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 mass Thursday, April 11, 13 Rigid Body Kinematics v = v +(! r ) A B ⇥ A/B a = a +(↵ r )+(! (! r )) A B ⇥ A/B ⇥ ⇥ A/B Useful Shortcuts for 2D planar motion ↵ = ↵k • ! = !k r = rxi + ryj ! r = r !i + r !j ⇥ − y x ↵ r = r ↵i + r ↵j ⇥ − y x ! (! r)= r !2i r !2j ⇥ ⇥ − x − y Thursday, April 11, 13 Rigid Body Dynamics Linear Motion: sum of the forces is the d(mv) F = ma = time rate of change of dt linear momentum Works for particles - and also works for rigid bodies if the acceleration is at the center of mass! F = maG Thursday, April 11, 13 Rigid Body Dynamics Rotational Motion: d(H ) sum of the moments is M = G = H˙ G dt G the time rate of change of angular momentum about the center of mass Thursday, April 11, 13 Rigid Body Dynamics Rotational Motion: sum of the moments is d(HG) MG = = IG↵ the time rate of change dt of angular momentum Mass Moment of Inertia: 2 2 IG = (miri )= (⇢r )dV i V X Z “Rotational Inertia” Resistance to angular acceleration Thursday, April 11, 13 Mass Moment of Inertia L m m Two spherical masses connected by a massless bar thin plate, height b, solid cylinder z length c, rotating length L radius R, around y-axis rotating around z- y axis Thursday, April 11, 13 Thursday, April 11, 13 Thursday, April 11, 13 Parallel Axis Theorem G O object rotating around axis d “O” NOT through the CG Find parallel axis through CG 2 IO = IG + md IG can read from tables Thursday, April 11, 13 Composite Bodies O d1 1 2 3 d2 d2 2 2 2 IO = IG1 + m1d1 + IG2 + m2d2 + IG3 + m3d3 IG can read from tables Thursday, April 11, 13 rigid body dynamics problems: 2D planar motion • Free Body Diagram! • 3 equations of motion: Fx = max Fy = may • problem constraints Mz = I↵ • mass moment of inertia calculation • can we solve? if not, need more eqns: • kinematics equations: connection between ↵, ! AND v, a Thursday, April 11, 13 refer to 2012 HW#7 for more details and solution Thursday, April 11, 13.