Propulsive Effects of a Rotating Mass 



^;h U 



5 □ 



7^ 



?^i< 



1.^ 



M 5" 



P^h) 



Fig. 4 - Arrangement 

 of the device in Fig. 3 

 to include one fixed and 

 two mobile toothed 

 wheels 



Fig. 5 - Arrangement 

 of the device in Fig. 3 

 so that the two propul- 

 sive effects atone shaft 

 turn are both symmet- 

 rical to the y axis 



toothed wheels have a negligible weight in respect to the weight of the mass m , 

 and finally that the passive resistances are null. With these simplifications, we 

 can write equations for the motion of the mass and obtain useful results. 



Let us begin with the case of the basic device indicated in Fig. 6, which, as 

 has already been mentioned, executes a rotatory motion of a mass around an 

 axis, with the latter rotating in turn around another axis. Let us refer the mo- 

 tion of the mass to the system of axes o, x, y, and z as previously indicated. 

 Let us assume the point Pq as the origin of the motion on the xy plane cor- 

 responding to the angle of rotation 9=0. At time t the two arms are turned 

 by ; therefore, from OAqPq we pass to OAF . If P' is the projection of P on the 

 plane zOx, we have ap' = ap sin ^ = r sin . The coordinates of P then are: 



= AC - AB 



- R cos 



AP' 



r cos = R sin^ - cos 



(1) 



z :^ DC + CP' = OA sin 5 + AP' cos ^ (R cos 9 + r) sine* . 



These expressions represent a trajectory whose projections on the three 

 coordinate planes have the forms indicated in Fig. 7. In Pj we have a check- 

 point. If the angular speed and acceleration are indicated with 9 = co and 

 = e , the components of the speed and those of the acceleration assume the 

 form 



v = (R sin 20 + r sin0)aj 



V = R sin0co (2) 



V = (R cos 2 <9 + r cos 9) co , 



1377 



