SME
Mechanical Vibrations
Damping
Use this for “A 2 kg mass on an 800 N/m spring has a damper with c = 16 N·s/m. Find ccr, ζ and the damped frequency ωd.”
Diagram
ωₙ
=
√(
k
m
)
=
√(
800
2
)
=
20
rad/s
ζ
=
c
2
√
km
ζ
=
16
2
√
800 × 2
=
0.2
ζ
<
1,
underdamped:
ωd
=
ωₙ
√
1 −
ζ
2
=
19.6
rad/s,
less
than
ωₙ;
the
crests
shrink
inside
±X
exp(−ζωₙt).
This is an example. Type your own values to start a new problem.
ccr (N·s/m)
=
2
×
k (N/m)
×
m (kg)
ζ
=
c (N·s/m)
÷
ccr (N·s/m)
ωd (rad/s)
=
ωₙ (rad/s)
×
1
−
ζ
2
D
Damping case
worked out (not typed)
Formulas
ccr
=
2
×
√
k
×
m
80
=
2
×
√
800 × 2
ζ
=
c
ccr
0.2
=
16
80
ωₙ
=
√(
k
m
)
20
=
√(
800
2
)
D
=
compare
ζ
with
1
0.2 < 1, so underdamped
ωd
=
ωₙ
×
√
1 −
ζ
2
19.5959
=
20
×
√
1 −
0.2
2
Assumptions
•
Viscous damping: the dashpot pushes back with −cẋ, so mẍ + cẋ + kx = 0.
•
ccr = 2√(km) is the least damping that stops the swing; ζ = c ÷ ccr.
•
ζ < 1 is underdamped and rings at ωd; ζ = 1 is critically damped and ζ > 1 overdamped, with no ωd.
Step-by-step
WE KNOW
m = 2 kg, k = 800 N/m, c = 16 N·s/m
FIND
critical damping (ccr), damping ratio (ζ), natural frequency (ωₙ), damping case (D), damped frequency (ωd)
Step 1 · Find critical damping (ccr)
Use ccr = 2 × √(k × m)
Critical
damping
makes
c
2
=
4km,
where
the
two
roots
of
ms²
+
cs
+
k
meet.
ccr
=
2
×
√
800 × 2
ccr
=
2
×
√
1
,600
ccr = 2 × 40
ccr
=
80
N·s/
m
Step 2 · Find damping ratio (ζ)
Use ζ = c ÷ ccr
The damping ratio is the damping as a fraction of critical.
ζ
=
16
80
ζ
=
0.2
Step 3 · Find natural frequency (ωₙ)
Use ωₙ = √(k ÷ m)
The undamped natural frequency, as if the dashpot were taken off.
ωₙ
=
√(
800
2
)
ωₙ
=
√
400
ωₙ = 20 rad/s
Step 4 · Find damping case (D)
Use D = compare ζ with 1
ζ
is
below
1:
the
roots
are
complex,
so
the
mass
rings
inside
a
shrinking
envelope.
0.2 < 1: underdamped
D
=
underdamped
Step 5 · Find damped frequency (ωd)
Use ωd = ωₙ × √(1 − ζ²)
Underdamped only: the dashpot slows the swing a little, so ωd < ωₙ.
ωd
=
20
×
√
1 −
0.2
2
ωd
=
20
×
√
1 − 0.04
ωd
=
20
×
√
0.96
ωd = 20 × 0.9798
ωd = 19.5959 rad/s
Check
Put the numbers back in. Both sides should match.
80
=
2
×
√
800 × 2
✓
0.2
=
16
80
✓
20
=
√(
800
2
)
✓
0.2 < 1, so underdamped ✓
19.5959
=
20
×
√
1 −
0.2
2
✓
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Logarithmic decrement: ζ from two crests
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