SME
Flight Mechanics, Stability & Control · Problem type
Damping and period from an eigenvalue
Use this for “A mode’s eigenvalues are λ = −0.5 ± 2i per second. Find ωₙ, ζ, the period and the time to half amplitude.”
Diagram
λ = −0.5 ± 2i: a conjugate pair.
|λ|
=
√
0.5
2
+
2
2
=
2.062
=
ωₙ.
This is an example. Type your own values to start a new problem.
λ
=
n
±
ω
i
ωₙ
Natural frequency
calculated · rad/s
ζ
Damping ratio
calculated
T
Period
calculated
t½
Time to halve (n < 0) or double (n > 0)
calculated
Formulas
ωₙ
=
√
n
2
+
ω
2
2.062
=
√(
(−0.5)
2
+
2
2
)
ζ
=
−
n
ωₙ
+0.2425
=
−
−0.5
2.062
T
=
2π
ω
3.142
=
2π
2
t
1
2
=
ln(2)
÷
|
n
|
1.386 = ln(2) ÷ |−0.5|
Assumptions
•
The mode’s eigenvalues are a complex pair λ = n ± iω; the motion is entcos(ωt).
•
n < 0 decays (stable); n > 0 grows, and t½ becomes the time to double.
•
ζ is the cosine of the root’s angle from the negative real axis.
Step-by-step
WE KNOW
n = −0.5 rad/s, ω = 2 rad/s
FIND
natural frequency (ωₙ), damping ratio (ζ), period (T), time to halve (n < 0) or double (n > 0) (t½)
Step 1 · Find natural frequency (ωₙ)
Use ωₙ = √(n² + ω²)
ωₙ is the root’s distance from the origin of the complex plane.
ωₙ
=
√(
(−0.5)
2
+
2
2
)
ωₙ
=
√
0.25 + 4
ωₙ
=
√
4.25
ωₙ = 2.062 rad/s
Step 2 · Find damping ratio (ζ)
Use ζ = −n ÷ ωₙ
The decay rate as a share of ωₙ: the cosine of the root’s angle from the negative real axis.
ζ
=
−
−0.5
2.062
ζ
=
+0.2425
Step 3 · Find period (T)
Use T = 2π ÷ ω
The imaginary part sets the swing: 2π radians.
T
=
2π
2
T
=
3.142
s
Step 4 · Find time to halve (n < 0) or double (n > 0): t½
Use t½ = ln(2) ÷ |n|
The
envelope
e
nt
halves
(or
doubles)
when
|
n
|t
=
ln
2.
t
1
2
=
ln(2)
÷
|−0.5|
t
1
2
=
0.6931
0.5
t
1
2
=
1.386
s
Check
Put the numbers back in. Both sides should match.
2.062
=
√(
(−0.5)
2
+
2
2
)
✓
+0.2425
=
−
−0.5
2.0616
✓
3.142
=
2π
2
✓
1.386 = ln(2) ÷ |−0.5| ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.