Geometric design, modelling and finite element analysis of spur gear pairs having normal teeth based on the modification of the pressure angle

Sándor Bodzás

doi:10.1556/1848.2021.00383

International Review of Applied Sciences and Engineering

Volume/Issue:: Volume 13: Issue 3

Geometric design, modelling and finite element analysis of spur gear pairs having normal teeth based on the modification of the pressure angle

Author:

Sándor Bodzás

Sándor Bodzás Department of Mechanical Engineering, University of Debrecen, Debrecen, Ótemető str. 2-4, H-4028, Debrecen, Hungary

Search for other papers by Sándor Bodzás in
Current site
Google Scholar
PubMed

bodzassandor@eng.unideb.hu https://orcid.org/0000-0001-8900-2800

View More View Less

Pages:: 285–297

Online Publication Date:: 24 Mar 2022

Publication Date:: 27 Oct 2022

Article Category:: Research Article

DOI:: https://doi.org/10.1556/1848.2021.00383

Keywords:: spur gear; CAD; TCA; pressure angle

Open access

Download PDF

Abstract

The aim of the thesis is the geometric design, CAD modelling and TCA of spur gear pairs having normal teeth based on the modification of the pressure angle. The first task is the geometric design and CAD modelling of the gear pairs (5 pairs are designed) where only the pressure angle is modified beside the constancy of the other initial parameters. The second task is to analyse the comparison possibility and the accuracy similarity of the 2D and the 3D models by the Hertz (equivalent) stress analysis. Finally, I give analysis of the maximum equivalent stress, normal stress and contact pressure for each pair while three teeth are rolling down on each other.

Abstract

1 Introduction

The TCA is a special sub area of the FEM analysis where the mechanical parameters are analysed on different types of toothed gear pairs on the tooth contact zone due to the loads [8].

The main property of the spur gear pairs having normal teeth is the application of the addendum modification that is why the tooth connection is established on the rolling circles [4–8, 11, 12]. Consequently, the base angle (α _p) is different from the pressure angle (α _w), which is interpreted between the common tangent line of the rolling circles and the common tangent line of the base circles (Fig. 1). The common tangent line of the base circles, which goes through the C main point, is called line of action. This line and the common tangent line of the rolling circles (r _w1 , r _w2 ) always from an α_w angle that is the pressure angle (Fig. 1). Due to the modification of the centre distance (a) this angle will be also modified based on Fig. 1. The centre distance can be calculated by the following formula [4–8, 11, 12]:

a = r_{w 1} + r_{w 2} = r_{p 1} \cdot \frac{c o s α_{p}}{c o s α_{w}} + r_{p 2} \cdot \frac{c o s α_{p}}{c o s α_{w}} = a_{0} \cdot \frac{c o s α_{p}}{c o s α_{w}}

(1)

where

r_{b} = r_{p} \cdot c o s α_{p} = r_{w} \cdot c o s α_{w}

(2)

Fig. 1.

Connection of the involute curves in case of different centre distances

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

The basic rack profile contains the base parameters of the normal section (circular pitch, whole depth, base profile angle and clearance). This profile has infinite number of teeth along a line. The base profile of an involute gear is standardized (Fig. 2).

Fig. 2.

Tool base profile in case of involute gear having normal teeth

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

The phenomenon when the tool center line and the tool reference line are not same is called gear having addendum modification. This process is also called addendum modification. This parameter can be calculated by the following formula [4–8, 11, 12].

x \cdot m

(3)

The x can be positive when the basic profile is moved from the gear axis (Fig. 3b). The x can be negative when the basic profile is moved to the gear axis (Fig. 3a). If x = 0, the tool center line and the tool reference line are the same. Such type of gear pair is called spur gear drive with no addendum modification [4–8, 11, 12, 13].

Fig. 3.

The connection possibilities of the tool base profile and the gear profile

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

In a general way the basic theorem of the gear connection is the multiplication of the relative velocity vector and the normal vector is zero on the contact points of the gears [7, 8]:

\vec{n_{1 R}} \cdot \vec{v_{1 R}} = \vec{n_{2 R}} \cdot \vec{v_{2 R}} = 0

(4)

2 Geometric design and the modelling of toothed gear pairs

The geometric establishment of the spur gear drive having normal teeth can be seen in Fig. 4. Addendum modification is used, thus the tooth connection occurs on the rolling circle diameters $(d_{p} \neq d_{w})$ [4–8, 11, 12]. The elementary and the normal center distances are not equal $(a_{0} \neq a)$ [4–8, 11, 12]. The initial parameters for the design process are z ₁ , z ₂ , $c_{0}^{'}$ , l, α _p and α _w. Gear I is the pinion that is assembled on the input shaft. This element drives Gear II, which is called gear (Fig. 4).

Fig. 4.

The geometric establishment of the spur gear drive having normal teeth

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

Based on the geometric formulas of the literature I have developed a computer software in Matlab programming language to determine the geometric parameters and the involute profiles of the pinion and the gear. The program can save the profile points of the gear pairs into a txt file. This file can be imported into the SolidWorks software where an interpolated B-spline can be fit on these points [1–3]. Considering the received geometric results and the profile curves, the computer aided (CAD) models of the elements can be built up. Finally, the connection-correct gear assembly can be made (Fig. 5) [1–3].

Fig. 5.

The CAD model of the gear pair (z ₁ = 20, z ₂ = 30, m = 6 mm, α _w = 21°)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

I have designed five gear pairs. The initial geometric parameters can be seen in Table 1. I have only modified the α _w pressure angle beside the constancy of the other initial geometric parameters.

Table 1.

The initial geometric parameters of the designed toothed gear pairs

Geometric parameters	Gear drive I	Gear drive II	Gear drive III	Gear drive IV	Gear drive V
α _w [°]	20	21	22	23	24
m [mm]	6
z ₁	20
z ₂	30
α _p [°]	20
$c_{0}^{'}$	0.2
u	1.5
l [mm]	50
c [mm]	1.2

The shape of the involute profile is unchanged since the base circle diameters are constant. The profile curve is generated from this circle. In spite of that, the shapes of the teeth are different because of the geometric parameters. The tooth connections take place on different d _w rolling circle diameters.

3 Comparative finite element analysis of 2D and 3D models

The aim of the analysis is to compare the accuracy similarity between the 2D and the 3D models beside the constancy of the load and boundary conditions by static structural analysis. The analysed model is the Gear drive III in Table 1. The material type can be seen in Table 2. Frictionless contact is defined between the connecting surfaces of the teeth. The load moment is 80 Nm, which can be selected by experience.

Table 2.

The applied material

Name	Steel
Linear Elastic	Isotropic Elasticity
Young's Modulus (E)	$210 \cdot 10^{3} M P a$
Poisson's Ratio (ν)	0.3

I have created the 2D model by SpaceClaim software, where the thickness (l = 50 mm tooth length) is considered for the analysis (Fig. 6).

Fig. 6.

The 2D model for the FEM analysis

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

Coordinate systems are adapted into the middle points of the gears while local coordinate systems are defined in the contact point of the teeth.

The main problem of the comparison is the mesh type and the element size. It is well-known that the smaller the mesh size is, the longer the calculation process takes. In addition, if we use different element sizes for the two models the results will be different [9, 10]. It is known that a much smoother mesh can be useable in the case of the 2D model than in the case of the 3D model. Because of the computer’s capacity, the mesh of the 3D model can be smoothed until a certain limit, which is bigger than in the case of the 2D model. Otherwise, the computer could not calculate the FEM problem. That is why I have to find an optimized mesh where the FEM results of the 2D and 3D models are comparable.

3.1 The analytic determination of the contact stress

The maximum Hertz stress (Equivalent stress) can be calculated in the case of toothed gear pairs by the following formula [11]:

σ_{H}^{2} = 0.35 \cdot \frac{F_{n}}{l} \cdot \frac{\frac{1}{ρ_{1}} + \frac{1}{ρ_{2}}}{\frac{1}{E_{1}} + \frac{1}{E_{2}}}

(5)

Consequently, the characteristic of the involute curve is that the common normal line of the connecting tooth curves touches the base circle diameters (d _b1, d _b2) from which the involute curve can be generated (Fig. 6).

The Pythagorean theorem can be useable for the O ₁ N ₁ C and the O ₂ N ₂ C rectangular triangles to determine the ρ ₁ and ρ ₂ involute curvatures on the common contact point which is the common tangent point of the rolling circles (Fig. 7):

{(\frac{d_{w 1,2}}{2})}^{2} = ρ_{1,2}^{2} + {(\frac{d_{b 1,2}}{2})}^{2} \to ρ_{1,2} = \sqrt{{(\frac{d_{w 1,2}}{2})}^{2} - {(\frac{d_{b 1,2}}{2})}^{2}}

(6)

ρ_{1} = 22.778 m m, ρ_{2} = 34.169 m m

Fig. 7.

The connection of involute curves

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

The load moment from the pinion affects the gear along the perimeter of the rolling circle diameter of the pinion. The circumferential force can be calculated by the following formula:

M = F_{c} \cdot \frac{d_{w 1}}{2} \to F_{c} = \frac{2 \cdot M}{d_{w 1}} = \dots = 1315.594 N

(7)

Based on Fig. 8 the normal force is

F_{n} = \frac{F_{c}}{\cos α_{w}} = \dots = 1418.914 N

(8)

Fig. 8.

The relation between the normal force and the circumferential force

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

Substituting the calculated results into (5) and extracting of square root from (5), the calculated maximum Hertz stress on the contact zone is $σ_{H} = 276.239 M P a$ .

3.2 Approach by Normal Lagrange formulation

I calculated the simulation result by Augmented Lagrange formulation. When I compared the calculated results for the 2D and 3D models I got different values. Having checked the received reaction force in the software and the calculated results, I found that they are identical. I applied a parametric study for generalized plain strain 2D behaviour. I got much lower results than what I had calculated manually.

That is the reason why I have to approach the problem by Normal Lagrange formulation. The Normal Lagrange formulation adds an extra degree of freedom (contact pressure) to satisfy contact compatibility. Consequently, instead of resolving contact force as contact stiffness and penetration, contact force (contact pressure) is solved explicitly as an extra degree of freedom. This process can enforce zero/nearly-zero penetration. Normal contact stiffness is not required. It requires Direct Solver, which can be more computationally expensive (Fig. 9) [14].

Fig. 9.

The concept of the Normal Lagrange formulation

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

I also run the convergence analysis to decide the applicable element size, and to estimate the sphere radius for meshing. I cannot get significant changes after the 7th step. The result can be seen in Fig. 10. The maximum equivalent stress is 288.36 MPa. The 2D behaviour is plain stress.

Fig. 10.

The results of the convergence analysis using Normal Lagrange

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

Based on Fig. 10, I can measure the element size (0.025 mm) in the red contact zone for the other analysis.

After that, I make parametric study on the 2D model to optimize the element size and the sphere radius for the meshing. The results can be seen in Fig. 11.

Fig. 11.

The results of the parametric study on the 2D model using Normal Lagrange in case of plain stress behaviour

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

I select the 7th line on Fig. 11 for setting it on the 3D model for meshing since I got the same result as with the calculation. The length is reduced to one tenth of the total length, that is why the load moment is 8 Nm. The element size inside the sphere is 0.025 mm. The number of divisions is 15 pieces. The results can be seen in Fig. 12. I got a lower result than the calculated and the received ones on the 2D model version.

Fig. 12.

The results on the 3D model using Normal Lagrange (7th line)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

I select another line, which is the 3rd line (Figs 10 and 13). The initial parameters are the same as in the previous case except for the sphere radius (1.1 mm). The results can be seen in Fig. 13. In the engineering practice, 10–12% error tolerance is acceptable. The results of the 2D model (Fig. 14) and the 3D model (Fig. 13) are within this acceptable zone.

Fig. 13.

The results on the 3D model using Normal Lagrange (1st line)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

Fig. 14.

The results on the 2D model using Normal Lagrange (1st line)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

I had the reaction forces calculated by the software (Fig. 15). This result and the calculated result are identical. Consequently, the key for the comparative simulation is the optimization of the element size to compare the results of the 2D and the 3D models. The software can calculate the reaction forces correctly, regardless of the meshing density, since the simulation force result and the calculation force result are the same. It had the same result in the case of Augmented Lagrange formulation too.

Fig. 15.

The reaction forces on the 2D model

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

The stress distributions on the contact surfaces on the pinion and the gear can be seen in Fig. 16.

Fig. 16.

The stress distributions on the contact surfaces

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

4 Comparative TCA in the function of the pressure angle

4.1 Making the CAD models

Three-three teeth connections are analysed on all of the pairs according to Table 1. In order to make the calculation easier, the CAD models have to be simplified on each pairs (Fig. 17). I created sub elements, which are inserted into the teeth to generate dense meshing on the contact surfaces.

Fig. 17.

The CAD models of the connecting elements (Gear drive III)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

4.2 Setting the mesh, the load and the boundary conditions

The applied material is selected based on Table 2. The analysis type is static structural. The half tooth length is used for the reduction of the calculation time. The applied load moment is 40 Nm with which the pinion is loaded.

Augmented Lagrange formulation is used for the analysis. The setting of the contacts can be seen in Fig. 18. Frictionless contact type is selected.

Fig. 18.

The setting of the contacts

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

I have defined cylindrical joints for the pinion and the gear. Only the rotations around the axis of rotations are enabled. The other degrees of freedom are fixed.

Dense meshing (Fig. 19) is used on the sub elements (element size 0.2 mm). The number of divisions is 5 along the tooth length for the sweep method. The free face mesh type is quadratic and triangular combined. Automatic meshing is used on the outside areas.

Fig. 19.

The establishment of the mesh

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

The number of both the initial sub steps and the minimum sub steps are set for 100. The maximum sub steps are set for 1,000 steps. Frictionless support is used on the head surfaces. Only the rotations around the axis of rotations are enabled. During the analysis, the three teeth are continuously rolling down on each other.

4.3 Evaluation of the results

The equivalent and the normal stress, as well as the contact pressure are calculated by the software for each pair during the rolling down. The simulations are calculated in the global coordinate system. The results of the equivalent stress can be seen in Fig. 20 on an intermediary status (t = 0.5 s). I have to use average values due to the limitations of the computer capacity and the meshing. The average results can be seen in Fig. 21. The lowest result is received in the case of α _w = 21°. The highest result is in the case of α _w = 20° (Fig. 21).

Fig. 20.

The equivalent stress results (t = 0.5 s)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

Fig. 21.

The average results of the maximum equivalent stresses

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

The results of the normal stress can be seen in Fig. 22 in an intermediary status (t = 0.5 s). The average results can be seen in Fig. 23. The lowest result is received in the case of α _w = 20°. The highest result is in the case of α _w = 22° (Fig. 23).

Fig. 22.

The distribution of the maximum normal stress (t = 0.5 s)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

Fig. 23.

The average results of the maximum normal stresses

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

The results of the contact pressure can be seen in Fig. 24 in an intermediary status (t = 0.5 s). The average results can be seen in Fig. 25. The lowest result is received in the case of α _w = 22°. The highest result is in the case of α _w = 23° (Fig. 25).

Fig. 24.

The distribution of the maximum contact pressure (t = 0.5 s)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

Fig. 25.

The average results of the maximum contact pressure

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2021.00383

5 Conclusion

This study looks into the field of CAD and TCA of spur gear pairs. The TCA is a subfield of the FEM. This area is the comprehensive research field of the connection analysis of various types of toothed gears where the tooth contact zone is analysed in mechanical and dynamical aspects based on different loads.

I have designed five types of spur gear pairs having normal teeth where only the pressure angle was different beside the constancy of the other initial parameters. I created a computer software in Matlab language to simplify the calculation and the designing process. Knowing the geometric parameters and the involute profile curves, the CAD models can be constructed. After that, the assembly and the beat examinations can be done by SolidWorks software to check the geometric accuracy.

Plenty of simulations can be performed by the Ansys software. Based on the references I worked on the Static Structural option which is mainly used in the field of connection analysis of toothed gear pairs.

I did comparative FEM analysis for a given geometry (α _w = 22°) among the Hertz (equivalent) stresses of the 2D and 3D models and the manual calculation. I approached the problem by Augmented Lagrange and Normal Lagrange formulations. The Normal Lagrange formulation proved better since the results were closer to the manual result. During the analysis I did parametric studies and convergence analyses to determine the comparative meshing possibility and the element size. Finally, I managed to get equal stress results, which are within 10–12% tolerance limit for the 2D and 3D models compared to the manual result. I also checked the similarity of the reaction forces on the FEM models and the manual calculation. I received the same results.

Considering the five types of gear pairs, I compared the effect of modification of the pressure angle on the mechanical parameters (maximum equivalent and normal stress, and maximum contact pressure). I applied three connecting teeth for each pairs. I divided the 1 s duration of rolling down for 100 pieces for which I got the results calculated. I made diagrams to evaluate the average results by MS Excel program.

I created a unique FEM model and analysis process to compare these gear pairs as a function of the modification of the pressure angle. By refining the meshing better results can be achieved but a stronger computer would be required for this purpose. The used computer capacity was the following: Processor: Intel(R) Core(TM) i5-8300H CPU @ 2.30 GHz, Memory (RAM) size: 16,0 GB, System type: 64 bit operation system, Windows version: Windows 10 Enterprise. It is well-known that by smoothing the element size of the meshing, the calculation time is increasing exponentially, which would require more developed computer apparatus too.

Acknowledgement

The work is supported by the EFOP-3.6.1-16-2016-00022 project. The project is co-financed by the European Union and the European Social Fund.

I appreciate the help and the simulation knowledge of assistant lecturer Dávid Huri.

I thank for the help of Péter Szurdoki, who is an English teacher. He reviewed the publication regarding the English grammar.

I would like to thank Professor Illés Dudás and Professor Alfonso Fuentes, who introduced this interesting TCA research field to me.

References

[1]↑
S. Bodzás , “Computer aided designing and modelling of spur gear pairs having normal and modified straight teeth,” Int. Rev. Appl. Sci. Eng., vol. 10, no. 2, pp. 157–163, 2019, 7 p. https://doi.org/10.1556/1848.2019.0019.
- Search Google Scholar
- Export Citation
[2]
S. Bodzás , “Analysis of the effect of the addendum modification coefficient for contact surfaces of spur gear,” Strojnicky Casopis/J. Mech. Eng., vol. 69, no. 1, pp. 5–16, 2019, 12 p. https://sciendo.com/doi/10.2478/scjme-2019-0001.
- Search Google Scholar
- Export Citation
[3]
S. Bodzás , “Tooth contact analysis of spur gear pairs having normal straight teeth in the function of the module,” Acta Facultatis Technicae Zvolen, vol. 23, no. 2, pp. 91–103, 2018, 13 p. https://ft.tuzvo.sk/sites/default/files/aft_2_2018_journal_0.pdf.
- Search Google Scholar
- Export Citation
[4]↑
J. Drobni , Gépelemek III. National Publisher, University of Miskolc, 1993, p. 231.
- Search Google Scholar
- Export Citation
[5]
Gy. Erney , Fogaskerekek. Budapest: Technical Publisher, 1983, p. 460.
- Search Google Scholar
- Export Citation
[6]
V. Goldfarb , E. Trubachev , and N. Barmina , Advanced Gear Engineering. Springer, 2018, p. 197, ISBN 978-3-319-60398-8.
- Search Google Scholar
- Export Citation
[7]↑
F. L. Litvin , A fogaskerékkapcsolás elmélete. Budapest: Technical Publisher, 1972.
- Search Google Scholar
- Export Citation
[8]↑
F. L. Litvin and A. Fuentes , Gear Geometry and Applied Theory. Cambridge University Press, 2004, p. 800, ISBN 978 0 521 81517 8.
- Search Google Scholar
- Export Citation
[9]↑
T. Mankovits and D. Huri , Modellezés és szimuláció (A lineáris rugalmasságtan és a végeselem módszer). University of Debrecen, lecture note, 2015, p. 183 [Online]. Available: https://mecheng.unideb.hu/sites/default/files/upload_documents/modellezesszimulacio_mankovits-huri.pdf. Accessed: Nov. 15, 2021.
- Search Google Scholar
- Export Citation
[10]↑
S. Moaveni , Finite Element Analysis, Theory and Application with ANSYS. Pearson Education Limited, 2015, p. 928, ISBN 10: 0-273-77430-1.
- Search Google Scholar
- Export Citation
[11]↑
Z. Terplán , Gépelemek IV. Budapest: Book Publisher, 1975, p. 220.
- Search Google Scholar
- Export Citation
[12]↑
I. Vörös , Gépelemek III., Fogaskerekek, 2nd Edition. Budapest: Book Publisher, 1958, p. 515.
- Search Google Scholar
- Export Citation
[13]↑
R. Golebski and A. Szarek , “Diagnosis of the operational gear wheel wear,” Tech. Gaz., vol. 24, no. 4, pp. 658–661, 2017, ISSN 1330-3651 (Print), ISSN 1848-6339 [Online]. Available: https://doi.org/10.17559/TV-20180321171428. Accessed: Nov. 16, 2021.
- Search Google Scholar
- Export Citation
[14]↑
Ansys Lecture 3: Introduction to Contact, Ansys Mechanical Structural Nonlinearities. Costumer Training Material, 2010 [Online]. Available: https://www.researchgate.net/profile/Musavir_Bashir/post/how_can_I_solve_such_type_of_problem_in_ansys_CFD/attachment/5ac9a4be4cde260d15d76e9f/AS%3A613009869258752%401523164350093/download/ansys+mech+structural.pdf. Accessed: Nov. 16, 2021.
- Search Google Scholar
- Export Citation

NOMENCLATURE

Symbol	Unit	Parameter
ρ ₁	[mm]	Curvature radius of the involute curve on the pinion
ρ ₂	[mm]	Curvature radius of the involute curve on the gear
σ _H	[MPa]	Hertz (Equivalent) stress
α _p	[°]	Base profile angle (α _p = 20°)
α _w	[°]	Pressure angle
$c_{0}^{'}$		Clearance factor ( $c_{0}^{'} = 0.25$ )
a	[mm]	Normal center distance
a ₀	[mm]	Elementary center distance
CAD		Computer Aided Design
d _a1	[mm]	Outside circle diameter of the pinion
d _a2	[mm]	Outside circle diameter of the gear
d _b1	[mm]	Base circle diameter of the pinion
d _b2	[mm]	Base circle diameter of the gear
d _f1	[mm]	Root circle diameter of the pinion
d _f2	[mm]	Root circle diameter of the gear
d _p1	[mm]	Pitch circle diameter of the pinion
d _p2	[mm]	Pitch circle diameter of the gear
d _w1	[mm]	Rolling circle diameter of the pinion
d _w2	[mm]	Rolling circle diameter of the gear
E ₁	[MPa]	Young modulus of the pinion
E ₂	[MPa]	Young modulus of the gear
F _c	[N]	Circumferential force
FEM		Finite Element Analysis
F _n	[N]	Normal force
h	[mm]	Whole depth
h _a	[mm]	Addendum
h _f	[mm]	Dedendum
l	[mm]	Tooth length
m	[mm]	Transverse module
p _p	[mm]	Circular pitch on the pitch circle
p _w	[mm]	Circular pitch on the rolling circle
r _p1	[mm]	Pitch circle radius of the pinion
r _p2	[mm]	Pitch circle radius of the gear
r _w1	[mm]	Rolling circle radius of the pinion
r _w2	[mm]	Rolling circle radius of the gear
s _p1	[mm]	Tooth (arc) thickness of the pinion on the pitch circle
s _p2	[mm]	Tooth (arc) thickness of the gear on the pitch circle
s _w1	[mm]	Tooth (arc) thickness of the pinion on the rolling circle
s _w2	[mm]	Tooth (arc) thickness of the gear on the rolling circle
TCA		Tooth Contact Analysis
u		Tooth ratio
x		Addendum modification coefficient
x ₁		Addendum modification coefficient of the pinion
x ₂		Addendum modification coefficient of the gear
z ₁		Number of teeth of the pinion
z ₂		Number of teeth of the gear
ω ₁	[1/s]	Angular velocity on the pinion
$\vec{n_{2 R}}$		Normal vector of the gear
$\vec{v_{2 R}}$		Relative velocity vector of the gear
$\vec{n_{1 R}}$		Normal vector of the pinion
$\vec{v_{1 R}}$		Relative velocity vector of the pinion

ProCite

RefWorks

Reference Manager

BibTeX

Zotero

EndNote

Save
Cite
Email this content

Share Link

Copy this link, or click below to email it to a friend
Email this content
or copy the link directly:

https://akjournals.com/view/journals/1848/13/3/article-p285.xml
The link was not copied. Your current browser may not support copying via this button.

Link copied successfully

Collapse
Expand

International Review of Applied Sciences and Engineering

Print ISSN:: 2062-0810

Online ISSN:: 2063-4269

Editorial Board

Mohammad Nazir AHMAD, Institute of Visual Informatics, Universiti Kebangsaan Malaysia, Malaysia

Murat BAKIROV, Center for Materials and Lifetime Management Ltd., Moscow, Russia

Nicolae BALC, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Umberto BERARDI, Toronto Metropolitan University, Toronto, Canada

Ildikó BODNÁR, University of Debrecen, Debrecen, Hungary

Sándor BODZÁS, University of Debrecen, Debrecen, Hungary

Fatih Mehmet BOTSALI, Selçuk University, Konya, Turkey

Samuel BRUNNER, Empa Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

István BUDAI, University of Debrecen, Debrecen, Hungary

Constantin BUNGAU, University of Oradea, Oradea, Romania

Shanshan CAI, Huazhong University of Science and Technology, Wuhan, China

Michele De CARLI, University of Padua, Padua, Italy

Robert CERNY, Czech Technical University in Prague, Prague, Czech Republic

Erdem CUCE, Recep Tayyip Erdogan University, Rize, Turkey

György CSOMÓS, University of Debrecen, Debrecen, Hungary

Tamás CSOKNYAI, Budapest University of Technology and Economics, Budapest, Hungary

Anna FORMICA, IASI National Research Council, Rome, Italy

Alexandru GACSADI, University of Oradea, Oradea, Romania

Eugen Ioan GERGELY, University of Oradea, Oradea, Romania

Janez GRUM, University of Ljubljana, Ljubljana, Slovenia

Géza HUSI, University of Debrecen, Debrecen, Hungary

Ghaleb A. HUSSEINI, American University of Sharjah, Sharjah, United Arab Emirates

Nikolay IVANOV, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

Antal JÁRAI, Eötvös Loránd University, Budapest, Hungary

Gudni JÓHANNESSON, The National Energy Authority of Iceland, Reykjavik, Iceland

László KAJTÁR, Budapest University of Technology and Economics, Budapest, Hungary

Ferenc KALMÁR, University of Debrecen, Debrecen, Hungary

Tünde KALMÁR, University of Debrecen, Debrecen, Hungary

Milos KALOUSEK, Brno University of Technology, Brno, Czech Republik

Jan KOCI, Czech Technical University in Prague, Prague, Czech Republic

Vaclav KOCI, Czech Technical University in Prague, Prague, Czech Republic

Imre KOCSIS, University of Debrecen, Debrecen, Hungary

Imre KOVÁCS, University of Debrecen, Debrecen, Hungary

Angela Daniela LA ROSA, Norwegian University of Science and Technology, Trondheim, Norway

Éva LOVRA, Univeqrsity of Debrecen, Debrecen, Hungary

Elena LUCCHI, Eurac Research, Institute for Renewable Energy, Bolzano, Italy

Tamás MANKOVITS, University of Debrecen, Debrecen, Hungary

Igor MEDVED, Slovak Technical University in Bratislava, Bratislava, Slovakia

Ligia MOGA, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Marco MOLINARI, Royal Institute of Technology, Stockholm, Sweden

Henrieta MORAVCIKOVA, Slovak Academy of Sciences, Bratislava, Slovakia

Phalguni MUKHOPHADYAYA, University of Victoria, Victoria, Canada

Balázs NAGY, Budapest University of Technology and Economics, Budapest, Hungary

Husam S. NAJM, Rutgers University, New Brunswick, USA

Jozsef NYERS, Subotica Tech College of Applied Sciences, Subotica, Serbia

Bjarne W. OLESEN, Technical University of Denmark, Lyngby, Denmark

Stefan ONIGA, North University of Baia Mare, Baia Mare, Romania

Joaquim Norberto PIRES, Universidade de Coimbra, Coimbra, Portugal

László POKORÁDI, Óbuda University, Budapest, Hungary

Roman RABENSEIFER, Slovak University of Technology in Bratislava, Bratislava, Slovak Republik

Mohammad H. A. SALAH, Hashemite University, Zarqua, Jordan

Dietrich SCHMIDT, Fraunhofer Institute for Wind Energy and Energy System Technology IWES, Kassel, Germany

Lorand SZABÓ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Csaba SZÁSZ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Ioan SZÁVA, Transylvania University of Brasov, Brasov, Romania

Péter SZEMES, University of Debrecen, Debrecen, Hungary

Edit SZŰCS, University of Debrecen, Debrecen, Hungary

Radu TARCA, University of Oradea, Oradea, Romania

Zsolt TIBA, University of Debrecen, Debrecen, Hungary

László TÓTH, University of Debrecen, Debrecen, Hungary

László TÖRÖK, University of Debrecen, Debrecen, Hungary

Anton TRNIK, Constantine the Philosopher University in Nitra, Nitra, Slovakia

Ibrahim UZMAY, Erciyes University, Kayseri, Turkey

Andrea VALLATI, Sapienza University, Rome, Italy

Tibor VESSELÉNYI, University of Oradea, Oradea, Romania

Nalinaksh S. VYAS, Indian Institute of Technology, Kanpur, India

Deborah WHITE, The University of Adelaide, Adelaide, Australia

International Review of Applied Sciences and Engineering
Address of the institute: Faculty of Engineering, University of Debrecen
H-4028 Debrecen, Ótemető u. 2-4. Hungary
Email: irase@eng.unideb.hu

Indexing and Abstracting Services:

DOAJ
ERIH PLUS
Google Scholar
ProQuest
SCOPUS
Ulrich's Periodicals Directory

2023
Scimago
Scimago H-index	11
Scimago Journal Rank	0.249
Scimago Quartile Score	Architecture (Q2) Engineering (miscellaneous) (Q3) Environmental Engineering (Q3) Information Systems (Q4) Management Science and Operations Research (Q4) Materials Science (miscellaneous) (Q3)
Scopus
Scopus Cite Score	2.3
Scopus CIte Score Rank	Architecture (Q1) General Engineering (Q2) Materials Science (miscellaneous) (Q3) Environmental Engineering (Q3) Management Science and Operations Research (Q3) Information Systems (Q3)
Scopus SNIP	0.751

International Review of Applied Sciences and Engineering
Publication Model	Gold Open Access Online only
Submission Fee	none
Article Processing Charge	1100 EUR/article
Regional discounts on country of the funding agency	World Bank Lower-middle-income economies: 50% World Bank Low-income economies: 100%
Further Discounts	Limited number of full waivers available. Editorial Board / Advisory Board members: 50% Corresponding authors, affiliated to an EISZ member institution subscribing to the journal package of Akadémiai Kiadó: 100%
Subscription Information	Gold Open Access

International Review of Applied Sciences and Engineering
Language	English
Size	A4
Year of Foundation	2010
Volumes per Year	1
Issues per Year	3
Founder	Debreceni Egyetem
Founder's Address	H-4032 Debrecen, Hungary Egyetem tér 1
Publisher	Akadémiai Kiadó
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	2062-0810 (Print)
ISSN	2063-4269 (Online)