SME
Orbital Mechanics
Spacecraft attitude dynamics
Use this for “A spinner with Iₛ = 300 and Iₜ = 200 kg·m² spins at 2 rad/s with a 0.1 rad/s wobble. Find H, the nutation angle and the precession rates.”
Diagram
H
2
=
(Hₛ)
2
+
(Hₜ)
2
:
H
=
√
600
2
+
20
2
=
600.3
kg·
m
2
/s.
Drawn to scale: Hₜ up from the end of Hₛ, H closing the triangle.
This is an example. Type your own values to start a new problem.
Iₛ
Spin-axis inertia
example · kg·m²
Iₜ
Transverse inertia
example · kg·m²
ωₛ
Spin rate
example
ωₜ
Transverse rate
example
Hₛ
Spin momentum
calculated · kg·m²/s
Hₜ
Transverse momentum
calculated · kg·m²/s
H
Angular momentum
calculated · kg·m²/s
θ
Nutation angle
calculated · °
λ
Body precession rate
calculated
ωₚ
Inertial precession rate
calculated
Formulas
Hₛ = Iₛ × ωₛ
600 = 300 × 2
Hₜ = Iₜ × ωₜ
20 = 200 × 0.1
H
=
√
Hₛ
2
+
Hₜ
2
600.3332
=
√
600
2
+
20
2
θ
=
tan⁻¹(
Hₜ
Hₛ
)
1.9092
=
tan⁻¹(
20
600
)
λ
=
(Iₛ
−
Iₜ)
×
ωₛ
Iₜ
+1
=
(300
−
200)
×
2
200
ωₚ
=
H
Iₜ
3.0017
=
600.3332
200
Assumptions
•
Axisymmetric body with no external torque, so the angular momentum H stays fixed in space.
•
λ > 0 (Iₛ > Iₜ, a flat spinner) turns the transverse rate with the spin; λ < 0 against it.
•
With energy loss (flexing, fuel slosh), only a spin about the largest inertia stays stable.
Step-by-step
WE KNOW
Iₛ = 300 kg·m², Iₜ = 200 kg·m², ωₛ = 2 rad/s, ωₜ = 0.1 rad/s
FIND
spin momentum (Hₛ), transverse momentum (Hₜ), angular momentum (H), nutation angle (θ), body precession rate (λ), inertial precession rate (ωₚ)
Step 1 · Find spin momentum (Hₛ)
Use Hₛ = Iₛ × ωₛ
Momentum about the spin axis: its inertia times its rate.
Hₛ = 300 × 2
Hₛ
=
600
kg·
m
2
/s
Step 2 · Find transverse momentum (Hₜ)
Use Hₜ = Iₜ × ωₜ
Momentum across the spin axis: the transverse inertia times the transverse rate.
Hₜ = 200 × 0.1
Hₜ
=
20
kg·
m
2
/s
Step 3 · Find angular momentum (H)
Use H = √(Hₛ² + Hₜ²)
The two parts are at right angles, so Pythagoras adds them.
H
=
√
600
2
+
20
2
H
=
√
360,000 + 400
H
=
√
360
,400
H
=
600.3332
kg·
m
2
/s
Step 4 · Find nutation angle (θ)
Use θ = tan⁻¹(Hₜ ÷ Hₛ)
θ
is
the
angle
from
the
spin
axis
to
H
:
opposite
it
lies
Hₜ,
beside
it
Hₛ.
θ
=
tan⁻¹(
20
600
)
θ
=
1.9092°
Step 5 · Find body precession rate (λ)
Use λ = (Iₛ − Iₜ) × ωₛ ÷ Iₜ
Euler’s equations turn the transverse rate round the spin axis at this rate, seen from the body.
λ
=
(300
−
200)
×
2
200
λ
=
100
×
2
200
λ
=
200
200
λ
=
+1
rad/s
Step 6 · Find inertial precession rate (ωₚ)
Use ωₚ = H ÷ Iₜ
The
spin
axis
circles
the
fixed
H
at
this
rate,
seen
from
space.
ωₚ
=
600.3332
200
ωₚ = 3.0017 rad/s
Check
Put the numbers back in. Both sides should match.
600 = 300 × 2 ✓
20 = 200 × 0.1 ✓
600.3332
=
√
600
2
+
20
2
✓
1.9092
=
tan⁻¹(
20
600
)
✓
+1
=
(300
−
200)
×
2
200
✓
3.0017
=
600.3332
200
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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