1 Two equal masses m are fastened to the ends of a straight

1. Two equal masses m are fastened to the ends of a straight rod of length 2l and of negligible weight. The rod is attached to the center of a vertical shaft of length L at an angle a as shown. a) (15 pts) If the vertical shaft rotates at a constant angular velocity du, find the dynamic reactions at the bearings. b) (5 pts) The system is to be dynamically balanced by the addition of two concentrated masses m1. These masses are to be located in planes at a distance a from the bearings (see picture). Show where these masses should be attached and find the radius at which they should be located

Solution

Notice that when w=w_c=p. w^{2less than or equal to 0}thencostheta 2=p²/w² this requires p/w less than 1 and relative equilibrium rate theta 2=Cos^-1 (p²/w^{2) regardless of} the massoffirst the rod in infinnestinally stable w is greaterthand p= w_{c. The small amplitue vibrationot frequency of the rod about the displaced relative equilibrium. State is Wd=W (1-p}⁴/w4)^1/2. . Sum if w less thand we the vertical relative equilibrium popinion of the rod is it\'s only infinitesimal lyrics stable equilibrium state . When the vertical shafts attains it\'s critical angular speed w=w_{c . When w is greater than Wc the displaced configuration of the rod at theta 2 = Cos^-1}(p²/w²) is it\'s only infinetsimally stble reative equillibium position.