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IMPACT OF THE INPUT SHAFT NUMBER OF REVOLUTIONS ON THE GEAR
PAIR TEETH BENDING STRESS IN A PLANETARY REDUCER
Sasko Mile *1,
Da ko Tase ski2
Me ludin Shabani3
Laza Jo ano 4
Zo an Dimi o ski5
*1,5 Facul y o Mechanical Enginee ing, Goce Delce Uni e si y, S ip, No h Macedonia
K s e Misi ko ,10A, 2000, S ip,
2Facul y o Mechanical Enginee ing, Ss Cy il and Me hodius Uni e si y in Skopje, No h Macedonia
3Facul y o Mecha onics, Uni e si y o Business and Technology, P ish ina, Koso o,
4 Ruen-Inox Au omobile, Kocani, No h Macedonia
ABSTRACT
Тhis pape aimed o examine he possibili ies o imp o e he cha ac e is ics o plane a y educe s,
conside ing hei wide applica ion in mechanical enginee ing. Fo his pu pose, i was i s
in es iga ed how he change in he speed o he inpu sha o he plane a y educe a ec s he
bending s ess in he oo s o he ee h o he gea pai s ha a e o med by he ee h o he d i ing
sha and he sa elli e gea s, a a cons an numbe o ee h and a cons an ansmission a io o he
plane a y gea . Then i was in es iga ed whe he and how he change in he numbe o ee h on he
d i ing sha a ec s he bending s ess o he ee h.
Keywo ds:
Plane a y Reduce , Bending S ess, Gea Pai Tee h, T ansmission Ra e,
INTRODUCTION
Mechanical ansmission is ans e o powe o mo ion om he d i ing o he d i en elemen
h ough a mechanical assembly called a ansmission. T ansmissions consis o d i ing and d i en
o a e elemen s, ans e is achie ed h ough hei di ec o indi ec con ac , o example, using a gea
pai , bel , chain. Changing he numbe o o a ions is accomplished by al e ing he diame e o he
d i ing and he d i en elemen s. Main cha ac e is ic o mechanical ansmissions is hei ansmission
a io. When ansmission a io is g ea e han 1 (i > 1), he numbe o o a ions dec eases, he o que
inc eases, and he ansmission is called a educe . Con e sely, when he ansmission a io is less
han 1 (i < 1), he numbe o o a ions inc eases, he ansmission is called a mul iplie . Plane a y
ansmissions belong o he g oup o mechanical ansmissions, i.e., gea mechanical ansmissions
and he mechanical ans e is achie ed by a leas one elemen ( he sa elli e gea ), o a ing no only
a ound i s own axis bu also a ound ano he axis. These mechanical ansmissions ind applica ion
when is a need o ans e o la ge o ces and a high numbe o o a ions, wi h he imposed
equi emen o a small olume and weigh o he ansmission. In plane a y ansmissions, he o ce
is dis ibu ed among mul iple plane a y gea s [1] [2].
Ad an ages o plane a y ansmissions in ela ion o non-plane a y ones a e; high compac ness
exp essed h ough small o e all dimensions and low olume, low pe iphe al speeds, and hus a small
le el o noise, small mass, wo o h ee imes less han ha o non-plane a y ansmissions, using he
p inciple o sepa a ion o o que (be ween se e al plane a y gea s), e y high e iciency–higheas o
all ansmissions, ela i ely good cinema ic abili ies, ela i ely low noise le el due o: lowe
pe iphe al speeds, smalle dynamic o ces due o smalle dimensions, high ansmission a io wi hin
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one deg ee, e c. Al hough he la ge numbe o ad an ages, he e a e also disad an ages: geome ic
calcula ions a e complex, he une en dis ibu ion o load be ween plane s is a complex and speci ic
p oblem whose solu ion is no easy, he ee h o he plane a y gea s a e olled less a o ably han in
o he gea s, hey ha e a complex cons uc ion, and a high isk o a de ec o he en i e ansmission,
high p oduc ion echnology is needed(due o he imposed quali y equi emen s). Plane a y
ansmissions ha e a numbe o capabili ies in sol ing ce ain asks in he ield o powe ansmission:
1.Achie ing a cons an ansmission a io can wo k as a educe - (wi h one deg ee o
eedom), i is mos commonly used when he ing is ixed, i.e. , less o en when , and
a ely when , and as a mul iplie - (wi h one deg ee o eedom) i is commonly used
when in hyd o u bines, less o en when , and a ely when . 2.Implemen a ion o
s epwise change o ansmission a io in single and double d i es. 3. Implemen a ion o con inuous
change in ansmission a io, di e en ial (wi h wo deg ees o eedom), achie ing a high angula
eloci y and o he applica ion. The designa ion o plane a y gea sys ems is done using he numbe s 1
and 2, and he le e s A and I. The numbe s 1 and 2 indica e whe he he gea sys em is single-s age o
wo-s age, while he le e A deno es cen al gea s wi h ex e nal ee h (sun gea ), and he le e I
deno es cen al gea s wi h in e nal ee h ( ing gea ). Simple plane a y gea sys ems a e o ype 1AI,
and hey ha e a single-s age plane a y se wi h one cen al gea wi h ex e nal ee h (sun gea ) and one
cen al gea wi h in e nal ee h. Plane a y gea sys ems can be used as educe s, mul iplie s (wi h one
deg ee o eedom), and di e en ials (wi h wo deg ees o eedom), depending on hei cons uc ion
[3].
Fig. 1: Plane a y gea ain 1AI
This plane a y gea sys em consis s o he ollowing main componen s: cen al gea wi h ex e nal
ee h 1(sun gea ), plane a y gea s 2, cen al gea wi h in e nal ee h 3( ing gea ), and ca ie S. The
cen al gea wi h ex e nal ee h is he d i ing elemen and is a ached o he inpu sha . I is coupled
wi h he plane a y gea s, which pe o m wo o a ions, one a ound hei axis and he o he a ound he
axis o he cen al gea s. The cen al gea wi h in e nal ee h is in con ac wi h he plane a y gea s bu
does no unde go o a ional mo emen as i se es as a eac ion membe in his case. The ca ie S is
a ached o he sha s o he plane a y gea s, and he ou pu sha is inse ed in o i , ansmi ing
o a ional mo emen o he ou pu sha as a esul o he o a ion o he plane s a ound he axes o he
cen al gea s. Since his is a educe , he numbe o o a ions o he ou pu sha will be less han he
numbe o o a ions o he inpu sha .
OBJECTIVES
The main objec i e o he s udy is o de e mine he impac o he change o he numbe o e olu ions
o he plane a y educe inpu sha on he gea pai (z1-z2) ee h bending s ess , when he numbe o
d i ing sha ee h is unchanged. Then he impac o he change o he numbe o d i ing gea ee h on
he bending s ess o he o he gea pai ee h is analyzed.
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METHODOLOGY
Kinema ic analysis o 1AI plane a y gea ain
The kinema ic analysis is pe o med wi h he supe posi ion me hod-Swamp ule. Acco ding o his
me hod, indi idual elemen s o he plane a y ansmission a e obse ed in di e en speci ied
mo emen s in ew se mo emen s. Numbe o o a ions and he numbe o e olu ions o he
plane a y educe gea s a e gi en in Table 1 [4].
Table 1: Kinema ic Analysis o Plane a y Gea T ain
Numbe o
mo emen s
Gea
1
Gea 2
Gea 3
Ca ie
S
1
+1
+1
+1
+1
2
-1
0
Sum o he
i s wo
mo ions
0
+1
3
0
4
0
Sum o he 3 d
and 4 h
mo ions
The numbe o o a ions o gea 1:
𝑛
1
=
𝑛
inpu
The numbe o o a ions o ca ie s:
𝑛
𝑠
=
𝑛
ou pu
F om he kinema ic analysis (Table 1), exp essions a e de i ed o de e mine he numbe o
o a ions o gea 2 (plane a y gea ) and he numbe o o a ions o gea 3 ( ing gea )
The o e all ansmission a io can be ob ained om he a io o he numbe o o a ions a he
inpu and ou pu o he plane a y gea ain. The di ec ion o o a ion o he ou pu sha , is
opposi e o he di ec ion o o a ion o he inpu sha , whe e he cen al gea wi h ex e nal ee h
(sun gea ) is loca ed.
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Fig. 2 : Di ec ion o o a ion o he ca ie S and gea 1
Since he cen al gea wi h in e nal ee h ( ing gea ) will be he eac ion membe , i s numbe o
o a ions will be equal o ze o. The o e all ansmission a io will be: i= n1/ n2
Coaxiali y condi ion
Using coaxiali y condi ion, which gua an ees equal spacing be ween he i s se (sun gea and
plane a y gea s) and he second se (plane a y gea s wi h ing gea ) he numbe o ee h on he cen al
gea wi h in e nal ee h is:
In his exp ession, i is assumed ha he gea s will ha e s aigh ee h and no p o ile shi ing (x=0).
The numbe o ee h on he cen al gea wi h ex e nal ee h (sun gea ) and plane a y gea s a e chosen
a bi a ily. Due o he high numbe o o a ions a he inpu , i is p e e able o choose a smalle
numbe o ee h o sun gea o educe he size and, consequen ly, he mass o he gea . Once we ha e
he numbe s o all ee h, we can es ablish he ansmission a ios be ween indi idual pai s o gea s. In
he case whe e he di ec ion o o a ion o he d i en gea is opposi e o he di ec ion o o a ion o he
d i ing gea , a nega i e sign is assigned o he ansmission a io [5].
The ansmission a io be ween gea 1 (sun gea ) and gea 2 (sa elli e gea s) is:
The ansmission a io be ween gea 2 (plane a y gea s) and gea 3 ( ing gea ) is:
Hence, he ac ual s anda d ansmission a io is:
Sizing o he gea s
The o al powe ansmi ed by he plane a y ansmission consis s o wo componen s: he powe a
he coupling and he o ce due o he olling o he ee h , also known as kinema ic powe [4].
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Fig. 3: Sha s o he plane a y ansmission
Inpu sha is sha A , [W]
Angula eloci y o he sha s: -1]
-1]
-1]
To que o he inpu sha A:
To que o sha B:
To que o sha C: [Nm]
Due o he di ision o he powe on and , he powe on he sha A is:
[W]
The powe s on sha B a e:
The powe s on sha C a e:
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Calcula ion o he o ien a ion module o he gea pai
Module o he gea pai s z1-z2 is calcula ed using he o mula [6]:
- oo h shape ac o , – s ess co ec ion ac o , - load dis ibu ion ac o in he on al c oss-
sec ion o s ess a he oo h oo , - load dis ibu ion ac o along he leng h o he oo h o s ess
a he oo h oo , - dis ibu ion ac o o he pe iphe al o ce along he engagemen o cen al gea s
wi h plane a y gea s, - ac o o uni o m loading o he machine in con inuous ope a ion wi h he
elec ic mo o , - oo h wid h ac o and - oo bending s ess [6]. As a ma e ial o he
manu ac u e o he cen al gea wi h ex e nal ee h, i is choiced induc ion-ha dened s eel.
Because he e a e h ee plane a y gea s in he plane a y ansmission, he o al powe is di ided in o 3
pa s:
The o que a he poin o con ac be ween gea 1 and one o he plane a y gea s 2 will be:
Calcula ion o he o ien a ion module o he gea pai
Because his plane a y ansmission is o he 1AI ype, all gea s need o ha e he same module o
achie e mu ual engagemen . I was necessa y o calcula e he o ien a ion module o he gea pai , and
hen a module ha mee s he equi emen s o bo h gea pai s was selec ed.
We ollow he same p ocedu e as wi h he gea pai , so he nex s ep is [6]:
– ans e se load ac o o con ac s ess, – ace load ac o con ac s ess
Because he e a e 3 plane a y gea s in he plane a y ansmission, he o al powe is di ided in o 3
pa s:
The o que a he poin o con ac be ween gea 3 and one o he plane a y gea s 2 will be:
Whe e he angula eloci y o he plane a y gea is , and i is:
Dimensions o he gea s
The wid h o he ee h is de e mined om he exp ession o he oo h wid h ac o .
The wid h o he ee h is
The cen e dis ance be ween he wo gea pai s is equal and is:
,
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Diame e s o gea 1 (cen al gea wi h ex e nal ee h)
Re e ence diame e : , Tip diame e : , Roo diame e :
Diame e s o gea 2 (plane a y gea s)
Re e ence diame e : Tip diame e : , Roo diame e :
Diame e s o gea 3 (cen al gea wi h in e nal ee h)
Re e ence diame e : , Tip diame e : , Roo diame e :
Inspec ion o ee h oo bending s ess
Gea pai
The ee h oo bending s ess is calcula ed acco ding o he equa ion [7]:
The o ce is:
Gea pai
The o ce is:
RESULTS AND DISCUSSION
Fo he gi en alues z1=25, z2=85 and z3=195, P=4.000[W] he plane a y educe ansmission a ios
we e i s calcula ed. Then, bending s esses in he oo s o he ee h o he gea pai z1-z2 we e
calcula ed o di e en numbe s o e olu ions o he inpu sha , o a numbe o e olu ions om
500 o 6,000 e olu ions (z2 and z3 a e no changed). Ob ained alues o z1=25 a e shown in able 2.
Based on he alues om able 2 is d awn he diag am p esen ed in he Figu e 4. This cu e ac ually
de e mines he gea pai z1-z2 bending s ess change depending on he numbe o e olu ions o he
inpu sha n1. Тhe equa ion o his cu e is y=58716/x and i was calcula ed using app oxima ion
me hod.
Table 2: Bending s ess o ee h o gea pai z1-z2
Z1
Numbe o e olu ions pe minu e o he Inpu Sha o Plane a y Reduce
500
750
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
21
117.57
78.38
58.78
39.19
29.39
23.52
19.59
16.80
14.70
13.06
11.77
10.69
9.8
23
104.75
69.83
52.38
34.92
26.19
20.95
17.46
14.96
13.09
11.64
10.48
9.52
8.73
25
93.99
62.66
46.99
31.33
23.50
18.80
15.66
13.43
11.75
10.44
9.4
8.54
7.83
27
85.89
57.26
42.94
28.63
21.47
17.18
14.31
12.27
10.74
9.54
8.59
7.81
7.16
29
78.91
52.6
39.45
26.3
19.73
15.78
13.15
11.27
9.86
8.77
7.89
7.17
6.58
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Fig. 4: Bending S ess o he Gea Pai (z1-z2) Tee h o Di e en Numbe o Re olu ions
o he Plane a y Inpu Sha
Nex calcula ions we e made o he cases when he numbe o ee h on he d i ing sha is changed
om z1=21, z1=23, z1=27 o z1=29. Fo each o hese cases he bending s esses we e calcula ed o
di e en inpu sha numbe s o e olu ions om 500 o 6000 e olu ions pe minu e.
Fig. 5: Bending S ess o he Gea Pai (z1-z2) Tee h o Di e en Numbe o Re olu ions
o he Plane a y Inpu Sha o Di e en Values o z1
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CONCLUSION
Wi h a cons an numbe o ee h in he plane a y educe (cons an ansmission a io) and cons an
powe o he inpu sha , wi h an inc ease in he numbe o e olu ions o he inpu sha , he bending
s ess in he oo s o he ee h o he gea pai z1-z2 dec eases. The deg ee o bending s ess educ ion
o a lowe numbe o e olu ions (up o 1000 e olu ions pe minu e) is he g ea es . Wi h he u he
inc ease in he numbe o e olu ions o he inpu sha , he educ ion o he bending s ess on he
ee h con inues bu wi h a educed in ensi y. Fo he numbe o e olu ions g ea e han 4000
e olu ions pe minu e he educ ion o he bending s ess is insigni ican . This ac can be used in
cases whe e i is necessa y o a ce ain plane a y educe o educe he bending s ess a he oo s o
he ee h.
F om he Diag ams in Figu e 5, i can be seen ha he educing he bending s ess in he ee h
(z1=21,23,27,29) also applies upon changing he numbe o ee h o he d i e gea . By inc easing he
numbe o ee h o he d i e gea , he bending s ess in he ee h o he same numbe o e olu ions o
he inpu sha dec eases and ice e sa.
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