SME
Kinematics of Mechanisms
Mobility (Gruebler's equation)
Use this for “A five-bar linkage has 5 links (counting the ground) and 5 pin joints. How many inputs does it need?”
Diagram
Gruebler: M = 3(n − 1) − 2j₁ − j₂ = 3 × 4 − 2 × 5 − 0 = 2.
M = 2: 2 inputs drive it.
This is an example. Type your own values to start a new problem.
M
=
3
×
(
n
−
1)
−
2
×
j₁
−
j₂
K
Kind of linkage
worked out (not typed)
Formulas
M
=
3
×
(
n
−
1)
−
2
×
j₁
−
j₂
+2 = 3 × (5 − 1) − 2 × 5 − 0
K
:
compare
M
with
0
+2 ≥ 1, so mechanism
Assumptions
•
The links move in a plane, so each free link has 3 freedoms: x, y and a turn.
•
A full joint (pin or slider) takes away 2 freedoms; a half joint (cam or slipping contact) takes 1.
•
The ground counts as one of the n links and never moves.
•
A pin joining three links counts as two full joints.
Step-by-step
WE KNOW
n = 5, j₁ = 5, j₂ = 0
FIND
mobility (M), kind of linkage (K)
Step 1 · Find mobility (M)
Use M = 3 × (n − 1) − 2 × j₁ − j₂
Every link but the ground has 3 freedoms; each full joint takes 2 and each half joint 1.
M
=
3
×
(5
−
1)
−
2
×
5
−
0
M
=
3
×
4
−
2
×
5
M
=
12
−
10
M
=
+2
Step 2 · Find kind of linkage (K)
Use K: compare M with 0
M
is
1
or
more:
the
linkage
moves,
and
it
needs
M
inputs
to
drive
it.
+2 ≥ 1: mechanism
K
=
compare
+2
with
0
K
=
mechanism
Check
Put the numbers back in. Both sides should match.
+2 = 3 × (5 − 1) − 2 × 5 − 0 ✓
+2 ≥ 1, so mechanism ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.