Bài giảng Kinematics and dynamics of machines - Chapter 9: CAM mechanism (Part 2) - Phạm Huy Hoàng
III. Cam synthesis
Determine cam profile and some important dimensions for:
d
- Smooth motion of follower (without self locking):
- Desired movement program of follower:d 2 s
2
or
dt2
2
dt
- Compactness.
Input data:
- Acceleration diagram of follower’s motion.
- Allowable pressure angle:
- Length of follower l (if needed).
- Maximum displacement of follower.
Cam design procedure:
1. Cooperate/Combine motions in machine
2. Choose acceleration diagram of follower’s motion.
3. Build desired movement program of follower.
4. Determine important dimension:
- Choose rotation center of cam. Determine distance
between rotation centers of cam and follower.
Determine lowest position of follower (smallest
displacement of follower) (knife-edge or roller follower)
- Determine radius of base circle (flat face follower)
- Synthesis of cam profile.
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1.
Shaping machine
Requirement:
Shaping machine
includes 2 motions:
- Shaping motion of tool:
quick return, rocker and
slider mechanisms.
- Interrupted motion of
table using cam, ratchet
and screw-nut
mechanisms.
2nd motion can not happen
when shaping.
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1. Cooperation of motions
in machine
V engine
V engine
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Requirement:
Intake, compression, power
and exhaust motions of two
pistons: two crank and slider
mechanisms.
Valves’ controlling: intake and
exhaust valves are controlled
by cam mechanisms.
Requirements:
Valve must be opened or
closed in time of the
corresponding periods.
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2. Choose acceleration diagram of follower
- Cam mechanism works
with high speed.
- Based on dynamic
characteristics.
- Speed may not be
changed suddenly.
Constant Velocity Motion
Advantages: driving torque does not
change.
Disadvantages:
- Hard impacts at the end of periods.
- Cam profile has sharp point.
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Constant Velocity Motion
Rarely used. Unsuitable with
high speed machine (more than
100 RPM).
Can be improved by using
curves at the end of periods.
Constant Acceleration Motion
In rising period, acceleration varies
increasingly then decreasingly.
Soft impact.
Radius of curvature of profile changes
suddenly and causes vibration.
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Harmonic Motion - Cosine
Far dwell causes soft impact.
Harmonic Motion - Sine
There is no impact, good choice for dynamics.
Evenly decreased acceleration
Soft impact as cosine.
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III. Cam design
3. Determine important dimensions
Knife edge oscillating follower:
Determine distance a between rotation centers of cam and
oscilating follower and initial position ψ0 based on the limit of
[]
pressure angle
Knife edge follower :
Determine eccentricity e and initial position S0 based on the
limit of pressure angle []
Flat-face follower:
Determine radius of base circle R0 for convex
profile of cam (curvature radius of cam profile is
always positive)
Knife-Edge Oscillating Follower
v
A2
n
HN
OH
tg
tg
A
d
d
H
v
acos () l l
asin ()
()
A1
N
l
n
B
O
a
OH OBsin asin
d
l
v
d
d
d
A2
OAN ~ v Av AN
OA
OA l
A1 A2
v
OA
A1
d
d
HN BH AN AB acos l
l
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v
A2
n
A
H
v
A1
N
l
n
B
O
a
Tìm a và ψ0 nhỏ nhất và thỏa điều
kiện sau đây với mọi 0 2
d
d
acos[ ()] l l
()
0
tg
tg[]
a[ ()]
0
Available area of cam’s rotation center
v
Ai2
n
A
i
v
Ai1
N
i
l
n
i
[]
B
a
O
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Available area of cam’s rotation center
Nj
Ai
Ni
Aj
A0
B
l
uj
ui
vj
vi
ˆ
T (u N v ) (u N v )
i i i
j j j
N
Available area of cam’s
rotation center
j
+
A
j
i
A
l
N
i
[]
i
B
[]
A
[]
0
j
0
[]
u
j
u
a
i
Selecting a base
radius for roller
followers has a
strong influence on
the pressure angle.
v
i
O
m
ˆ
u N v )
(T
i
i i
v
j
i0
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Knife-Edge Follower
n
S e
1
vA2
tan
2
2
vA1
S R e
0
S
t
t
R0
R
S
Determine minimum e and
S0 for
0 2
n
dS
() e
d
e
tg
tg[]
S S()]
0
v
Ai2
B
n
N
A
i
i
n
[]
[]
S
v
i
Ai1
O
I
i
e
H
i
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+
N
j
A
j
S
max
A
i
N
i
A
Selecting a base radius for
roller followers has a strong
influence on the pressure
0
[]
[]
S
angle.
0
O
u
T
v
e
Flat-faced follower
Convex profile
n
A
OAH ~ v Av
A1
n
AH S cos
B
x (R S cos)cos S cos sin
0
x
A:
v
v
A1
A2
y (R S cos)sin S cos cos
n
A
0
v
S cos
t
A
H
t
S
y
n
A
0
x
Curvature radius
y
3/ 2
2
2
dx dy
d
d
O
2
2
dx d y dy d x
R
0
2
2
d
d
d
d
R S cos S cos
0
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n
B
23
r
a
a
A2
Curvature radius
a
A3
t
t
A
OIL ~ a Aa
A3
A2
R S cos
0
// nn
KI LI cos
2
2
LI /OI a / a (S )/OI
A2
A3
O
L
K
I
LI S; IA KA IK
R S cos S cos
0
n
Convex profile – Positive curvature radius
R S cos S cos 0
0
0
R h
0 max
R h
0 max
10mm
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Knife-edge Oscillating follower
B
i
l
i
a
A
i
A
0
l
i
0
B
0
0
O
a
Knife-edge follower
B
A
i
A
0
S
i
H
i
S
0
i
H
0
O
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Roller Oscillating follower
A
C
l
B
a
O
- Initially draw the pitch profile.
- Choose radius of roller.
A
C
B
O
- Substract the value of the
follower roller radius to each
point to obtain the cam
surface profile (real profile).
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* How to choose the radius of roller: smaller than
smallest positive curvature radius of pitch profile
Smaller
Biiger
Equal
Undercutting: The roller follower radius r is larger than the
smallest positive (convex) local radius. No sharp corners for
an acceptable cam design.
smallest positive (convex) local radius
Traslating Roller Follower
- Initially draw the pitch profile.
- Choose radius of roller.
B
- Substract the value of the
follower roller radius to each
point to obtain the cam
A
C
surface profile (real profile).
O
O
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Flat-Faced Followers
Si
Si
S0
S0
i
i
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B
i
S
i
A
3
A
i
A
2
A
1
i
O
A
0
R
0
Addition!
Analytical synthesis of cam profile
Notations:
: rotation angle of cam
θ : parameter which creating profile of roller or flat end
R0: radius of base circle
r: roller radius
a: distance between rotation centers of cam and oscillating follower
l: length of follower
S: motion of follower
: rotation of follower
e: eccentricity
: differentiation of H
H
Obtain the family of roller circles
x x(,)
(C ) :
,
y y(,)
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Cam profile is the envelope (closed curve) of the
family of roller circles
x x(,)
y y(,)
y
B
r
l
(B) :
A
x y
x y
0
C
a
O
x
x acos l cos( ) r cos
y asin l sin( ) r sin
(C ) :
,
x x(,)
y y(,)
(B) :
x y
x y
0
P(x, y) (C )
,
x acos l cos( ) r cos
x y
x y
B P(x, y) :
P(x, y) : y asin l sin( ) r sin
0
asin l( 1)sin( )
1
tan
acos l( 1)cos( )
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