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MEAM 211

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Kinematics of Planar Rigid Bodies

Chapter 6

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Planar motion

C

D

A

BA

B

Does the coupler/car rotate? Translate?

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Planar Rigid Body Motion

For any two points (say A, B) fixed to a rigid body

Rigid body constraint

Position of A and B:

Velocity of A and B:

( )AB

ABABAB dt

ddt

d vvrrrv −=−

== //

1

1

Fig 6.6 needs to be corrected

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1

1

Expression for vB - vA

For any two points (say A, B) fixed to a rigid body

( )AB

ABABAB dt

ddt

d vvrrrv −=−

== //

θ

( )jir θ+θ= sincos// ABAB r

( )dtd

⎟⎠⎞

⎜⎝⎛ θ+θ= jir sincos// dt

ddtdr

dtd

ABAB

constant ( )θθ+θ−= &jiv cossin// ABAB r

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1

1

Definition of Angular Velocity

θ

( )θθ+θ−= &jiv cossin// ABAB r

( )jikv θ+θ×θ= sincos// ABAB r&

Can rewrite as

kθ=ω &

Define angular velocity, w

ABAB // rv ×= ω

So the relative velocity for points A, B

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Kinematics of Planar Rigid Bodies: Key Fact!Relative velocity between anytwo points fixed on any rigid body, vQ/P

A

BP

QrQ/P

b1

b2

θP

PP Q

BQQ dt

d/

//

rr

v ×ω==

body, B

angular velocity of the rigid body

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piston (slider)

frame

θ2

x

y

θ3

r1

r2

r3

r4 θ4O

P

Q

R

Slider Crank Linkage Velocity AnalysisBefore, by solving velocity equations

Alternative method: solve by writing vector equations representing rigid body constraints

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Example

piston (slider)

frame

θ2

x

y

θ3

r1

r2

r3

r4 θ4O

P

Q

R

OQOQ /2/ rv ×ω=

Given crank angular velocity, ω2, solve for piston velocity, vP

vP in this directionQPQP /3/ rv ×ω=

QPQP /vvv +=Magnitude of ω3 unknownbut direction is known

Magnitude of ω3, vP unknown

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Example

θ2

x

y

θ3

r1

r2

r3

r4

O P

Q

Given crank angular velocity, ω2 = 1 rad/s, solve for piston velocity, vP

=4.0 =6.95

=60 deg

θ3 = -30 deg

Solve closure equations to get:

r1 = 8.02

All dimensions in cm.

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Examples: Transmissions

Gears Spur gearsHelical gearsHypoid gears

Gear reductionsGear trainsWormPlanetaryHarmonic

Chain & Chain Drives

Decrease (increase) speeds

Need gears/transmissions to:

Increase (decrease) torques

Transmissionτ1 , ω1

τ2 , ω2

input

output

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Spur and Helical Gears

Spur gear Loud: Each time a gear tooth engages a tooth on the other gear, the teeth collide, and this impact makes a noise Wear and tear

Helical gearsContact starts with point contact to line contact

Crossed helical gearsShaft angles need not be parallel

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Rolling Contact

Contact pointsP1 and P2, coincident instantaneously

C

A

B

P1

P2

n

contactnormal

O

r1

r2

dtd

P1

1

rv =

dtd

P2

2

rv =

Body A rolls on body B vP1 = vP2

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Modeling of Gears

Pitch circles

The kinematics of rotation of a pair of meshing gears can be modeled as a rotation of the corresponding pitch circles.

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Rack and PinionSimilar to a wheel on a ground with friction

But positive engagementRack is a gear with infinite pitch circle radiusConverts rotary motion to linear motion

Linear speed Proportional to pinion speed

v = rp ω

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Analysis of Spur GearsPinion, PGear, GNumber of teeth, nRadius, rAngular velocity, ω

nP , rP nG , rGG

P

G

P

P

G

nn

rr

==ωω−

1

2

3

The maximum reduction in a single stage is limited!

To get higher reductionMultiple stagesBut…

lead to bulky package and weightSpur gears have high wear and tear and are noisy

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Analysis of Planetary GearsSimple Example

Ring gear, RSun gear, SCarrier arm, CPlanet gear, PFrame, F

[Waldron and Kinzel, 1999]

If carrier is stationary…

S

P

P

Srr−

=ωω

S

R

R

Srr

−=ωω

But suppose the ring gear is stationary and the carrier is not stationary

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Analysis of Planetary GearsSimple Example

Ring gear, RSun gear, SCarrier arm, CPlanet gear, PFrame, F

[Waldron and Kinzel, 1999]

If rP = 2, rS = 2, rR = 6,and ωR = 0:

4=ωω

C

S Assume positive counter clockwise directions

[stationary]

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Planetary Gears

90-95% efficiency30-50 Nm3 lbs20 arc min

Machine Design, Aug 2000

http://www.apexdyna.com/