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