MECH2210 Experiment 4: 3D Gyroscope Dynamics Lab Report PDF

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This is a lab report on 3D gyroscope dynamics for MECH2210....

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Faculty of Engineering, Architecture and Information Technology A S S E S S ME NT C O V E R S HE E T

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MECH2210: Intermediate Mechanical & Space Dynamics Laboratory: Formal Reports(2) & Logbook entries(4) Cour s eCoor di nat or :Pr of es s orPaul Meehan Cour s eOff er i ngDet ai l s :Semes t er2,2020;STLUCI A( FD)

Due Date & Time: For malRepor t s2t eac hi ngweek saf t erexper i mentper f or med,16: 00&Logbookdue Thur sweek1216: 00

Declaration:

Wehav er eadt hePl agi ar i s m Sec t i onoft hec our s epr ofi l eandagr eet hatt hi swor ki s ourownandt hatanywor korc ol l abor at i onwi t hot hers t udent si sac k nowl edged.We hav eal s oac k nowl edgedal l ot hersour c esus edi nt hi sas s es s mentt as k . J ia FUXIN ( 46206181)

Si gnat ur e:

Harrison MEIER ( 45836572)

Si gnat ur e:

Tian ZHAOYANG ( 45711026)

Si gnat ur e:

524562

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MECH 2210 INTERMEDIATE MECHANICAL & SPACE DYNAMICS EXPERIMENT 4: 3D GYROSCOPIC DYNAMICS

GROUP MEMBERS: JIA FUXIN 4620618 TIAN ZHAOYANG 45711026 HARRISON MEIER 45836572 28 October 2020

Abstract This report presents the results of theoretical and experimental investigation of a gyroscope in various configurations. It helps to obtain a clear understanding of 3D rigid body dynamics such as gyroscopic forces and precessions. This experiment investigates the dynamics of rotating masses and forces as well as how gyroscopes performs under forced and free precession. Relation between spin rate and precession rate will be analysed using a free hanging gyroscope and a more complex digital analyse on how precession affects the gyroscope will be done on a motorised gyroscope and computer.

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Table of Contents 1.0 Introduction 2.0 Objectives 3.0 Theory 3.1 Equations 3.2 Definitions 4.0 Apparatus 5.0 Procedure 6.0 Results 6.1 Part A: String Suspended Gyroscope 6.2 PART B1: Free Precession 6.3 PART B2: Forced Precession 7.0 Uncertainty Analysis 7.1 Part A: String Suspended Gyroscope Error Analysis 7.2 Part B1: Free Precision Error Analysis 7.3 Part B2: Forced Precision Error Analysis 8.0 Discussion 9.0 Conclusion 10.0 References

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Appendix A-Super Precision Gyroscope Appendix B-Gimbals Kit for the Super Precision Gyroscope Appendix C-Digital Tachometer Appendix D-Control Moment Gyroscope Appendix E- Calculations Appendix F- Uncertainty Calculations Appendix F – Uncertainty Calculations

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1.0 Introduction A gyroscope is a device used for measuring orientation and angular velocity. It is a spinning wheel or disc in which the spin axis is free to rotate in any orientation. It is usually used to determine the dynamics of a rotating body. The concept of gyroscope is used in many areas, for example, spinning top, frisbee and in designs of spacecrafts, rockets where stability is to be considered.

2.0 Objective The key objective of this report is to recognise the aspects regarding 3D rigid body dynamics, with emphasis on gyroscopic forces and gyroscopes which test for both forced and free precession.

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3.0 Theory This experiment makes use of gyroscope to determine 3D dynamic such as gyroscopic forces and precession. Gyroscopic motion is the tendency of a rotating object to maintain the orientation of its rotation. A rotating object possesses angular momentum, and this momentum must be conserved or else the rotating effect will be interrupted. The object will resist any change in its axis of rotation, as a change in orientation will result in a change in angular momentum. To determine the precession and other parameters required for this experiment, a 3D motion calculation known as Euler’s Angles is used. For all spinning objects there will be a mass moment inertia and a transverse moment of inertia. I =¿ mass moment of inertia of area (spin axis, z axis) I 0 = transverse mass moment of area (x and y axis) H x =I x ω x =I 0 θ´ H y =I y ω y =I 0 ψ´ sin(θ) H z =I z ω z=I ( ψ´ cos ( θ ) + p) We also consider the application of moment (force at a distance) can be expressed as: dH ¿ + Ω× H dt xyz dH =¿ M= dt Combine the equation, we get the following expressions for moments, which is known as Euler’s angle equation:

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2 ´ cos ( θ) + p)sin(θ) M x =I 0 ( ´θ−ψ´ sin (θ ) cos ( θ) ) + I ψ´ ( ψ

´ ψ´ cos ( θ )+ p) ´ sin ( θ ) +2 ψ´ θ´ cos(θ) ) + I θ( M y =I 0( ψ ψ (¿cos ( θ ) + p) d M y =I ¿´ dt When the rate of precession is constant, steady, it implies no nutation and constant spin rate since spin and precession are inversely proportional. ´ cosθ + p ) −I 0 (ψ´ cosθ ) ] M x = ψ´ sinθ [ I ( ψ M y =0

M z =0 When the angle of precession is 90 ° to the precession axis the Euler equation will simplifies to the following form: M x =Ip ψ´ If no torque is applied to the gyroscope, free precession occurs, then the precession rate will be expressed as: ´ψ =

Ip (I 0−I ) cosθ

Where the relationship between I and I 0 dictates what type of precession occurs. I 0> I : Direct precession I =I 0 : No precession I 0< I : Retrogressive precession

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3.1 Equations Steady Precession Equation： ´ cosθ + p ) −I 0 (ψ´ cosθ ) ] --------------------------------------------------(1) M x = ψ´ sinθ [ I ( ψ M y =0

M z =0 When

θ=90 ° ,

M x =Ip ψ´ -----------------------------------------------------------------------------------------(2) M x =F g L=m gyro∗g∗L ---------------------------------------------------------------------(3)

´ψ=

Mx 2 π --------------------------------------------------------------------------------------(4) = t Ip

p=

RPM∗2 π 60

p=

RPM∗2 π 60

------------------------------------------------------------------------------------(5)

When no torque is applied or free precession: ´ψ =

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Ip -------------------------------------------------------------------------------------(6) (I 0−I ) cosθ

3.2 List of Variables Table 1 Definitions of Symbols used in report Symbol Mx θ I

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p ´ψ Fg

Definition Momentum along x-axis Angle Spin axis Mass Moment of Inertia Spin Rate Precession Rate Gravitational Force

m gyro g

Mass of Gyroscope gravity

L t I0

Arm Length Time Average transverse moment of Inertia

SI Unit Kgm/s Degree° kilogram / metre2 Radians/second Radians/second Newton metre2 kilogram2 Kilogram metre second2 Metre Seconds 2 kilogram / metre 2 kg m

3.3 Definitions Forced Precession Forced precession is known as the change in motion and axis of the body when external torque is applied. Free Precession Free precession is known as the change in motion and axis of the body when no external torque is applied. Nutation Nutation is the rocking motion in the axis of rotation of a large axially symmetrical object. Gyroscopic Moment/ Force Gyroscopic moment is the rate of change of angular momentum that is applied to the gyroscope and gyroscopic force is the force exerted by the gyroscopic body when resisting the applied couple. Stability of Torque-Free Motion of a Rigid Body Stability of torque-free motion of a rigid body shows the position of the body when free motion is occurring, and rigid body makes rotation about its principal axis of inertia.

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4.0 Apparatus Part A: (More information in Appendixes A-C)

Figure 1 Apparatus used for experiment Part A 1. Precision Gyroscope 2. Electric Motor 3. Battery Pack 4. Extensions 5. String 6. Digital Tachometer 7. Stopwatch Part B:

Figure 2 Apparatus used for experiment Part B 1. Control Moment Gyroscope System 11

5.0 Procedure Part A – String Suspended Gyroscope 1. One of the ‘ball’ ends was screwed into the opposite side of the gyroscope and connected to the motor. The motor will be pushed onto the axle of the gyroscope while it is OFF. The gyroscope spinning was commenced with the motor and the speed was taken to full speed. The tachometer was used to monitor the spin speed. The full speed was noted, and the motor was removed. Hold both ends of the string and put the ball end of the Gyroscope into the string.

Figure 2 Picture of gyroscope attached to a string in experiment Part A 2. The Gyroscope was raised with the string and the behaviours was observed and noted. Any precession that took place had been measured; rate, direction and angle between precession and spin axes. Lower the Gyroscope onto the table and secure with your hand. The spin speed was measured again using tachometer while the spin rate is determined by recording the time taken for the gyroscope to complete one revolution. 3. Repeat step 1 and 2 for spin speed between 2800 to 8500 RPM.

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4. All the required measurement of the apparatus was taken to determine the spin axis mass moment of inertia of the gyroscope shaft/disk based on your experimental measurement of precession. The mass of the gyroscope was directly measured.

Part B – Control Moment Gyroscope Free Precession 1. Turn on the power of the ECP machine and open the program ECP MV-E2Usr32 in the program files menu. Axis 3 break was assured to ON and Axis 4 break was assured OFF. The gyro was set up gently by hand and the rotor (Body D) was at 90 degrees (Vertical) and gimbal (Body C) was horizontal. The file ‘mech2210gyro.alg’ was loaded from the disk through the control algorithm in the setup menu.

Figure 3 3D Gyroscope Dynamics

2. The current gyroscope configuration was set to zero as a reference using the Utility menu on the computer by choosing zero position. By pressing the Setup menu go to the control algorithm and load from disk mech2210gyro.alg. The sample period was lowered to 0.000884 samples per second then click on Implement Algorithm. The OK was clicked which interfaced the program with drive motors. 13

3. The rotor speed was adjusted to 200rpm in the Initialized rotor speed from the command menu. Trajectory 1 from the command menu was clicked when the rotor speed achieved steady. To provide an impulse torque about Axis 2 and 4 seconds of recording, click on impulse and followed by setup. The following data are used: Table 2 Setup data used for control moment gyroscope system for free precession in Part B1 Impulse amplitude

Pulse width

Reps

Dwell time

16000

4000(ms)

1

50(ms)

4. After inputting the above data, the impulsive response of the gyroscope was executed from command menu with normal data sampling. The precession was observed and noted as direct or retrograde. To examine the various encoder reading for the movement about the different axes, the setup plot was selected from the plotting menu when the recording is finished. An appropriate measurement is chosen and allows you to examine the rate of precession and click plot data. The axis can be rescale using plotting menu. To calculate the precession rate base on the measurement, click on export raw data from data menu and save it for analysis. 5. Once the gyro has stopped spinning, go to command menu and disable the rotor speed. 6. Repeat step 2 to 5 for different spin speed; 400,600 and 800rpm with the gyroscope rotated to 45 degrees.

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Forced precession 1. The procedures of Free Precession were repeated with a step torque instead of an impulsive torque to investigate force precession. Repeat step 2 to 7 by changing the following parameters of Trajectory 1. from command menu. Click on impulse then setup and the parameter were changed to blow: Table 3 Setup data used for control moment gyroscope system for forced precession in Part B2 Impulse amplitude

Pulse width

Reps

Dwell time

1000

4000(ms)

1

0(ms)

The decaying oscillation and then the steady precession about axis 4 were noted after execution. Now the applied moment can be calculate using the measurements of the steady force precession.

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6.0 Results 6.1 Part A: String Suspended Gyroscope Practical This experiment was conducted using a range of diverse spin rates between 2800 and 8500 rpm. The precession rate was also determined by using a stopwatch to time how long it took for one revolution to be completed. The data collected from Part A of this practical is shown in Table 1 below: Through the speed (rpm) and time (s) collected, the spin rate and precession rate of the gyroscope were calculated as shown in Appendix E. Table 4 Practical data obtained for free precession in Part A Speed (rpm)

Time (s)

Spin Rate ( p ) (rad/s)

Precession Rate ( ψ ) (rad/s)

Spin axis mass moment of inertia ( kg m2 ¿

2800

2.00

293.22

3.14

6.05 ×10−5

3800

2.30

397.94

2.73

5.12 × 10−5

4800

2.82

502.65

2.23

4.97 × 10−5

5100

2.99

534.07

2.1

4.96 ×10−5

5600

3.46

586.43

1.82

5.23 × 10−5

6800

3.90

712.09

1.61

6.1 × 10−5

7200

4.53

753.98

1.39

5.33 × 10−5

7400

4.72

774.93

1.33

5.4 ×10

8000

4.43

837.76

1.42

4.69 ×10−5

8500

5.43

890.12

1.16

5.41 × 10

Average Spin axis mass moment of inertia

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−5

−5

5.33 × 10−5

Theoretical The theoretical mass moment of inertia was calculated using the mass of the gyroscope. The calculation of the theoretical mass moment of inertia of the gyroscope. Appendix E demonstrates that we can predict the expected precession rate for the different spin rates as shown in the table below: Table 5 Theorical data obtained for free precession in Part A Speed (rpm)

Spin Rate ( p ) (rad/s)

Precession Rate ( ψ ) (rad/s)

2800 293.22 3.68 3800 397.94 2.71 4800 502.65 2.15 5100 534.07 2.02 5600 586.43 1.84 6800 712.09 1.51 7200 753.98 1.43 7400 774.93 1.39 8000 837.76 1.29 8500 890.12 1.21 Average Spin axis mass moment of inertia

Spin axis mass moment of inertia 5.16 5.16 5.15 5.16 5.16 5.18 5.17 5.17 5.15 5.17 5.16

−5

×10 −5 ×10 ×10−5 −5 ×10 ×10−5 −5 ×10 ×10−5 −5 ×10 −5 ×10 −5 ×10 ×10−5

A comparison between the experimental and theoretical precession rates are shown in the table below: Table 6 Comparison of experimental and theoretical data Speed (rpm)

2800 3800 4800 5100 5600 6800 7200 7400 17

Spin Rate ( p ) (rad/s) 293.22 397.94 502.65 534.07 586.43 712.09 753.98 774.93

Experimental Precession Rate ( ψ ) (rad/s) 3.14 2.73 2.23 2.1 1.82 1.61 1.39 1.33

Theoretical Precession Rate ( ψ ) (rad/s) 3.68 2.71 2.15 2.02 1.84 1.51 1.43 1.39

8000 8500

837.76 890.12

1.42 1.16

1.29 1.21

Below portrays a graph comparing the predicted precession rate and the actual measured precession rate.

Figure 5 Plot comparing experimental and theoretical spin rate and precession rate relationship

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6.2 PART B1: Free Precession During free precession, there is no moment or external force acting on the gyroscope, so only gravity at its centre of mass. Hence the gyroscope under free precession. Practical Table 7 depicts the data which was collected from the free precession part of the experiment. The speeds ranging from 200 – 800 rpm and the period were converted to rad/s. Table 7 Practical data obtained for free precession Part B1

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Speed (rpm)

Spin Rate ( p ) (rad/s)

Period ( τ ) (s)

Precession Rate ( ψ ) (rad/s)

200

20.94

0.435

14.44

400

41.89

0.222

28.30

600

62.83

0.159

39.52

800

83.78

0.127

49.47

Theoretical Table 8 shows an average transverse moment of inertia which is derived from formula using the theoretical spin axis moment of inertia, the spin rate and the practical precession rate. Table 8 Theorical data obtained for free precession Part B1 Speed

Spin Rate

Nutation

Nutation

Spin axis

Precession

Transverse

(rpm)

( p )

Angle

Angle

moment of

rate (

moment of

(rad/s)

(counts)

inertia (I) (

´ψ ¿

inertia (

2 kg m )

(rad/s)

I o¿ ( 2

200

20.94

1518

0.390897

0.0273

14.44

kg m ) 0.066888

400

41.89

744

0.191586

0.0273

28.30

0.067710

600

62.83

482

0.124119

0.0273

39.52

0.070702

800

83.78

353

0.0909

0.0273

49.47

0.073534

Average transverse moment of inertia

0.069709

Base on the calculation, we found the transverse moment of inertia. And the experimental transverse moment of inertia was used to calculate the theoretical precession rate. The following table is shown the different between the experimental and theoretical precession rate.

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Table 9 portrays a comparison between the precession rate collected through the experiment and a predicted precession rate through calculations. Table 9 Comparison of Experimental vs Theoretical Precession Rate for Part B1 Speed (rpm)

Spin Rate ( p ) (rad/s)

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Practical Precession rate ( ´ψ ¿

Theoretical Precession rate (

(rad/s)

´ψ ¿

200

20.94

14.44

(rad/s) 15.62

400

41.89

28.30

28.83

600

62.83

39.52

39.83

800

83.78

49.47

49.68

Figure 6 Graph showing the practical and theoretical precession rate relationship for free precession in Part B1 6.3 PART B2: Forced Precession From part B1 the same mass moment of inertia (0.0273 kg m2 ) and the average transverse moment of inertia (0.069709 kg m2 ) were used to determine the moment values. We used encoder 2 and 4 value from the data to calculate the nutation angle and the experimental precession rate. The calculation can be seen from appendix. The period is constant which is 3.984. Practical Table 10 Experimental data obtained for forced precession in Part B2 Speed (rpm)

Spin Rate ( p ) (rad/s)

Encoder 2 (counts)

Encoder 4 (counts) 1508

200

20.94

879

400

41.89

322

794

600

62.83

187

569

800

83.78