SME
Machine Design · Problem type
First critical speed of a shaft
Use this for “A disc sags its shaft 0.1 mm at rest. Find the first critical speed in rad/s and in rpm.”
Diagram
The disc’s weight sags the shaft 0.1 mm (drawn far larger).
Its
stiffness
k
holds
the
weight,
kδ
=
mg,
so
it
whirls
at
√(
k
m
)
=
√(
g
δ
).
ω
=
√
9.81 m/
s
2
÷ 1 ×
10
-4
m
=
313.2
rad/s;
nc
=
60ω
2π
=
2,991
rpm.
This is an example. Type your own values to start a new problem.
ω (rad/s)
=
9.81
m/s²
δ (mm)
nc (rpm)
=
60
×
ω (rad/s)
2π
Formulas
ω
=
√(
9.81
δ
)
313.2092
=
√(
9.81
0.0001
)
nc
=
60
×
ω
2π
2,990.9275
=
60
×
313.2092
2π
Formulas are worked in m. The step-by-step shows the conversions.
Assumptions
•
One heavy disc on a light shaft; δ is the static sag at the disc under its own weight.
•
The shaft’s stiffness k holds the weight, kδ = mg, so ω = √(k ÷ m) = √(g ÷ δ) with g = 9.81 m/s².
•
Keep the running speed well away from nc (a common rule: below about 0.7nc).
•
Several discs: Rayleigh’s ω = √(gΣWδ ÷ ΣWδ²); this page takes one disc.
Step-by-step
WE KNOW
δ = 0.1 mm
FIND
critical speed in rad/s (ω), critical speed in rpm (nc)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m).
δ = 0.1 mm = 0.0001 m (1 m = 1000 mm)
Step 2 · Find critical speed in rad/s (ω)
Use ω = √(9.81 ÷ δ)
The
shaft
whirls
hardest
at
the
disc’s
natural
frequency
√(
k
m
),
and
kδ
=
mg
makes
k
m
=
g
δ
.
ω
=
√(
9.81
0.0001
)
ω
=
√
98
,100
ω
=
313.209195
rad/s
Step 3 · Find critical speed in rpm (nc)
Use nc = 60 × ω ÷ (2π)
One turn is 2π radians and one minute is 60 seconds.
nc
=
60
×
313.209195
2π
nc = 2,990.927 rpm
Check
Put the numbers back in. Both sides should match.
313.209195
=
√(
9.81
0.0001
)
✓
2,990.927
=
60
×
313.209195
2π
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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