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IYPT 2010 Austria, I. R. IranIYPT 2010 Austria, I. R. Iran
Problem No.
1
ROTATING SPRING
Reporter: Reza M. Namin
16
IYPT 2010 Austria, I. R. Iran2
The problem• A helical spring is rotated about one of its ends around
a vertical axis.
• Investigate the expansion of the spring with and without an additional mass attached to it’s free end.
IYPT 2010 Austria, I. R. Iran3
Main approach• Theory
– Background– Theory base– Developing the equations– Numerical solution
• Experiment– Setup– Parameters, results and comparison
• Conclusion
IYPT 2010 Austria, I. R. Iran4
Theory - Background
• Act of a spring due to tensile force:– Hook's law: F = k ∆L
• F: Force parallel to the spring• k: Spring constant• ∆L: Change of length
– A spring divided to n parts:• F = n k ∆L• μ = k L remains constant
• Circular motion– a = r ω2
• a: Acceleration• r: Distance from the rotating axis• ω: Angular velocity
IYPT 2010 Austria, I. R. Iran5
Theory - Base
• Effective parameters:– ω: Angular velocity– λ : Spring liner density = m / l – M: Additional mass– μ : Spring module = k l– l, l1, l2: Spring geometrical properties
M
l1 l l2
IYPT 2010 Austria, I. R. Iran6
Theory - Base
• Looking for the stable condition in the rotating coordinate system– Accelerated system → figurative force
• Acting forces:– Gravity– Spring tensile force– Centrifugal force
IYPT 2010 Austria, I. R. Iran7
Theory – Developing the equations• Approximation in mass attached
conditions:– Considering the spring to be
weightless:
l M
ω
x
y
Fs
Fc
Mg
l
xFF
xmF
llkF
sc
c
s
2
0 )(
20
mk
kllapx
k
mgll 0min
),max( minlll apx
IYPT 2010 Austria, I. R. Iran8
Theory – Developing the equations• Exact theoretical description:
– Problem: The tension is not even all over the spring…
– Solution: Considering the spring to be consisted of several small springs.
M
IYPT 2010 Austria, I. R. Iran9
Theory – Numerical solution
• Numerical method– Finite-volume approximation:
• Converting the continuous medium into a discrete medium
– Transient (dynamic unsteady) method– Programming developed with QB.
M
dlllT
dlllT
xxmcf
ygmw
iiii
iiii
ii
i
)(
)(
ˆ
ˆ
,1,1
,1,1
2
w
Ti-1
Ti+1
fc
lm
nll /
IYPT 2010 Austria, I. R. Iran10
Theory – Numerical solution
Mesh independency check
80 85 90 95 100 105 110 115 1200.15
0.2
0.25
0.3
0.35
0.4
0.45
n = 5
n = 10
n = 15
Angular velocity (s-1)
Sp
rin
g le
ngt
h (
m)
n: Number of mesh points
As n increases, the result will approach to the correct answer
IYPT 2010 Austria, I. R. Iran11
Theory – Numerical solution
Tension in different points of the spring with different additional mass amounts:
0 2 4 6 8 10 12 14 16 180
0.2
0.4
0.6
0.8
1
1.2
1.4
m = 0m = 2.5gm = 5g
Position in the spring (cm)
Ten
sion
(N
)
IYPT 2010 Austria, I. R. Iran12
Experiment• Finding spring properties
– Direct measurement: Mass & lengths– Suspending weights with the spring to
measure k and μ
• Changing the angular velocity, measuring the expansion– Change of the angular velocity with different
voltages– Measuring the angular velocity with
Tachometer– Measuring the length of the rotating spring
using a high exposure time photo
IYPT 2010 Austria, I. R. Iran13
Experiment setup
The motor, connection to the spring and the sensor sticker
IYPT 2010 Austria, I. R. Iran17
Experiments
Suspending weights with the spring
Finding k and using that to find μ
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.120
0.5
1
1.5
2
2.5
f(x) = 33.782655313683 x − 1.84774791757864R² = 0.999226158045033
Spring length (m)
Att
ache
d w
eigh
t (N
)
→K = 33.78 N/m
→μ = K l = 1.824 N
IYPT 2010 Austria, I. R. Iran19
Experiments
Measurement of length in different angular velocities
Comparison with the numerical theory
10 20 30 40 50 60 70 80 900
5
10
15
20
25
ExperimentsNumerical result
Angular velocity (Rad / s)
Spri
ng le
ngth
(cm
)
λ =0.148 kg/mμ =1.824 Nl1 =1.5 cml2 = 1.7 cm
IYPT 2010 Austria, I. R. Iran20
Experiments
Comparing the shape of the rotating spring in theory and experiment
0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000
-0.1000
-0.0900
-0.0800
-0.0700
-0.0600
-0.0500
-0.0400
-0.0300
-0.0200
-0.0100
0.0000
λ =0.103 kg/mμ =0.369 Nl = 16.3 cml1 =1 cmω = 120 RPM
IYPT 2010 Austria, I. R. Iran21
Experiments
Investigation of the l-ω plot within different initial lengths
100 150 200 250 300 3500
5
10
15
20
25
30
35
40
45Experiment: l = 16.3Experiment: l = 14Experiment: l = 11.5Experiment: l = 7.2Numerical: l = 16.3Numerical: l = 14Numerical: l = 11.5Numerical: l = 7.2
Angular velocity (RPM)
Sp
rin
g le
ngt
h (
cm)
λ =0.103 kg/mμ =0.369 Nl1 =1 cm
IYPT 2010 Austria, I. R. Iran22
Experiments
Comparison between the physical experiments, numerical results and theoretical approximation within different additional masses
0 50 100 150 200 250 300 350 400 4500
5
10
15
20
25
30
35
40
45Exp M = 0
Exp M = 5g
Exp M = 10g
Exp M = 15g
Num M = 0
Num M = 5g
Num M = 10g
Num M = 15g
Appx M = 0
Appx M = 5g
Appx M = 10g
Appx M = 15g
Angular velocity (RPM)
Sp
rin
g le
ngt
h (
cm)
λ =0.103 kg/mμ =0.369 Nl = 5.4cml1 =1 cm
IYPT 2010 Austria, I. R. Iran23
Conclusion• According to the comparison
between the theories and experiments we can conclude:– In case of weightless spring
approximation:
20
mk
kllapx
k
mgll 0min
),max( minlll apx
IYPT 2010 Austria, I. R. Iran24
Conclusion
• In general, the numerical method may be used to achieve precise description and evaluation.
• Some of the results of the numerical method are as follows:
IYPT 2010 Austria, I. R. Iran25
Conclusion
Numerical solution results
Change of the spring hardness
0 50 100 150 200 250 300
-0.0999999999999994
5.82867087928207E-16
0.100000000000001
0.200000000000001
0.300000000000001
0.400000000000001
0.500000000000001
0.600000000000001miu = 0.1
miu = 0.2
miu = 0.3
miu = 0.5
Angular velocity (RPM)
Sp
rin
g le
ng
th (
m)
λ =0.1 Nl = 10 cml1 =1 cm
IYPT 2010 Austria, I. R. Iran
0 50 100 150 200 250 300 3500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Landa = 0.05
Landa = 0.1
Landa = 0.15
Landa = 0.2
Conclusion
Numerical solution results
Change of spring density
μ =0.3 Nl = 10 cml1 =1 cm
IYPT 2010 Austria, I. R. Iran
0 50 100 150 200 2500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
l = 8cm
l = 10cm
l = 12cm
l = 14cm
Angular velocity (RPM)
Sp
rin
g len
gth
(m
)Conclusion
Numerical solution result
Change of initial length
λ =0.2 kg/mμ =0.3 Nl1 =1 cm
IYPT 2010 Austria, I. R. Iran
Conclusion
Numerical solution results
Change in additional mass
0 20 40 60 80 100 120 140 160 1800
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5m = 0
m = 5g
m = 10g
m = 15g
Angular velocity (RPM)
Sp
rin
g len
gth
(m
)
λ =0.2 kg/mμ =0.3 Nl = 10cml1 =1 cm