8/12/2019 Lecture1 Jen RigidBodies

1/19

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!) andhow 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

8/12/2019 Lecture1 Jen RigidBodies

2/19

Rigid Bodies

BEFORE: points/particles - no rotation, just translation

RIGID BODIES: arbitrary shaped bodies that do notdeform - must consider translation and rotation

2Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

3/19

x

y

x

y d2

d2d1

d1

Rigid Bodies

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

springs: notrigid

3Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

4/19

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

8/12/2019 Lecture1 Jen RigidBodies

5/19

i

j

counter-clockwise

rotation about O

O

Pfind motion of P

(velocity and

acceleration)

rp

vp

rp= r(cosi+ sinj)

vp= r(sini+ cosj)

rp vp = 0

Rotation about a fixed point

r = distance b/t 2 points = constant8Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

6/19

i

j

counter-clockwise

rotation about O

O

Pfind motion of P

(velocity and

acceleration)

rp

vp

rp= r(cosi+ sinj)

vp= r(sini+ cosj)

Rotation about a fixed point

rp vp = 0

apt

apn

ap=r

(sin

i+cos

j) r2

(cos

i+sin

j)9Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

7/19

i

j

counter-clockwise

rotation about O

O

Pfind motion of P

(velocity and

acceleration)

rp

vp

rp= r(cosi+ sinj)

vp= r(sini+ cosj)

Rotation about a fixed point

rp vp = 0

apt

apn

ap=r

(sin

i+cos

j) r2

(cos

i+sin

j)

|v| =r =r

|at| =r =r

|an| =r 2=r2 =|v|/2

10Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

8/19

Example 1: 2 gears

rA

rB

A B

B

A

what are B and B?

A an A are prescri e

11Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

9/19

Example 1: 2 gears

rA

rB

A B

B

A

what are B and B?

A an A are prescri e

|vp|=rAA=rBB

B = (rA/rB)A

B = (rA/rB)A

Solution: take velocity at point P where gears

intersect:

P

12Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

10/19

What if you have multiple gears?

PQ

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

8/12/2019 Lecture1 Jen RigidBodies

11/19

Traditional 3-gear Manual Transmission

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

Spinning Levers (1936)

14Friday, April 5, 13

http://www.youtube.com/watch?v=JOLtS4VUcvQhttp://www.youtube.com/watch?v=JOLtS4VUcvQ

8/12/2019 Lecture1 Jen RigidBodies

12/19

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 towheels

ICE powers plate

of planetary

gears

MG1powers

sun gear

15Friday, April 5, 13

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

8/12/2019 Lecture1 Jen RigidBodies

13/19

Fixed Point Rotation

in Vector Form

i

j

O

P

k

r

r = vector from axis of rotation to point P

v

",#= angular velocity and accel vectors

",#always point along axis of rotation

RHR: curl fingers in direction of rotation,

thumb points in direction of "

"

16Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

14/19

Fixed Point Rotation

in Vector Form

i

j

O

P

k

r

v "

v = r

a= r+ ( r)

",#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

8/12/2019 Lecture1 Jen RigidBodies

15/19

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 theinstant when "=2 rad/sec

18Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

16/19

General Kinematics:2D planar motion

motion is decomposed intotranslation and rotation

x

y

19Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

17/19

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 theinstant when "=2 rad/sec

given: "= -2 k, #= 4 k, r = 0.4i + 0.3j

i

j

draw coordinate system aligned with body

solve: v = "x r

a ="

x ("

x r) ="

x v20Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

18/19

j

O

k

i

rB

rA

rA/BA

B

ArA/B

rA/B =|rA/B|(cosi+ sinj)

rA = rA/B + rB

rA = rB+|rA/B|(cosi+ sinj)vA =

vB+|

rA/B|(

sini+

cosj)

translation of B + rotation of A

about point B

rA/B=location of A relative to B

21Friday, April 5, 13

8/12/2019 Lecture1 Jen RigidBodies

19/19

a

A =a

B+ ( r

A/B) + (

( r

A/B))

vA = vB+ ( rA/B)

kinematics of point A:

depends on kinematics of B, plusrotational velocity and

acceleration

j

O i

rB

rA

rA/BAB

B

ArA/B