© AFM, EDP Sciences 2019

1 Introduction

Straight bevel gears are applied widely in machinery (in vehicles, tool machines, for medical tools and machines etc.). They are used to connect shafts whose axes intersect in some angles, thus the meshing surfaces form a cone on which teeth are shaped (Fig. 1) [1–6].

One of the oldest process for producing gears applied even today is the Reinecker − Bilgram process. In this case, the vertical shaft of the machine intersects the centre of the pitch surface. The cutting tool makes an alternating movement while the operating pitch cone of the bevel gear unwinds the axoid of the deriving pitch cone (Fig. 2) [1,3–8].

During another production process of Heidenreich–Harbeck, the machine uses 2 cutting tools. This method is more advantageous because in this case the machine is moved in the axes perpendicular to the axes of the pitch surface. The machine is not set according to the λ dedendum angle, and the grip of the tool is stricter (Fig. 3) [1,3–8].

The paths of the cutting tools are not parallel during production, but they intersect in 2⋅θ angle, this is called planing angle, which can be calculated in the following way [5] (Fig. 4):$$\text{arc}\theta =\frac{0.25\cdot m\cdot \pi +R_e\cdot \sin \lambda \cdot \tan {\alpha }_0}{R_e\cdot \cos \lambda }$$(1)

Fig. 1 Angles and diameters of bevel gear.

Fig. 2 Reinecker–Bilgram principle [6].

Fig. 3 Heidenreich–Harbeck principle [6].

Fig. 4 Definition of planing angle.

2 Defining the contact points

The profile curve ${\displaystyle \overrightarrow{r_{1R}}}$ of the bevel gear wheel is made as a result of the movement of the machining tool and the unwinding movement for the K_1R (x_1R, y_1R, z_1R) rotating coordinate system related to the driver gear wheel.

I am looking for the surface connected to K_2R coordinate system related to ${\displaystyle \overrightarrow{r_{1R}}}$. I use the fact that the 2 surfaces during their movements coincide. Taking into consideration the following correlation [4,9]:$${\varphi }_{2R}=i_{21}\cdot {\varphi }_{1R}$$(2)

I can state that the movement can be described with (ϕ₁) movement parameter [4,9,10].

The relative velocity between the 2 surfaces can be determined by the transformation between the rotating K_1R coordinate system of the driver wheel and the rotating K_2R coordinate system of the driven wheel [4,9]:$${{\displaystyle \overrightarrow{v_{2R}}}}^{\left(12\right)}=\frac{d}{\mathit{d}t}{\displaystyle \overrightarrow{r_{2R}}}=\frac{d}{\mathit{d}t}\left(M_{\mathbf{2}R,\mathbf{1}R}\right)\cdot {\displaystyle \overrightarrow{r_{1R}}}\text{.}$$(3)Transformation matrices between the K_1R and K_2R coordinate systems (Fig. 5):$$M_{\mathbf{2}R,\mathbf{1}R}=M_{\mathbf{2}R,\mathbf{2}S}\cdot M_{\mathbf{2}S,\mathbf{1}S}\cdot M_{\mathbf{1}S,\mathbf{1}R}=\left[\begin{array}{cccc}\hfill -\cos \left({\varphi }_{1R}\right)\cdot \sin \left({\varphi }_{2R}\right)\hfill & \hfill -\cos \left({\varphi }_{1R}\right)\cdot \cos \left({\varphi }_{2R}\right)\hfill & \hfill \text{sin}\left({\varphi }_{1R}\right)\hfill & \hfill 0\hfill \\ \hfill -\sin \left({\varphi }_{1R}\right)\cdot \text{sin}\left({\varphi }_{2R}\right)\hfill & \hfill -\sin \left({\varphi }_{1R}\right)\cdot \cos \left({\varphi }_{2R}\right)\hfill & \hfill -\text{cos}\left({\varphi }_{1R}\right)\hfill & \hfill 0\hfill \\ \hfill \text{cos}\left({\varphi }_{2R}\right)\hfill & \hfill -\sin \left({\varphi }_{2R}\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$$(4) $$M_{\mathbf{1}R,\mathbf{2}R}=M_{\mathbf{1}R,\mathbf{1}S}\cdot M_{\mathbf{1}S,\mathbf{2}S}\cdot M_{\mathbf{2}S,\mathbf{2}R}=\left[\begin{array}{cccc}\hfill -\sin \left({\varphi }_{1R}\right)\cdot \cos \left({\varphi }_{2R}\right)\hfill & \hfill -\sin \left({\varphi }_{1R}\right)\cdot \sin \left({\varphi }_{2R}\right)\hfill & \hfill \text{cos}\left({\varphi }_{1R}\right)\hfill & \hfill 0\hfill \\ \hfill -\cos \left({\varphi }_{1R}\right)\cdot \text{cos}\left({\varphi }_{2R}\right)\hfill & \hfill -\cos \left({\varphi }_{1R}\right)\cdot \sin \left({\varphi }_{2R}\right)\hfill & \hfill -\text{sin}\left({\varphi }_{1R}\right)\hfill & \hfill 0\hfill \\ \hfill \text{sin}\left({\varphi }_{2R}\right)\hfill & \hfill -\cos \left({\varphi }_{2R}\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \mathbf{1}\hfill \end{array}\right]$$(5)

Taking into consideration the correlation between the velocity vectors of the relative movement in K_1R and K_2R coordinate systems [4,9]:$${{\displaystyle \overrightarrow{v_{1R}}}}^{\left(12\right)}=M_{\mathbf{1}R,\mathbf{2}R}\cdot {{\displaystyle \overrightarrow{v_{2R}}}}^{\left(12\right)}$$(6)in K_1R coordinate system, the relative velocity vector, based on (3) is:$${{\displaystyle \overrightarrow{v_{1R}}}}^{\left(12\right)}=M_{\mathbf{1}R,\mathbf{2}R}\cdot \frac{d}{\mathit{d}t}\left(M_{\mathbf{2}R,\mathbf{1}R}\right)\cdot {\displaystyle \overrightarrow{r_{1R}}}\text{,}$$(7)where P_1k is the kinematic mapping matrix [4,9]:$$P_{\mathbf{1}k}=M_{\mathbf{1}R,\mathbf{2}R}\cdot \frac{d}{\mathit{d}t}\left(M_{\mathbf{2}R,\mathbf{1}R}\right)$$(8)

On the tooth surfaces of the meshing teeth, contact points mutually covering each other can be determined by solving the connection equation − which expresses the 1st Law of Contact − and the vector-scalar function simultaneously [4,9]:$${\displaystyle \overrightarrow{n_{1R}}}\cdot {{\displaystyle \overrightarrow{v_{1R}}}}^{\left(12\right)}={\displaystyle \overrightarrow{n_{2R}}}\cdot {{\displaystyle \overrightarrow{v_{2R}}}}^{\left(12\right)}={\displaystyle \overrightarrow{n}}\cdot {{\displaystyle \overrightarrow{v}}}^{\left(12\right)}\text{.}$$(9)

Fig. 5 Connection between the coordinate systems of the bevel gear drive pair.

3 Designing the computer program

Input data for designing the drive pair are: m module, z₁ number of teeth of the driver bevel gear wheel, z₂ number of teeth of the driven gear wheel, c ^* root clearance factor and the α₀ angle of contact [1,3–8].

For designing, the necessary geometrical correlations from technical literature was also used, which can be seen below based on Figures 1 and 5 [1,3–8]:

• the largest pitch circle diameters:

$$d_{\mathrm{01,2}}=m\cdot z_{\mathrm{1,2}}\text{,}$$(10)

• half pitch angle of the driver gear wheel:

$${\delta }_{01}=\text{atan}\left(\frac{z_{\mathbf{2}}}{z_{\mathbf{1}}}\right)\text{,}$$(11)

• half pitch angle of the driven gear wheel:

$${\delta }_{02}=90°-{\delta }_{01},$$(12)

• effective pitch surface radius:

$$R_e=\sqrt{{\left(d_{01}/2\right)}^2+{\left(d_{02}/2\right)}^2}\text{,}$$(13)

• addendum on the largest diameter:

$$f_0=m\text{,}$$(14)

• dedendum on the largest diameter:

$$l_0=\left(1+c^{*}\right)\cdot m\text{,}$$(15)

• the largest tip circle diameters:

$$d_{f\mathrm{1,2}}=\left(z_{\mathrm{1,2}}+2\cdot \text{cos}\left({\delta }_{\mathrm{01,2}}\right)\right)\cdot m\text{,}$$(16)

• the largest root circle diameters:

$$d_{a\mathrm{1,2}}=z_{\mathrm{1,2}}\cdot m-2\cdot l_0\cdot \text{cos}({\delta }_{\mathrm{01,2}})\text{,}$$(17)

• face width:

$$b=\frac{1}{3.5}\cdot R_e\text{,}$$(18)

• dedendum angle:

$$\lambda =\text{atan}\left(\frac{l_0}{R_e}\right)\text{,}$$(19)

• tip cone angles

$${\delta }_{f\mathrm{1,2}}={\delta }_{\mathrm{01,2}}+\lambda \text{,}$$(20)

• root cone angles:

$${\delta }_{l\mathrm{1,2}}={\delta }_{\mathrm{01,2}}-\lambda \text{.}$$(21)

The output of my computer program is some calculated geometrical data of the bevel gear, and the profile curves (Fig. 6) in case of the smallest and largest diameters. Knowing the above mentioned manufacturing technologies [1,3–8] and the path of the cutting tool this program can determine the shape of the gear profiles.

The program saves the profiles of the bevel gear in .txt file format. Onto the given profile set of dots an interpolating B-spline can be fitted [11]. With the geometrical parameters of the drive pair and its profiles, the CAD model of the bevel gear was made by SolidWorks designing software (Fig. 7).

Fig. 6 Profile curves of the designed bevel gear pair (m = 15 mm, z1 = 20, z2 = 30, Σ = 90°). (a) Profile curves of the driver gear. (b) Profile curves of the driven gear.

Fig. 7 CAD model of the bevel gear pair having straight teeth (m  = 15 mm, z1 = 20, z2 = 30, Σ = 90°).

4 LTCA analysis

The aims of the so called “Loaded Tooth Contact Analysis” (LTCA) are to model the connection of toothed drive pairs and to simulate the model by given loads and boundary conditions. During the LTCA in case of this given geometrical bevel gear (Figs. 6 and 7, Tab. 1) normal stress, normal elastic strain and normal deformation values were analysed on the surface of the driven and driver gear wheels [12–21].

4.1 Material quality, FE mesh (finite element)

The property of the material used for our bevel gear is given in Table 2.

During defining the FE mesh in the contact zone, sphere volume (37 mm radius) including dense triangles (1 mm mesh dimension) has been applied on the toothed area (Fig. 8). Automatic meshing was used on the outside areas. The friction coefficient in the tooth contact zone is µ = 0.01, because I considered the liquid and rolling frictions on the contact zone.

Fig. 8 FE mesh.

4.2 Loads and boundary conditions

Four coordinate systems have been defined for LTCAs: K_s − absolute static, K_s1–static related to the driver wheel, K_s2–static, related to the driven wheel and K_c– coordinate system in the tooth contact zone.

During the analyses, the gear having less number of teeth (driver gear) drives the gear having more number of teeth. Five degrees of freedom were fixed of the driver gear, only rotation along the rotational axis was allowed. The driver gear was loaded by M = 200–400 Nm torque by 50 Nm steps. Six degrees of freedom of the driven gear were fixed (Fig. 9).

Fig. 9 Setting loads and boundary conditions.

Fig. 10 Defining normal stress.

4.3 Normal stress analyses

In a fixed P point of the body, ${\displaystyle \stackrel{\rho }{{\to }_n}}$ signals the stress vector awakening on the ${\displaystyle \overrightarrow{n}}$ normal surface elements. It can be divided into a normal directional component and one perpendicular to it [22–24] (Fig. 10):$${\displaystyle \overrightarrow{{\rho }_n}}={\sigma }_n\cdot {\displaystyle \overrightarrow{n}}+{\displaystyle \overrightarrow{{\tau }_n}}\text{,}$$(22)where the normal stress is [22–24]:$${\sigma }_n={\displaystyle \overrightarrow{n}}\cdot {\displaystyle \overrightarrow{{\rho }_n}}\text{.}$$(23)

Normal stress partitions and average normal stress values have been detected as an effect of the different loads, on the tooth surfaces of the driver and driven gear (Fig. 11).

Results are shown in the diagram of Figure 12. In absolute value, as an effect of the increasing load moment, normal stress values also increase on the tooth surfaces of the driver and driven gear.

The shapes of the two diagrams are the same and they are parallel. The normal stresses of the driven gear are a little higher than the normal stresses of the driver gear in absolute value. Because of the good comparison the same limits were set.

Fig. 11 Normal stress values. (a) M = 200 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-2.054$ MPa (top), driven gear, ${\displaystyle \overline{{\sigma }_n}}=-2.164$ MPa (bottom). (b) M = 250 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-2.62$ MPa, driven gear, ${\displaystyle \overline{{\sigma }_n}}=-2.709$ MPa. (c) M = 300 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-3.203$ MPa, driven gear, ${\displaystyle \overline{{\sigma }_n}}=-3.297$ MPa. (d) M = 350 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-3.796$ MPa, driven gear, ${\displaystyle \overline{{\sigma }_n}}=-3.893$ MPa. (e) M = 400 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-4.402$ MPa, driven gear, ${\displaystyle \overline{{\sigma }_n}}=-4.495$ MPa.

Fig. 11 Continued.

Fig. 12 Load − normal stress diagram.

4.4 Normal elastic strain analyses

The state of a body's deformation can be signalled by deformation tensor, which can be determined from the displacement vector as seen below [22–24]:$$A=\frac{1}{2}\left({\displaystyle \overrightarrow{u}}°{\displaystyle \overrightarrow{\nabla }}+{\displaystyle \overrightarrow{\nabla °}}{\displaystyle \overrightarrow{u}}\right)\text{,}$$(24) $$A=\left[\begin{array}{ccc}\hfill \frac{\partial u}{\partial x}\hfill & \hfill \frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\hfill & \hfill \frac{1}{2}\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right)\hfill \\ \hfill \frac{1}{2}\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)\hfill & \hfill \frac{\partial v}{\partial y}\hfill & \hfill \frac{1}{2}\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)\hfill \\ \hfill \frac{1}{2}\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)\hfill & \hfill \frac{1}{2}\left(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\right)\hfill & \hfill \frac{\partial w}{\partial z}\hfill \end{array}\right]\text{,}$$(25)where$$\begin{array}{cc}\hfill {\epsilon }_x=\frac{\partial u}{\partial x}\hfill & \hfill {\gamma }_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\hfill \\ \hfill {\epsilon }_y=\frac{\partial v}{\partial y}\hfill & \hfill {\gamma }_{yz}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\hfill \\ \hfill {\epsilon }_z=\frac{\partial w}{\partial z}\hfill & \hfill {\gamma }_{zx}=\frac{\partial w}{\partial z}+\frac{\partial u}{\partial x}\hfill \end{array}$$(26)

Based on the values of ε_x, ε_y, ε_z elastic strain per unit length and unit vectors related to them, the FE program calculates the normal direction resultant (ε_n ), which is perpendicular to the surface. As an effect of the loads on the surface of the driver and driven gear, normal elastic strain partitions and average normal elastic strain values occur (Fig. 13).

The results are shown in the diagram of Figure 14. You can see that, in absolute value, as an effect of increasing loads, normal elastic strain values also increase on the tooth surfaces of the driver and driven gears.

The shapes of the two diagrams are the same and they are parallel. The normal elastic strains of the driven gear are a little higher than the normal elastic strains of the driver gear in absolute value. Because of the good comparison the same limits were set.

Fig. 13 Normal elastic strain values as a result of increasing loads. (a) M = 200 Nm: driver gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000114 (top), driven gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000127 (bottom). (b) M = 250 Nm: driver gear,${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000141, driven gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000154. (c) M = 300 Nm: driver gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000168, driven gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000183. (d) M = 350 Nm: driver gear,${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000197, driven gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000214. (e) M = 400 Nm: driver gear,${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000227, driven gear,${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000246.

Fig. 13 Continued.

Fig. 14 Load − normal elastic strain diagram.

4.5 Normal deformation analyses

As an effect of a load, the points of a solid body move, and the beginning state of the body differs from the state after full load application [22–24].

For instance, the position of an arbitrary P point of the body is given by the following placement vector, in the state before the load was applied [22–24]:$${\displaystyle \overrightarrow{r_P}}=x_P{\displaystyle \overrightarrow{e_x}}+y_P{\displaystyle \overrightarrow{e_y}}+z_P{\displaystyle \overrightarrow{e_z}}\text{.}$$(27)

Signals P′ the state of P point after loading, the related placement vector is ${\displaystyle \overrightarrow{r_P}},$ the displacement vector is u_P. Based on Figure 15 you can state the following correlations [22–24]:$${\displaystyle \overrightarrow{r_P\text{'}}}={\displaystyle \overrightarrow{r_P}}+{\displaystyle \overrightarrow{u_P}}\text{.}$$(28)

The displacement vector is the function of the position of the points of the body before loading, i.e. the placement vector [22–24]:$${\displaystyle \overrightarrow{u}}={\displaystyle \overrightarrow{u}}\left({\displaystyle \overrightarrow{r}}\right)\text{.}$$(29)

The sum of the displacement vectors related to the points of the body is called displacement field, where the displacement vector is [22–24]:$${\displaystyle \overrightarrow{u}}=u_x{\displaystyle \overrightarrow{e_x}}+u_y{\displaystyle \overrightarrow{e_y}}+u_z{\displaystyle \overrightarrow{e_z}}\text{.}$$(30)

The coordinates of the displacement:$$\begin{array}{c}\hfill u_x=u_x\left(x,y,z\right)\hfill \\ \hfill u_y=u_y\left(x,y,z\right)\hfill \\ \hfill u_z=u_z\left(x,y,z\right)\hfill \end{array}\}$$(31)

As an effect of the different loads, on the tooth surfaces of the driver and driven gear, the following normal deformation values (in this case x direction deformation) and average normal deformation values (the average of the x direction values) occur (Fig. 16).

Calculated results are shown in Figure 17. You can see that if the loads are increased the values of normal deformation also increase on the tooth surfaces of the driver and driven gear.

The shapes of the two diagrams are almost parallel. The normal deformations of the driver gear are higher than the normal deformations of the driven gear in absolute value. The teeth are in contact that is why the presages of this specific are different. Because of the good comparison the same limits were set.

Fig. 15 Definition of displacement vector.

Fig. 16 Normal deformation values. (a) M = 200 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0654 mm (top), driven gear, ${\displaystyle \overline{u_x}}=$ 0.00166 mm (bottom). (b) M = 250 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0644 mm, driven gear, ${\displaystyle \overline{u_x}}=$ 0.00207 mm. (c) M = 300 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0636 mm, driven gear, ${\displaystyle \overline{u_x}}=$ 0.00247 mm. (d) M = 350 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0629 mm, driven gear, ${\displaystyle \overline{u_x}}=$ 0.00286 mm. (e) M = 400 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0621 mm, driven gear, ${\displaystyle \overline{u_x}}=$ 0.00325 mm.

Fig. 16 Continued.

Fig. 17 Load–normal deformation diagram.

5 Conclusion

Straight bevel gears are widely used in machinery where the axes of the two shafts intersect, and the faces of the gears themselves form a cone. They are used as load transmission equipment, to change the number of rotation etc. Their further development (from constructional and technological point of view) and their LTCA based on different mechanical properties are highly needed.

Based on the geometrical and production analyses of the straight bevel gear pair, I developed a computer program to be able to design the drive pair. This program calculates all the geometrical parameters of the drive pair and draws the profile curves (which occur during production), on the smallest and largest diameters.

The given profile curves can be analyzed from constructional point of view and they are important for making our CAD model.

After that a CAM (Computer-Aided Manufacturing) software can be used to simulate the total production process of the drive pair.

It is important to have the CAD model for LTCA as well. These analyses are good for analyzing the mechanical properties on the driver and driven gear (such as stress, strains, elastic strain). LTCA are also important for having the appropriate constructional planning, to be able to choose the appropriate raw material, and to get important information about loads.

Normal stress, normal deformation and normal elastic strain values have been analyzed on the tooth surfaces of the driver and driven gear pair by different load torques. The results have been averaged, analyzed and shown in diagrams.

Nomenclature

${{\displaystyle \underset{v}{\to }}}_{\mathbf{1}R}^{\left(12\right)}$ : Relative velocity vector (mm/min⁻¹)

${\displaystyle \overrightarrow{n_{\mathbf{1}R}}}$ : Normal vector of the surface in K_1R coordinate system

${\displaystyle \overrightarrow{r_{\mathbf{1}R}}}$ : Placement vector of the moving points of the profile curve

${\displaystyle \overrightarrow{r_{\mathbf{2}R}}}$ : Placement vector of moving point of conjugated surface

${\displaystyle \overrightarrow{r_P}}$ : Placement vector

${\displaystyle \overrightarrow{{\rho }_n}}$ : Normal stress vector (MPa)

${\displaystyle \overrightarrow{{\tau }_n}}$ : Shear stress vector (MPa)

${\displaystyle \overrightarrow{\nabla }}$ : Differential operator

M_2R,1R, M_1R,2R : Translation matrices between K_1R and K_2R coordinate systems

P_1k : Matrix of kinematic mapping

c ^* : Root clearance factor

${\displaystyle \overrightarrow{u}}$ : Displacement vector (mm)

σ_n : Normal stress (MPa)

2 ⋅ θ : Angle of planning (°)

A : Deformation tensor

b : Face width (mm)

d₀ : Largest pitch circle diameter (mm)

d_f : Largest tip circle diameter (mm)

d_l : Largest root circle diameter (mm)

f₀ : Largest addendum (mm)

i₂₁, i : Transmission

j_s : Clearance at flank (mm)

K_1R (x _1R, y _1R, z _1R): Rotational coordinate system related to the driver gear wheel

K_1S (x_1S, y _1S, z _1S): Static coordinate system related to the driver gear wheel

K_2R (x _2R, y _2R, z _2R): Rotational coordinate system related to the driven gear wheel

K_2S (x _2S, y _2S, z _2S): Static coordinate system related to the driven gear wheel

l₀ : Largest dedendum (mm)

m : Module (mm)

M: Load moment (Nm)

R_e : Effective pitch surface radius (mm)

S_ax : Pitch circle face width on the largest diameter (mm)

t : Circular pitch on the largest pitch circle diameter (mm)

z₁, z₂ : Number of teeth

α₀ : Angle of contact (°)

δ₀ : Pitch angle (°)

δ_f : Tip cone angle (°)

δ_l : Root cone angle (°)

ε_n : Normal elastic strain (mm)

ε_x, ε_y, ε_z : Elastic strain values per unit length (mm)

λ : Dedendum angle (°)

μ : Friction coefficient

Σ : Shaft angle (°)

ϕ : Addendum angle (°)

ϕ_1R, ϕ_2R : Angular displacement (°)

E : Distance between the vertex of cones (mm)

u, v, w : Coordinates of the displacement vector (mm)

$\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}$ : Parts of the differential operator

γ _xy, γ _yx, γ_xz : Shear specific elongation (mm)

x, y, z : Coordinates (mm)

Acknowledgments

This research was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The publication itself is partly supported by the EFOP-3.6.1-16-2016-00022 project, and it is co-financed by the European Union and the European Social Fund.

References

All Tables

All Figures

	Fig. 1 Angles and diameters of bevel gear.
In the text

	Fig. 2 Reinecker–Bilgram principle [6].
In the text

	Fig. 3 Heidenreich–Harbeck principle [6].
In the text

	Fig. 4 Definition of planing angle.
In the text

	Fig. 5 Connection between the coordinate systems of the bevel gear drive pair.
In the text

	Fig. 6 Profile curves of the designed bevel gear pair (m = 15 mm, z1 = 20, z2 = 30, Σ = 90°). (a) Profile curves of the driver gear. (b) Profile curves of the driven gear.
In the text

	Fig. 7 CAD model of the bevel gear pair having straight teeth (m  = 15 mm, z1 = 20, z2 = 30, Σ = 90°).
In the text

	Fig. 8 FE mesh.
In the text

	Fig. 9 Setting loads and boundary conditions.
In the text

	Fig. 10 Defining normal stress.
In the text

Fig. 11 Normal stress values. (a) M = 200 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-2.054$ MPa (top), driven gear, ${\displaystyle \overline{{\sigma }_n}}=-2.164$ MPa (bottom). (b) M = 250 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-2.62$ MPa, driven gear, ${\displaystyle \overline{{\sigma }_n}}=-2.709$ MPa. (c) M = 300 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-3.203$ MPa, driven gear, ${\displaystyle \overline{{\sigma }_n}}=-3.297$ MPa. (d) M = 350 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-3.796$ MPa, driven gear, ${\displaystyle \overline{{\sigma }_n}}=-3.893$ MPa. (e) M = 400 Nm: driver gear, ${\displaystyle \overline{{\sigma }_n}}=-4.402$ MPa, driven gear, ${\displaystyle \overline{{\sigma }_n}}=-4.495$ MPa.

In the text

	Fig. 11 Continued.
In the text

	Fig. 12 Load − normal stress diagram.
In the text

Fig. 13 Normal elastic strain values as a result of increasing loads. (a) M = 200 Nm: driver gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000114 (top), driven gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000127 (bottom). (b) M = 250 Nm: driver gear,${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000141, driven gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000154. (c) M = 300 Nm: driver gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000168, driven gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000183. (d) M = 350 Nm: driver gear,${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000197, driven gear, ${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000214. (e) M = 400 Nm: driver gear,${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000227, driven gear,${\displaystyle \overline{{\epsilon }_n}}=$ −0.0000246.

In the text

	Fig. 13 Continued.
In the text

	Fig. 14 Load − normal elastic strain diagram.
In the text

	Fig. 15 Definition of displacement vector.
In the text

Fig. 16 Normal deformation values. (a) M = 200 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0654 mm (top), driven gear, ${\displaystyle \overline{u_x}}=$ 0.00166 mm (bottom). (b) M = 250 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0644 mm, driven gear, ${\displaystyle \overline{u_x}}=$ 0.00207 mm. (c) M = 300 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0636 mm, driven gear, ${\displaystyle \overline{u_x}}=$ 0.00247 mm. (d) M = 350 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0629 mm, driven gear, ${\displaystyle \overline{u_x}}=$ 0.00286 mm. (e) M = 400 Nm: driver gear, ${\displaystyle \overline{u_x}}=$ −0.0621 mm, driven gear, ${\displaystyle \overline{u_x}}=$ 0.00325 mm.

In the text

	Fig. 16 Continued.
In the text

	Fig. 17 Load–normal deformation diagram.
In the text