Manufacturing of spur gears having normal teeth on different pressure angles by module disc milling cutter

Sándor Bodzás

doi:10.1556/1848.2022.00418

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International Review of Applied Sciences and Engineering

Volume/Issue:: Volume 13: Issue 3

Manufacturing of spur gears having normal teeth on different pressure angles by module disc milling cutter

Author:

Sándor Bodzás

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

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

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Pages:: 321–334

Online Publication Date:: 05 Apr 2022

Publication Date:: 27 Oct 2022

Article Category:: Research Article

DOI:: https://doi.org/10.1556/1848.2022.00418

Keywords:: spur gear; pressure angle; manufacturing; analysis; gear; pinion

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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

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]

\begin{matrix} x = r_{b} \cdot sin φ - r_{b} \cdot φ \cdot cos φ \\ y = r_{b} \cdot cos φ + r_{b} \cdot φ \cdot sin φ \end{matrix}}

(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

(2)

The following two equations can also be derived on Fig. 1. [10, 21]:

ρ = r_{b} \cdot φ

(3)

ρ = r_{b} \cdot tan \propto

(4)

Based on (3) and (4)

φ = tan \propto

(5)

Substituting (5) into (2) and expressing inv

α

inv α = tan \propto - \propto

(6)

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 (α _p ).

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} \cdot cos α_{p} = r_{w} \cdot cos α_{w}

(7)

r_{w} = r_{p} \cdot \frac{cos α_{p}}{cos α_{w}}

(8)

The centre distance is

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

(9)

The elementary centre distance is [10, 11, 19, 21]

a_{0} = r_{p 1} + r_{p 2} = \frac{d_{p 1} + d_{p 2}}{2}

(10)

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.

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} \cdot m_{a x}

(11)

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.

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} = \frac{p_{p}}{2} - 2 \cdot x \cdot m_{a x} \cdot \tan α_{p} = m_{a x} \cdot (\frac{π}{2} - 2 \cdot x \cdot \tan α_{p})

(12)

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} = \frac{p_{p}}{2} + 2 \cdot x \cdot m_{a x} \cdot \tan α_{p} = m_{a x} \cdot (\frac{π}{2} + 2 \cdot x \cdot \tan α_{p})

(13)

The

σ

angle is (Fig. 6) [10, 11, 21]

σ = inv α_{p} + \frac{s_{p}}{2 \cdot r_{p}} = inv α + \frac{s}{2 \cdot r}

(14)

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 \cdot r \cdot (\frac{s_{p}}{2 \cdot r_{p}} + inv α_{p} - inv α)

(15)

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 ( $\vec{v_{c}}$ ) and linear ( $\vec{v_{f}}$ ) 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.

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.

The structure of the dividing head

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

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.

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.

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
$\sum^{} x$	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.

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.

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 ( $\vec{v_{c}}$ ) and linear $(\vec{v_{f}})$ 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.

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 = \sqrt{{(\frac{D}{2})}^{2} - {(\frac{D}{2} - h)}^{2}} = \sqrt{D \cdot h - h^{2}}

(16)

The

φ

_c angle of contact is (Fig. 14a EBD triangle)

φ_{c}^{°} = a tan (\frac{m}{\frac{D}{2} - h})

(17)

The i arc of contact is (Fig. 14a)

i = φ_{c}^{°} \cdot \frac{D \cdot π}{360 °}

(18)

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.

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}

(19)

It means

h \cdot w_{a} \cdot f_{z} = i \cdot w_{a} \cdot h_{m}

(20)

h_{m} = \frac{h \cdot f_{z}}{i}

(21)

Substituting (18) into (21) the h _m medium chip thickness is

h_{m} = h \cdot f_{z} \cdot \frac{360 °}{φ^{°} \cdot D \cdot π}

(22)

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} = \frac{d_{b}}{d_{a}} \to α_{a} = \dots

(23)

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}

(24)

inv α_{p} = \tan α_{p} - α_{p}

(25)

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} \cdot (\frac{s_{p}}{d_{p}} + inv α_{p} - inv α_{a}) \cdot d_{a} - \frac{j_{s}}{2}

(26)

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} \cdot π = z \cdot t_{p} \to \frac{d_{p}}{t_{p}} = \frac{z}{π}

(27)

d_{a} \cdot π = z \cdot t_{a} \to \frac{d_{a}}{t_{a}} = \frac{z}{π}

(28)

Based on (27) and (28) the circular pitch on the outside circle is

t_{a} = \frac{d_{a} \cdot t_{p}}{d_{p}}

(29)

Based on (26) and (29) the w _a tooth thickness on the outside diameter is (Fig. 16)

w_{a} = t_{a} - s_{a}

(30)

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} \cdot h_{m} \cdot w_{a}

(31)

The t tooth pitch means the peripheral distance between two neighbouring teeth on the tool [1–9, 12, 14–16]:

t = \frac{D \cdot π}{z_{t}}

(32)

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 = \frac{i}{t} = \frac{φ^{°}}{360} \cdot z_{t}

(33)

Considering the ψ switch number the total cutting force along the i arc of contact is

F_{c} = ψ \cdot F_{c 1}

(34)

Substituting (22), (31) and (33) into (34)

F_{c} = k_{c} \cdot w_{a} \cdot h \cdot f_{z} \cdot \frac{z_{t}}{D \cdot π}

(35)

The rotational cutting speed is [1–9, 12, 14–16]

v_{c} = D \cdot π \cdot n

(36)

The feed speed is [1–9, 12, 14–16]

v_{f} = f_{z} \cdot z \cdot n

(37)

Knowing of the cutting force (35) and the cutting speed (36) the cutting power is

P_{c} = F_{c} \cdot v_{c} = k_{c} \cdot w_{a} \cdot h \cdot f_{z} \cdot z_{t} \cdot n

(38)

Based on Fig. 14, the overall machining time that is needed for the manufacturing of all of the teeth is

T = \frac{L}{v_{f}} \cdot z = \frac{m + x_{1 o} + l_{m} + x_{2 o} + m}{v_{f}} \cdot z

(39)

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) [°]
Manufacturing parameters	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) [°]
Manufacturing parameters	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.

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.

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.

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.

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.

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.

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.

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.

References

[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.
- Search Google Scholar
- Export Citation
[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.
- Search Google Scholar
- Export Citation
[5]↑
P. De Vos , and J. E. Stahl , Metal Cutting, Theories in Practices. Seco, 2014, p. 183.
- Search Google Scholar
- Export Citation
[6]↑
K. Gyáni , Gépgyártástechnológia alapjai I./Basic Studies of Manufacturing Processes I. Budapest: Tankönyvkiadó, 1980, p. 128.
- Search Google Scholar
- Export Citation
[7]↑
L. Gribovszki , Gépipari megmunkálások/Manufacturing Processes. Budapest: Tankönyvkiadó, 1977, p. 454.
- Search Google Scholar
- Export Citation
[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.
- Search Google Scholar
- 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.
- Search Google Scholar
- Export Citation
[11]↑
S. P. Radzevich , Dudley’s Handbook of Practical Gear Design and Manufacture. 3rd ed, CRC Press, 2016, p. 656, ISBN 9781498753104.
- Search Google Scholar
- Export Citation
[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. 111–140, https://doi.org/10.5772/47043.
- Search Google Scholar
- Export Citation
[14]↑
L. Fridrik , Forgácsolás I. (Forgácsoláselmélet)/Cutting processes I. (Cutting theorem). Miskolci Egyetemi Kiadó, 2011, p. 205.
- Search Google Scholar
- Export Citation
[15]↑
F. Klocke , Manufacturing Processes I. Cutting: RWTH Aachen University, 2011, p. 504, ISBN 978-3-642-26839-7.
- Search Google Scholar
- Export Citation
[16]↑
M. Horváth , and S. Markos , Gépgyártástechnológia/Manufacturing Processes. Budapest: Műegyetemi Kiadó, 1998, p. 513.
- Search Google Scholar
- Export Citation
[17]↑
H. B. Kief and H. A. Roschiwal , CNC Handbook. Mc Graw Hill, 2011, p. 451, ISBN 978-0-07-179948-5.
- Search Google Scholar
- Export Citation
[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.
- 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. 141–155.
- Search Google Scholar
- Export Citation
[21]↑
Z. Terplán , Gépelemek IV./Machine Elements IV., Kézirat. Budapest: Tankönyvkiadó, 1975, p. 220.
- Search Google Scholar
- Export Citation

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
$\vec{v_{c}}$	[m min⁻¹]	Real cutting speed
$\vec{v_{f}}$	[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
$\sum^{} x$		Sum of the addendum modification coefficients

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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
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International Review of Applied Sciences and Engineering
Language	English
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Founder	Debreceni Egyetem
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