Contact Forces of Spur Gearing
DANIELA GHELASE, LUIZA TOMULESCU
“DUNAREA DE JOS” University of Galati
ROMANIA
Abstract : - In this paper, on the basis of the bending produced by the normal force which is applied in contact
point of teeth flanks, the contact force of external spur gearing was determined. For the calculus methodology
presented in this paper, a computer program has been developed.
Key-Words: - contact forces, spur gearing, tooth pair of gearing.
1 Introduction
As has been known, during the meshing, the
contact point moves from the tooth tip to its root and
from the conjugated gear tooth root to its tip (from
the initial point of contact to the final point of
contact). The normal force is mobile on the tooth
flank, it changes continuously the position respect to
the tooth fixing zone. Calculating the contact forces
in conditions as near as possible by the reality, the
gears dimension designing can be developed more
exactly.
Fig.1 Geometric parameters at the contact point
2 Bases of design
In this paper are presented the following bases
of design:
a) The involute tooth is considered to be a beam
which is fixed at one end in the body of gear.
b) The gear tooth is a non-uniform cantilever beam,
as shown in figure 2.
EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2022.2.10
Daniela Ghelase, Luiza Tomulescu
E-ISSN: 2944-9006
68
Volume 2, 2022
c) It is taken into consideration only the bending
produced by the normal force.
d) Gears are rigid bodies except for their teeth.
e) The load is uniformly distributed along the
tooth width.
In figure 1 is presented an external spur gearing,
where:
Pc-contact point
T1T2-line of contact
P1P2-length of path of contact
[ ]
[ ]
θ γ β
θ γ β
1 1 1
2 2 2
0
0
∈ +
∈ +
,
,
- angles corresponding to
length of path of contact.
The following elements are known:
-normal module of tooth-mn, [mm];
-helix angle-
β
d
=0
-number of gear 1-z1
-number of gear 2-z2
-basic rack:
- pressure angle-
- whole depth-
2 25
.
⋅
m
n
, [mm];
3 Equation System of Forces
First of all, on the basis of the known data and
figure 1, the following parameters will be
determined:
-base radius of gear:
( )
)2,1(;cos
2
0=
⋅
=jwhere
zm
Rjn
bj
α
(1)
-root radius of gear:
−⋅= 25,1
2
j
nf
z
mR j
(2)
-tip radius of gear:
+⋅= 1
2
j
nv
z
mR j (3)
-gear center distance:
( )
2
21
21
zzm
AOO n
+⋅
== (4)
-length line of contact:
( )
( )
021021
sin()
2
)tan(
21
αα
zz
m
RRTT
n
bb
+=⋅+=
(5)
3.1 The Calculation of Angles
γ β
j j, .
In 211 PTO∆
2
11
2
2121 TOPOPT −=
0
21
1
1
arctan
αβ
−=
b
R
PT
(6)
In
122 PTO∆
0
12
2
2
arctan
αγ
−=
b
R
PT
and the following relations may be used:
PTPTPP 2121 −=
PTPTPP 1212 −=
2121 PPPPPP +=
122111 PTTTTP −=
1
11
01 arctan
b
R
TP
−=
αγ
2
22
02 arctan
b
R
TP
−=
αβ
(7)
3.2 The Calculation of The Radius of The
Contact Point Pc
The angle θ1 takes values in
[ ]
111 ,0
βγθ
+∈
.
For a known value of θ1, the radius Rc1 is given by:
( )
110
cos 1
1
θγα
+−
=b
c
R
R (8)
( )
111
22
111 cos211
θ
⋅⋅⋅−+= ccC RPORPOPP
111
22
11
2
1
11 2
arccos 1
POPP
RPOPP
OPP
c
cc
c⋅⋅
−+
=
112121
OPPOPP
cc
−−−=
γγπ
The angle θ2 takes values in [0,γ2+β2]. For a
known value of θ2 the radius Rc2 is given by:
)cos(2211
2
1
2
222 OPPPPRPPRR ccvcvc
⋅⋅⋅−+= (9)
By means of Rc1and Rc2, the teeth depth h1and
h2in the contact point Pc can be determined:
jj fcj RRh −=
(10)
Without of the teeth pair with the contact in Pc,
another teeth pairs there are in contact too. The
contact points there are on the line P1P2 at the
distance Pb from point Pc. For each teeth pair which
there are in contact it is possible to calculate the
depth hji down to contact point. In the contact point
Pi, a contact force Fi appears between the flanks.
Under the influence of contact force Fi, the teeth
EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2022.2.10
Daniela Ghelase, Luiza Tomulescu
E-ISSN: 2944-9006
69
Volume 2, 2022
deform with the common dimension:
ii
i21
δδδ
+=
(11)
If the deflection is given by:
( )
21;
,orjfF jjiizhij =⋅=
δ
(12)
than, the equations system of forces will be:
=⋅
===
∑11
21
MRF bi
i
δδδ
(13)
Fig.2
Tooth deflection in the direction of the applied
Fi
Table 1
The system (13) has “n” equations, the unknowns
being:F1, F2, ....Fn
In assumption that the beam has a variable cross-
sectional area, in accordance with involute profile
(figure 2), the deflection δji is calculated with the
following formula:
( )
ji
jii
ji k
F
ε
δ
2
cos⋅
= (14)
where: kji [N/mm] is the elasticity constant of tooth
deformed due to contact force Fi applied to depth hji ;
1
kji
is the compliance of tooth “j” due to contact
force Fi.
Own studies [4],[5] have established that, in this
case, the compliance expression is given by:
Bzh
k
jji
ji
⋅⋅⋅⋅=
−− 03885,113633.310
10362.5
1
(15)
where:
z-teeth number of the gear “j”.
B-tooth face width, [mm].
εji is the angle between the force Fi and the normal
line to the symmetry axis of tooth “j” from the “i”
teeth pair which is in contact.
By means of the computer program, the equations
system (15) was solved. In table 1 are presented the
numerical values of the contact forces and bending
deflections, having the following parameters:
mn=10 mm; B=10 mm; α0=200; z1=40; z2=60;
F=20000N; M1=3758,77Nm; γ1=7,3850; β1=6,4990;
Numerical Results
Nr
crt θ1
Number of
contact
F
i
δji
δi
[mm]
[o]
teeth pairs (i)
[N]
δ
1i
δ
2i
1
0,93
2
5770,42
0,000167
0,009189
0,009356
14229,58
0,007688
0,001668
2
1,85
2
5711,42
0,000246
0.010853
0.011099
14288,58
0,009890
0,001209
3
2,78
2
5762,85
0,000361
0,013005
0,013366
14237,15
0,012522
0,000844
4
3,7
2
5883,02
0,000527
0,015697
0,016223
14116,98
0,015657
0,000566
5
4,63
2
6042,89
0,000761
0,018985
0,019745
13957,11
0,019383
0,000362
EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2022.2.10
Daniela Ghelase, Luiza Tomulescu
E-ISSN: 2944-9006
70
Volume 2, 2022
Table 1 (continuation)
To be repeat to θ1=13,884
4 Conclusion
The analytical modeling presented in this paper
permits to establish the numerical values of the
contact forces for external spur gearing. The real
tooth profile and the geometry elements which are
specific for these gear pairs permit to compute the
contact forces and the deflection of a teeth pair with
a high accuracy. As part of further work on this
research program, the contact or Hertzian deflection
will be taken in consideration.
The authors would like to acknowledge the S.C.
FLEX CONSULTING S.R.L. Braila for their
support that contributed for publishing this paper.
References
[1] ATANASIU, V.: Specific Aspects of The
Deflection of Meshing Teeth for Cylindrical
Gears with Smaller Number of Pinion Teeth. –
9th Word Congress on the Theory of Machine
and Mechanics in Design, Milano, 1995, p.610-
614.
[2] BUZDUGAN, GH.: Rezistenta Materialelor. –
Editura Tehnica., Bucuresti, 1980.
[3] GAFITEANU, M.: Organe de Masini. –
Editura Tehnica, Bucuresti, 1983.
6
5,55
2
6223,15
0,001085
0,022934
0,024019
13776,85
0,023801
0,000219
7
6,48
1
20000
0,004770
0,086160
0,090930
8
7,4
1
20000
0,006442
0,100396
0,106839
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2022.2.10
Daniela Ghelase, Luiza Tomulescu
E-ISSN: 2944-9006
71
Volume 2, 2022