Top

You may have played in your childhood with such a cone-shaped object and kept it spinning by lashing at it with the end of a whip. It made marvelous jumps, seemingly defying the law of gravity, as long as it spun fast enough. Now you realize that it is its angular momentum which stabilizes it.

Euler’s law governs the dynamics of the top. We derive its specialized form by considering the reference point R a point of body B, which implies that for any particle i, DBs— 0. Furthermore, R is also the reference point 7, thus Sr/ — sи — 0. Starting with the terms of Eq. (6.35), we modify them like before, except this time we cannot take advantage of the simplifications brought about by the c. m:

Term (1):

D’faSiRD’siR) = DI[miSu(DBsiI + 12B/s,,)]

/ І

= J2 D‘(mi Si, nBISu) = D1 (if u)BI)

І

Term (2):

YiDI(miSiaDIsRI)= 0

І

because sRi = s/j = 0.

Term (3):

J^D^miSaD’sR,) = £ D^S,^) = 0

І І

because v‘R = 0.

Term (4):

^D/(miSwD/s«)= 0

І

because Sw = S// = 0.

Term (5):

X>*/i) = £>,//,) = /я/

2 І

Term (6):

= «и/ = 0

І

because Srj = Sn = 0.

Only Terms (1) and (5) remain. Euler’s law for a body spinning about a fixed point / is

DI(lfa>BI)=mI (6.47)

Compare both formulations, Eqs. (6.36) and (6.47). They are distinguishable only by the reference points. In both cases, whether it is the c. m. or a body/inertial reference point, Euler’s law assumes the same simple form.

Example 6.10 Force-Free Top

A moment free symmetric body spins about its minor principal MOI axis and is supported at the bottom of its spin axis. Its MOI in body coordinates is

h oo 0 /2 0 0 0 h

where the minor MOI is in the first direction and the two others are equal. The angular velocity of the top is a constant p0. Its attitude equations are derived from Eq. (6.47) by transforming the rotational derivative to the body frame В and expressing the terms in body coordinates }B

~dco‘

dt

With the angular velocity [a)BI]B = [po q r] the equations are in body coordinates

1 о о		"о"		I V 1 0 0 _		1 О 0 ^Г 1 _		Ро		"о~
о о		<7	+	"5 0 1 о		о о		q		0
1 о о		г		1 1 о О I__		1 о о to4 1		г		0

and evaluated

hq – (h ~ h)por = 0

Ы + (h – h)poq = 0

These are two coupled linear differential equations with pitch rate q and yaw rate r as state variables. The terms (h — h)Por and (h — h)Poq model the gyroscopic coupling between the pitch and yaw axes. You should be able to verify the oscillatory solution

h ~ h

q = Ao sinfiwot), r = Aq cos (coot) with o)q =———— po

h

Ao depends on the initial conditions.