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

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Abstract

The aim of this study is the manufacturing analysis of five spur gear pairs where the initial geometric parameters are the same only the pressure angle is different. Firstly, the gears must be designed and modelled. After that, I analyse the modification of this geometric parameter for the manufacturing parameters of the pinion and the gear in the case of gear cutting by module disc milling cutter. Using this technology the one tooth cutting can repeat from tooth to tooth in the function of the number of teeth. I would like to find correlations between the pressure angle and the manufacturing parameters. For this purpose, I define the initial technological parameters and calculate necessary technological parameters for the manufacturing process in a general way. I also define the manufacturing parameters for the given gear geometries. This analysis is practical and theoretical at the same time since the results and the process can help the manufacturing engineers to develop the gear manufacturing processes and applying my results for similar manufacturing problems.

Abstract

The aim of this study is the manufacturing analysis of five spur gear pairs where the initial geometric parameters are the same only the pressure angle is different. Firstly, the gears must be designed and modelled. After that, I analyse the modification of this geometric parameter for the manufacturing parameters of the pinion and the gear in the case of gear cutting by module disc milling cutter. Using this technology the one tooth cutting can repeat from tooth to tooth in the function of the number of teeth. I would like to find correlations between the pressure angle and the manufacturing parameters. For this purpose, I define the initial technological parameters and calculate necessary technological parameters for the manufacturing process in a general way. I also define the manufacturing parameters for the given gear geometries. This analysis is practical and theoretical at the same time since the results and the process can help the manufacturing engineers to develop the gear manufacturing processes and applying my results for similar manufacturing problems.

1 Introduction

The involute profile is selected based on experience of geometric, tooth connection and load transmission. It can be generated by constructive and mathematical way. The involute curve is always generated from the base circle of the gear [10, 11, 19, 21].

The parametric equation of the involute curve is (Fig. 1) [10, 21]
x = r b sin φ r b φ cos φ y = r b cos φ + r b φ sin φ }
Fig. 1.
Fig. 1.

The generation of the involute curve by mathematical way

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The polar angle is the inv α . The φ angle is
φ = inv α + α
The following two equations can also be derived on Fig. 1. [10, 21]:
ρ = r b φ
ρ = r b tan
Based on (3) and (4)
φ = tan
Substituting (5) into (2) and expressing inv α
inv α = tan

The common normal line, which is the common tangent line of the base circles, has to go through on the C main point (Fig. 2) [10, 11, 19, 21]. The connection always takes place on this line. This line is called line of action. This line and the common tangent line of the rolling circles (r w1 , r w2 ) always include an α w angle [21]. This angle is called pressure angle. If the centre distance is modified from a to a′ the α w will be also modified to α′ w . The pressure angle that belongs to the pitch circle radius (r p ) is called base profile angle p ).

Fig. 2.
Fig. 2.

Connection of the involute curves in the case of different centre distances

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

Starting from the pitch and the rolling circle radiuses, the radius of the base circle from which the arc can be generated is [10, 11, 19, 21]
r b = r p cos α p = r w cos α w
r w = r p cos α p cos α w
The centre distance is
a = r w 1 + r w 2 = r p 1 cos α p cos α w + r p 2 cos α p cos α w = a 0 cos α p cos α w
The elementary centre distance is [10, 11, 19, 21]
a 0 = r p 1 + r p 2 = d p 1 + d p 2 2

1.1 The properties of spur gear having normal teeth

The basic rack gear tooth profile contains the base parameters of the normal section (circular pitch, whole depth, basic rack gear tooth profile angle and clearance). This profile has infinite number of teeth along a line. The basic rack gear tooth profile of an involute gear is standardized (Fig. 3) [10, 11, 19, 21].

Fig. 3.
Fig. 3.

Tool basic rack gear tooth profile in the case of involute gear having normal teeth

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

During the manufacturing process, the pitch circle of the gear is rolled down on the tool centre line without slip [4, 5, 7, 8, 12, 15, 16]. The tool centre line and the tool reference line can be different because the pitch circle of the gear can be rolled down any parallel lines of the tool centre line [1, 2, 5, 11, 12, 21]. The phenomenon when the tool centre line and the tool reference line are not same is called gear having addendum modification [10, 11, 21]. This process is also called addendum modification. This parameter can be calculated by the following formula [10, 11, 21]
x 1 m a x

The x is positive when the basic profile is moved from the gear axis (Fig. 4b). The x is negative when the basic profile is moved to the gear axis (Fig. 4a). If x=0, the tool centre line and the tool reference line are the same. This type of gear pair is called x-zero gear drive [10, 11, 21].

Fig. 4.
Fig. 4.

The connection possibilities of the tool basic rack gear tooth profile and the gear profile

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

1.2 Determination of the tooth thickness in general way

Based on Fig. 5a the tooth thickness is [10, 11, 21]
s p = p p 2 2 x m a x tan α p = m a x ( π 2 2 x tan α p )
Fig. 5.
Fig. 5.

The correlation between the addendum modification and the tooth thickness on the pitch circle on the basic rack gear tooth profile

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

Based on Fig. 5b the tooth thickness is [10, 11, 21]
s p = p p 2 + 2 x m a x tan α p = m a x ( π 2 + 2 x tan α p )
The σ angle is (Fig. 6) [10, 11, 21]
σ = inv α p + s p 2 r p = inv α + s 2 r
Fig. 6.
Fig. 6.

The correlations of the tooth thicknesses on the pitch circle (r p ) and an arbitrary (r) circle

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The s tooth thickness is on an arbitrary circle is
s = 2 r ( s p 2 r p + inv α p inv α )

1.3 Manufacturing of spur gear by module disc milling cutter

Spur and helical gears could be manufactured by plain milling technology on a horizontal knee type milling machine (Figs 8 and 9) or a CNC milling machine (Fig. 9) [1, 3–5, 8, 9, 12, 15, 16]. The profile of the module disc milling cutter is the same as that of the tooth space [1, 2, 4, 7–9, 11, 12]. The tool is doing rotation ( v c ) and linear ( v f ) motions at the same time. This linear motion could also be provided by the workpiece (Fig. 7). After one tooth is ready the division form tooth to tooth could be made possible by a dividing head (Fig. 8) in a classical way. The milling process can restart again [1–4, 9, 11, 12, 15, 16].

Fig. 7.
Fig. 7.

Manufacturing of spur gear by module disc milling cutter [3, with permission from Debrecen University Press]

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

Fig. 8.
Fig. 8.

The structure of the dividing head

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

Fig. 9.
Fig. 9.

Spur gear manufacturing by a CNC milling machine

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The geometric shape of the module disc milling cutter can be seen in Fig. 10. The geometry of the tool depends on the number of teeth and the module of the gear [1, 2, 9, 12, 15, 16].

Fig. 10.
Fig. 10.

The geometry of the module disc milling cutter

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

2 Geometric design and modelling of spur gears having normal teeth

Knowing the references' recommendations [10, 11, 13, 19, 20, 21] and the initial gear parameters all of the other geometric parameters can be calculated by MATLAB software, which was created by me. The formulas for the gear design were programmed into this software. The output parameters of this program are the calculated geometric parameters, the profile curves of the elements and a txt file that contains the point coordinates of the profile points. The involute profile curves on the pinion and the gear in the case of α w = 23° as an example can be seen in Fig. 11.

Fig. 11.
Fig. 11.

The profiles of the gear pairs ( α w = 23°)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

Geometrically, the shape of the involute curves is similar to both gear pairs since the base circle diameters are the same for each gear pair. The differences are the arc length between the root and the outside circles. The tooth connections take place on different d w rolling circle diameters [10, 11, 21]. The calculated geometric parameters of the designed gear pairs can be seen in Table 1.

Table 1.

The calculated geometric parameters of the gear pairs

Geometric parameters Gear drive I Gear drive II Gear drive III Gear drive IV Gear drive V
m ax [mm] 6
z 1 20
z 2 30
α p [°] 20
c 0 0.2
u 1.5
l m [mm] 50
α w [°] 20 21 22 23 24
d p1 [mm] 120
d p2 [mm] 180
d b1 [mm] 112.763
d b2 [mm] 169.144
d w1 [mm] 120 120.785 121.618 122.501 123.434
d w2 [mm] 180 181.178 182.428 183.752 185.151
x 0 0.167 0.353 0.559 0.786
a [mm] 150 150.982 152.023 153.126 154.293
a 0 [mm] 150
y 0 0.163 0.337 0.521 0.715
h’ [mm] 12 11.976 11.901 11.770 11.576
h a [mm] 6 5.988 5.950 5.885 5.788
c [mm] 1.2
p p [mm] 18.849
p w [mm] 18.849 18.973 19.103 19.242 19.389
h f [mm] 7.2 7.188 7.150 7.085 6.988
h [mm] 13.2 13.176 13.101 12.970 12.776
d a1 [mm] 132 132.761 133.520 134.271 135.011
d a2 [mm] 192 193.154 194.329 195.522 196.728
d f1 [mm] 105.6 106.409 107.317 108.331 109.458
d f2 [mm] 165.6 166.802 168.126 169.582 171.175
j s [mm] 0.942 0.948 0.955 0.962 0.969
x 1 0 0.065 0.134 0.208 0.286
x 2 0 0.102 0.218 0.351 0.499
s p1 [mm] 9.424 9.710 10.014 10.335 10.674
s p2 [mm] 9.424 9.871 10.380 10.957 11.608
s w1 [mm] 8.953 9.005 9.045 9.071 9.082
s w2 [mm] 8.953 9.019 9.103 9.208 9.336

Knowing the geometric parameters of the gear pairs the CAD models and the assembly can be done by SolidWorks software. These models are important for the manufacturing simulations (CAM) [3, 17, 18] and the tooth contact analysis (TCA) [10]. The CAD models of a gear pair ( α w = 23°) can be seen in Fig. 12.

Fig. 12.
Fig. 12.

The CAD model of a gear pair (z 1 =20, z 2 =30, max=6 mm, α w =23°, Gear drive IV)

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The effect of the pressure angle on the geometric parameters can be seen in the diagrams of Fig. 13.

Fig. 13.
Fig. 13.

The effect of the modification of the pressure angle on the geometric parameters

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

3 Analysis of the manufacturing parameters

3.1 Determination of the technological parameters by general way

The gear cutting process by module disc milling cutter is applicable in a conventional way (gear cutting on horizontal knee type milling machine) or in a modern way (application of CNC machine) too [1, 3–5, 8, 9, 12, 15, 16]. Since the basis of the modern way is the conventional way, I analyse this gear cutting technology in the conventional way. The gear parameters are changing (tooth thickness, diameters, pressure angle, etc.), consequently the tool geometry must be also changed [1, 2, 7–9, 11, 12, 15, 16]. Gears having different geometries need different cutting tools in geometric aspects.

The cutting process for one tooth can be seen in Fig. 14. This process must be repeated in the function of the number of teeth. The gear is fixed into a clamping device. After the cutting of one tooth the gear has to be divided according to the circular pitch. The tool has two motions: rotation ( v c ) and linear ( v f ) motions at the same time [1, 2, 4–9, 11, 12, 15, 16]. The ‘E’ middle point of the tool is controlled [17, 18]. We need to provide safety distance before and after the cutting process. These distances are called overruns (x 1 and x 2 ). The initial parameters of the execution of the technology are the following: the tool and workpiece material, the workpiece (pinion and gear) geometry, the tool geometry, the feed for one edge (f z ), the adjustable number of revolution (n) and the overruns (x 1 , x 2 ).

Fig. 14.
Fig. 14.

The cutting process for one tooth

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

Knowing the h whole depth and the D tool diameter the m distance is (Fig. 14a EBD triangle)
m = ( D 2 ) 2 ( D 2 h ) 2 = D h h 2
The φ c angle of contact is (Fig. 14a EBD triangle)
φ c ° = a tan ( m D 2 h )
The i arc of contact is (Fig. 14a)
i = φ c ° D π 360 °

The separated chip volume can be seen in Fig. 15. The widest distance is the w a tooth space on the outside circle of the gear.

Fig. 15.
Fig. 15.

The approximation by volume constancy

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The chip thickness is continuously changing along the i arc of contact, that is why this parameter will be considered by average value, which is called h m medium chip thickness [1, 2, 3, 6, 7, 12, 14]. The V h volume, which has a complex shape, can be approximated by the V 1 and V 2 prism volumes (Fig. 15) because of the simplification of the calculation and design process:
V h = V 1 = V 2
It means
h w a f z = i w a h m
h m = h f z i
Substituting (18) into (21) the h m medium chip thickness is
h m = h f z 360 ° φ ° D π
Considering Fig. 6 and formulae (12)(15) w a tooth space on the outside circle can be determined. A side view for the manufacturing process can be seen in Fig. 16. The blue milling cutter is milling one tooth. The α a angle can be calculated from the OFG triangle:
cos α a = d b d a α a =
Fig. 16.
Fig. 16.

Determination of the w a tooth space on the outside circle

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

Based on (6)
inv α a = tan α a α a
inv α p = tan α p α p
Based on (15) and considering the j s backlash that we have to provide between the teeth [10, 11, 12, 19, 20, 21] the s a tooth thickness on the d a outside circle is (Fig. 16)
s a = d a ( s p d p + inv α p inv α a ) d a j s 2
The perimeter of the given circle has to be equal with the multiplication of the number of teeth and the given pitch, which is interpreted on the given circle of the gear [10, 11, 21]:
d p π = z t p d p t p = z π
d a π = z t a d a t a = z π
Based on (27) and (28) the circular pitch on the outside circle is
t a = d a t p d p
Based on (26) and (29) the w a tooth thickness on the outside diameter is (Fig. 16)
w a = t a s a
Knowing the specific cutting force (k c ) from the material property, the h m and the w a the cutting force for one edge of the tool can be determined:
F c 1 = k c h m w a
The t tooth pitch means the peripheral distance between two neighbouring teeth on the tool [1–9, 12, 14–16]:
t = D π z t
The ψ switch number means the number of the working teeth along the i arc of contact (Fig. 14) [1–9, 12, 14–16]:
ψ = i t = φ ° 360 z t
Considering the ψ switch number the total cutting force along the i arc of contact is
F c = ψ F c 1
Substituting (22), (31) and (33) into (34)
F c = k c w a h f z z t D π
The rotational cutting speed is [1–9, 12, 14–16]
v c = D π n
The feed speed is [1–9, 12, 14–16]
v f = f z z n
Knowing of the cutting force (35) and the cutting speed (36) the cutting power is
P c = F c v c = k c w a h f z z t n
Based on Fig. 14, the overall machining time that is needed for the manufacturing of all of the teeth is
T = L v f z = m + x 1 o + l m + x 2 o + m v f z

3.2 Manufacturing design and analysis for the designed gear pairs

Knowing the geometric parameters of the gear pairs, the geometric and manufacturing formulas I tried to find correlations between the geometric formulas and the manufacturing parameters for the concrete cases and analyse the results. The geometric parameters are found in Table 1. The initial manufacturing parameters are found in Table 2. According to the subchapter 3.1., I made an Excel table to determine the manufacturing parameters for each gear pair. The results can be seen in Tables 3 and 4 for the pinions and the gears.

Table 2.

The initial manufacturing parameters

Manufacturing parameters Value
D [mm] 120
z t 18
f z [mm] 0.06
l m [mm] 50
k c [N mm−2] 5,000
x 1o , x 2o [mm] 3
Table 3.

Manufacturing parameters for the pinions

Manufacturing parameters Pressure angles ( α w) [°]
20 21 22 23 24
v c [m min−1] 45.216
m [mm] 37.546 37.516 37.423 37.258 37.012
φ c [°] 38.759 38.722 38.608 38.407 38.107
i [mm] 40.568 40.529 40.409 40.199 39.885
h m [mm] 0.019522 0.019505 0.019452 0.019358 0.019219
w a [mm] 17.037 17.288 17.527 17.750 17.951
F c1 [N] 1663.068 1686.138 1704.787 1718.105 1725.060
ψ 2.028 2.026 2.020 2.010 1.994
F c [N] 3373.353 3416.913 3444.468 3453.295 3440.237
P c [W] 2542.159 2574.985 2595.751 2602.403 2592.562
v f [mm min−1] 129.6
T [min] 20.230 20.221 20.192 20.141 20.065
Table 4.

Manufacturing parameters for the gears

Manufacturing parameters Pressure angles ( α w) [°]
20 21 22 23 24
v c [m min−1] 45.216
m [mm] 37.546 37.516 37.423 37.258 37.012
φ c [°] 38.759 38.722 38.608 38.407 38.107
i [mm] 40.568 40.529 40.409 40.199 39.885
h m [mm] 0.019522 0.019505 0.019452 0.019358 0.019219
w a [mm] 16.144 16.392 16.601 16.762 16.864
F c1 [N] 1575.952 1598.717 1614.679 1622.500 1620.632
ψ 2.028 2.026 2.020 2.010 1.994
F c [N] 3196.647 3239.757 3262.407 3261.133 3231.980
P c [W] 2408.993 2441.481 2458.550 2457.590 2435.620
v f [mm min−1] 129.6
T [min] 30.346 30.332 30.288 30.212 30.098

The correlation between the pressure angle ( α w ) and the angle of contact ( φ c ) can be seen in Fig. 17. The diagram is the same for the cohesive connecting pinions and gears. The shape of the diagram is parabola. The angle of contact is exponentially decreasing while the pressure angle is increasing.

Fig. 17.
Fig. 17.

The correlation between the pressure angle and the angle of contact

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The correlation between the pressure angle ( α w ) and the arc of contact (i) can be seen in Fig. 18. The diagram is the same for the cohesive connecting pinions and gears. The shape of the diagram is parabola. The arc of contact is exponentially decreasing while the pressure angle is increasing. The highest result is in the case of α w = 20°. The lowest result is in the case of α w = 24°.

Fig. 18.
Fig. 18.

The correlation between the pressure angle and the arc of contact

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The correlation between the pressure angle ( α w ) and the cutting force for one edge (F c1 ) can be seen in Fig. 19. I got higher results for the pinion than for the gear. The results are increasing in the function of the pressure angle in the case of the pinion. The highest result is received in the case of α w = 24°. The shape of the diagram is parabola. The higher the pressure angle, the higher the cutting force for one edge on the tool tooth.

Fig. 19.
Fig. 19.

The correlation between the pressure angle and the cutting force for one edge on the tool

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The highest result is received in the case of α w = 23° on the gear. The results are continuously increasing in the function of the increasing pressure angle until α w = 23°. The shape of the diagram is a parabola until this main point. The lowest result is in the case of α w = 20°.

The correlation between the pressure angle ( α w ) and the total cutting force (F c ) can be seen in Fig. 20. I got higher results for the pinion than for the gear. The results are continuously increasing in the function of the increasing pressure angle until α w = 23° in both cases. The shapes of the diagrams are a parabola until these main points. The highest results are in the case of α w = 23° in both cases. The lowest results are in the case of α w = 20° in both cases.

Fig. 20.
Fig. 20.

The correlation between the pressure angle and the total cutting force on the tool

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The correlation between the pressure angle ( α w ) and the switch number ( ψ ) can be seen in Fig. 21. I got the same results for the cohesive, connecting pinion and gear. The shape of the diagram is a parabola. The higher the pressure angle, the lower the switch number. The highest result is received in the case of α w = 20°. The lowest result is received in the case of α w = 24°.

Fig. 21.
Fig. 21.

The correlation between the pressure angle and the switch number

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The correlation between the pressure angle ( α w ) and the cutting power (P c ) can be seen in Fig. 22. I got higher results for the pinion than for the gear. The shapes of the diagrams are a parabola until α w = 23° in both cases. The results are continuously increasing in the function of the increasing pressure angle until this main point. The highest results are in the case of α w = 23° in both cases. The lowest results are in the case of α w = 20° in both cases.

Fig. 22.
Fig. 22.

The correlation between the pressure angle and the cutting power

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

The correlation between the pressure angle ( α w ) and the machining time (T) can be seen in Fig. 23. The shapes of the diagrams are a parabola in both cases. I got higher results for the gear than for the pinion. The main reason is the higher number of teeth around the perimeter of the gear (Table 1). The higher the pressure angle, the less the machining time in both cases. I got the highest results in the case of α w = 20° for the cohesive, connecting pairs. I got the lowest results in the case of α w = 24° for the cohesive, connecting pairs.

Fig. 23.
Fig. 23.

The correlation between the pressure angle and the machining time

Citation: International Review of Applied Sciences and Engineering 13, 3; 10.1556/1848.2022.00418

4 Consclusion

The aim of this study is to find correlations between the modified geometric parameter, that is the pressure angle and the manufacturing parameters in the case of gear cutting by module disc milling cutter for the pinion and the gear. This technology can be executed in a conventional way (using a horizontal knee type milling cutter) or a computer numerical controlled way (using of a CNC milling machine). In this work, I determined the necessary technological parameters for both cases in a general way.

Firstly, I designed five types of connecting gear pairs where the difference between the initial parameters was the pressure angle beside the constancy of the other initial geometric parameters. I made a computer program in MATLAB language to enhance the design time and process for the output geometric parameters and the involute profile points. The received geometric results can be imported into the SolidWorks designer software where the CAD models can be generated for the tooth contact analysis (TCA) and the computer aided manufacturing (CAM) analysis. The CAM analysis is important for making CNC programs for CNC machines among other things if we choose CNC manufacturing for the gears.

I selected the conventional manufacturing process for the gears since this is the oldest way for which the newest methods are built up. I determined all of the necessary technological parameters for the manufacturing design in a general way. After that, I chose initial parameters with concrete values for the manufacturing design. I made an Excel table to determine the manufacturing parameters for the pinion and the gear. Considering the results, I made diagrams for the possible correlations of the analysed technological parameters and the pressure angle. I determined the consequences.

This study is theoretical and practical at the same time. In a theoretical way, there are a lot of ways to continue this research. In a practical way, this study can help the manufacturing engineers to design such manufacturing technologies for spur gears since my developed process is general. The received formulas are useable for the manufacturing of different types of spur gears by module disc milling cutter in a conventional or CNC way.

Obviously, changing the tooth space geometrically a different type of module disc type milling cutter is needed. It is another field how it is possible to design the tool for these manufacturing problems.

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 would like to thank Mr. Zoltán Géresi manufacturing engineer (University of Debrecen, Department of Mechanical Engineering) for the experimental manufacturings.

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    F. Klocke , Manufacturing Processes I. Cutting: RWTH Aachen University, 2011, p. 504, ISBN 978-3-642-26839-7.

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    M. Horváth , and S. Markos , Gépgyártástechnológia/Manufacturing Processes. Budapest: Műegyetemi Kiadó, 1998, p. 513.

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    H. B. Kief and H. A. Roschiwal , CNC Handbook. Mc Graw Hill, 2011, p. 451, ISBN 978-0-07-179948-5.

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    Gy. Mátyási , and Gy. Sági , Számítógéppel támogatott technológiák, CNC, CAD/CAM, /Computer Aided Manufacturing Technologies, CNC, CAD/CAM. Budapest: Műszaki Kiadó, 2021, p. 422, ISBN 978-963-16-6048-7.

    • Search Google Scholar
    • Export Citation
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    T. Nieszporek , P. Boral , and R. Golebski , “An analysis of gearing,” MATEC Web of Conferences, vol. 94, 2017, https://doi.org/10.1051/matecconf/20179407006.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [20]

    M. Rackov , M. Cavic , M. Pencic , I. Knezevic , M. Veres , and M. Tica , “Reducing of scuffing phenomenon at HCR spur gearing,” International Conference on Advanced Manufacturing Engineering and Technologies, Springer, 2017, pp. 141155.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [21]

    Z. Terplán , Gépelemek IV./Machine Elements IV., Kézirat. Budapest: Tankönyvkiadó, 1975, p. 220.

Nominations

Symbol Unit Parameter
φ [°] The sum of the involute angle and the angle between the arbitrary radius and the base circle radius
α [°] Angle between the arbitrary radius and the base circle radius
ρ [mm] Curvature radius of the involute curve
σ [°] Angle between the basic circle radius and the tooth centre line
ψ Switch number
α a [°] Angle between the outside circle radius and the base circle radius
φ c [°] Angle of contact
v c [m min−1] Real cutting speed
v f [mm min−1] Feed speed
c 0 Clearence factor ( c 0 = 0.25 )
a [mm] Normal centre distance
a 0 [mm] Elementary centre distance
c [mm] Clearance
C Main point
CAD Computer Aided Design
CAM Computer Aided Manufacturing
CNC Computer Numerical Control
D [mm] Diameter of the module disc milling cutter
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
F c [N] Total cutting force
F c1 [N] Cutting force for one edge
f z [mm] Feed for one edge
h [mm] Whole depth
h’ [mm] Working depth
h a [mm] Addendum
h f [mm] Dedendum
h m [mm] Medium chip thickness
i [mm] Arc of contact
inv α [°] Arbitrary involute angle (polar angle)
inv α a [°] Involute angle of the outside circle
inv α p [°] Involute angle of the pitch circle
inv α w [°] Involute angle of the rolling circle
j s [mm] Backlash
k c [N mm−2] Specific cutting force
l m [mm] Tooth length
m ax [mm] Transverse module
m [mm] Distance between the tool centre line and the corner point of the workpiece
n [min−1] Adjustable number of revolution
O 1 , O 2 Middle points of the pinon and the gear
P c [W] Cutting power
p p [mm] Circular pitch on the pitch circle
p w [mm] Circular pitch on the rolling circle
r [mm] Arbitrary radius of the involute curve
r b [mm] Base circle radius
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 a [mm] Tooth (arc) thickness on the outside circle
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
t [mm] Tooth pitch on the module disc milling cutter
T [min] Machining time
t a [mm] Tooth pitch on the outside circle
TCA Tooth Contact Analysis
t p [mm] Tooth pitch on the pitch circle
u Tooth ratio
V 1 , V 2 [mm3] Approximate chip volumes
V h [mm3] Separated chip volume
w a [mm] Tooth space on the outside circle
x c Addendum modification coefficient
x, y Coordinates of the given point on the involute curve
x 1, x 2 Addendum modification coefficient of the pinion and the gear
x 1o , x 2o [mm] Overruns
y Centre distance increment
z 1 Number of teeth of the pinion
z 2 Number of teeth of the gear
z t Number of teeth around the perimeter of the module disc milling cutter
α p [°] Base profile angle
α w [°] Pressure angle
x Sum of the addendum modification coefficients

  • [1]

    L. Bálint , A forgácsoló megmunkálás tervezése/Design of Chip Separation Processes. Budapest: Műszaki Könyvkiadó, 1961, p. 860.

    • Search Google Scholar
    • Export Citation
  • [2]

    J. Bali , Forgácsolás/Cutting Processes. Budapest: Tankönyvkiadó, 1988, p. 538.

  • [3]

    S. Bodzás , Manufacturing Processes I. Debrecen: University of Debrecen, Debrecen University Press, 2021, p. 203, ISBN 978-963-318-907-8.

    • Search Google Scholar
    • Export Citation
  • [4]

    J. T. Black , and R. A. Kohser , Materials and Processes in Manufacturing, 10th ed, United States of Amerika, p. 1033, ISBN 978-0470-05512-0.

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  • [5]

    P. De Vos , and J. E. Stahl , Metal Cutting, Theories in Practices. Seco, 2014, p. 183.

  • [6]

    K. Gyáni , Gépgyártástechnológia alapjai I./Basic Studies of Manufacturing Processes I. Budapest: Tankönyvkiadó, 1980, p. 128.

  • [7]

    L. Gribovszki , Gépipari megmunkálások/Manufacturing Processes. Budapest: Tankönyvkiadó, 1977, p. 454.

  • [8]

    H. N. Gupta , R. C. Gupta , and A Mittal , Manufacturing Processes, 2nd ed, New Age International Publishers, 2009, p. 194, ISBN 978-81-224-2844-5.

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    • Export Citation
  • [9]

    K. Gupta , N. K. Jain , and R. Laubscher , Advanced Gear Manufacturing and Finishing, Classical and Modern Processes. Academic Press, Elsevier, p. 230, ISBN 978-0-12-804460-5.

    • Search Google Scholar
    • Export Citation
  • [10]

    F. L. Litvin , and A., A. Fuentes , Gear Geometry and Applied Theory. Cambridge University Press, 2004, p. 800, ISBN 978 0 521 81517 8.

  • [11]

    S. P. Radzevich , Dudley’s Handbook of Practical Gear Design and Manufacture. 3rd ed, CRC Press, 2016, p. 656, ISBN 9781498753104.

  • [12]

    I. Dudás , Gépgyártástechnológia III./Manufacturing Processes III. Budapest: Műszaki Kiadó, 2011, p. 538, ISBN 978-963-16-6531-4.

    • Search Google Scholar
    • Export Citation
  • [13]

    L. Dudás , New way for the innovation of gear types. Engineering the Future, 2010, pp. 111140, https://doi.org/10.5772/47043.

  • [14]

    L. Fridrik , Forgácsolás I. (Forgácsoláselmélet)/Cutting processes I. (Cutting theorem). Miskolci Egyetemi Kiadó, 2011, p. 205.

  • [15]

    F. Klocke , Manufacturing Processes I. Cutting: RWTH Aachen University, 2011, p. 504, ISBN 978-3-642-26839-7.

  • [16]

    M. Horváth , and S. Markos , Gépgyártástechnológia/Manufacturing Processes. Budapest: Műegyetemi Kiadó, 1998, p. 513.

  • [17]

    H. B. Kief and H. A. Roschiwal , CNC Handbook. Mc Graw Hill, 2011, p. 451, ISBN 978-0-07-179948-5.

  • [18]

    Gy. Mátyási , and Gy. Sági , Számítógéppel támogatott technológiák, CNC, CAD/CAM, /Computer Aided Manufacturing Technologies, CNC, CAD/CAM. Budapest: Műszaki Kiadó, 2021, p. 422, ISBN 978-963-16-6048-7.

    • Search Google Scholar
    • Export Citation
  • [19]

    T. Nieszporek , P. Boral , and R. Golebski , “An analysis of gearing,” MATEC Web of Conferences, vol. 94, 2017, https://doi.org/10.1051/matecconf/20179407006.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [20]

    M. Rackov , M. Cavic , M. Pencic , I. Knezevic , M. Veres , and M. Tica , “Reducing of scuffing phenomenon at HCR spur gearing,” International Conference on Advanced Manufacturing Engineering and Technologies, Springer, 2017, pp. 141155.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [21]

    Z. Terplán , Gépelemek IV./Machine Elements IV., Kézirat. Budapest: Tankönyvkiadó, 1975, p. 220.

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Editor-in-Chief: Ákos, Lakatos University of Debrecen (Hungary)

Founder, former Editor-in-Chief (2011-2020): Ferenc Kalmár University of Debrecen (Hungary)

Founding Editor: György Csomós University of Debrecen (Hungary)

Associate Editor: Derek Clements Croome University of Reading (UK)

Associate Editor: Dezső Beke University of Debrecen (Hungary)

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  • 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 Ryerson University Toronto Canada

    Ildikó BODNÁR University of Debrecen Debrecen Hungary

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    Fatih Mehmet BOTSALI Selçuk University Konya Turkey

    Samuel BRUNNER Empa Swiss Federal Laboratories for Materials Science and Technology

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    Constantin BUNGAU University of Oradea Oradea Romania

    Michele De CARLI University of Padua Padua Italy

    Robert CERNY Czech Technical University in Prague Czech Republic

    György CSOMÓS University of Debrecen Debrecen Hungary

    Tamás CSOKNYAI Budapest University of Technology and Economics Budapest Hungary

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    Anna FORMICA IASI National Research Council Rome

    Alexandru GACSADI University of Oradea Oradea Romania

    Eric A. GRULKE University of Kentucky Lexington United States

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    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

    Imra KOCSIS University of Debrecen Debrecen Hungary

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    Antal PUHL University of Debrecen Debrecen Hungary

    Roman RABENSEIFER Slovak University of Technology in Bratislava Bratislava Slovak Republik

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    Ibrahim UZMAY Erciyes University Kayseri Turkey

    Tibor VESSELÉNYI University of Oradea Oradea Romania

    Nalinaksh S. VYAS Indian Institute of Technology Kanpur India

    Deborah WHITE The University of Adelaide Adelaide Australia

    Sahin YILDIRIM Erciyes University Kayseri Turkey

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

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2021  
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Scopus
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47
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International Review of Applied Sciences and Engineering
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International Review of Applied Sciences and Engineering
Language English
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2010
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per Year
3
Founder Debreceni Egyetem
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