SME
Kinematics of Mechanisms
Cams
Use this for “A cam lifts its follower 20 mm with simple harmonic motion over 120° of turn at 300 rpm. Find s, v and a at 45°.”
Chart
s(45) ≈ 6.173
Level h = 20 meets the curve at θ = 120
This is an example. Type your own values to start a new problem.
s (mm)
=
h (mm)
2
×
(1
−
cos(180°
×
θ
°
÷
β
°))
v (mm/s)
=
π
×
h (mm)
×
ω (rpm)
2
×
βᵣ (rad)
×
sin(180°
×
θ
°
÷
β
°)
a
Acceleration
calculated · m/s²
Formulas
βᵣ
=
β
×
π
180
2.0944
=
120
×
π
180
s
=
(
h
2
)
×
(1
−
cos(180°
×
(
θ
β
)))
6.1732
=
(
20
2
)
×
(1
−
cos(180°
×
(
45
120
)))
v
=
(π
×
h
×
ω
2 × βᵣ
)
×
sin(180°
×
(
θ
β
))
435.368
=
(π
×
20
×
31.4159
2 × 2.0944
)
×
sin(180°
×
(
45
120
))
a
=
(
π
2
×
h
×
ω
2
2 ×
βᵣ
2
)
×
cos
180° × (
θ
÷
β
)
1,000
+8.4981
=
(
π
2
×
20
×
31.4159
2
2 ×
2.0944
2
)
×
cos
180° × (45 ÷ 120)
1,000
Formulas are worked in rad/s. The step-by-step shows the conversions.
Assumptions
•
The rise starts at θ = 0 and ends at θ = β; the cam turns at a steady speed ω.
•
The follower moves like the shadow of a point going steadily round half a circle.
•
The acceleration jumps at both ends of the rise, which shakes a fast cam; cycloidal motion avoids it.
Step-by-step
WE KNOW
θ = 45°, h = 20 mm, β = 120°, ω = 300 rpm
FIND
rise in radians (βᵣ), displacement (s), velocity (v), acceleration (a)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to rad/s).
ω = 300 rpm = 31.4159 rad/s (1 rad/s = 9.5493 rpm)
Step 2 · Find rise in radians (βᵣ)
Use βᵣ = β × π ÷ 180
The speed formulas need the rise angle in radians: 180° is π rad.
βᵣ
=
120
×
π
180
βᵣ = 2.094395 rad
Step 3 · Find displacement (s)
Use s = (h ÷ 2) × (1 − cos(180° × (θ ÷ β)))
Over
the
rise
180°
θ
β
runs
from
0°
to
180°,
so
the
follower
climbs
half
a
cosine
wave
of
height
h
.
s
=
(
20
2
)
×
(1
−
cos(180°
×
(
45
120
)))
s
=
20
2
×
(1
−
cos(180°
×
0.375))
s
=
20
2
×
(1
−
cos(67.5°))
s
=
20
2
×
(1
−
0.3827)
s
=
20
2
×
0.6173
s
=
10
×
0.6173
s
=
6.1732
mm
Step 4 · Find velocity (v)
Use v = (π × h × ω ÷ (2 × βᵣ)) × sin(180° × (θ ÷ β))
Differentiate
s
by
time:
the
angle
grows
at
ω
,
so
β
must
be
in
radians
here.
v
=
(π
×
20
×
31.4159
2 × 2.094395
)
×
sin(180°
×
(
45
120
))
v
=
20π
×
31.4159
2 × 2.0944
×
sin(180°
×
0.375)
v
=
20π
×
31.4159
4.189
×
sin(67.5°)
v
=
20π
×
31.4159
4.189
×
0.9239
v
=
1,974
4.189
×
0.9239
v
=
471.2
×
0.9239
v
=
435.368
mm/
s
Step 5 · Find acceleration (a)
Use a = (π² × h × ω² ÷ (2 × βᵣ²)) × cos(180° × (θ ÷ β)) ÷ 1,000
Differentiate
once
more;
÷
1000
turns
mm/
s
2
into
m/
s
2
.
It
is
+
in
the
first
half,
−
in
the
second.
a
=
(
π
2
×
20
×
31.4159
2
2 ×
2.094395
2
)
×
cos
180° × (45 ÷ 120)
1,000
a
=
π
2
×
20
×
31.4159
2
2 ×
2.0944
2
×
cos(180° × 0.375)
1,000
a
=
π
2
×
20
×
31.4159
2
2 × 4.3865
×
cos(67.5°)
1,000
a
=
π
2
×
20
×
31.4159
2
8.773
×
cos(67.5°)
1,000
a
=
9.8696
×
20
×
986.96
8.773
×
0.38268
1,000
a
=
197.392
×
986.96
8.773
×
0.38268
1,000
a
=
194,820
8.773
×
0.38268
1,000
a
=
22,207
×
0.38268
1,000
a
=
8,498.2
1,000
a
=
+8.4981
m/
s
2
Check
Put the numbers back in. Both sides should match.
2.094395
=
120
×
π
180
✓
6.1732
=
(
20
2
)
×
(1
−
cos(180°
×
(
45
120
)))
✓
435.368
=
(π
×
20
×
31.4159
2 × 2.094395
)
×
sin(180°
×
(
45
120
))
✓
+8.4981
=
(
π
2
×
20
×
31.4159
2
2 ×
2.094395
2
)
×
cos
180° × (45 ÷ 120)
1,000
✓
Topic check
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