THE REMAINDER OF h*

The remainder of h* ordinarily comes from the motion of hinged parts and from elastic deformation, although there are other kinds of possible relative motion, such as fuel sloshing which is important in liquid-fueled rockets (ref. 5.14). This total remainder is denoted by he. We now show that it is possible always to choose a set of body-axes FB for which he vanishes. These are termed “mean axes” by Milne (ref. 5.2).

Consider two centroidal reference frames FBi and FB^ for which the angular momenta are

hBl = RBlR£l dm + SBl(aBi+ 2 hr‘Bi (a)

*’ (5.4,25)

Ьд2 = J dm + + Ao>)Ba + 2 hr*B2 (6)

Here the summations are the contributions of spinning rotors, R in the inte­grals represents the residual relative motion, and Ato is the angular velocity of FB relative to FBl. The first term of (5.4,256) can be transformed as

Подпись: [Єв2 — J THE REMAINDER OF h*

140 Dynamics of atmospheric flight follows:

THE REMAINDER OF h*H /gil dm “bj 11 /;, Ato^R^ dm Applying (5.4,17) to the last term, we get

he_B; = l^Bjj RbjRbj dm + в у AwjBjj (5.4,26)

It follows that the angular momentum b. eBi of the distortional relative motion vanishes in FB^ if

j dm + -&bx AtoBi = 0

or if

Awjjj = —SBfdm (5.4,27)

Equation (5.4,27) provides the condition that the axis system FB must satisfy if the angular momentum hBj referred to it is to have the form

К=-*1^в + 1Ъив (6.4,28)

І

This condition will be met when FB has the orientation required by TjBiBJt) that satisfies the differential equation [see (4.6,6)]

(5.4,29)

It is not necessary actually to solve (5.4,27 and 29) for LgiBa in order to make use of mean axes. Our concern here is simply to establish their ex­istence. We note that when the body axes are mean axes, the following relations must hold for the distortional motion. Since the origin is the mass center,

J* x’ dm —jy1 dm = J"s’ dm = 0 (a)

and from (5.4,23) (5.4,30)

J” (yz’ — y’z) dm = j (zx! — z’x) dm = J* (xy’ — xy) dm = 0 (6)

in which the prime denotes the distortional component of the velocity relative to FB. The use of mean axes, and the consequent elimination of distortional contributions to h* has the effect of eliminating the main inertial

coupling between the distortional degrees of freedom and those of the rigid body. Some coupling still remains through jP however, see (5.6,7).

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