Space Station Precession Problem

The space station precessesabout ny at a rate of one

revolution per hour.

Determine the spin rate about the space station’s

axis of symmetry.

Draw space and body cones and give numerical

values for θ and γ.

R30 °L

nx

nynz

Problem 6.5.3 Solution

R

30°L

nx

ny

nz

The space station can be modeled as a cylinder of:

height h = 30 m

radius R = 3 m (of the cross-section) and

mass = 1000 kg.

r

axial

transverse

h

( )22

transverse

2

axial

312

mI

mr2

1I

hr +=

=

Cylinder

r

axial

transverse

h

( )22

transverse

2

axial

312

mI

mr2

1I

hr +=

=

CylinderMass Properties of Cylinder:

Moments of Inertia of the cylinder about the center of mass:

Iaxial = C = ½ (1000 kg) (3 m)2

= 4500 kg m2

Itransverse = A = (1000 kg) { 3 (3 m)2

+ (30m)2 }

= 77250 kg m2

12

1

Moments of Inertia of the cylinder about the center of mass:

Iaxial = C = ½ (1000 kg) (3 m)2

= 4500 kg m2

Itransverse = A = (1000 kg) { 3 (3 m)2

+ (30m)2 }

= 77250 kg m2

12

1

Solution

θ = 30°

tan γ = tan θ = tan 30°= 0.0336

γ = 1.93°

77250

4500

A

C

γ

γ − θθ

γ

h

ωωωω

φ&

φ&

ψ&

θ − γ

γ

γ − θθ

γ

h

ωωωω

φ&

φ&

ψ&

θ − γ

( ) ( ) ( )γθγθ −==

−° sinφ

sinψ

180sin&&ω

hour

rev14 φ

hour

rev 1 ψ ==>= &&

( ) ( ) ( )

( ) ( ) ( )°−°=

°=

°−°

−==

−°

1.9330sinφ

1.93sinψ

30180sin

sinφ

sinψ

180sin

&&

&&

ω

ω

γθγθ