The coordinate system by cutting the worm wheel is the same like shown in fig. 1. The description of one side of the tooth surface is presented. The worm wheel tooth surface is generated by the hob cutter model (Fig. 6), because the thread length of tool should be longer than in the worm. The presented gear hob model is without intermediate cutting edges. The continuous generative surface of the hob cutter between the extreme cutting edges is established. It is noted that worm wheel surface is divided into three regions (Fig. 5) [5]. Region II is the envelope to the family of contact lines of the globoidal worm gear. Region I and III is formed by a first cutting edge of worm hob cutter (Fig. 6) [5]. One extreme cutting edge of the tool forms one side of worm wheel tooth and the second edge forms the another flank.
Figure 5 Illustrative figure of tooth side of globoidal worm wheel with marked region I, II, III.
Figure 6 Illustrative figure of tool model with marked extreme cutting edges.
The condition of existence of an envelope is represented by the equation of meshing:
$$\begin{array}{}n\cdot v=0\end{array}$$(22)
where:

n (n_{x}, n_{y},n_{z}) normal vector to the surface,

v (v_{x}, V_{y}, V_{z}) tangent vector.
Relationship between rotation of worm wheel
$\begin{array}{}{\phi}_{2}^{{}^{\prime}}\end{array}$ to rotation of tool
$\begin{array}{}{\phi}_{1}^{{}^{\prime}}\end{array}$ is given by homogenous matrix
$\begin{array}{}{M}_{{2}^{\prime}{1}^{\prime}}^{{}^{\prime}}\end{array}$:
$$\begin{array}{}{M}_{{2}^{\prime}{1}^{\prime}}^{{}^{\prime}}={M}_{{2}^{\prime}2}^{{}^{\prime}}\cdot {M}_{21}\cdot {M}_{{11}^{\prime}}^{{}^{\prime}}\end{array}$$(23)
The developed form of homogenous matrix
$\begin{array}{}{M}_{{2}^{\prime}{1}^{\prime}}^{{}^{\prime}}\end{array}$ is presented:
$$\begin{array}{}{M}_{{2}^{\prime}{1}^{\prime}}^{{}^{\prime}}=\left[\begin{array}{cccc}\mathrm{cos}({\phi}_{1}^{{}^{\prime}})& \mathrm{sin}({\phi}_{1}^{{}^{\prime}})& 0& 0\\ \mathrm{cos}({\phi}_{2}^{{}^{\prime}})sin({\phi}_{1}^{{}^{\prime}})& \mathrm{cos}({\phi}_{2}^{{}^{\prime}})\cdot \mathrm{cos}({\phi}_{1}^{{}^{\prime}})& \mathrm{sin}({\phi}_{2}^{{}^{\prime}})& a\cdot \mathrm{cos}({\phi}_{2}^{{}^{\prime}})\\ sin({\phi}_{1}^{{}^{\prime}})sin({\phi}_{2}^{{}^{\prime}})& cos({\phi}_{1}^{{}^{\prime}})sin({\phi}_{2}^{{}^{\prime}})& \mathrm{cos}({\phi}_{2}^{{}^{\prime}})& a\cdot sin({\phi}_{2}^{{}^{\prime}})\\ 0& 0& 0& 1\end{array}\right]\end{array}$$(24)
The normal vector
$\begin{array}{}{n}_{1}^{(2{}^{\prime})}\end{array}$ can be calculated on the basis of worm tooth surface description. It is expressed as:
$$\begin{array}{}{\displaystyle {n}_{1}^{(2{}^{\prime})}=\left[\begin{array}{c}{n}_{x1}^{(2{}^{\prime})}\\ {n}_{y1}^{(2{}^{\prime})}\\ {n}_{z1}^{(2{}^{\prime})}\end{array}\right]={L}_{{2}^{\prime}{1}^{\prime}}\cdot (\frac{\mathrm{\partial}{r}_{1}^{({1}^{\prime})}}{\mathrm{\partial}{\phi}_{1}}\times \frac{\mathrm{\partial}{r}_{1}^{({1}^{\prime})}}{\mathrm{\partial}u})}\end{array}$$(25)
where: L_{2′1′} – is the matrix of transformation from 1′ do 2′.
$$\begin{array}{}{L}_{{2}^{\prime}{1}^{\prime}}=\left[\begin{array}{ccc}cos({\phi}_{1}^{{}^{\prime}})& sin({\phi}_{1}^{{}^{\prime}})& 0\\ cos({\phi}_{2}^{{}^{\prime}})\cdot sin({\phi}_{1}^{{}^{\prime}})& \mathrm{cos}({\phi}_{2}^{{}^{\prime}})\cdot \mathrm{cos}({\phi}_{1}^{{}^{\prime}})& sin({\phi}_{1}^{{}^{\prime}})\\ sin({\phi}_{1}^{{}^{\prime}})sin({\phi}_{2}^{{}^{\prime}})& cos({\phi}_{1}^{{}^{\prime}})sin({\phi}_{2}^{{}^{\prime}})& cos({\phi}_{2}^{{}^{\prime}})\end{array}\right]\end{array}$$(26)
L_{2′1′} is obtained by crossing out the last row and the last column of the homogeneous matrix of transformation (24).
In equation (25) the partial derivative of position vector
$\begin{array}{}{r}_{1}^{({1}^{\prime})}\end{array}$ to surface parameterφ_{1} and u is calculated. Partial derivative
$\begin{array}{}\frac{\mathrm{\partial}{r}_{1}^{({1}^{\prime})}}{\mathrm{\partial}{\phi}_{1}}\end{array}$ is obtained by substituting in equation (21)for φ_{2} = φ_{1} · i(φ_{2} – the auxiliary surface parameter). The expression
$\begin{array}{}\frac{\mathrm{\partial}{r}_{1}^{({1}^{\prime})}}{\mathrm{\partial}u}\end{array}$ can be calculated after extending the equation (21) by the parametric equation of tooth profile x_{1}(u), y_{1}(u)and z_{1}(u).
Tangent vector can be calculated based on kinematics of worm gear machining. Tangent vector is given by the following expression:
$$\begin{array}{}{\displaystyle {v}_{1}^{(2{}^{\prime})}=\left[\begin{array}{c}{v}_{x1}^{(2)}\\ {v}_{y1}^{(2{}^{\prime})}\\ {v}_{z1}^{(2{}^{\prime})}\end{array}\right]=\frac{d{r}_{1}^{({2}^{\prime})}}{d{\phi}_{2}^{{}^{\prime}}}=\frac{d{M}_{{2}^{\prime}{1}^{\prime}}^{{}^{\prime}}}{d{\phi}_{2}^{{}^{\prime}}}\cdot {r}_{1}^{({1}^{\prime})}}\end{array}$$(27)
The derivative
$\begin{array}{}\frac{d{M}_{{2}^{\prime}{1}^{\prime}}^{{}^{\prime}}}{d{\phi}_{2}^{{}^{\prime}}}\end{array}$ in the eq. (27) is calculated by substituting in eq. (24) for
$\begin{array}{}{\phi}_{1}^{{}^{\prime}}=\frac{{\phi}_{2}^{{}^{\prime}}}{i}.\end{array}$ In the general equation (22) of gear meshing the eq. of surface normal vector
$\begin{array}{}{n}_{1}^{({2}^{\prime})}\end{array}$ (25) and tangent vector
$\begin{array}{}{v}_{1}^{({2}^{\prime})}\end{array}$ (27) are introduced:
$$\begin{array}{}{\displaystyle {n}_{1}^{({2}^{\prime})}\cdot {v}_{1}^{({2}^{\prime})}=\left[\begin{array}{c}{n}_{x1}^{({2}^{\prime})}\\ {n}_{y1}^{(2{}^{\prime})}\\ {n}_{z1}^{(2{}^{\prime})}\end{array}\right]\cdot \left[\begin{array}{c}{v}_{x1}^{({2}^{\prime})}\\ {v}_{y1}^{(2{}^{\prime})}\\ {v}_{z1}^{(2{}^{\prime})}\end{array}\right]=0}\end{array}$$(28)
After solving the eq. (28), for given parameters u the solutions set of p_{1} is obtained. Substituting the solutions to eq. (21) the lines of contact between worm and worm wheel in
$\begin{array}{}{x}_{1}^{{}^{\prime}}{y}_{1}^{{}^{\prime}}{z}_{1}^{{}^{\prime}}\end{array}$ coordinate system are received (Fig. 7):
$$\begin{array}{}{r}_{cl}^{({1}^{\prime})}{r}_{1}^{({1}^{\prime})}({\phi}_{1},u)\end{array}$$(29)
Figure 7 Contact lines shown in
$\begin{array}{}{s}_{1}^{{}^{\prime}}\end{array}$
coordinate system.
Worm surface is in tangent with worm wheel surface at every instant of two lines. One contact line lies in the central plane of worm drive. It is constant and straight. The other contact line is curvilinear and is moving to the first on each worm wheel tooth being in mesh with worm. These lines generate that part of worm wheel, which is described as region II. Region II is generated as the envelope to the family of surface Σ_{1}. This part of worm wheel
$\begin{array}{}({r}_{2\mathrm{\_}Region\mathrm{\_}II}^{({1}^{\prime})})\end{array}$ can be obtained by rotation the worm with specified value in rotation range from 0 to 2π (0 ≤ r ≤ 2π) and determining the contact lines.
These lines are selected, which are not lying in the axial section of worm. Then the selected contact lines should be brought to the one tooth side of worm wheel, as shown as example in Fig. 8.
Figure 8 Region II presented in
$\begin{array}{}{s}_{1}^{{}^{\prime}}\end{array}$ coordinate system.
In the developed algorithm contact lines are determined for one tooth side of worm wheel. The tool is set in the base position (in this position the extreme edge of the tool is in the central plane). For this purpose, by solving the equation (28) the rotation parameter
$\begin{array}{}{\phi}_{1}^{{}^{\prime}}\end{array}$ includes the angle of rotation
$\begin{array}{}{\phi}_{1\mathrm{\_}tool\mathrm{\_}base}^{{}^{\prime}},\text{\hspace{0.17em}ie.\hspace{0.17em}}{\phi}_{1}^{{}^{\prime}}={\phi}_{1\mathrm{\_}tool\mathrm{\_}base}^{{}^{\prime}}+r.\end{array}$ The solution of equation (28) for the contact line which doesn’t lie in the axial plane of the tool is inserted into (29). The position of the tool
$\begin{array}{}{\phi}_{1}^{{}^{\prime}}\end{array}$ at which the contact line is determined, is taken into account:
$$\begin{array}{}{r}_{cl}^{\ast ({1}^{\prime})}={M}_{{11}^{{}^{\prime}}}^{{}^{\prime}}\cdot {r}_{cl}^{({1}^{\prime})}\end{array}$$(30)
In the matrix
$\begin{array}{}{M}_{{11}^{{}^{\prime}}}^{{}^{\prime}}\end{array}$ the expression
$\begin{array}{}{\phi}_{1}^{{}^{\prime}}={\phi}_{1\mathrm{\_}tool\mathrm{\_}base}^{{}^{\prime}}\end{array}$ is substituted. Contact line from equation (30) is transformed to base position:
$$\begin{array}{}{r}_{2regio{n}_{II}}^{({1}^{\prime})}(i,j)={M}_{12}\cdot {M}_{{22}^{\prime}}^{{}^{\prime}}\cdot {M}_{21}\cdot {r}_{cl}^{\ast ({1}^{\prime})}\end{array}$$(31)
In the equation (31) in the matrix
$\begin{array}{}{M}_{{2}^{\prime}2}^{{}^{\prime}}\end{array}$ the expression
$\begin{array}{}{\phi}_{2}^{{}^{\prime}}=r\cdot i\end{array}$ is substituted. The set of determined and transformed contact lines represents the region II of worm wheel tooth
$\begin{array}{}({r}_{2\mathrm{\_}region\mathrm{\_}II}^{({1}^{\prime})}).\end{array}$ Index (i, j) by the position vector determines the numerical representation of the surface and refers to the indexes of the solution table (i, j  natural numbers).
Region I and III is formed by a first cutting edge of worm hob cutter. It is equivalent with the extreme contact line$\begin{array}{}({r}_{cl\mathrm{\_}2}^{({1}^{\prime})}\end{array}$ from Fig. 7), lying in central plane. The alternative is transformation of axial profile of tool
$\begin{array}{}{r}_{1({\phi}_{1}={\phi}_{1p})}^{({1}^{\prime})}\end{array}$ using the equation (21) and then the next transformation of the profile to central plane:
$$\begin{array}{}{r}_{1({\phi}_{1}={\phi}_{1p})}^{\ast ({1}^{\prime})}={M}_{{11}^{\prime}}^{{}^{\prime}}\cdot {r}_{1({\phi}_{1}={\phi}_{1p})}^{({1}^{\prime})}\end{array}$$(32)
In the matrix
$\begin{array}{}{M}_{{11}^{\prime}}^{{}^{\prime}}\end{array}$ the expression
$\begin{array}{}{\phi}_{1}^{{}^{\prime}}={\phi}_{1\mathrm{\_}tool\mathrm{\_}base}^{{}^{\prime}}\end{array}$ is substituted. The surface generated by the extreme cutting edge represented in coordinate system of tool
$\begin{array}{}{S}_{1}^{{}^{\prime}}\end{array}$ is obtained by applying the following equation:
$$\begin{array}{}{r}_{2}^{\ast ({1}^{\prime})}={M}^{\ast}{\phantom{\rule{thinmathspace}{0ex}}}_{{1}^{\prime}1}\cdot {r}_{cl\mathrm{\_}2}^{\ast ({1}^{\prime})}\end{array}$$(33)
or
$$\begin{array}{}{r}_{2}^{\ast ({1}^{\prime})}={M}^{\ast}{\phantom{\rule{thinmathspace}{0ex}}}_{{1}^{\prime}1}\cdot {r}_{1({\phi}_{1}={\phi}_{1p})}^{\ast ({1}^{\prime})}\end{array}$$
where:
$\begin{array}{}{r}_{cl\mathrm{\_}2}^{\ast ({1}^{\prime})}\end{array}$ – contact line
$\begin{array}{}{r}_{cl\mathrm{\_}2}^{({1}^{\prime})}\end{array}$ after taking equation (30) into consideration.
In eq. (33) in the matrix
$\begin{array}{}{M}^{\ast}{\phantom{\rule{thinmathspace}{0ex}}}_{{1}^{\prime}1}\end{array}$ the range of parameter φ_{1} is selected to obtain the worm wheel tooth surface of a given width (φ_{1p} ≤ φ_{1} ≤ φ_{1k}). In the fig. 9 the surface generated by extreme cutting edge of tool on the basis of eq. (33) is plotted.
Figure 9 Surface generated by extreme cutting edge of the tool.
From the surface shown in Fig. 9 region I and III have to be separated. The two contact lines in the area of the extreme cutting edge of the tool are the boundaries of the regions (Fig. 7). For region I there is the contact line, which doesn’t lie in the axial plane of tool
$\begin{array}{}({r}_{cl\mathrm{\_}1}^{\ast ({1}^{\prime})}={r}_{cl}^{\ast ({1}^{\prime})}(i,1))\end{array}$, marked in Fig. 7). For region III there is the contact line lying in the axial plane
$\begin{array}{}({r}_{cl\mathrm{\_}2}^{\ast ({1}^{\prime})}={r}_{cl}^{\ast ({1}^{\prime})}(i,2)\end{array}$ marked in Fig. 7). The separated region I and III of the worm wheel tooth surface generated during machining through the extreme cutting edge of the tool is shown in Fig. 10.
Figure 10 The separated region I and III of the worm wheel tooth surface generated during machining through the extreme edge of the tool.
The algorithm for selecting the region I
$\begin{array}{}({r}_{2\mathrm{\_}region\mathrm{\_}I}^{({1}^{\prime})})\end{array}$ consists in checking the condition:
$$\begin{array}{}{r}_{2\phantom{\rule{thinmathspace}{0ex}}{x}_{{1}^{\prime}}}^{\ast ({1}^{\prime})}(i,j)<{r}_{cl\mathrm{\_}{x}_{{1}^{\prime}}}^{\ast ({1}^{\prime})}(i,1)\end{array}$$(34)
where:
$\begin{array}{}{r}_{2\phantom{\rule{thinmathspace}{0ex}}{x}_{{1}^{\prime}}}^{\ast ({1}^{\prime})}(i,j)\end{array}$ – element of coordinate table x_{1′}, of worm wheel surface
$\begin{array}{}{r}_{2}^{\ast ({1}^{\prime})}\end{array}$ determined on the basis on equation (33), generated during machining through the extreme edge of the tool,
$\begin{array}{}{r}_{cl\mathrm{\_}{x}_{{1}^{\prime}}}^{\ast ({1}^{\prime})}(i,1)\end{array}$ – element of coordinate table x_{1′} of contact line
$\begin{array}{}{r}_{cl\mathrm{\_}1}^{\ast ({1}^{\prime})}\end{array}$ (Fig. 10), i, 1 – natural numbers.
Equation (34) specifies the range of coordinates (i, j) of the table for region I. It can be expressed as:
$$\begin{array}{}{r}_{2\mathrm{\_}region\mathrm{\_}I}^{({1}^{\prime})}(i,j)={r}_{2}^{\ast ({1}^{\prime})}(i,j)\end{array}$$(35)
where: i, j are satisfying the condition (34).
The algorithm for selecting the region III
$\begin{array}{}({r}_{2\mathrm{\_}region\mathrm{\_}III}^{({1}^{\prime})})\end{array}$ is analogous and consists in checking the condition:
$$\begin{array}{}{r}_{2\phantom{\rule{thinmathspace}{0ex}}{x}_{{1}^{\prime}}}^{\ast ({1}^{\prime})}(i,j)>{r}_{cl\mathrm{\_}{x}_{{1}^{\prime}}}^{\ast ({1}^{\prime})}(i,2)\end{array}$$(36)
where:
$\begin{array}{}{r}_{cl\mathrm{\_}{x}_{{1}^{\prime}}}^{\ast ({1}^{\prime})}(i,2)\end{array}$ – element of coordinate table x_{1′} of contact line
$\begin{array}{}{r}_{cl\mathrm{\_}2}^{\ast ({1}^{\prime})}\end{array}$ (Fig. 10).
Equation (36) specifies the range of coordinates (i, j) of the table for region III. It can be expressed as
$$\begin{array}{}{r}_{2\mathrm{\_}region\mathrm{\_}III}^{({1}^{\prime})}(i,j)={r}_{2}^{\ast ({1}^{\prime})}(i,j)\end{array}$$(37)
where: i, j are satisfying the condition (36).
The worm wheel tooth surface is generated by the combination of the regions I, II, III (Fig. 11):
Figure 11 The worm wheel tooth surface of globoidal worm drive generated by the tool with straight profile.
$$\begin{array}{}{r}_{2}^{({1}^{\prime})}={r}_{2\mathrm{\_}region\mathrm{\_}I}^{({1}^{\prime})}\cup {r}_{2\mathrm{\_}region\mathrm{\_}II}^{({1}^{\prime})}\cup {r}_{2\mathrm{\_}region\mathrm{\_}III}^{({1}^{\prime})}\end{array}$$(38)
The worm wheel tooth profile in region I and III is straight and in the middle part – region II is concave.
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