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# hw08sols - AAE 250 Homework 8 Solutions...

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Unformatted text preview: AAE 250 Homework 8 Solutions http://nsgsun.aae.uiuc.edu/AAE250 1. Solution (a) The constraint equation is simply x 2 + y 2 + z 2 = b 2 This constraint expresses the fact that the mass is located at a distance b from the main satellite. (b) The constraint is holonomic, since it can be expressed in the form φ ( q 1 ,q 2 ,q 3 ,t ) = 0 , where q 1 = x, q 2 = y, and q 3 = z are the three coordinates, and φ is given by φ ( q 1 ,q 2 ,q 3 ,t ) = x 2 + y 2 + z 2- b 2 The system is actually rheonomic. Greenwood says that a scleronomic sys- tem satis es the conditions that (1) none of the constraint equations contain explicit functions of time, and (2) the transformation equations (6-1) express the Cartesian x 's as functions of the generalized coordinates ( q 's) only, not time. Obviously the constraint does not involve time, but the transformation equations do. The transformation equations express the relationsship between the generalized coordinates (which in our case are ( x,y,z ) , the coordinates relative to the moving coordinate system Oxyz ) and Cartesian coordinates, attached to an inertial frame. Since the coordinate system Oxyz is moving with respect to an inertial frame with a velocity which is a known function of time, the transformation equations between the Cartesian and generalized coordinates will be, in general time-dependent....
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