SME
Orbital Mechanics · Problem type
Reaction-wheel slew time
Use this for “A reaction wheel gives 0.1 N·m to a craft with I = 800 kg·m². How long does a 30° rest-to-rest slew take, and what momentum must the wheel hold?”
Chart
Shaded: ∫ from 0 to 129.4 of ω(t) dt = 0.5236
This is an example. Type your own values to start a new problem.
θrad (rad)
=
θ (°)
×
π
÷
180
t (s)
=
2
×
θrad (rad)
×
I (kg·m²)
τ (N·m)
ωₘₐₓ
Peak rate
calculated
Hw
Wheel momentum
calculated · kg·m²/s
Formulas
θrad
=
θ
×
π
180
0.5236
=
30
×
π
180
t
=
2
×
√(θrad
×
I
τ
)
129.4417
=
2
×
√(0.5236
×
800
0.1
)
ωₘₐₓ
=
τ
×
t
2 ×
I
0.00809
=
0.1
×
129.4417
2 × 800
θrad
=
0.5
×
ωₘₐₓ
×
t
0.5236 = 0.5 × 0.00809 × 129.4417
Hw
=
τ
×
t
2
6.4721
=
0.1
×
129.4417
2
Assumptions
•
Rest to rest about one principal axis, with no other torque on the craft.
•
Full torque one way for half the time, then the other way (bang-bang): the fastest slew.
•
The wheel must store Hw at the halfway point without saturating.
Step-by-step
WE KNOW
I = 800 kg·m², τ = 0.1 N·m, θ = 30°
FIND
angle in radians (θrad), slew time (t), peak rate (ωₘₐₓ), wheel momentum (Hw)
Step 1 · Find angle in radians (θrad)
Use θrad = θ × π ÷ 180
The motion rules work in radians: π radians make 180°.
θrad
=
30
×
π
180
θrad = 0.5235988 rad
Step 2 · Find slew time (t)
Use t = 2 × √(θrad × I ÷ τ)
Each
half
turns
θ
2
=
1
2
(
τ
I
)
(
t
2
)
2
from
rest,
so
t
2
=
4
θ
I
τ
.
t
=
2
×
√(0.5235988
×
800
0.1
)
t
=
2
×
√(
418.9
0.1
)
t
=
2
×
√
4
,189
t
=
2
×
64.72
t
=
129.4417
s
Step 3 · Find peak rate (ωₘₐₓ)
Use ωₘₐₓ = τ × t ÷ (2 × I)
The
craft
speeds
up
at
τ
I
for
half
the
slew
time,
so
the
peak
is
at
the
halfway
point.
ωₘₐₓ
=
0.1
×
129.4417
2 × 800
ωₘₐₓ
=
0.1
×
129.4417
1,600
ωₘₐₓ
=
12.944
1,600
ωₘₐₓ = 0.00809 rad/s
Step 4 · Find wheel momentum (Hw)
Use Hw = τ × t ÷ 2
The wheel takes up the momentum the craft gains: the torque times the half time.
Hw
=
0.1
×
129.4417
2
Hw
=
12.944
2
Hw
=
6.4721
kg·
m
2
/s
Check
Put the numbers back in. Both sides should match.
0.5235988
=
30
×
π
180
✓
129.4417
=
2
×
√(0.5235988
×
800
0.1
)
✓
0.00809
=
0.1
×
129.4417
2 × 800
✓
0.5235988 = 0.5 × 0.008090108 × 129.4417275 ✓
6.4721
=
0.1
×
129.4417
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.