Lecture 3: rigid body dynamics - Brown University

1 Lecture 3: rigid body dynamics kinematics example: rolling no slip rotational equation of motion mass moment of inertia solving rigid body dynamics problems dynamics example: pulley with massThursday, April 11, 13 rigid body Kinematics Useful Shortcuts for 2D planar motionaA=aB+( rA/B)+(! (! rA/B))vA=vB+(! rA/B) = k!=!kr=rxi+ryj! r= ry!i+rx!j r= ry i+rx j! (! r)= rx!2i ry!2jThursday, April 11, 13 rigid body DynamicsF=ma=d(mv)dtLinear Motion:sum of the forces is the time rate of change of linear momentumWorks for particles - and also works for rigid bodies if the acceleration is at the center of mass!

2 F=maGThursday, April 11, 13 rigid body DynamicsRotational Motion:sum of the moments is the time rate of change of angular momentumMG=d(HG)dt= HGabout the center of massThursday, April 11, 13 rigid body DynamicsRotational Motion:sum of the moments is the time rate of change of angular momentumMG=d(HG)dt=IG Mass Moment of Inertia: Rotational Inertia Resistance to angular accelerationIG=Xi(mir2i)=ZV( r2)dVThursday, April 11, 13 Mass Moment of InertiaLmTwo spherical masses connected by a massless barmzsolid cylinder length L radius R, rotating around z-axisthin plate, height b, length c, rotating around y-axisyThursday, April 11, 13 Thursday, April 11, 13 Thursday, April 11, 13 Parallel Axis TheoremGOdIO=IG+md2object rotating around axis O NOT through the CGFind parallel axis through CGIG canreadfromtablesThursday, April 11, 13 Composite BodiesOd1d2d2123IO=IG1+m1d21+IG2+m2d22+I G3+m3d23 IGcanreadfromtablesThursday, April 11, 13rigid body dynamics problems.

3 2D planar motion Free body Diagram! 3 equations of motion: problem constraints mass moment of inertia calculation can we solve? if not, need more eqns: kinematics equations: connection between Fx=maxFy=mayMz=I ,!ANDv,aThursday, April 11, 13refer to 2012 HW#7 for more details and solutionThursday, April 11, 13