1 Introduction

The most commonly used tooth profile of cylindrical gears in industrial engineering is the involute. The involute gearing has certain limits for external cylindrical gears, such as a low minimum number of teeth and high contact pressure near the base circle. Special gearings can avoid these disadvantages by an adjusted geometry and achieve an improvement of the gear characteristics for certain applications. Modern manufacturing capabilities also allow the advantages of special gears to be implemented economically and precisely. These processes include 5‑axis milling [1] and additive manufacturing [2].

The eccentric cycloid gearing (EC gearing) is a special gearing which uses an eccentric circular arc profile on one gear. This allows extremely low numbers of teeth to be achieved down to the single-tooth pinion, see Fig. 1a. The mating gear has an equidistant of a roulette as the flank profile according to the law of gearing. Due to the circular arc profile, the shape of the roulette is a cycloidal curve. The cycloidal curve is also called a trochoid [3]. Figure 1a shows a closed form of the cycloidal curve which represents the flanks of the gear for the meshing with a single-toothed pinion.

Fig. 1
figure 1

Different EC gearings with the same center distance (a : z1 = 1, z2 = 6. b : z1 = 2, z2 = 6; c : z1 = 12, z2 = 15)

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Based on the descriptions of the geometry, a general definition of EC gearing can be given. The EC gearing is defined by the meshing of a gear with a circular arc profile with a gear with a tooth profile of an equidistant of a trochoid. For external cylindrical gears, this definition is applied to the transverse section. The following study refers only to this type of gears. The gear with the circular arc profile is called an arc gear, the gear with the profile of an equidistant of a trochoid a cycloid gear.

With this definition, the different number of teeth of the arc gear z1 can be chosen. Figure 1b shows an arc gear with z1 = 2, Fig. 1c a gearing with z1 = 12. The number of teeth of the mating cycloid gear is z2 = 6 for Fig. 1a and b and z2 = 15 for Fig. 1c.

Figure 1 shows the different geometries possible for an EC gearing with the same center distance a. For the possibility of comparing the EC gearing with the established involute gearing, a detailed description of the geometry is necessary. This allows the characteristics of comparable gears with different profiles to be examined and evaluated.

2 State of the art

A gearing consisting of a gear with circular arc profile and a gear with the tooth profile of an equidistant of a trochoid is already described in early gear design literature. Authors of standard literature such as Niemann and Winter [4] and Linke et al. [5] describe the basic principle of this gearing under the name cylindrical lantern gear. The exact geometry and gear characteristics are not described in detail. Another type of gearing similar to the EC gearing is the Grisson gearing [6]. In this case, an extremely low number of teeth of the cycloid gear is usually selected with a higher number of teeth of the arc gear. In this case, as well, only the structure is described in the literature without a detailed analysis of the geometry.

One of the first descriptions of the geometry of a gearing with the name EC gearing is given by Stanovskoy et al. [7]. Here, the structure of an EC gearing with a single-toothed arc gear is described. Contact patterns of this EC gearing with a single-toothed pinion and deviations of the center distance are shown by Kazakyavichyus et al. [8]. Li et al. [9] describe the details of the geometry of an EC gearing with a single-toothed arc gear and analyze the undercutting conditions. Li et al. [9] study the tooth contact in terms of the radius of curvature, the contact lines, and the surface of action. They also present results of a loaded tooth contact analysis performed with the finite element method. Batsch et al. [10] apply flank modifications to the EC gearing and determine the ease-off and transmission errors. They also provide comparisons to involute gearing and Wildhaber-Novikov gearing. Stanovskoy et al. describe in different patents [11,12,13,14] the structure of the EC gearing in different designs and applications. A detailed analysis of the geometry is not performed.

The additive manufacturing of a multiple-toothed EC gearing together with a basic description of the geometry is shown by Batsch [15]. Bubenchikov et al. [16] describe a profile cutter for the manufacturing of an EC gearing. Shcherbakov et al. [17] use envelopes for the manufacturing of the EC gearing with a milling process. A possible application for EC gearing is shown by Dubov et al. [18] in the transmission of a geokhod for boring.

For the description of the geometry of the EC gearing, many different methods can be used. The method by Johann and Scheurle [19] can be used in particular to determine the geometry of special gearings. Based on the given geometry of a gear, the geometry of the mating gear is determined with the aid of differential equations derived from the law of gearing. Zimmer et al. [20] show the application of this method to the geometry of an EC gearing with the known geometry of a single-tooth arc gear. An independent description of the two gears is not possible with the method by Johann and Scheurle [19]. Another method of describing the trochoid and its equidistant is shown by Lehmann [21]. The analytical derivation of the curve geometry allows an independent description of the two meshing flanks. Lehmann [21] does not show the complete geometry of a gear, but limits to the curves.

Overall, the literature shows that a complete description of all partial geometries of an arbitrary EC gearing is currently not available. Therefore, this study presents a parameter-based description of the EC gearing, which can be used for further analyses. The developed geometry is used to determine various load-free characteristics.

3 Analytical description of the geometry

For an analytical description of the EC gearing, the individual parts of the geometry must be considered. The first step is to determine the flank geometry of the arc gear. This is represented by a circular arc profile. According to Fig. 2 the vector \(\overrightarrow{\mathrm{p}_{\mathrm{PA}}}\) from the center of the arc gear O1 to the point on the circular arc profile PA can be calculated using the position of the center of the arc profile OA. Therefore, it results using the eccentricity e of the circular arc profile to the gear origin and its arc radius rA:

$$\overrightarrow{\mathrm{p}_{\mathrm{PA}}}=\left(\begin{array}{c} \mathrm{x}_{\mathrm{PA}}\\ \mathrm{y}_{\mathrm{PA}} \end{array}\right)=\mathrm{e}\cdot \left(\begin{array}{c} \cos \left(\Uptheta _{\mathrm{A}}\right)\\ \sin \left(\Uptheta _{\mathrm{A}}\right) \end{array}\right)-\mathrm{r}_{\mathrm{A}}\cdot \left(\begin{array}{c} \cos \left(\Uptheta _{\mathrm{A}}+\upvarphi _{\mathrm{A}}\right)\\ \sin \left(\Uptheta _{\mathrm{A}}+\upvarphi _{\mathrm{A}}\right) \end{array}\right)$$
(1)

xPA is the x‑component, yPA the y‑component of the vector \(\overrightarrow{\mathrm{p}_{\mathrm{PA}}}\). θA is the angle of arc position, φA the angle of arc profile. With Eq. 1, the transverse section of the single-tooth arc gear from Fig. 1a is completely described.

Fig. 2
figure 2

Structure of the circular arc profile of the arc gear

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For multi-tooth gears, the description must be extended. Figure 3 shows the transverse section of an arc gear with three teeth. The right and left flanks of one tooth in this case are described by the same circle. This represents a special case, which is used as a starting point for further considerations. The teeth are connected by a circular fillet, which is common for optimizing the load capacity of gears [22] and has advantages over trochoidal fillets especially for small numbers of teeth [23]. The transition between the circular arc of the flank and the fillet profile is tangential and is defined by the starting angle of the arc profile φA,s, see Fig. 3. The arc profile is limited towards the tooth tip by the ending angle of the arc profile φA,e. The tip radius ra1 can be calculated with:

$$\mathrm{r}_{\mathrm{a}1}=\mathrm{e}-\mathrm{r}_{\mathrm{A}}\cdot \cos \left(\upvarphi _{\mathrm{A}{,}\mathrm{e}}\right)$$
(2)
Fig. 3
figure 3

Structure of the arc gear with multiple circular arcs

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The reference diameter d1 of the arc gear is defined by the reference radius r1. The reference radius r1 is identical to the eccentricity e:

$$\mathrm{d}_{1}=2\cdot \mathrm{r}_{1}=2\cdot \mathrm{e}$$
(3)

All centers of the arc profiles of one arc gear must be on the reference circle to fulfil the law of gearing. This ensures a uniform transmission ratio. The module is a common reference parameter for involute gears, which is also implemented as a reference parameter for the EC gearing. The module m is defined in accordance with the transverse module from ISO 21771 [24]:

$$\mathrm{m}=\frac{\mathrm{d}_{1}}{\mathrm{z}_{1}}=\frac{2\cdot \mathrm{e}}{\mathrm{z}_{1}}$$
(4)

With the same underlying circle for both flanks, the radius of the arc profile determines the tooth thickness. A radius rA,equal can be determined with the law of cosines where the tooth thickness is equal to the space width. This radius rA,equal is adjusted by the factor rA* to the intended arc radius rA:

$$\mathrm{r}_{\mathrm{A}}={\mathrm{r}_{\mathrm{A}}}^{\mathrm{*}}\cdot \mathrm{r}_{\mathrm{A}{,}\text{equal}}={\mathrm{r}_{\mathrm{E}}}^{\mathrm{*}}\cdot \mathrm{e}\cdot \sqrt{2-2\cdot \cos \left(\frac{\uppi }{2\cdot \mathrm{z}_{1}}\right)}$$
(5)

With this definition of the arc radius factor, its influence on the geometry can be determined. The factor must be greater than zero. An upper limit depends on the remaining geometry parameters. With a factor rA* = 1.0 the tooth thickness at the reference circle is equal to the space width.

The assumption made above that the two flanks of one tooth are described by the same circle is generalized in the following. The two flanks are thus described independently of each other. Figure 4 shows an arc gear, where each flank is described by its own circular arc profile. The condition remains that the center of each profile must be located on the reference circle. In the following, symmetrical gear teeth are required, so the radius of each arc profile must be identical. The angle between the flank centers OA,l and OA,r is called tooth thickness angle φs1 and is composed of two separate influences:

$$\upvarphi _{\mathrm{s}1}=\upvarphi _{\mathrm{rA}}+\upvarphi _{\mathrm{j}1}$$
(6)
Fig. 4
figure 4

Structure of the arc gear with varying circular arc profiles

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φrA is the angle between the arc profiles without backlash and φj1 is the backlash angle. With the separation of these two angles, the backlash angle can be applied to the arc gear alone. The definition of the backlash angle φj1 is according to ISO 21771 [24]. φrA describes the angle between the arc profiles of the left and the right flank of one tooth which is needed to adjust a certain tooth thickness without backlash. The tooth thickness on the reference circle can be adjusted with the factor st*. The following equation results from the law of cosines:

$$\upvarphi _{\mathrm{rA}}=2\cdot \arccos \left(\frac{2\cdot \mathrm{e}^{2}-{\mathrm{r}_{\mathrm{A}}}^{2}}{2\cdot \mathrm{e}^{2}}\right)-{\mathrm{s}_{\mathrm{t}}}^{\mathrm{*}}\cdot \frac{\uppi }{\mathrm{z}_{1}}$$
(7)

According to this definition, the tooth thickness factor is in the range of 0 to 2. The influence on the geometry can be shown by means of individual extreme cases for backlash-free gearings (φj1 = 0): In case of st* = 0, the tooth thickness at the reference circle is zero. The space width at the reference circle, on the other hand, becomes zero for the boundary case of st* = 2. For st* = 1, the tooth thickness is equal to the space width at the reference circle. Thus, the factor enables an adjustment of the gearing according to a complementary gearing [5, 25]. If the tooth thickness factor fits the following equation, the centers of the flanks are identical (φrA = 0):

$${\mathrm{s}_{\mathrm{t}}}^{\mathrm{*}}=\frac{2\cdot \mathrm{z}_{1}}{\uppi }\cdot \arccos \left(\frac{2\cdot \mathrm{e}^{2}-{\mathrm{r}_{\mathrm{A}}}^{2}}{2\cdot \mathrm{e}^{2}}\right)$$
(8)

The tooth root is designed as a circular fillet as described above. With the description of the flank geometry shown, the distance qF1 of the center of the circular fillet profile OF1 to the gear center O1 can be calculated using the law of sines:

$$\mathrm{q}_{\mathrm{F}1}=\mathrm{e}\cdot \frac{\sin \left(\upvarphi _{\mathrm{A}{,}\mathrm{s}}\right)}{\sin \left(\uppi \cdot \left(\frac{\mathrm{z}_{1}-1}{\mathrm{z}_{1}}\right)-\upvarphi _{\mathrm{A}{,}\mathrm{s}}-\frac{\upvarphi _{\mathrm{rA}}}{2}\right)}$$
(9)

The determined distance qF1 can be used to calculate the fillet radius rF1 of a tangential circle using the law of cosines:

$$\mathrm{r}_{\mathrm{F}1}=\sqrt{\mathrm{e}^{2}+{\mathrm{q}_{\mathrm{F}1}}^{2}-2\cdot \mathrm{e}\cdot \mathrm{q}_{\mathrm{F}1}\cdot \cos \left(\frac{\uppi }{\mathrm{z}_{1}}+\frac{\upvarphi _{\mathrm{rA}}}{2}\right)}-\mathrm{r}_{\mathrm{A}}$$
(10)

With the description of the fillet geometry, the root diameter df1 as well as the root radius rf1 can be determined as follows:

$$\mathrm{d}_{\mathrm{f}1}=2\cdot \mathrm{r}_{\mathrm{f}1}=2\cdot \left(\mathrm{q}_{\mathrm{F}1}-\mathrm{r}_{\mathrm{F}1}\right)$$
(11)

The geometry of the arc gear is thus described completely in terms of parameters. According to the method of Lehmann [21], the geometry of the circular arc profile of the arc gear is used to generate the flank geometry of the cycloid gear. Figure 5 shows the flank contact in detail. The transmission ratio i is crucial for the meshing of the two gears of the EC gearing. This is determined by the number of teeth [4]:

$$\mathrm{i}=\frac{\mathrm{z}_{2}}{\mathrm{z}_{1}}$$
(12)
Fig. 5
figure 5

Structure of the cycloid gear based on the meshing with the arc profile

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The geometry of a cycloidal curve is strongly determined by the trochoid ratio λ, which is defined with the pitch radius of the arc gear rw1 according to [26]:

$$\uplambda =\frac{\mathrm{e}}{\mathrm{r}_{\mathrm{w}1}}$$
(13)

The trochoid ratio determines whether the resulting trochoid is a normal (λ = 1), shortened (λ < 1), or elongated (λ > 1) cycloid. For the EC gearing, usually only the trochoid with λ < 1 is used.

The pitch radius of the cycloid gear rw2 can be determined with the transmission ratio i:

$$\mathrm{r}_{\mathrm{w}2}=\mathrm{i}\cdot \mathrm{r}_{\mathrm{w}1}$$
(14)

This results in the calculation of the pitch radius of the arc gear rw1 with the center distance a:

$$\mathrm{r}_{\mathrm{w}1}=\frac{\mathrm{a}}{1+\mathrm{i}}$$
(15)

The center distance a can therefore be determined as follows:

$$\mathrm{a}=\mathrm{r}_{\mathrm{w}1}+\mathrm{r}_{\mathrm{w}2}=\mathrm{e}\cdot \frac{1+\mathrm{i}}{\uplambda }$$
(16)

For the design of EC gearings, the specification of the center distance a and the trochoid ratio λ is more convenient. Therefore, the module m can be determined as follows:

$$\mathrm{m}=2\cdot \mathrm{a}\cdot \frac{\uplambda }{\mathrm{z}_{1}+\mathrm{z}_{2}}$$
(17)

The generation of the cycloidal curve is initiated by the angle of revolution ζ, see Fig. 5. This angle describes the revolution of the arc profile around the center O2 of the stationary cycloid gear. According to the rolling condition, a rotation of the circular arc profile around the center O1 results. The rotation angle κ can thus be determined from the transmission ratio i according to:

$$\upkappa =\mathrm{i}\cdot \upzeta$$
(18)

The point vector \(\overrightarrow{\mathrm{p}_{\mathrm{OA}}}\) is determined by vector addition:

$$\overrightarrow{\mathrm{p}_{\mathrm{OA}}}=\left(\begin{array}{c} \mathrm{x}_{\mathrm{OA}}\\ \mathrm{y}_{\mathrm{OA}} \end{array}\right)=\mathrm{a}\cdot \left(\begin{array}{c} -\sin \left(\upzeta \right)\\ \cos \left(\upzeta \right) \end{array}\right)-\mathrm{e}\cdot \left(\begin{array}{c} -\sin \left(\upzeta +\upkappa \right)\\ \cos \left(\upzeta +\upkappa \right) \end{array}\right)$$
(19)

xOA is the x‑component, yOA the y‑component of the vector \(\overrightarrow{\mathrm{p}_{\mathrm{OA}}}\). The point OA moves on a trochoid. The point PC moves on an equidistant curve to this trochoid. The point vector \(\overrightarrow{\mathrm{p}_{\mathrm{PC}}}\) is determined by vector addition with the vector \(\overrightarrow{\mathrm{O}_{\mathrm{A}}\mathrm{P}_{\mathrm{C}}}\) which is normal to the trochoid and points in the direction of the pitch point C:

$$\overrightarrow{\mathrm{p}_{\mathrm{PC}}}=\left(\begin{array}{c} \mathrm{x}_{\mathrm{PC}}\\ \mathrm{y}_{\mathrm{PC}} \end{array}\right)=\overrightarrow{\mathrm{p}_{\mathrm{OA}}}+\overrightarrow{\mathrm{O}_{\mathrm{A}}\mathrm{P}_{\mathrm{C}}}=\mathrm{a}\cdot \left(\begin{array}{c} -\sin \left(\upzeta \right)\\ \cos \left(\upzeta \right) \end{array}\right)-\mathrm{e}\cdot \left(\begin{array}{c} -\sin \left(\upzeta +\upkappa \right)\\ \cos \left(\upzeta +\upkappa \right) \end{array}\right)-\mathrm{r}_{\mathrm{A}}\cdot \left(\begin{array}{c} -\sin \left(\upzeta -\upxi \right)\\ \cos \left(\upzeta -\upxi \right) \end{array}\right)$$
(20)

xPC is the x‑component, yPC the y‑component of the vector \(\overrightarrow{\mathrm{p}_{\mathrm{PC}}}\). The contact angle ξ describes the rotation of the normal vector around the pitch point C and can be calculated using trigonometry:

$$\upxi =\arctan \left(\frac{\mathrm{e}\cdot \sin \left(\upkappa \right)}{\mathrm{r}_{\mathrm{w}1}-\mathrm{e}\cdot \cos \left(\upkappa \right)}\right)=\arctan \left(\frac{\uplambda \cdot \sin \left(\upkappa \right)}{1-\uplambda \cdot \cos \left(\upkappa \right)}\right)$$
(21)

Due to the alignment of the vector \(\overrightarrow{\mathrm{O}_{\mathrm{A}}\mathrm{P}_{\mathrm{C}}}\) in the direction of the pitch point C, the law of gearing is always fulfilled.

Figure 6 shows the structure of the cycloid gear with different circular arc profiles for the individual flanks of the arc gear. Here, two different trochoid equidistants are formed, representing the two flanks. The backlash angle is not considered here for the two generating circular arc profiles. This means that the resulting EC gearing has backlash. The complete cycloid gear can be generated from the two curves of the trochoid equidistants. The necessary boundaries for this can be derived from the arc gear.

Fig. 6
figure 6

Structure of the cycloid gear meshing with an arc gear with varying arc profiles

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The reference diameter of the cycloid gear d2 can be calculated with the module m:

$$\mathrm{d}_{2}=\mathrm{m}\cdot \mathrm{z}_{2}$$
(22)

The limitation of the flank curves is determined by the tip clearance c of the gearing. The tip clearance c is defined by a tip clearance factor c* and the module m:

$$\mathrm{c}=\mathrm{c}^{\mathrm{*}}\cdot \mathrm{m}$$
(23)

For involute gears, the tip clearance factor is usually c* = 0.25 [5]. Thus, the root diameter df2 can be calculated as follows:

$$\mathrm{d}_{\mathrm{f}2}=2\cdot \left(\mathrm{a}-\frac{\mathrm{d}_{\mathrm{a}1}}{2}-\mathrm{c}\right)$$
(24)

Using this root diameter, a tangential circle with radius rF2 can be fitted to the two trochoid equidistants. The tip diameter da2 is calculated analogous to the root diameter df2:

$$\mathrm{d}_{\mathrm{a}2}=2\cdot \left(\mathrm{a}-\frac{\mathrm{d}_{\mathrm{f}1}}{2}-\mathrm{c}\right)$$
(25)

The geometry of the cycloid gear in the transverse section is therefore completely defined. In order to generate helical EC gearing the helix angle β is introduced. According to Linke et al. [5], the helix angle of the cycloid gear β2 is the negative value of the helix angle of the arc gear β1:

$$\upbeta _{2}=-\upbeta _{1}$$
(26)

This equation can be applied to all cylindrical gears. According to ISO 21771 [24], the overlap angle φβ defines the relative angle of the two faces of a gear. The angle φβ1 of the arc gear is given by the facewidth b:

$$\upvarphi _{\upbeta 1}=2\cdot \mathrm{b}\cdot \frac{\tan \left(\upbeta _{1}\right)}{\mathrm{d}_{1}}$$
(27)

The convention here is that a positive helix angle for external gears results in a positive overlap angle about the z‑axis [4, 5]. A positive helix angle thus leads to right-handed gear teeth for external gears. It follows thus for the angle φβ2 of the cycloid gear:

$$\upvarphi _{\upbeta 2}=2\cdot \mathrm{b}\cdot \frac{\tan \left(\upbeta _{2}\right)}{\mathrm{d}_{2}}=-2\cdot \mathrm{b}\cdot \frac{\tan \left(\upbeta _{1}\right)}{\mathrm{d}_{2}}=-\frac{\upvarphi _{\upbeta 1}}{\mathrm{i}}$$
(28)

Now the complete geometry of the three-dimensional EC gearing is defined. All necessary parameters for the EC gearing can be summarized in Table 1.

Table 1 Set of parameters to describe the EC gearing
Full size table

The parameters from this table can be used to create arbitrary EC gearings.

4 Meshing of the EC gearing

The details of the tooth contact of the EC gearing with the corresponding path of contact are shown in Fig. 7. The rotation direction of the gears is represented by the angular velocities ω1 and ω2. The black line around the pitch point C shows the path of contact when the arc gear is driving. Point A shows the start of the meshing, point E the end of the meshing. The dotted line represents the path of action and thus the maximum possible path of contact. The limitation occurs at the respective tip circles. The grey line around the pitch point C shows the path of contact when the cycloid gear is driving. The path of contact is mirrored to the case of the driving arc gear. The two paths of action together give the shape of a distorted lemniscate, which is typical for any EC gearing.

Fig. 7
figure 7

Details of the meshing between the arc gear and the cycloid gear

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The mathematical description of the path of action is similar to the description of the trochoid equidistant and is achieved by vector addition. In contrast to Eq. 20, the angle ζ is neglected. It follows for the point vector \(\overrightarrow{\mathrm{p}_{\mathrm{poa}}}\) from the center of the cycloid gear O2 to the path of action:

$$\overrightarrow{\mathrm{p}_{\mathrm{poa}}}=\left(\begin{array}{c} \mathrm{x}_{\mathrm{poa}}\\ \mathrm{y}_{\mathrm{poa}} \end{array}\right)=\mathrm{a}\cdot \left(\begin{array}{c} 0\\ 1 \end{array}\right)-\mathrm{e}\cdot \left(\begin{array}{c} -\sin \left(\upkappa \right)\\ \cos \left(\upkappa \right) \end{array}\right)-\mathrm{r}_{\mathrm{A}}\cdot \left(\begin{array}{c} -\sin \left(-\upxi \right)\\ \cos \left(-\upxi \right) \end{array}\right)$$
(29)

xpoa is the x‑component, ypoa the y‑component of the vector \(\overrightarrow{\mathrm{p}_{\mathrm{poa}}}\). The path of action can also be referenced to the center of the arc gear O1 shifting it by the center distance a.

For further analyses, an exemplary EC gearing is chosen. The parameters of this gearing can be seen in Table 2.

Table 2 Exemplary set of parameters of the EC gearing for following calculations
Full size table

The geometry generated with this can be used to represent the meshing of the EC gearing. Figure 8 shows four different meshing positions with the arc gear driving and rotating counterclockwise. Figure 8a shows the beginning of the meshing, Fig. 8d shows the end.

Fig. 8
figure 8

Sequence of meshing of the exemplary EC gearing

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Especially noticeable in Fig. 8 is the strongly curved path of contact at the end of the meshing. The reason for this is the osculation of the two tooth flanks in this area.

The two-dimensional calculation of the path of contact can also be used for the three-dimensional determination of the line of contact. Thereby, a width-variable angle ζ(z) must be used for the calculation of the angles κ and ξ. The angle ζ(z) depends on the facewidth coordinate z and is determined analogous to Eq. 28:

$$\upzeta \left(\mathrm{z}\right)=\upzeta _{\mathrm{s}}+\mathrm{z}\cdot \frac{2\cdot \tan \left(\upbeta \right)}{\mathrm{i}\cdot \mathrm{d}_{2}}$$
(30)

ζs represents the starting angle of the gear face. The contact lines thus determined depend on the meshing position and can be projected onto the flanks of the arc gear and cycloid gear. Figure 9 shows the sequence of the line of contact projected onto the flank of the arc gear with the configuration of Fig. 8. The strong curvature of the path of contact at the end of the meshing results in the curvature of the line of contact at the tooth tip.

Fig. 9
figure 9

Sequence of the line of contact on the arc gear flank (view on the flank surface)

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Figure 10 shows the sequence of the line of contact projected onto the flank of the cycloid gear with the configuration of Fig. 8. Here, the curvature of the contact line at the tooth root is a result of the curvature of the path of contact at the end of meshing.

Fig. 10
figure 10

Sequence of the line of contact on the cycloid gear flank (view on the flank surface)

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5 Determination of load-free characteristics

The calculated geometry of the EC gearing can be used to determine certain characteristics. In this study, only load-free characteristics, which are determined on conjugate and non-deformed flanks are considered. The most important characteristics are the contact conditions, represented by pressure angle, sliding and radius of curvature.

5.1 Pressure angle

The transverse pressure angle αt describes the angle of the normal vector on the flanks in relation to the common tangent of the pitch circles, see Fig. 5. The transverse pressure angle can be calculated with:

$$\upalpha _{\mathrm{t}}=\frac{\uppi }{2}-\upxi$$
(31)

The pressure angle describes the direction of the contact force in the overall transmission. A changing value therefore leads to changing reaction forces in the gear components, e.g., in the bearings. A large pressure angle also results in the radial forces from the tooth contact becoming exceptionally large.

Figure 11 shows the transverse pressure angle for the EC gearing with the configuration of Fig. 8. A minimum pressure angle is shown in the area of the meshing position of Fig. 8c. The maximum pressure angle is at the end of the meshing.

Fig. 11
figure 11

Transverse pressure angle over the path of contact of the exemplary EC gearing

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5.2 Sliding factor

The sliding velocity vg of the gearing can be calculated with the velocities of each flank at the contact point. The velocities of the flank can be determined using the angular velocities. Here, a three-dimensional view is necessary. The vector of angular velocity of the arc gear \(\overrightarrow{\upomega _{1}}\) with the angular velocity ω1 is defined as follows:

$$\overrightarrow{\upomega _{1}}=\left(\begin{array}{c} 0\\ 0\\ \upomega _{1} \end{array}\right)$$
(32)

Analogous to this, the vector of angular velocity of the cycloid gear \(\overrightarrow{\upomega _{2}}\) considering the transmission ratio i:

$$\overrightarrow{\upomega _{2}}=\left(\begin{array}{c} 0\\ 0\\ -\frac{\upomega _{1}}{\mathrm{i}} \end{array}\right)$$
(33)

These vectors can be used to determine the velocity of the flanks. For the velocity \(\overrightarrow{\mathrm{v}_{1}}\) of points on the flank of the arc gear, the path of action must be referenced to the center of the arc gear. It follows:

$$\overrightarrow{\mathrm{v}_{1}}=\left(\begin{array}{c} \mathrm{v}_{1{,}\mathrm{x}}\\ \mathrm{v}_{1{,}\mathrm{y}}\\ \mathrm{v}_{1{,}\mathrm{z}} \end{array}\right)=\overrightarrow{\upomega _{1}}\times \left(\overrightarrow{\mathrm{p}_{\mathrm{poa}}}-\left(\begin{array}{c} 0\\ \mathrm{a}\\ 0 \end{array}\right)\right)=\left(\begin{array}{c} 0\\ 0\\ \upomega _{1} \end{array}\right)\times \left(-\mathrm{e}\cdot \left(\begin{array}{c} -\sin \left(\upkappa \right)\\ \cos \left(\upkappa \right)\\ 0 \end{array}\right)-\mathrm{r}_{\mathrm{A}}\cdot \left(\begin{array}{c} -\sin \left(-\upxi \right)\\ \cos \left(-\upxi \right)\\ 0 \end{array}\right)\right)$$
(34)

v1,x is the x‑component, v1,y the y‑component and v1,z the z‑component of the vector \(\overrightarrow{\mathrm{v}_{1}}\). Analogous for the mating gear, the velocity \(\overrightarrow{\mathrm{v}_{2}}\) of points on the flank of the cycloid gear is:

$$\overrightarrow{\mathrm{v}_{2}}=\left(\begin{array}{c} \mathrm{v}_{2{,}\mathrm{x}}\\ \mathrm{v}_{2{,}\mathrm{y}}\\ \mathrm{v}_{2{,}\mathrm{z}} \end{array}\right)=\overrightarrow{\upomega _{2}}\times \overrightarrow{\mathrm{p}_{\mathrm{poa}}}=\left(\begin{array}{c} 0\\ 0\\ -\frac{\upomega _{1}}{\mathrm{i}} \end{array}\right)\times \left(\mathrm{a}\cdot \left(\begin{array}{c} 0\\ 1\\ 0 \end{array}\right)-\mathrm{e}\cdot \left(\begin{array}{c} -\sin \left(\upkappa \right)\\ \cos \left(\upkappa \right)\\ 0 \end{array}\right)-\mathrm{r}_{\mathrm{A}}\cdot \left(\begin{array}{c} -\sin \left(-\upxi \right)\\ \cos \left(-\upxi \right)\\ 0 \end{array}\right)\right)$$
(35)

v2,x is the x‑component, v2,y the y‑component and v2,z the z‑component of the vector \(\overrightarrow{\mathrm{v}_{2}}\).

The circumferential velocity vt is the Euclidian norm of the vector \(\overrightarrow{\mathrm{v}_{\mathrm{C}}}\), which is the velocity of the pitch point C. This results in the following for vt:

$$\mathrm{v}_{\mathrm{t}}=\left| \overrightarrow{\mathrm{v}_{\mathrm{C}}}\right| =\left| \overrightarrow{\upomega _{1}}\times \overrightarrow{\mathrm{O}_{1}\mathrm{C}}\right| =\left| \left(\begin{array}{c} 0\\ 0\\ \upomega _{1} \end{array}\right)\times \left(-\begin{array}{c} 0\\ \mathrm{r}_{\mathrm{w}1}\\ 0 \end{array}\right)\right| =\upomega _{1}\cdot \mathrm{r}_{\mathrm{w}1}$$
(36)

The velocity v1/2,t in the direction of the common tangent of the flanks is a part of the velocity calculated in Eq. 34 and 35. The common tangent can be determined using the transverse pressure angle αt.

In the case of a driving arc gear, the tangential velocity v1/2,t is given by:

$$\mathrm{v}_{1/2{,}\mathrm{t}}=\left\langle \overrightarrow{\mathrm{v}_{1/2}}{,}\left(\begin{array}{c} \sin \left(\upalpha _{\mathrm{t}}\right)\\ \cos \left(\upalpha _{\mathrm{t}}\right)\\ 0 \end{array}\right)\right\rangle =\mathrm{v}_{1/2{,}\mathrm{x}}\cdot \sin \left(\upalpha _{\mathrm{t}}\right)+\mathrm{v}_{1/2{,}\mathrm{y}}\cdot \cos \left(\upalpha _{\mathrm{t}}\right)$$
(37)

The velocity in the direction of the common normal is zero according to the law of gearing [19]. The sliding velocity vg can be determined from the difference between the tangential velocity of the driving (vdriving,t) and driven (vdriven,t) gear:

$$\mathrm{v}_{\mathrm{g}}=\mathrm{v}_{\text{driving}{,}\mathrm{t}}-\mathrm{v}_{\text{driven}{,}\mathrm{t}}$$
(38)

In the case of a driving arc gear:

$$\mathrm{v}_{\mathrm{g}}=\left(\mathrm{v}_{1{,}\mathrm{x}}-\mathrm{v}_{2{,}\mathrm{x}}\right)\cdot \sin \left(\upalpha _{\mathrm{t}}\right)+\left(\mathrm{v}_{1{,}\mathrm{y}}-\mathrm{v}_{2{,}\mathrm{y}}\right)\cdot \cos \left(\upalpha _{\mathrm{t}}\right)$$
(39)

With these calculated velocities, the sliding factor Kg can be determined:

$$\mathrm{K}_{\mathrm{g}}=\frac{\mathrm{v}_{\mathrm{g}}}{\mathrm{v}_{\mathrm{t}}}$$
(40)

The sliding factor Kg is thus a dimensionless quantity for assessing the sliding velocity vg of a gearing. The sliding factor can be used to evaluate the gearing in terms of heating and scuffing [5]. Figure 12 shows the sliding factor for the EC gearing with the configuration of Fig. 8. A correlation between the curvatures of the path of action and the course of the sliding factor can be seen.

Fig. 12
figure 12

Sliding factor over the path of contact of the exemplary EC gearing

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5.3 Radius of curvature

The radius of curvature is needed to calculate the flank pressure according to the theory of Hertz. It is defined that a positive radius corresponds to a convex flank and a negative radius to a concave flank. The EC gearing is defined in the transverse section, the radii of curvature are also determined in the transverse section. The radius of curvature of the arc gear ϱ1,t is specified with the radius of the arc profile rA:

$$\upvarrho _{1{,}\mathrm{t}}=\mathrm{r}_{\mathrm{A}}$$
(41)

The radius of curvature of the cycloid gear ϱ2,t can be calculated as the radius of curvature of a plane curve. This curve is the trochoid equidistant of the cycloid gear. The derivative of the trochoid equidistant with respect to the angle ζ can be simplified with the derivative of the trochoid with respect to the angle ζ. Some transformations result in the following equation:

$$\upvarrho _{2{,}\mathrm{t}}=\frac{\left(\left(\frac{\partial \mathrm{x}_{\mathrm{PC}}}{\partial \upzeta }\right)^{2}+\left(\frac{\partial \mathrm{y}_{\mathrm{PC}}}{\partial \upzeta }\right)^{2}\right)^{\frac{3}{2}}}{\frac{\partial \mathrm{x}_{\mathrm{PC}}}{\partial \upzeta }\cdot \frac{\partial ^{2}\mathrm{y}_{\mathrm{PC}}}{\partial \upzeta ^{2}}-\frac{\partial ^{2}\mathrm{x}_{\mathrm{PC}}}{\partial \upzeta ^{2}}\cdot \frac{\partial \mathrm{y}_{\mathrm{PC}}}{\partial \upzeta }}=\frac{\left(\left(\frac{\partial \mathrm{x}_{\mathrm{OA}}}{\partial \upzeta }\right)^{2}+\left(\frac{\partial \mathrm{y}_{\mathrm{OA}}}{\partial \upzeta }\right)^{2}\right)^{\frac{3}{2}}}{\frac{\partial \mathrm{x}_{\mathrm{OA}}}{\partial \upzeta }\cdot \frac{\partial ^{2}\mathrm{y}_{\mathrm{OA}}}{\partial \upzeta ^{2}}-\frac{\partial ^{2}\mathrm{x}_{\mathrm{OA}}}{\partial \upzeta ^{2}}\cdot \frac{\partial \mathrm{y}_{\mathrm{OA}}}{\partial \upzeta }}-\mathrm{r}_{\mathrm{A}}=\mathrm{a}\cdot \frac{\left(1+\uplambda ^{2}-2\cdot \uplambda \cdot \cos \left(\upkappa \right)\right)^{\frac{3}{2}}}{1+\uplambda ^{2}\cdot \left(1+\mathrm{i}\right)-\uplambda \cdot \left(2+\mathrm{i}\right)\cdot \cos \left(\upkappa \right)}-\mathrm{r}_{\mathrm{A}}$$
(42)

Figure 13 shows the radii of curvature for the EC gearing with the configuration of Fig. 8. It can be seen that the curvature of the arc gear is uniform. The curvature of the cycloid gear, on the other hand, shows strong variations over the meshing. At the beginning of the meshing the flank is convex, at the end it is concave. Near the meshing position c, the radius of curvature becomes infinite, so there is an inflection point of the flank geometry. This inflection point is an asymptote of Eq. 42. The position of this asymptote can be calculated with:

$$\upzeta =\frac{1}{\mathrm{i}}\cdot \arccos \left(\frac{1+\uplambda ^{2}\cdot \left(1+\mathrm{i}\right)}{\uplambda \cdot \left(2+\mathrm{i}\right)}\right)$$
(43)
Fig. 13
figure 13

Transverse radius of curvature of both gears over the path of contact of the exemplary EC gearing

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The equivalent radius of curvature ϱe,t evaluates the contact condition for a calculation with respect to the theory of Hertz. The equivalent radius of curvature can be calculated with:

$$\upvarrho _{\mathrm{e}{,}\mathrm{t}}=\frac{\upvarrho _{1{,}\mathrm{t}}\cdot \upvarrho _{2{,}\mathrm{t}}}{\upvarrho _{1{,}\mathrm{t}}+\upvarrho _{2{,}\mathrm{t}}}$$
(44)

Figure 14 shows the equivalent radius of curvature for the EC gearing with the configuration of Fig. 8. The ordinate of the graph shown in Fig. 14 is a logarithmic scale. It can be seen that a minimum of the equivalent radius of curvature exists, which is not directly recognizable from Fig. 13. At the end of the meshing, a large equivalent radius of curvature is evident, which can be attributed to the osculation of the flanks.

Fig. 14
figure 14

Transverse equivalent radius of curvature over the path of contact of the exemplary EC gearing

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6 Conclusion

A detailed description of the geometry of the special gearings is necessary to determine and classify its characteristics. This allows potentials to be identified in comparison to the established involute gearing. This paper shows a parameter-based definition of the geometry of the EC gearing. The parameters are used for the complete description of the gear geometry and the determination of various characteristics derived from it. Options for setting a clearance for practice-relevant gearings are included.

The developed flank geometry is used to determine the meshing conditions. The sequence of the meshing in the transverse section and on the flank surface is shown. The shape of the path of action and the lines of contact are discussed. Various load-free characteristic values are determined based on an exemplary EC gearing. The specifics of the pressure angle, sliding factor, and radii of curvature are discussed.

Further work will focus on the determination of load-dependent characteristics. The deformation of the gearing can be considered using analytical approaches [27]. For example, the elastic consideration of the gearing can allow load-related transmission errors to be determined [28]. Comparable geometries from practice can be used to assess the applicability of the EC gearing. Other types of gears, such as the worm gear [29], can be used here as well.

7 Nomenclature

The nomenclature is shown in Table 3.

Table 3 Nomenclature
Full size table