Post on 01-Aug-2020
Rigid Body Motion Outline
• Kinematics - describes motion (position, vel, accel) without regard for the forces that cause it (hw7 due 4/12)
• Dynamics - describes the causes of motion (forces!) and how they influence the motion (hw8 due 4/26)
• 2 Projects:• Trifilar Pendulum, vibrations and rigid body motion (4/19)
• Solar Car, rigid body motion and power transmission (5/7)
1Friday, April 5, 13
Rigid Bodies• BEFORE: points/particles - no rotation, just translation
• RIGID BODIES: arbitrary shaped bodies that do not deform - must consider translation and rotation
2Friday, April 5, 13
x
y
x
yd2
d2d1
d1
Rigid Bodies• Between any 2 points on the body, the distance is fixed:
springs: not rigid
3Friday, April 5, 13
2D Kinematics: Rotation about a fixed point
i
j
counter-clockwise rotation about “O”
O
Pfind motion of “P”
(velocity and acceleration)
7Friday, April 5, 13
i
j
counter-clockwise rotation about “O”
O
Pfind motion of “P”
(velocity and acceleration)
rp
vp
rp = r(cos✓i + sin✓j)
vp = r✓̇(�sin✓i + cos✓j)
rp · vp = 0
Rotation about a fixed point
r = distance b/t 2 points = constant8Friday, April 5, 13
i
j
counter-clockwise rotation about “O”
O
Pfind motion of “P”
(velocity and acceleration)
rp
vp
rp = r(cos✓i + sin✓j)
vp = r✓̇(�sin✓i + cos✓j)
Rotation about a fixed point
rp · vp = 0apt
apn
ap = r✓̈(�sin✓i + cos✓j)� r✓̇
2(cos✓i + sin✓j)
9Friday, April 5, 13
i
j
counter-clockwise rotation about “O”
O
Pfind motion of “P”
(velocity and acceleration)
rp
vp
rp = r(cos✓i + sin✓j)
vp = r✓̇(�sin✓i + cos✓j)
Rotation about a fixed point
rp · vp = 0apt
apn
ap = r✓̈(�sin✓i + cos✓j)� r✓̇
2(cos✓i + sin✓j)
|v| = r✓̇ = r!
|at| = r✓̈ = r↵
|an| = r✓̇2 = r!2 = |v|/2
10Friday, April 5, 13
Example 1: 2 gears
rArB
!A !B↵B
↵A
what are !B and ↵B?!A and ↵A are prescribed
11Friday, April 5, 13
Example 1: 2 gears
rArB
!A !B↵B
↵A
what are !B and ↵B?!A and ↵A are prescribed
|vp| = rA!A = rB!B
!B = (rA/rB)!A
↵B = (rA/rB)↵A
Solution: take velocity at point P where gears intersect:
P
12Friday, April 5, 13
What if you have multiple gears?
P Q
rA rBrC
!B = (rA/rB)!A
!C = (rB/rC)!B
!C = (rB/rC)(rA/rB)!A
!C = (rA/rC)!A
velocity at point P:velocity at point Q:
13Friday, April 5, 13
Traditional 3-gear Manual Transmission
http://www.youtube.com/watch?v=JOLtS4VUcvQ
Spinning Levers (1936)
14Friday, April 5, 13
http://eahart.com/prius/psd/
Prius Power Split Device (PSD)
ICE: Internal Combustion Engine (gasoline)MG1: Electric Motor/Generator 1MG2: Electric Motor/Generator 2
http://eahart.com/prius/psd/
MG2 powers outer ring: connected to wheels
ICE powers plate of “planetary” gears
MG1 powers “sun” gear
15Friday, April 5, 13
Fixed Point Rotationin Vector Form
i
j
O
P
k
r
r = vector from axis of rotation to point Pv
ω, α = angular velocity and accel vectorsω, α always point along axis of rotation
RHR: curl fingers in direction of rotation, thumb points in direction of ω
ω
16Friday, April 5, 13
Fixed Point Rotationin Vector Form
i
j
O
P
k
r
v ωv = ! ⇥ ra = ↵⇥ r + ! ⇥ (! ⇥ r)
ω, α are the same at all points on a rigid body
a, v vary at each point and depend on distance r from rotation axis
tangential normal
17Friday, April 5, 13
Example 2: simple rotation in vector form
B
A
0.4 m
0.3 m
Rotates clockwise about B at an angular velocity decreasing at a rate of 4 rad/sec.
Find v and a at A at the instant when ω=2 rad/sec
18Friday, April 5, 13
General Kinematics: 2D planar motion
motion is decomposed into translation and rotation
x
y
19Friday, April 5, 13
Example 2: simple rotation in vector form
B
A
0.4 m
0.3 m
Rotates clockwise about B at an angular velocity decreasing at a rate of 4 rad/sec.
Find v and a at A at the instant when ω=2 rad/sec
given: ω = -2 k, α = 4 k, r = 0.4i + 0.3j
ij
draw coordinate system aligned with body
solve: v = ω x r
a = ω x (ω x r) = ω x v
20Friday, April 5, 13
j
O
k
i
rB
rA
rA/B AB
B
ArA/B
✓
rA/B = |rA/B |(cos✓i + sin✓j)
rA = rA/B + rB
rA = rB + |rA/B |(cos✓i + sin✓j)vA = vB + |rA/B |(�✓̇sin✓i + ✓̇cos✓j)
translation of B + rotation of A about point B
rA/B=location of A relative to B
21Friday, April 5, 13
aA = aB + (↵⇥ rA/B) + (! ⇥ (! ⇥ rA/B))
vA = vB + (! ⇥ rA/B)
kinematics of point A:
depends on kinematics of B, plus rotational velocity and
acceleration
j
O i
rB
rA
rA/B AB
B
ArA/B
✓
22Friday, April 5, 13