How To Pump A How To Pump A Swing?Swing?

Tareq Ahmed MokhiemerTareq Ahmed Mokhiemer

Physics DepartmentPhysics Department

Contents

• Introduction to the swing physics and different pumping schemes

• Pumping a swing from a standing position

• Qualitative understanding

• Pumping from seated position

• Qualitative understanding

• Conclusion

What is meant by “pumping a swing? ”

Repetitive change of the rider’s position and/or orientation relative to the suspending rod.

How to get a swing running from a standing position?

By standing and squatting at the lowest point

• The motion of the child is a modeled by the variation with r with time r(t)

• This is equivalent to a parametric oscillator

)cos(2

1)(

2

1 2.2

.

mgrrmrmL

sin2....

grr

)sin(0 trr

• Let

• By scanning against pumping frequencies

amplification was found to occur at ~2

• Constant pumping frequencie Succession of amplification and attenuation.

l

g

50 100 150 200 250

-0.4

-0.2

0.2

0.4

320 340 360 380

-0.6

-0.4

-0.2

0.2

0.4

0.6

Another (naïve) model for r(t)

)(0 Coslr

-0.6 -0.4 -0.2 0.2 0.4 0.6

1.325

1.35

1.375

1.4

1.425

1.45

50 100 150 200

-0.04

-0.02

0.02

0.04

No Amplification !!

Expected result !

A more realistic model

-1 -0.5 0.5 1

1.6

1.8

2.2

2.4

50 100 150 200

-0.03

-0.02

-0.01

0.01

0.02

0.03

• For each initial velocity a threshold for the steepness of r(t) at θ=0.

50 100 150 200

-0.001

-0.0005

0.0005

0.001

Unexpected result!!

How does pumping occur physically?

Two points of view The conservation of angular

momentum Conservation of energy

conservation of angular momentum

BBBB mrmr.

2'

.2

'

2

'

.

.

)1(BB

B

r

))('cos1()(2

10'

2'

.2

' nmgrnmr BBB

)(

)1(

))('cos1(

))1('cos1(2

'

.

2'

.

0

0

n

n

n

n

B

B

8

0

0 )1())('cos1(

))1('cos1(

Brn

n

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1

Conservation of energy

At the mid-point: Gravitational force

Centrifugal force

xmgW .

xl

mvW

2

At the highest pointOnly gravitational force

xmgW .

)21)(()2

1( 22

l

xnvnv

2

0

0 )21())('cos1(

))1('cos1(

l

x

n

n

10 20 30 40 50

0.1

0.2

0.3

0.4

0.5

0.6

Another scheme for pumping from a standing position

The swinger pumps the swing by leaning forward and backward while standing

Pumping a swing from a seated position

))(())(( 31312211 CoslCoslmCoslCoslmCoslm

Potential Energy=

)]()(2)([2

1

)]()(2)([2

1

2

1

...

312

..2

3

2.2

13

....

212

..2

2

2.2

12

2.2

11

Cosllllm

Cosllllmlm

kinetic energy =

The equation of motion

M I1g Sin(φ(t)) + N g Sin(φ(t)+θ(t))-I1φ’’(t)

–I2 (φ’’(t)+ θ’’(t))-2 I2 N θ’’(t) Cos(θ(t))-I1 N θ’’(t) Cos(θ(t))==0

A Surprise

100 200 300 400

-1

-0.5

0.5

1

Θ(t) is either 0.5 rad when φ is gowing or -0.5 when φ is decreasing

The oscillation grows up linearly!!

The growth rate is proportional to the steepness of the frequency of the swinger’s motion

100 200 300 400

-1.5

-1

-0.5

0.5

1

1.5

Θ(t) is changes between 0.7 rad and -0.7 rad

A special case:

03322 ImIm

The Lagrangian reduces to:

)()(2

1

2

11

2..

2

2.

1 gCosMlII

..

21

..

21 )()( ISinMglII

And the equation of motion is

A driven Oscillator.

10 20 30 40 50 60 70

-0.3

-0.2

-0.1

0.1

0.2

0.3

Θ(t) changes sinusoidally

Pumping occurs at approximately the natural frequency not double the frequency.

Pumping from a seated position…

•More efficient in starting the swing from rest position

•With the same frequency of the swinger motion, the oscillation grows faster in the seated pumping.

Conclusion

• Exponential growth• Parametric Oscillator

Standing positionSeated position

• Linear growth• Driven Oscillator• Efficient in starting the

swing from rest