SME
Kinematics of Mechanisms · Problem type
Transmission angle at a crank position
Use this for “A crank-rocker has crank a = 40, coupler b = 120, rocker c = 90 and ground d = 100 mm. Find the transmission angle at θ₂ = 60°, or its least value (θ₂ = 0°).”
Diagram
Three sides fix the triangle; the law of cosines gives each angle.
Law of sines: c/sin B = b/sin D = BD/sin γ ≈ 120.41
Crank a = 40 mm
Ground link d = 100 mm
Crank angle θ₂ = 60°
This is an example. Type your own values to start a new problem.
(
BD (mm)
)
2
=
(
a (mm)
)
2
+
(
d (mm)
)
2
−
2
×
a (mm)
×
d (mm)
×
cos(
θ₂
°)
cos(
γ
°)
=
(
)
2
+
(
)
2
−
(
)
2
2
×
b (mm)
×
c (mm)
Formulas
BD
=
√(
a
2
+
d
2
−
2
×
a
×
d
×
cos(
θ₂
°))
87.178
=
√(
40
2
+
100
2
−
2
×
40
×
100
×
cos(60°))
γ
=
cos⁻¹(
b
2
+
c
2
− BD²
2 ×
b
×
c
)
46.3845
=
cos⁻¹(
120
2
+
90
2
−
87.178
2
2 × 120 × 90
)
Assumptions
•
θ₂ is the crank’s angle from the ground line, measured at the crank pivot A.
•
γ is the angle between the coupler and the rocker, at their shared pin C.
•
Keep γ between 40° and 140° for smooth running; near 0° or 180° the coupler can’t turn the rocker.
Step-by-step
WE KNOW
θ₂ = 60°, a = 40 mm, b = 120 mm, c = 90 mm, d = 100 mm
FIND
diagonal (BD), transmission angle (γ)
Step 1 · Find diagonal (BD)
Use BD = √(a² + d² − 2 × a × d × cos(θ₂°))
The law of cosines on triangle ABD: crank, ground and the diagonal between their free ends.
BD
=
√(
40
2
+
100
2
−
2
×
40
×
100
×
cos(60°))
BD
=
√
1,600 + 10,000 − 2 × 40 × 100 × 0.5
BD
=
√
1,600 + 10,000 − 80 × 100 × 0.5
BD
=
√
1,600 + 10,000 − 8,000 × 0.5
BD
=
√
1,600 + 10,000 − 4,000
BD
=
√
7
,600
BD = 87.178 mm
Step 2 · Find transmission angle (γ)
Use γ = cos⁻¹((b² + c² − BD²) ÷ (2 × b × c))
The law of cosines on triangle BCD, turned round for the angle at C.
γ
=
cos⁻¹(
120
2
+
90
2
−
87.178
2
2 × 120 × 90
)
γ
=
46.3845°
Check
Put the numbers back in. Both sides should match.
87.178
=
√(
40
2
+
100
2
−
2
×
40
×
100
×
cos(60°))
✓
46.3845
=
cos⁻¹(
120
2
+
90
2
−
87.178
2
2 × 120 × 90
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.