Bài giảng Kinematics and dynamics of machines - Chapter 10: Gears and spur gear (Part 1) - Phạm Huy Hoàng
CHAPTER 10
GEARS AND SPUR GEAR
Pham Huy Hoang
I. Introduction
1. Concept and definition
A gear or cogwheel is a rotating machine part
having cut teeth, or cogs, which mesh with
another toothed part to transmit torque.
Requirement: constant angular-velocity ratio
(medium or instantaneous/regular)
Shape of teeth (profile) guarantees the instantaneous
constant constant angular-velocity ratio during meshing
2. Classification:
a. Planar gear mechanism (parallel axes)
a. Planar gear mechanism (cont.)
b. Spatial gear mechanism
3. Application: transmission of angular velocity between two
shafts (parallel axes: spur gears or helical gears with parallel
axes; intersected axes (bevel gear) or crossing axes: helical
gear without parallel axes and worm gear). gear parallel).
II. Fundamental lawof toothed gearing
1. Requirement: constant angular velocity ration during
meshing
O
1
1
1
i
12
const
1
2
v
M 2
2
2. Fundamental law:
The normal to conjugate
M
N
1
L
1
v
M1
P
profiles at the contact point M
intersects the line of centers
O1O2 at a fixed point P.
1
L
2
N
2
2
L1, L2: conjugate
P: instant center / pitch point
N1N2 : line of action
2
O
2
hc v
nn M
hc v
nn M
O
1
1
2
1
v
cos v
cos
M 2
M
1
1
2
1
O M. .cos O M. .cos
1
1
1
2
2
2
O N . O N .
1 1 1
2 2 2
v
M 2
2
M
O N
2 2
O P
N
1
2
1
i
12
L
1
v
O N
O P
M1
P
2
1 1
1
1
L
2
N
2
O1P O2 P O1O2 const
2
i12 const
Point P must be fixed
2
O
2
3. Involute profile
a. Involute
Equation of involute
M N
r ( )
0
NM
0
tan
tan
r
r
r
0
0
0
r
0
r
cos
y
N
In polar coordinate system (r,θ):
M (x, y)
r
r
0
r
x
M
O
0
M :
cos
tan inv()
r
0
i
i( / 2)
OM ON NM r e e
0
0
i
i( / 2)
r e M N e
0
y
i
i( / 2)
r e r e
0
0
N
M
(
x
,
y
)
r
x
M
O
0
r
0
In cartesian coordinate system (x,y):
x r (cos sin )
0
M :
y r (sin cos )
0
Involute characteristics
+ There is not any part of
involute inside base circle.
N
M
1
M
2
r
0
0
M
1
O
0
2
M
+ A line that is normal to involute at any point is also tangent to
base circle, and inversely, a line that is tangent to base circle is
also normal to involute.
+
+ N is the curvature
center of involute i at Mi.
N
M
1
M
2
r
0
0
M
1
O
0
2
+
M
b. Involute profile adapts
O
1
fundamental law of meshing
1
The normal nn to
r
base circle 1
voøng cô sôû 1
01
both involute profiles
L1, L2 is also tangent
to both base circles
(O1, r01) and (O2, r02).
normal
L
1
N
1
M
vboaønsge cciôrcsleô2û 2
P
Two base circle are
fixed, therefore the
center line O1O2 and
the common normal
are fixed.
L
N
2
2
center
line
r
02
P is fixed
2
O
2
c. Change of center distance
O
O
1
1
1
1
base circle 1
base circle 1
voøng cô sôû 1
r
r
01
01
voøng cô sôû 1
L
1
L
1
N
1
N
1
M
M
base circle 2
voøng cô sôû 2
P
base circle 2
voøng cô sôû 2
P
L
L
N
2
2
2
N
2
r
02
r
02
2
O
2
2
O
2
O
2
III. Gear terminology
• Base circle (Oi, r0i).
• Instant center/Pitch
2
r
r
02
l2
center P:
v
v
P
P
1
2
• Pitch circle (Oi, rli).
pitch
circle 2
N
2
• Line of action N1N2.
(theoretical meshing
segment)
l
v
M 2
pitch
circle 1
instant
center
N
1
• Pressure angle α.
r
01
r
l1
• Pressure angle at instant
center (pressure angle on
pitch circle) αl.
1
O
1
Width of space
Tooth thickness
2
base circle 2
voøng cô sôû 2
N
2
addvenodønumgcñircænleh1 1
voøng ñænh
B
P
addendum
circle 2
A
2
N
1
1
• Addendum circle (Oi, rei).
• Dedendum circle (Oj, rij).
• Meshing segment AB.
• Pressure angle α.
voøng cô sôû 1
base circle 1
t
x
w
s
x
x
t
l
t
n
t
0
• Pitch on base circle t0. (angular step on base circle)
• Pitch on normal tn. (step on normal line)
• Pitch on circle (Oi, x) : tx.
• Tooth thickness Sx and width of space (tooth space) Wx.
N2
B
P
A
N1
17 trang
30581
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