf{\alpha }\end{array}\right)\cdot \sin {\phi }_1\\ +\mathrm{i}\cdot \left(\begin{array}{l}{\mathrm{p}}_{\mathrm{a}}\cdot \sin \mathsf{\alpha }\\ -{\mathrm{p}}_{\mathrm{r}}\cdot \cos \mathsf{\alpha }\\ \cdot {\cos \mathsf{\gamma }}^{12}\end{array}\right)\cdot {\phi }_1\cdot \sin {\phi }_1\\ -\left(\begin{array}{l}\mathrm{i}\cdot \mathrm{c}\cdot \cos \mathsf{\alpha }\cdot {\cos \mathsf{\gamma }}^{12}\\ +{\mathrm{a}}_0+{\mathrm{p}}_{\mathrm{r}}\\ +\mathrm{i}\cdot {\mathrm{z}}_{\mathrm{ax}}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\end{array}\right)\cdot \cos {\phi }_1\\ +\left(\begin{array}{l}\mathrm{i}\cdot {\mathrm{p}}_{\mathrm{a}}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ +{\mathrm{p}}_{\mathrm{r}}\end{array}\right)\cdot {\phi }_1\cdot \cos {\phi }_1\\ \begin{array}{c}\\ \end{array}\end{array}\\ \begin{array}{l}-\mathrm{i}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \sin {\phi }_1\\ +\mathrm{i}\cdot \sin \mathsf{\alpha }\cdot \cos {\phi }_1\end{array}& \begin{array}{l}\\ -\mathrm{i}\cdot \sin \mathsf{\alpha }\cdot \sin {\phi }_1\\ -\mathrm{i}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ \cdot \cos {\phi }_1\\ \end{array}& \begin{array}{l}0\\ \end{array}& \begin{array}{l}-\left(\begin{array}{l}{\mathrm{p}}_{\mathrm{a}}+\\ \mathrm{i}\cdot {\mathrm{p}}_{\mathrm{r}}\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\end{array}\right)\cdot {\phi }_1\\ -\mathrm{i}\cdot {\mathrm{a}}_0\cdot \cos \mathsf{\alpha }\cdot {\sin \mathsf{\gamma }}^{12}\\ -\mathrm{i}\cdot \mathrm{c}\cdot \sin \mathsf{\alpha }+{\mathrm{p}}_{\mathrm{a}}+{\mathrm{z}}_{\mathrm{ax}}\\ \begin{array}{c}\\ \end{array}\end{array}\\ 0& 0& 0& 0\end{array}\right]\end{array}$$

(15)

The equation of meshing can be written in the following form

$$n_{1F}\cdot v_{1F}=f\left(\eta ,\vartheta ,{\phi }_1\right)=0$$

(16)

The tooth surface Σ₂ 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

$$\left.\begin{array}{l}f\left(\eta ,\vartheta ,{\phi }_1\right)=0\\ r_{1\mathrm{F}}=r_{1\mathrm{F}}\left(\eta ,\vartheta \right)\\ r_{2\mathrm{F}}=M_{2\mathrm{F},1\mathrm{F}}\cdot r_{1\mathrm{F}}\end{array}\right\}$$

(17)

The geometric location of those points of the l_φ1 contact curves occurring for any φ₁ parameter should be determined on the generator surface Σ₁, which results in singular points on the generated gear tooth surface Σ₂.

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 $r=r\left(\eta ,\vartheta \right)$interpreted on a simply connected region of the plane $\left(\eta ,\vartheta \right)$, where

(a) 

the mapping defined by$r=r\left(\eta ,\vartheta \right)$is topological

(b) 

$r=r\left(\eta ,\vartheta \right)$is continuously differentiable

(c) 

vectors$\partial r/\partial \eta$and$\partial r/\partial \vartheta$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 Σ₁ that create the singular points on the regulated surface Σ₂, 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 Σ₁, but the generated surface Σ₂ 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

$$M_{xy}=\left(\begin{array}{ccc}\frac{\partial x_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial x_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fx}}^{\left(12\right)}\\ \frac{\partial y_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial y_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fy}}^{\left(12\right)}\\ \frac{\partial f}{\partial \eta }& \frac{\partial f}{\partial \vartheta }& -\frac{\partial f}{\partial {\phi }_1}\cdot \frac{\partial {\phi }_1}{\partial t_{}}\end{array}\right)$$

(18)

be${\mathsf{\Delta }}_{xy}$in the mathematical kinematical model.

Definition 4.

Let the value of the determinant of matrix

$$M_{yz}=\left(\begin{array}{ccc}\frac{\partial y_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial y_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fy}}^{\left(12\right)}\\ \frac{\partial z_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial z_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fz}}^{\left(12\right)}\\ \frac{\partial f}{\partial \eta }& \frac{\partial f}{\partial \vartheta }& -\frac{\partial f}{\partial {\phi }_1}\cdot \frac{\partial {\phi }_1}{\partial t_{}}\end{array}\right)$$

(19)

be ${\mathsf{\Delta }}_{yz}$in the mathematical kinematical model.

Definition 5.

Let the value of the determinant of matrix

$$M_{zx}=\left(\begin{array}{ccc}\frac{\partial z_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial z_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fz}}^{\left(12\right)}\\ \frac{\partial x_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial x_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fx}}^{\left(12\right)}\\ \frac{\partial f}{\partial \eta }& \frac{\partial f}{\partial \vartheta }& -\frac{\partial f}{\partial {\phi }_1}\cdot \frac{\partial {\phi }_1}{\partial t_{}}\end{array}\right)$$

(20)

be ${\mathsf{\Delta }}_{zx}$in the mathematical kinematical model.

Definition 6.

Let the value of the determinant of matrix

$$M_{\eta \vartheta }=\left(\begin{array}{ccc}\frac{\partial x_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial x_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fx}}^{\left(12\right)}\\ \frac{\partial y_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial y_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fy}}^{\left(12\right)}\\ \frac{\partial z_{1\mathrm{F}}^{}}{\partial {\eta }_{}}& \frac{\partial z_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}& -v_{1\mathrm{Fz}}^{\left(12\right)}\end{array}\right)$$

(21)

be${\mathsf{\Delta }}_{\eta \vartheta }$in the mathematical kinematical model.

The contact point C is located on both the generator surface Σ₁ and the generated surface Σ₂ 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 Σ₁ and as point C⁽²⁾ fitted as to the generated surface Σ₂. 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

$$v^{\left(12\right)}=v^{\left(1\right)}-v^{\left(2\right)}$$

(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 Σ₁ generates a singular point on the meshed surface Σ₂, then using the equality of the absolute velocity vectors of the contact points Cⁱ the following equation must be fulfilled:

$$v^{\left(2\right)}=v^{\left(1\right)}+v^{\left(12\right)}=0$$

(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 $f\left(\eta ,\vartheta ,{\phi }_1\right)=0$ according to the time parameter t from Equation (6) helps to filter the points on Σ₁, which generates singular points on Σ₂ as follows:

$$\frac{d}{dt}\left[f\left(\eta ,\vartheta ,{\phi }_1\right)\right]=0$$

(24)

Theorem 1.

In order for the surface Σ₁ with regular points to create singular points on the surface Σ₂ enveloped by it, the sufficient condition is the fulfilment of the next equation

$$F\left(\eta ,\vartheta ,{\phi }_1\right)={\mathsf{\Delta }}_{xy}^2+{\mathsf{\Delta }}_{yz}^2+{\mathsf{\Delta }}_{zx}^2=0$$

(25)

Proof of Theorem 1.

Equations (23) and (24) in the frame K_1F of the mathematical kinematical model result in the forms

$$\frac{\partial r_{1\mathrm{F}}^{}}{\partial {\mathsf{\eta }}_{}}\cdot \frac{\partial \mathsf{\eta }}{\partial \mathrm{t}}+\frac{\partial r_{1\mathrm{F}}^{}}{\partial {\vartheta }_{}}\cdot \frac{\partial \vartheta }{\partial \mathrm{t}}=-v_{1\mathrm{F}}^{\left(12\right)}$$

(26)

$$\frac{\partial f}{\partial {\mathsf{\eta }}_{}}\cdot \frac{\partial \mathsf{\eta }}{\partial \mathrm{t}}+\frac{\partial f}{\partial {\vartheta }_{}}\cdot \frac{\partial \vartheta }{\partial \mathrm{t}}=-\frac{\partial f}{\partial {\phi }_1}\cdot \frac{\partial {\phi }_1}{\partial \mathrm{t}}$$

(27)

contributing to the elimination of points on the surface Σ₁, that generate singular points.

□

From Equations (26) and (27) prescribing $\mathrm{d}{\phi }_1/\mathrm{d}t=1\mathrm{rad}/1\sec$, an overdetermined system of four linear equations arises with two unknowns, which are $\mathrm{d}\eta /\mathrm{d}t$ and $\mathrm{d}\vartheta /\mathrm{d}t$. This system leads to the matrix G_4×3 with rank r = 2 and a certain solution for the unknowns

$$G_{4\times 3}=\left(\begin{array}{cccc}\frac{\partial x_{1\mathrm{F}}^{}}{\partial \eta }& \frac{\partial x_{1\mathrm{F}}^{}}{\partial \vartheta }& -v_{1\mathrm{Fx}}^{\left(12\right)}& \\ \frac{\partial y_{1\mathrm{F}}^{}}{\partial \eta }& \frac{\partial y_{1\mathrm{F}}^{}}{\partial \vartheta }& -v_{1\mathrm{Fy}}^{\left(12\right)}& \\ \frac{\partial z_{1\mathrm{F}}^{}}{\partial \eta }& \frac{\partial z_{1\mathrm{F}}^{}}{\partial \vartheta }& -v_{1\mathrm{Fz}}^{\left(12\right)}& \\ \frac{\partial f}{\partial \eta }& \frac{\partial f}{\partial \vartheta }& -\frac{\partial f}{\partial {\phi }_1}\cdot \frac{\partial {\phi }_1}{\partial t}& \end{array}\right)$$

(28)

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 $M_{\eta \vartheta }$, which respectively take the value ${\mathsf{\Delta }}_1=0$, ${\mathsf{\Delta }}_2=0$, ${\mathsf{\Delta }}_3=0$ and ${\mathsf{\Delta }}_4=0$.

The ${\mathsf{\Delta }}_4=0$ yields to the equation of meshing [9], and it is fulfilled for the contact points of surfaces Σ₁ and Σ₂, which are taken into account during the tests. Thus only the equalities ${\mathsf{\Delta }}_1=0$, ${\mathsf{\Delta }}_2=0$ and ${\mathsf{\Delta }}_3=0$ need to apply to determine the singularity conditions of the surface, which procedure yields to the sufficient condition $F\left(\eta ,\vartheta ,{\phi }_1\right)=0$.

The simultaneous fulfilment of equations

$$r_{1F}=r_{1F}\left(\eta ,\vartheta \right) f\left(\eta ,\vartheta ,{\phi }_1\right)=0F\left(\eta ,\vartheta ,{\phi }_1\right)=0$$

(29)

determines the points on the surface Σ₁ that form singular points on the surface Σ₂.

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.

3. 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 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.

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

$$\left.\begin{array}{l}{x}_{1\mathrm{F}}=-\eta \cdot \sin \vartheta \\ y_{1\mathrm{F}}=\eta \cdot \cos \vartheta \\ z_{1\mathrm{F}}=\mathrm{p}\cdot \vartheta -\sqrt{{\rho }_{\mathrm{ax}}^2-{(K-\eta )}^2}\end{array}\right\}{r}_{1\mathrm{F}}$$

(30)

The normal vectors of this worm surface can be described as follows

$$\left.\begin{array}{l}{\mathrm{n}}_{1\mathrm{Fx}}=-\mathsf{\eta }\cdot \sin \vartheta \cdot \frac{\mathrm{K}-\mathsf{\eta }}{\sqrt{{\mathsf{\rho }}_{\mathrm{ax}}^2-{(\mathrm{K}-\mathsf{\eta })}^2}}+\mathrm{p}\cdot \cos \vartheta \\ {\mathrm{n}}_{1\mathrm{Fy}}=\mathsf{\eta }\cdot \cos \vartheta \cdot \frac{\mathrm{K}-\mathsf{\eta }}{\sqrt{{\mathsf{\rho }}_{\mathrm{ax}}^2-{(\mathrm{K}-\mathsf{\eta })}^2}}+\mathrm{p}\cdot \sin \vartheta \\ n_{1\mathrm{Fz}}=\mathsf{\eta }\end{array}\right\}{n}_{1\mathrm{F}}$$

(31)

Based on (4) the coordinates of the relative velocity vector $v_{1\mathrm{F}}^{\left(12\right)}$ are calculated in the form

$$\left.\begin{array}{l}{v}_{1\mathrm{Fx}}^{(12)}=-\mathsf{\eta }\cdot \cos \vartheta \cdot \left(1+i_{21}\cdot \cos \gamma \right)+\\ \begin{array}{cc}& \end{array}{i}_{21}\cdot \sin \gamma \cdot \sin {\phi }_1\cdot \left(p\cdot \vartheta -\sqrt{{\mathsf{\rho }}_{\mathrm{ax}}^2-{(\mathrm{K}-\mathsf{\eta })}^2}+c+p\cdot {\phi }_1\right)-i_{21}\cdot a\cdot \cos \gamma \cdot \cos {\phi }_1\\ v_{1\mathrm{Fy}}^{(12)}=-\mathsf{\eta }\cdot \sin \vartheta \cdot \left(1+i_{21}\cdot \cos \gamma \right)+\\ \begin{array}{cc}& \end{array}{i}_{21}\cdot \sin \gamma \cdot \cos {\phi }_1\cdot \left(p\cdot \vartheta -\sqrt{{\mathsf{\rho }}_{\mathrm{ax}}^2-{(\mathrm{K}-\mathsf{\eta })}^2}+c+p\cdot {\phi }_1\right)-i_{21}\cdot a\cdot \cos \gamma \cdot \sin {\phi }_1\\ v_{1\mathrm{Fx}}^{(12)}=-\mathsf{\eta }\cdot i_{21}\cdot \sin \gamma \cdot \cos \left(\vartheta +{\phi }_1\right)-i_{21}\cdot a\cdot \sin \gamma +p\end{array}\right\}{v}_{1\mathrm{F}}^{(12)}$$

(32)

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

$$\left.\begin{array}{c}\partial x_{1\mathrm{F}}/\partial \eta =-\sin \vartheta \\ \partial y_{1\mathrm{F}}/\partial \eta =\cos \vartheta \\ \partial z_{1\mathrm{F}}/\partial \eta =-\left(K-\eta \right)/\sqrt{{\rho }_{\mathrm{ax}}^2-{(K-\eta )}^2}\end{array}\right\}\begin{array}{c}\\ \mathrm{and}\\ \end{array}\left.\begin{array}{c}\partial x_{1\mathrm{F}}/\partial \vartheta =-\eta \cdot \cos \vartheta \\ \partial y_{1\mathrm{F}}/\partial \vartheta =-\eta \cdot \sin \vartheta \\ \partial z_{1\mathrm{F}}/\partial \vartheta =\mathrm{p}\end{array}\right\}$$

(33)

The element $\partial f/\partial \eta$ of the matrix G_4×3 is presented in the equation

$$\begin{array}{l}\frac{\partial f}{\partial \eta }=\frac{\left(\mathrm{K}-2\cdot \mathsf{\eta }\right)\cdot {\rho }_{ax}^2-{(\mathrm{K}-\mathsf{\eta })}^3}{\sqrt{{\left({\mathsf{\rho }}_{\mathrm{ax}}^2-{(\mathrm{K}-\mathsf{\eta })}^2\right)}^3}}\cdot \left[\sin \gamma \cdot \cos \left(\vartheta +{\phi }_1\right)\cdot z_{ax}+a\cdot \cos \gamma \cdot \sin \left(\vartheta +{\phi }_1\right)\right]\\ -p_a\cdot \cos \gamma -2\cdot \eta \cdot \sin \gamma \cdot \cos \left(\vartheta +{\phi }_1\right)-a\cdot \sin \gamma \end{array}$$

(34)

And the element $\partial f/\partial \vartheta$ of the matrix G_4×3 is presented in the following equation

$$\begin{array}{l}\frac{\partial f}{\partial \vartheta }=\eta \cdot \frac{(\mathrm{K}-\mathsf{\eta })}{\sqrt{{\left({\mathsf{\rho }}_{\mathrm{ax}}^2-{(\mathrm{K}-\mathsf{\eta })}^2\right)}^3}}\cdot \left[\sin \gamma \cdot \sin \left(\vartheta +{\phi }_1\right)\cdot z_{ax}+a\cdot \cos \gamma \cdot \cos \left(\vartheta +{\phi }_1\right)\right]+\\ p_a\cdot \sin \gamma \cdot \cos \left(\vartheta +{\phi }_1\right)\cdot z_{ax}+p_a\cdot a\cdot \cos \gamma \cdot \sin \left(\vartheta +{\phi }_1\right)+{\eta }^2\cdot \sin \gamma \cdot \sin \left(\vartheta +{\phi }_1\right)\end{array}$$

(35)

Similarly to the previous calculations, the element $\partial f/\partial {\phi }_1^{}$ of the matrix G_4×3 is as follows

$$\begin{array}{l}\frac{\partial f}{\partial {\phi }_1}=\eta \cdot \frac{(\mathrm{K}-\mathsf{\eta })}{\sqrt{{\left({\mathsf{\rho }}_{\mathrm{ax}}^2-{(\mathrm{K}-\mathsf{\eta })}^2\right)}^3}}\cdot \left[-\sin \gamma \cdot \sin \left(\vartheta +{\phi }_1\right)\cdot z_{ax}+a\cdot \cos \gamma \cdot \cos \left(\vartheta +{\phi }_1\right)\right]+\\ p_a\cdot \sin \gamma \cdot \cos \left(\vartheta +{\phi }_1\right)\cdot z_{ax}+p_a\cdot a\cdot \cos \gamma \cdot \sin \left(\vartheta +{\phi }_1\right)+{\eta }^2\cdot \sin \gamma \cdot \sin \left(\vartheta +{\phi }_1\right)\end{array}$$

(36)

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 output part of the table contains the coordinates of the meshing knotes K_A and K_B, as well as the value of the meshing opening angles β_A and β_B.

The equations represented by matrix G_4×3 allow our computer program to determine the points of curve L on surface Σ₁, which generate singular points on the Σ₂. The qualification of procedure is shown in the case of a particular cylindrical worm gear drive with circle arc profile in worm axial section [11,18].

In case of the worm gear drive with geometrical and precision parameters z₁ = 3, m = 12.5, γ₀ = 21°2′15′′, ρ_ax = 50, ${\overline{S}}_{a1}$ = 10 to set addendum to the reference line on tooth, ${\overline{S}}_{n1}=13\begin{array}{c}+0.0\\ -0.125\end{array}$ size of worm tooth chord, a = 280, d₀₁ = 97.5, H = 117.809722, δ_ax = 24°31′10′′, z₂ = 35, F_r1 = ±0.017, f_p1 = ±0.016, f_γ = ±0.018, f_f = 0.08 [11], the contact points are presented in projection on the coordinate plane [x_1F, y_1F] in Figure 4.

The geometric location of the nodes K_A and K_B occurring for each value of φ₁ are the lines L_A and L_B. The position of L_A and L_B curves is illustrated in Figure 5.

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 Σ₂, it is sufficient to delimit the worm surface Σ₁ 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.

4. 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.

5. 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.