1 ME 302 DYNAMICS OF MACHINERY GEAR FORCE ANALYSIS Dr. Sadettin KAPUCU © 2007 Sadettin Kapucu.
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Transcript of 1 ME 302 DYNAMICS OF MACHINERY GEAR FORCE ANALYSIS Dr. Sadettin KAPUCU © 2007 Sadettin Kapucu.
1
ME 302 DYNAMICS OF ME 302 DYNAMICS OF MACHINERYMACHINERY
GEAR FORCE ANALYSIS
Dr. Sadettin KAPUCU
© 2007 Sadettin Kapucu
2Gaziantep University
GEAR FORCE ANALYSISGEAR FORCE ANALYSIS
Gears are used to transmit force and motion from one shaft to the other.
addendum circle
addendum
dedendum
dedendum circle
clerance
tooth thickness
space width
Circular pitch
top land
face width
face
flank
bottom land
pitch circle
Base circle
3Gaziantep University
Spur GearSpur Gear In theory, one blank transmit force and motion via the
friction force occurring at the pitch point. There must be no slip between the two blanks at the pitch points. Blanks roll but do not slip with respect to each other. Input /output relationship for circular gear is linear. So:
1
2
3
r2r3
VI23
I23I13I12
Pitch point
Pitch Circle
2 3 2
,
223 * rVI
3
,
323 * rVI
32
323322
r
rr*r*
...
4Gaziantep University
Spur GearSpur Gear Magnitudes of friction forces are generally small. So, if we want to
transmit larger amounts of forces, friction becomes inadequate and slip occurs. To prevent slip, we make the joint between links 2 and 3 “form closed”. This is obtained by putting teeth around the periphery of the gear blanks. For fitting these details, we need some space. We simply separate the gear blanks apart a bit.
1
2
3
r2
r3I13
Pitch Circle
FG
This separation causes the transmitted force to at an angle called “pressure angle”, denoted by . Pressure angle is standardized;
In imperial system 20In international system 5.18
5Gaziantep University
Spur GearSpur Gear
1
2
3
r2
r3I13
Pitch CircleFGFr
FGFt
Fr
Ft
Characteristic dimension for the tooth is either the number of teeth on the blank or the length of the portion of the pitch circle within the tooth body.
)inch(diametercirclePitch
teeth#PpitchDiametral d
teeth#
)mm(diametercirclePitchModule
6Gaziantep University
Helical GearHelical GearTo improve the force carrying capacity of the gears the teeth are cut in a helix. This increase the tooth thickness, so helical gears are stronger. Also they operate with less noise.
Fa
Ft
Fr
FG
Ft
Fa
gear axis of rotation
FaFt
Fr
FG
Force acting normal to the tooth surface, hence it makes an angle of (helix angle) with the gear axis of rotation and with the common tangent.
traG FFFF
tanFF ta
tanFF tr
7Gaziantep University
Bevel GearBevel GearBevel gears are conical in shape and used to couple the shafts not parallel but intersecting. Point of intersection of shafts is called the apex. Gear force acts as distributed over the whole tooth thickness, but we can assume a resultant single force acting on the mid point of the tooth thickness.
resultant forceacts at the midpoint of the gearthikchness
traG FFFF
sintanFF ta
costanFF tr
8Gaziantep University
Example 1Example 1The gear train shown in the figure is composed of 6 diametral pitch spur gears and 20 degrees pressure angle. Link 2 is the driving gear, delivering 25 Hp at a CCW speed of 900 rpm. Gear 3 is an idler and gear 4 carries the external load. Draw freebody diagrams of the gears, show all the forces acting and calculate their magnitudes.
2
3
4
18 T
36 T
20 T
Given:o20
6pitchDiametral
HpPower 25
9Gaziantep University
Example 1Example 12
3
4
18 T
36 T
20 T2
4
18 T
20 T
3
36 T
F’3t
F’3r
F3x
F3yF3t
F3r
F4t
F4r
F2r
F2t
F2x
F2y
F4x
F4y
4
2
x
y+
3
10Gaziantep University
Example 1Example 12
3
4
18 T
36 T
20 T2
18 T
F2r
F2t
F2x
F2y
4
x
y+
From second gear freebody diagram:
txtxx FFFFF 2222 0;0
ryyry FFFFF 2222 0;0
2222222 *0*;0 rFrFM tt
11Gaziantep University
Example 1Example 12
3
4
18 T
36 T
20 T
3
36 T
F’3t
F’3r
F3x
F3yF3t
F3r
x
y+
From third gear freebody diagram:
trxtrxx FFFFFFF 233233 0;0
trytryy FFFFFFF 423423 0;0
232333323333 **0**;0 rFrFrFrFM tttt
3
12Gaziantep University
Example 1Example 12
3
4
18 T
36 T
20 T
4 20 T F4t
F4rF4x
F4y
4
x
y+
From fourth gear freebody diagram:
txtxx FFFFF 4444 0;0
rytry FFFFF 4444 0;0
24424444 *0*;0 rFrFM tt
Radius of the gears can be calculated following formulas:
d
d
P
teeth#ddiametercirclePitch
diametercirclePitch
teeth#PpitchDiametral
2
"5,16*2
18
*2
##2
222 r
P
teethr
P
teethd
d
"36*2
36
*2
#3
33 r
P
teethr
"67,16*2
20
*2
#4
44 r
P
teethr
13Gaziantep University
Example 1Example 1Gear forces of the second gear become:
lbrn
PowerF t 14,1167
5,1*900**2
12*25*33000
***2
12**33000
222
lbFFF rtr 8,42420tan*14,1167tan* 222
Speed of the fourth gear is:
rpmnTT
TTn 810900*
36*20
18*36*
*
*2
43
324
Then, gear forces of the fourth gear become:
lbrn
PowerF t 3,1168
67,1*810**2
12*25*33000
***2
12**33000
444
lbFFF rtr 22,42520tan*3,1168tan* 444
Using the equationsunknowns become:
lbF x 14,11672
lbF y 8,4242 inlb.7,17502 lbF x 23,15013 lbF y 18,13703
inlb.03 lbF x 22,4254 lbF y 3,11684 inlb.97,19464
2
4
18 T
20 T
3
36 T
F’3t
F’3r
F3x
F3yF3t
F3r
F4t
F4r
F2r
F2t
F2x
F2y
F4x
F4y
4
4
x
y+