### Advance Dynamics: Modeling and Analysis by A. Frank D'Souza By A. Frank D'Souza

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Case 2: e = I, E = 0. 79), we get the lowest energy. 79) is conserved. A point of the orbit where drfdO 0 is called an apsis. For an open orbit such as a hyperbola or parabola, there is only one apsis but for an ellipse there exist two apsides. The shortest distance from the force center to one of the apsides is called the pericentron and the longer one is called the apocentron. 77). Letting v. Oz - G(m, ~rpfl + mz) 0 rP rpO in the foregoing equation, it is seen that v. scape velocity, is the minimum velocity for which an open orbit is obtained.

_ (ro dt dt = ~. + - X ;) _ dro X r + ro dr X dt = ~. 65) as ~ ~. :.. >. ;:.. II. lo. - X (Wn;F X X (WB/F X ;) Hence, it follows that a ~. 7) by letting point. 65) by replacing a 1 by a. and\ letting r r 0. The reason for selecting a body coordinate system is that the inertia matrix. lnsome cases, a rigid· body possesses axial symmetry with the result that two of its principal moments of inertia are equal. In such cases, it is possible to choose a that is different from the coordinate system which has an angular velocity angular velocity of the body, and yet the inertia matrix remains constant with respect to such axes as discussed later.

28) where, as shown in Chapter 2, the rotation transformation matrix [C) is made up of direction cosines between the x 1 y 1 z 1 and xyz axes. 24) is the angular momentum of the body about the center of mass. 26) In order to determine the principal axes and the principal moments of inertia, we consider the rotational transformation of the coordinate system. Consider two systems of coordinates xyz and x 1 y 1z 1 which have the same common origin 0 but are rotated with respect to each other as shown in Fig. 