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