AMME2500
Gyroscopic Systems Laboratory
By: Joon Sunwoo, 510438814
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Introduction and Aims:
A gyroscope is a device that preserves its orientation by utilizing the principle of angular
momentum conservation. It consists of a spinning mass known as a flywheel, which is
mounted on a base at its pivotal point. The flywheel can move freely in any direction,
enabling it to maintain its orientation despite any movement in the base. The gyroscope's
motion involves three types:
Axial spin: This refers to the rotation of the flywheel around its central spin axis, denoted as
𝜔𝑠 in this experiment.
Precession: It involves the rotation of the entire gyroscope around the axis positioned through
its pivotal point, denoted as 𝜔𝑝 .
Nutation: It describes a slight irregular motion in the rotation axis, denoted as 𝜔𝑛 .
Gyroscopes find applications in various fields, such as navigation systems in cars, planes, and
rockets, as well as computer pointing devices that enable control of the mouse cursor while it
is in mid-air. They are also used in the production of racing cars and motorbikes to maintain
engine orientation and balance. In addition, gyroscopes serve as stabilizers in ships and
certain monorails to keep the body upright. Essentially, the properties of a gyroscope make it
an ideal tool for navigation and maintaining the orientation of an object.
The aim of this lab was to observe and obtain insight into gyroscopic precession and nutation.
Method
Apparatus
Figure (1) left: Fly wheel and support & Figure (2) right: Motor
Figure (3) Left: Strobe & Figure (4) right: Flywheel and support diagram
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Procedure
Experiment 1 – Measuring Decay in flywheel speed over time
1. The flywheel was held onto the motor, letting the flywheel speed up to 3000rpm
2. Then the flywheel was placed onto the stand ensuring it was facing the strobe and the
strobe was set to 2800rpm
3. Thus, the frequency of the flywheel could be indicated when the black dot on the
flywheel appeared to stand still as it matched the frequency of the strobe, therefore,
when this occurred, 2800 rpm was reached and the stopwatch was started, and
reduced the strobe frequency to 2600rpm
4. When the flywheel reached 2600rpm a lap on the stopwatch was added to record the
time and reduced the frequency to 2400, 2200, 2000 and 1800 recording time as it
reached each frequency
5. This test was repeated 2 more times to ensure reliability of results
Experiment 2 - Measuring the precession of the flywheel
1. The centre of mass of the flywheel was recorded by adjusting the pin and balancing it
on the support and marking it with a pen
2. Then the pin on the flywheel was adjusted to have an eccentricity of 10mm away
from the flywheel
3. The fly wheel was sped up to around 3000rpm using the motor and measured with the
strobe to wait until the flywheel frequency reduced to 2500rpm
4. The flywheel was placed on the support as soon as 2500rpm was reached, the
stopwatch was started, and the flywheel was released on the stand and the starting
position marked
5. The stopwatch was stopped when the flywheel made one revolution from its starting
position
6. This process was repeated to get one additional time recording for the same
eccentricity to improve reliability
7. Then the whole process was repeated for eccentricity of 20, 30, 40mm away from
flywheel and also 10, 20, 30 and 40mm closer to the flywheel, ensuring to run through
twice for each eccentricity
Results
Experiment 1
Time [s]
Freq [rpm] T1
2800
2600
2400
2200
2000
1800
T2
0
10.21
12.99
12.67
16.15
17.2
T3
0
10.95
14.09
15.08
12.9
16.8
Average
0
11.82
11.21
18.75
18.64
20.71
Table (1): Time taken for Flywheel to decay to each frequency
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0
10.99
12.76
15.5
15.9
18.24
Experiment 2
𝟐
Eccentricity Precession Period [s]
𝝎𝒑 [rad/s] 𝑰𝒆𝒙𝒑
𝒛𝒛 [kg𝒎 ]
L [mm]
T1
T2
Average
10
23.72
25.65
24.69
0.25
0.0037
20
12.3
11.62
11.96
0.53
0.0036
30
7.64
7.47
7.555
0.83
0.0034
40
5.72
6.18
5.95
1.06
0.0035
-10
21.2
22.69
21.95
0.29
0.0033
-20
11.07
11.67
11.37
0.55
0.0034
-30
7.82
7.94
7.88
0.8
0.0035
-40
5.85
6.03
5.94
1.06
0.0035
Table (2): Precession Period for each Eccentricity with calculated angular velocity and
moment of inertia
𝒆𝒙𝒑
The moment of Inertia around the z-axis was calculated using this formula, 𝑰𝒛𝒛 =
|𝐿𝑊|
𝝎𝒛 𝝎𝒑
𝑒𝑥𝑝
With mean 𝐼𝑧𝑧
= 0.0035.
Figure (5): Eccentricity vs Precession Period Plot,
Note: eccentricity of 0 results in no precession to simplify it is plotted as 0
Eccentricity vs Precession Angular Velocity
Angular Velocity [rad/s]
1.2
-60
1
0.8
0.6
0.4
0.2
0
-40
-20
0
20
40
Eccnetricity [mm]
Figure (6): Eccentricity vs Precession Angular Velocity
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60
Discussion
Experiment 1 showed, that with time the decay of the flywheel frequency decelerated as it
took longer each time to decay 200 rpm. As shown by the Table 1, 2800 down to 2600 rpm
took 10.99 seconds whereas decaying from 2000 to 1800 rpm took 18.24 seconds. This is
because at higher speeds there are greater forces acting on the flywheel that oppose its motion
such as air resistance and friction these forces are greater at higher velocities such that the fly
wheel decelerates at a faster rate.
In the Experiment 2, the average moment of inertia calculated was 0.0035 kgm^2, this is
0.0005 kgm^2 lower than the theoretical value of 0.004 kgm^2. Showing an percentage error
of 12.5%. There are many sources of error that could lead to this error. One major one was
assuming that the flywheel rotation speed of 2500 rpm was constant. In reality, this value
would have been decaying during the precession period just like in experiment 1, thus the
experimental data would deviate from theoretical due to this. Further when measuring the
revolution it relied on human reaction to stop the stopwatch as it passed the starting point this
would result in an inaccuracy of up to 1s that will skew results. Further inaccuracies could
have been made when measuring the eccentricity as the centre of mass may not have been
exactly centre and the eccentricity measurements with a ruler may also have been inaccurate
due to human error.
As noticed during experiment 2, when eccentricity was away from the flywheel, the
precession was clockwise, but when it when eccentricity was towards the flywheel, the
precession was counter clockwise. This is due the torque produced by gravity on the flywheel
that would be opposite depending on the eccentricity.
Figure (7) left & Figure (8) right
Figure 7 shows the eccentricity as positive and thus, due to right hand rule to torque goes into
the page causing the flywheel to rotate clockwise, whereas Figure 8 shows the eccentricity as
towards the flywheel and as such the torque goes out the page causing a counter clockwise
precession. As figure 5 & 6 show a larger eccentricity result in a higher torque generated by
gravity thus a faster precession period.
Conclusion
In conclusion, it was found that higher rpm decays faster than lower rpm due to larger
opposing forces generated by large rpms. Also that the toque produced by gravity results in
the opposite spin precessions and that larger eccentricity results in greater precession angular
velocity due greater torque produced by gravity.
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