PERPUSTAKAAN UMP

11111111111111111 0000087370

S

1t1iC)

(iJf:1t J3F

Fundamental Study on Bending Strength of Thin-

Rimmed Helical Gears with Web Arrangements

2O141

DAING MOHAMAD NAFIZ BIN DAING IDRIS

iRfl

1

J. OWN

2.1 *

2.2

2.2.1

2.2.2 Jji'J

2.3 :m 2.3.1

2.3.2 -r92)•

2.3.3 9 J•

2.3.4 1-9J5•

2.4 ±

3* rn t 1 jc7)

3 1

.1 'T

3.2

3.2.1

3.2.2 JJ

3.3 zzm

3.3.1 rJ)rn5fJh 3.3.2

3.3.3

3.3.4 1

) A ^i

.).'I I'Il

%41 co (,l=io, 20, 300)

4 . 1 !l4Z1

4.2

4.2.1

4.2.2 tJj'J

4.3

4.3.1

4.3.2 EJJ Q)f

4.3.3 f7

4.3.4 QTL

5

8

15

17

17

22

23

24

24

26

27

29

37

38

41

43

43

44

45

50

55

60

'I

4.3.5 J ' 1 tJ IL O)J 72

4.3.6 i'iI f,iL' 73

4.4 1 75

5 *

5.1 76

5.2

5.2.1 76

5.2.2 79

5.3

5.3.1 82

5.3.2 J'. 82

5.3.3 J. 85

5A 11±

.'t ;Ic

90

Af 4-f92

93

31111-1 4 ik 111

116

- 11 -

(Nomenclature)

a :rP,NTEM (Center distance)

b : ikt (Face width)

r- (Web thickness)

d : R (Reference diameter)

da : (Tip diameter)

db : (Base diameter)

E : - -' 7' *1L (Young's modulus)

F, RJJ)J (Circumferential load)

F1 : (Bending fatigue limit circumferential load)

L : -1i (Length of path of contact)

it : y Y (Transverse backlash)

At : y -' (Transverse base backlash)

Mn (Normal module)

m t : *E- —iI' (Transverse module)

N U (Number of load cycles)

Phi : y (Transverse base pitch)

r : (Reference radius)

ra : R1 (Tip radius)

rb : 94M R (Base radius)

1. 9 AJ (Rim thickness)

X : (Coordinate axis)

y : )*I (Coordinate axis)

z : i1k (Number of tooth)

() (Number of tooth of driving gear)

Z2 : %i) i*c1)1 (Number of tooth of driven gear)

aa : (Tip angle)

j (Normal pressure angle)

at : IiM i$JIf (Transverse standard pressure angle)

awt 9- 1 'flE)JI (Pressure angle at the pitch cylinder)

180 R MIN ±Oa UL (Helix angle on pitch cylinder)

A 1I (Base helix angle)

' (Transverse contact ratio)

6,6 -i " (Overlap ratio)

Ey (Total contact ratio)

C : (Tangential angle)

- 111 -

or : &5GJJ (Root stress)

5iJJ (Compressive root stress)

I 15ZJJ (Tensile root stress)

(Suffixes)

I: Ib () ffi* (Driving gear, test gear)

2 : (4) i* (Driven gear, supporting gear)

c : )I (Compression)

max JOz (Maximum)

mm : /J\ (Minimum)

solid : {*i' (Solid gear)

t 91 , * i (Tension; transverse section)

- iv -

'&*LS& 1- PDB

-C /M" <J k <

:, !)

uu,

ft 6b

ft±,1C "13t:), L+Z

(9,

9M4 N,' - 7L/ N <'J'FH

gp - -o \ 53 7 4 fl.8)

1 j'VC1i \ 1.9)(1.12)

H 1.13)(1.15)

U-CILL Z cJ:

VC 1i. 'J\I1 -g id• /J\ "6119, L, -C, BEM FEM -J

i*

U, Bibel Handschuh 1.2O)

9 J Q)J, Lewicki I21) 9 JQ)-\

Uf cfqi^ t,

*t U(9,

tbL7.

gf< (9 nil

(9 3pE1.22)

U, Wellauer 1.23)(124)

" - c *dtY Niemann (I.25)(1.26)

, fl:- i: -v . r T127),

1.28)(1.31) -

. .

-, rn t CI

VC1i \. -,

- 1 -

(1.29),(1.31)

(1.36)(l 37)

c8.13j(I.32)(1.38)-1.39)

job i1 • •

1.32), (1.39)(1 .45)

irn'J

(1.46)(1.53)- \iJU

1) j<

_rc< 7 1) J1.4)(1.21) 9 .AJ 9 i5

(FEM) :ztb

M0t 7

<JjtL7.

1.54) 1.55) ,j . /j\ft . 11

3D-FEM

}tF-°I 'tJJU 'LC l ' ' t, American Gear Manufacturers Association

Lt 6 T,

O)l'L 9 Jz

Q)9J

=2O0 f ur U-c,

)LJ U-, c jQ),

1 ( 4) , 9 JJ 3OU-(Y7 7J LCY

, =2000

U,jJ0)Th,

L )L, 1*

JJ1,

7.

-2-

U=1O13O0 *

t UEJJ IJ U, t c7) ,

:, wt . rnico WiJ

rnrr U, 'J : 1* U

9 J . 1

EL

1.1

-3-

1 -

QJf

• Jj:3

IL (--L j7

L-J .l)•

Ij] 1 JD

PUR

Fig. 1.1 Research flowchart

-4-

2.1 *

')J <,

< M1 \ 7; . U U, I TIM J:t Ilk

'. i_), C1i\ \ 5 :4i . Q)i,

2.3H2.7) -vc 9, WE \ U WIJ

*Q)0t

Jj9

(FEM)

U, Q))f

W Le

Lj/) 9 L7J ,

j- 1) tJ Q:-c : Ut-Lt **, 2Q)9

5jQ)Th, jk i'z

1 :

2.2

2.2.1$

= 200, rnz1 iz2 = 36/24, iJb = 30 mm135 mm,

UtL8O = 20m : 't

S45C . () 5 Lij (*, i) *, m, a -3

Z2 24 )), rnu*< b35mm

2.2 tt, tQ)I j. 2.1

A TV L^z l, tJ bUj, 1 2.2

-5-

5jL ^ 900 ci-- H'1U, I1 2.2 bl>' l J1'

Q)ii 9 5 2.2.

24i t 1. Q), Q)iIH JIS4 (VT

Table 2.1 Dimensions and calculated results of test helical gear pairs

Normal module Mn 4

Normal pressure angle an 200

Number of teeth z1 /Z2 36 / 24*

Face width bmm30/35*

Transverse backlash f tm 640

Helix angle fib 200

Transverse module m1 4.257

Transverse pressure angle at 21.1730

Base helix angle fib 18.7470

Reference diameter d, mm 153.242

d2 mm 102.162*

Tip diameter d,,, mm 161.242

d,,2 MM 110.162*

Base diameterdblmm 142.896

db2mm 95.264*

Transverse base pitch Pbi 12.470

Transverse base backlash fbi 596.796

Pressure angle at the pitch cylinder

, 22.0330

Center distance a mm 128.511

Length of path of contact L 16.815

Transverse contact ratio 1.348

Overlap contact ratio ep 0.817

Total contact ratio e, 2.165

Material S45C

Accuracy JIS Grade 4 (N8)

Driven (Mating / supporting) gear

Table 2.2 Chemical compositions of gear material(mass %)

Material C Si Mn P S

F 0.42' 0.15— 0.60---'0.030 0.035 S45C

0.48 0.35 0.90

(a) Test gear (b) Supporting gear

Test gear

(c) Gear pairs in mesh

Fig. 2.1 Photographs of test gear pairs used in measurement of root stresses

7-

T3

Part PA2

T T2

0) 0)

90° Pa PO

Gear part sign P0 PA I PA2 I PA3

Gear structure Solid Symmetric web

Rim thickness mmoo 2mn** 1.5 Mn

Web thickness b mm - 7.5 5

*00: solid gear **Mn: normal module

Fig. 2.2 Shape and geometry of test gear (With symmetric web)

2.2.2iZi]

2.3 *ll1tt.

! *(®) M(®) 1©) 7(©) v' l:j- /. 1frt i, 9 <

'/J,640jtm i yxqf)H#, I12.4L

64Ojm 4

a = 105-150 mm . J 2.5 JtJZ

10H 7

z ,

2.1ja z 128.511 mm1Uit. L/C, WjifLMAX-4J:, 2.7

) 9

-8-

I hydraulic pump

Fixing flange © Test gear © Supporting gear ® Torque pick-up © Torque generator

5 , - C j 9 1Ct t

t U fiMi. )ftW< *tit)

q-5 Uti.

2.8 7Ft 9

tO= 300 (0:

0) Hofer f' —' 0.2mm 6mm 5t

-v 1,5 9 3 m vPA 9 ) LC, FiJJJ F1/b = 196, 392, 588 N/mm

Ut1I,

9 2.2 0--PA3)0Y T0

--T3 2.9N:, ±_ 0) (UMM ©—UTI N:PMEA 10 0){ 20 MMAL Uft).

U, (P) 5. M2.10 N:, * It

5 - 7. 5 , ''* U FiL'

(Z)

JII \,

z {9 fWL-. U, 11:

2.11 ,

2.11 5, UMM

, JJ = 20°4øJ

Fig.2.3 Static load test machine

S

Pick gauge

gear

Fig. 2.4 Photograph of setting of backlash

Center distance

Test gear (zi=36)

©, ©, : Tooth number

0: invaa - zflVcj

Fig. 2.5 Schematic illustration of setting of meshing position (z1 /z2 = 36/24)

-10-

Reading telescopeScale

Test gear Supporting gear

t te

Fig. 2.6 Photograph of setting of pitch point meshing position

upporring gear 1L

t te

(a) Red lead thickness is thick

(b) Red lead thickness is thin

Fig. 2.7 Photograph of tooth bearing adjustment by using red lead

- 11 -

(r'imrrssive

Hofer's critical section

No. 1, 2, 3, 4, 5: Strain gages

Fig. 2.8 Positions of strain gages

1W

Pbf 12.47 mm (P)

epbf=l6.8lmm

Beginning of End of engagement engagement

(Root) (Tip)

Fig. 2.9 Contact lines on the plane of action used in measurement of root stresses

-12-

Torque initiation part

Supporting gear (driven gear)

I

L

I

, V, Strain measurement 1

device

Test gear (driving gear)

Hydraulic pump

Fig. 2.10 Photographs of static bending test machine and peripherals

Dial gauge

Fig. 2.11 Photograph of tooth bearing by means of root stresses measurement of Part P0 (Solid gear)

- 13 -

Start

Preparation of test gear (Line scribing, strain gages bonding etc.)

I Setting of pitch point meshing Gear pairs assembly I position and tooth bearing

Setting of backlash], I

NO

YES

I_

rMeasurement of center distance a ]

YES

ITooth contact marking check by using red lead

Equal along

trace

YES - Tooth bearing by means o measurement of root stres

Root stress measurement of solid helical gear part during static load test (Circumferential load F,/b = 196, 392, 588 N/mm)

Comparison of measured and calculated root stress by Kubo-Umezawa method

Measured Calculated NO Adjust the position of supporting gear pedestals

YES

Root stress measurement of thin-rimmed helical gears part during static load test (Fe/b 196, 392, 588 N/mm)

End

Fig. 2.12 Flowchart of root stresses measurement of thin-rimmed helical gears (,0020°)

Adjust the position of supporting gear pedestals ]

Adjust tooth tip distance of test gear pairs

-14-

2.3 JZU

2.3.1 Ittc)Zi]

F1 2.13, 2.14 , F1 2.2 Part P0 (—I*i') iLR)JJ F1/b = 196, 392, 588 N/mm

(b=30mm: *)

UC7Th. 2.13, 2.14 fo),

-D -t:. 2.2) jy

Acute side (A. side), Obtuse side (Ob.

side) LC1I' . . 1 2.13 , 9

FI 2.13 Jz 9 Ob. side i 1 f U

M 2.14 It, 'Jz 10 iQ)E

LI 2.14 1 9 fflIJ f

U

5Z JJ1 ,< 2

(0. P.: Outer point) ^2

UU,

t1f'

9 .A, E7 I1 2.14

(>2) U, acmg 1'i:W 1i) [i

(P)] M 2.9

Qy;;, Part PO 2.13

t)L Ut, Pa1.tPA1*PA3

- 15 -

-800 fi0=20 [I,= b/b = 11 Trailing side Calculated -600

-400Solid gear /(Ob.side)

- FIb = 196N/mm -200 I- 392N/mm -800

m I - 588N/mm -600 Measured -400 ________ . F/b = 196N/mm

a -200 j I • 392N/mm - 800 • 588N/mm -600 A E

Middle A. side: Acute side of face width -400 ________

•1 • Ob.side: Obtuse side of face width

-200 Gn

5 -800 A: Beginning of engagement -600 B: End ofengagement -400 P: Pitch point

200• a • • • - P• Transverse base pitch

-800 r I -600

Le ading side I

-

-200 - • - - •

(A. side)

p

Positions of engagement 3

Ptg tooth pairs No. of me

Fig. 2.13 Calculated root stresses and measured root stresses after

tooth bearing by using red lead (P0, flo=20°)

800 1/ 20 [ç= , b./b = 11 Trailing side Calculated 600 Solid gear /(Ob. side) -400

-

- F1 1b 196N/mm 200 r • I 392 N/mm 800 -800 588 N/mm

-600 Measured -400 • F1 /b 196 N/mm

a 200 ____________ 392N/mm -800 E 588N/mm

b'-600

(0 -400E

Middle Aside: Acute side offaCewidth -200 U - Ob.side: Obtuse side offace Width -800

- A: Beginning of engagement -600 B: End of engagement -400 P: Pitch point -200 _________ Transverse base pitch -800 r 4: Position of max. stress -600 Ocmax Leading side -400 ______ (A. side) -200

Positions of engagement

Ptmeg tooth paira No.of

Fig. 2.14 Calculated root stresses and measured root stresses after tooth bearing by means of measurement of root stresses (P0, ).=20°)

-16-

2.3.2 (Z&(t1)1

I1 2.15 , 9 1 = 2m,, (ma : ¶7rtN b lb = 0.25 (b: i, II 2.2

1) PartPAl) , l 1.5m, blb = 0.25, 0.17 (11 2.2 ci) PartPA2, PA3)

., F1/b = 196, 392, 588 N/mm 9 iEJjQ)

11 2.14, 2.15 '9, Wii 9

im rns ' <

9 J1k <'j)m

1 2.15(b), (c) 9, li 5 Q', Il 2.15(a

9 9 1)7. il 2.15 c1 Q) ,

l\4j (P)Z '9 U 1 (P clZ '9 i° J) PA2

50PA3 P 19 t-tU-CJVN UR-DfL7))- 9;^- 1^- V fA-L Wt (P J, '9 n,, ft N 4E A

I?. PA2 U PA3 O) 9 tWl

< ?J . PAl, PA2 131UPA3

I1 2.9 ci)1 hQ)*© (PAl) © (PA2,PA3)

2.3.3 *IZt1)b

I1 2.16 , II 2.14 Fe/b = 196, 392, 588 N/mm &ci) P0

. M2.17, 2.15 1lF1/b= 196,392,588N/mm1 PA1,PA2LtYPA3

i2.16,2.17

II 2.16, 2.17 2.16,2.17Q ft

*ci) p {Q)W

L12.17 J, 9, 91J, 1) J4-,

It., , Ob.

side

-17-

Positions of engagement

Pt F No. of meshing tooth pairs

(a) PAl (l = 2m, b lb = 0.25)

Positions of engagementri3

Pt No. of i tooth pairs

0

mesh

(b) PA2 (l = 1.5m,, b lb = 0.25)

80 1f. =20 [IU,= 2m, b/b = 0.25 (PAl)] -600 FSymmetncweb gear -400

-200

-800 S S S

-600

-400

200 CL

-800

-600 A -400

-200

-800

-600

-400

-200

-800

100 Ocmax

-400

-200 F-

Trailing side-800 /3=20 [ç = 1 .5mg, b lb = 0.25 (PA2)] Trailing side

,10b.de)600

-400Symmetncweb gear b. side)

-200

-800

-600

-400

ca -200

-800 E E A

E E

Middle-400 Middle -200

- .0 - S-- • U)-800

_____________________________________

2 IX -600 :9

v—-400 B LL

-200 U) Li

-800

-600 cycmax

Leadingside -400 Leadingside (A. side) -200 - (A. side)

800 / =20 [i 1 .5m0, b/b = 0.17 (PA3)] Trailing side 600 Symmetricweb gear 10b. side)

-400

-200/ Measured

-800

- S S

—.— FIb= 196N/mm -600 . --- 392N/mm -400 -.-- 588N/mm -200

Aside: Acute side of face width -800

-600E

A EOb.side: Obtuse side of face width

-400 Middle A: Beginning of engagement -200 B: End of engagement -800 P: Pitch point -600

pbf Transverse base pitch -400 B 0 4: Position of max. stress -200 Co

LI..

-8

-600

00Ocmax

-400 Leading side -200 (A. side)

Positions of engagement 3 LI0

No. of meshing tooth pairs

(c)PA3 (1, = 1.5m,blb0.17)

Fig. 2.15 Measured root stresses of thin-rimmed helical gears with symmetric web arrangement (J3o20°)

-18-

Lp.l____ A

LIIIII!

11,

End Middle End (A. side) (Ob. side)

(i)F,/b= 196N/mm

(U F1/b196 N/mm I=°° ' b,,/b= 1 z2/z1=24/36

d-200 -

8 b=30 mm 6

-100 o

11

End Middle End (A. side) (Ob. side)

(i )F,/b= 196 N/mm

LJ" r _ Is]

I

m=4 cc F/b=392 N/mm m=4 z2 / z1 24/36 /,,=oo, b/b = I z2/ z1=24/36

-30O -fi2O0 b=30 mm b=30 mm

-20C--- Jj IcT-r k —ioc

(U

Cl)

0 0

End Middle End (A. side) (Ob. side)

( ii) F,/b = 392 N/mm

500 I

F,/b=588 N/mm m=4 Ir, 00 , b/b = 1 z2/ z1=24/36

WO----J^8--Ao=20*

b=30 mm

-1

End Middle End

(A. side) (Ob. side)

(ii)F/b = 392 N/mm

-500 I CU I F,/b=588 N/mml

MO

=4

I i000,b,.,/b=1 z21z124/36 I

t-400 fl0=2O

I I I b3Ommma U) a) I I -300

0 0 Ix ras

-100

.i—t_S—t-----I_I 01 - j End Middle End End Middle End

(A. side) (Ob. side) (A. side) (Ob. side) (iii)F,/b588N/mm (iii)F,/b588N/mm

(a) Calculated (b) Measured

Fig. 2.16 Root stresses distributions of solid helical gear (P0, /30=20)

-19-