rigid body motion outline - brown university · rigid body motion outline • kinematics -...

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


x

y

x

yd2

d2d1

d1

Rigid Bodies• Between any 2 points on the body, the distance is fixed:

springs: not rigid


2D Kinematics: Rotation about a fixed point

i

j

counter-clockwise rotation about “O”

O

Pfind motion of “P”

(velocity and acceleration)


i

j


O



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


O



rp

vp




rp · vp = 0apt

apn

ap = r✓̈(�sin✓i + cos✓j)� r✓̇

2(cos✓i + sin✓j)


i

j


O



rp

vp




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


Example 1: 2 gears

rArB

!A !B↵B

↵A

what are !B and ↵B?!A and ↵A are prescribed


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


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:


Traditional 3-gear Manual Transmission

http://www.youtube.com/watch?v=JOLtS4VUcvQ

Spinning Levers (1936)




http://eahart.com/prius/psd/

Prius Power Split Device (PSD)

ICE: Internal Combustion Engine (gasoline)MG1: Electric Motor/Generator 1MG2: Electric Motor/Generator 2


MG2 powers outer ring: connected to wheels

ICE powers plate of “planetary” gears

MG1 powers “sun” gear




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


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


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


General Kinematics: 2D planar motion

motion is decomposed into translation and rotation

x

y


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


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


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

✓


rigid body motion outline - brown university · rigid body motion outline • kinematics -...

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