i

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Fig. 16-4. Variation-of-Parameters Formulation Applied to Motion of a Point Mass in One Dimen-

Fig. 16-4. Variation-of-Parameters Formulation Applied to Motion of a Point Mass in One Dimen-

differentiate the backward solution, Eq. (16-90):

Equation (16-88) is then substituted into Eq. (16-91), yielding dx0

Note that the right sides of these equations vanish when F=0 because x0 and v0 are constants in this case. Finally, we substitute the forward equations of motion on the right sides of Eq. (16-92) to eliminate x and v, and obtain the final equations dxn

These equations must be integrated to obtain x^t) and v^t), and then x(t) and v(t) are given by Eq. (16-89).

The equations of motion in the variation-of-parameters approach, Eqs. (16-93), are generally more complicated than the original equations of motion, Eqs. (16-88). They have the advantage, however, that the right sides are smaller than the right sides of the original equations if the forces are small. Thus, larger integration steps can be taken if the equations are integrated numerically (see Section 17.1). Comparison of Eqs. (16-88a) and (16-93a) shows that the variation-of-parameters approach will be useful if

that is, if the impulse of the applied force over the time interval considered is much less than the momentum of the particle.

We now consider the attitude dynamics problem in the axial symmetry case. The parameters to be varied are the initial values of both the Euler symmetric parameters and the components of the angular momentum vector along the body principal áxes. The foward solutions are Eq. (16-86) and

Ll(r)=LOIcos2a + La2sin2a (16-94a)

L2(r)=L02cos2a-L01sin2a (16-94b)

where a is given by Eq. (16-87e). These are obtained by multiplying Eqs. (16-60) by the principal moments of inertia along the three axes. The backward solutions are obtained from Eqs. (16-86) and (16-94) py interchanging L with Lq and q with q^ and changing the sign of t (and thus of a and /?). Differentiating the backward solutions and substituting the forward equations of motion on the right-hand sides, as in the example above, yields the variation-of-parameters equations of motion for the axial symmetry case:

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