SME
Mechanical Vibrations · Problem type
Logarithmic decrement: ζ from two crests
Use this for “A 2 kg mass on an 800 N/m spring swings from 10 mm to 2 mm in 5 cycles. Find δ, ζ and c.”
Diagram
ωₙ
=
√(
k
m
)
=
√(
800
2
)
=
20
rad/s
ζ
=
c
2
√
km
ζ
=
4.09
2
√
800 × 2
=
0.0512
ζ
<
1,
underdamped:
ωd
=
ωₙ
√
1 −
ζ
2
=
20
rad/s,
less
than
ωₙ;
the
crests
shrink
inside
±x₀
exp(−ζωₙt).
δ
=
(
1
n
)
ln(
x₀
xₙ
)
=
2πζ
÷
√
1 −
ζ
2
=
0.322:
each
crest
is
e⁻ᵟ
of
the
one
before.
This is an example. Type your own values to start a new problem.
δ
=
(1
÷
n
)
×
ln(
x₀ (mm)
÷
xₙ (mm)
)
ζ
=
δ
÷
4
×
π²
+
δ
2
c (N·s/m)
=
2
×
ζ
×
k (N/m)
×
m (kg)
Formulas
δ
=
(
1
n
)
×
ln(
x₀
xₙ
)
0.3219
=
(
1
5
)
×
ln(
10
2
)
ζ
=
δ
÷
√
4 ×
π
2
+
δ
2
0.05116
=
0.3219
÷
√
4 ×
π
2
+
0.3219
2
c
=
2
×
ζ
×
√
k
×
m
4.093
=
2
×
0.05116
×
√
800 × 2
Assumptions
•
Viscous damping and underdamped (ζ < 1): every crest is the same fraction e−δ of the one before.
•
x₀ and xₙ are crests on the same side, n whole cycles apart.
•
c = ζ × ccr with ccr = 2√(km), for the mass and spring that made the record.
Step-by-step
WE KNOW
x₀ = 10 mm, xₙ = 2 mm, n = 5, m = 2 kg, k = 800 N/m
FIND
logarithmic decrement (δ), damping ratio (ζ), damping constant (c)
Step 1 · Find logarithmic decrement (δ)
Use δ = (1 ÷ n) × ln(x₀ ÷ xₙ)
n
cycles
shrink
the
crest
by
e
−
nδ
,
so
ln(
x₀
xₙ
)
=
nδ.
δ
=
(
1
5
)
×
ln(
10
2
)
δ
=
1
5
×
ln(5)
δ
=
1
5
×
1.6094
δ
=
0.2
×
1.6094
δ
=
0.3219
Step 2 · Find damping ratio (ζ)
Use ζ = δ ÷ √(4 × π² + δ²)
δ
=
2πζ
÷
√
1 −
ζ
2
,
solved
for
ζ
:
square
both
sides
and
collect
ζ
2
.
ζ
=
0.3219
÷
√
4 ×
π
2
+
0.3219
2
ζ
=
0.3219
÷
√
4 × 9.87 + 0.1036
ζ
=
0.3219
÷
√
39.48 + 0.1036
ζ
=
0.3219
÷
√
39.5836
ζ
=
0.3219
6.292
ζ
=
0.05116
Step 3 · Find damping constant (c)
Use c = 2 × ζ × √(k × m)
The
damping
is
ζ
times
the
critical
damping
ccr
=
2
√
km
.
c
=
2
×
0.05116
×
√
800 × 2
c
=
2
×
0.05116
×
√
1
,600
c
=
2
×
0.05116
×
40
c
=
0.1023
×
40
c
=
4.093
N·s/
m
Check
Put the numbers back in. Both sides should match.
0.3219
=
(
1
5
)
×
ln(
10
2
)
✓
0.05116
=
0.3219
÷
√
4 ×
π
2
+
0.3219
2
✓
4.093
=
2
×
0.05116
×
√
800 × 2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.