EXAMINATION OF LATERAL DEVIATION OF A PROJECTILE SUBJECTED TO
INTERNAL GYROSCOPIC FORCES
A Thesis by
Christopher M. Umstead
Bachelor of Sciences, LeTourneau University, 2006
Submitted to the Department of Mechanical Engineering
and the faculty of Graduate School of
Wichita State University
in partial fulfillment of
the requirements for degree of
Master of Science
December 2009
i
©Copyright 2009 by Christopher M. Umstead
All Rights Reserved
ii
EXAMINATION OF LATERAL DEVIATION OF A PROJECTILE SUBJECTED TO
INTERNAL GYROSCOPIC FORCES
The following faculty members have examined the final copy of this thesis for form and content,
and recommend that it be accepted in the partial fulfillment of the requirement for the degree of
Master of Science with a major in Mechanical Engineering.
Hamid M. Lankarani, Committee Chair
Brian Driessen, Committee Co-Chair
Christian Wolf, Committee Member
iii
ACKNOWLEDGEMENTS
I would like to thank Dr. Lankarani for his willingness to work with me and for
continually motivating me. Also, I am grateful to Cessna for my time of employment, for their
generosity in fund my further education and for a flexible work schedule.
An expression of gratitude is offered to my Co-chair, Dr. Dressien and to Dr. Wolf for
sitting on my thesis committee. Recognition is also due to my local church, WEFC, and to my
parents for their continuous support and encouragement through the years.
iv
ABSTRACT
The need for lateral travel can arise in a simple projectile either due to the target moving
or to a transverse external force that moves the projectile. By the use of internal moving parts, a
force in the lateral direction can be created, thus changing the flight trajectory of the projectile.
Therefore, the possibility of the use of an internal gyroscopic force derived from a rotating disk
could be used to counteract such activity. In this study, three internal swing masses attached at
the end of a mass-less rod are located on the points of an equilateral triangle centered about the
major axis of the projectile. The rotation of mass generates a normal acceleration and a
subsequent force and torque upon the projectile. The force from combined movements of the
masses can be used, to a small degree, to control the movement of the projectile.
A model for this movement is derived mathematically and simulated on the computer
using commercial-available software and a developed code. For the cases examined, the
projectile moved cyclically about the axis or deviated with a relatively even slope away from the
axis. With these models, a parametric study is conducted with predetermined changes.
The parametric study focused on actuating each mass in sequence or changing the initial
weight of the mass and starting position in relation to the projectile. Two different simulation
methods are used. A MATLAB program is created for the developed model for the equations of
motion to simulate the flight of the projectile. Also a model is created in the MSC Adams
software to simulate the projectile flight. The results from the simulation for the project are
compared for differences between physical conditions and simulation software used.
The results of this study show that the projectile could move in a lateral direction in a
controlled manner. But the overall movement is minimal when compared to the size of the
v
projectile or the distance traveled. While this application may not be suited for use in small
projectiles, it may hold merit for use with larger craft or re-entry vehicles.
vi
TABLE OF CONTENTS
Chapter
1
INTRODUCTION .............................................................................................................. 1
1.1
1.2
1.3
2
2.3
Physical Projectile Properties ............................................................................ 10
Solid Core Projectile Model............................................................................... 11
2.2.1
Solid Core Projectile Dynamics .......................................................... 11
2.2.2
Solid Core Projectile Adams’ Model .................................................. 12
Concept Projectile Model .................................................................................. 13
2.3.1
Concept Projectile Dynamics .............................................................. 14
2.3.2
Concept Projectile Computer Simulation ........................................... 15
2.3.3
Concept Projectile Dynamic Computer Model ................................... 16
CONCEPT MODEL PARAMETRIC STUDY ................................................................ 18
3.1
3.2
3.3
4
Background and motivation ................................................................................. 1
Literature Review................................................................................................. 2
Objectives ............................................................................................................ 8
1.3.1
Objective of Mathematical Model of Projectiles .................................. 8
1.3.2
Objective of Computer Dynamic Models of Projectiles ....................... 9
SIMULATION MODELS ................................................................................................ 10
2.1
2.2
3
Page
Parametric ω Change Study ............................................................................... 19
Parametric Weight Change Study ...................................................................... 20
3.2.1
Parametric Weight Change Computer Simulation .............................. 22
3.2.2
Parametric Weight Change Adams’ Model Simulation...................... 27
Parametric Phase Change Study ........................................................................ 31
3.3.1
Parametric Phase Change Computer Simulation ................................ 33
3.3.2
Parametric Phase Change Adams’ Model........................................... 35
RESULTS AND COMPARISONS OF PARAMETRIC STUDY ................................... 37
4.1
4.2
Comparison of Reaction to Parametric Mass Change ....................................... 37
4.1.1
Computer Simulation Comparison of Reaction to Mass Change ....... 37
4.1.2
Adams’ Simulation Comparison of Reaction to Mass Change .......... 39
4.1.3
Comparison of Adams Model with the Computer Simulation for
Parametric Masses Changes ............................................................................... 42
Comparison of Reaction to Parametric Phase Change ...................................... 46
4.2.1
Computer Simulation Comparison of Reaction to Phase Change ...... 46
4.2.2
Adams’ Simulation Comparison of Reaction to Phase Change.......... 48
4.2.3
Comparison of Adams’ Model with Computer Simulation for
Parametric Phase Change ................................................................................... 50
vii
TABLE OF CONTENTS (continued)
Chapter
5
Page
CONCLUSION ................................................................................................................. 51
5.1
5.2
Improvements .................................................................................................... 52
Areas for Further Research ................................................................................ 52
REFERENCES ............................................................................................................................. 54
APPENDICES .............................................................................................................................. 56
APPENDIX A COMPUTER SIMULATION CODE ...................................................... 57
APPENDIX B SELECTED COMPUTER SIMULATION GRAPHS ............................. 65
APPENDIX C SELECTED Adams SIMULATION GRAPHS ....................................... 71
viii
LIST OF TABLES
Table
Page
1. ω Conditions and Diagrams .................................................................................................... 20
2. Mass Conditions 1................................................................................................................... 21
3. Mass Conditions 2................................................................................................................... 21
4. Mass Conditions 3................................................................................................................... 21
5. ω and Phase Condition 1 ......................................................................................................... 32
6. ω and Phase Condition 2 ......................................................................................................... 33
ix
LIST OF FIGURES
Figure
Page
1. Projectile Sketch of Traditional and Concept Idea ................................................................... 1
2. Personal Motivation for Study .................................................................................................. 2
3. Cross Range verses Time of a Missile with a Translating Mass Perpendicular to the Major
Axis [3] ..................................................................................................................................... 3
4. Missile with Translating Mass Perpendicular to the Major Axis [3] ........................................ 4
5. Projectile Lateral Movement versus Offset of a Projectile with a Translating Mass
Perpendicular to the Major Axis [3] ......................................................................................... 4
6. Projectile with Translating Mass Parallel to the Major Axis [4] .............................................. 5
7. Deviation of a Missile with a Translation Mass along the Major Axis [4]............................... 5
8. Reentry Vehicle Trajectory with Three Slowly Rotating Masses [2] ....................................... 6
9. Cross Range Offset verses Range of a Projectile with a Single Internal Rotating Disk [1] ..... 7
10. Sketch of Projectile with Axis ................................................................................................ 10
11. Sketch of Solid Core Projectile ............................................................................................... 11
12. Adams’ Model of Solid Core Projectile.................................................................................. 12
13. Solid Core Projectile Flight of Elevation versus Distance Traveled ...................................... 13
14. Sketch of Concept Projectile ................................................................................................... 14
15. Rotation Point Diagram .......................................................................................................... 14
16. Adams’ Concept Model .......................................................................................................... 17
17. Rotation Masses in Adams’ Concept Model .......................................................................... 17
18. Parametric Study Flow Chart .................................................................................................. 19
19. Computer Simulation Mass Condition 1 and All ω Conditions ............................................. 23
x
LIST OF FIGURES (continued)
Figure
Page
20. Computer Simulation Mass Condition 1 and ωD Condition .................................................. 23
21. Computer Simulation Mass Condition 2 and All ω Conditions ............................................. 24
22. Computer Simulation Mass Condition 2 and ωD Condition .................................................. 25
23. Computer Simulation Mass Condition 3 and All ω Conditions ............................................. 26
24. Computer Simulation Mass Condition 3 and ωD Condition .................................................. 26
25. Adams’ Simulation Mass Condition 1 and All ωConditions .................................................. 27
26. Adams’ Simulated Deviation for Phase 1 Condition, Mass 1 Conditions for ωD .................. 28
27. Adams’ Simulation Mass Condition 2 and All ωConditions .................................................. 29
28. Adams’ Simulated Deviation for Phase 1 Condition Mass 2 Conditions for ωD ................... 29
29. Adams’ Simulation for Mass Condition 3 and All ωConditions ............................................ 30
30. Adams’ Simulated Deviation for Phase 1, Condition Mass 3 Conditions for ωD .................. 30
31. All Masses 90 Degrees as Specified in Phase Condition 1 ..................................................... 31
32. Masses in Equilateral Triangle as Specified in Phase Condition 2 ......................................... 31
33. Computer Simulation of Mass Condition 1 Phase Condition 2 .............................................. 34
34. Computer Simulation of Mass Condition 1, Phase Condition 2 for ωD ................................. 34
35. Adams’ Simulation of Mass Condition 1 Phase Condition 2 ................................................. 35
36. Adams’ Simulation of Mass Condition 1 Phase Condition 2 for ωD ..................................... 36
37. MATLAB Mass Comparison of ωB ....................................................................................... 38
38. MATLAB Mass Comparison of ωD ....................................................................................... 38
39. MATLAB Mass Comparison of ωE ....................................................................................... 39
40. MATLAB Mass Comparison of ωF ....................................................................................... 39
xi
LIST OF FIGURES (continued)
Figure
Page
41. Adams Mass Comparison of ωB............................................................................................. 40
42. Adams Mass Comparison of ωD ............................................................................................ 41
43. Adams Mass Comparison of ωE ............................................................................................. 41
44. Adams Mass Comparison of ωF ............................................................................................. 42
45. Adams verse MATLAB Mass1 of ωB and ωE ....................................................................... 43
46. Adams verse MATLAB Mass2 of ωB and ωE ....................................................................... 44
47. Adams verse MATLAB Mass3 of ωB and ωE ....................................................................... 45
48. Adams verse MATLAB Mass 1, 2, and 3 of ωE .................................................................... 46
49. MATLAB Phase 1 and 2 Comparison Mass1 of ωB and ωE ................................................. 47
50. MATLAB Phase 1 and 2 Comparison Mass1 of ωD and ωE ................................................. 48
51. Adams’ Phase 1 and 2 Comparison Mass 1 of ωB, ωE, and ωF ............................................ 49
52. Adams’ Phase 1 and 2 Comparison Mass 1 of ωD and ωF .................................................... 49
53. Computer Simulation versus Adams’ Phase 2, Mass 1 of ωB, ωE, and ωF ........................... 50
54. Maximum Deviation Comparison for all Mass Conditions, all ω Conditions, for Phase 1, both
Adams' and MATLAB ............................................................................................................ 51
55. Maximum Deviation Comparison for Mass 1, all ω Conditions, for Phase 1 and 2, both
Adams' and MATLAB ............................................................................................................ 52
56. Computer Simulation Phase1 Mass 1, 2, and 3 for ωB .......................................................... 65
57. Computer Simulation Phase1 Mass 1, 2, and 3 for ωC .......................................................... 66
58. Computer Simulation Phase1 Mass 1, 2, and 3 for ωD .......................................................... 66
59. Computer Simulation Phase1 Mass 1, 2, and 3 for ωE ........................................................... 67
60. Computer Simulation Phase1 Mass 1, 2, and 3 for ωF ........................................................... 67
xii
LIST OF FIGURES (continued)
Figure
Page
61. Computer Simulation Phase2 Mass 1 and 2 for ωB ............................................................... 68
62. Computer Simulation Phase2 Mass 1 and 2 for ωC ............................................................... 69
63. Computer Simulation Phase2 Mass 1 and 2 for ωE ................................................................ 70
64. Computer Simulation Phase2 Mass 1 and 2 for ωF ................................................................ 70
65. Adams Simulation Phase1 Mass 1, 2, and 3 for ωB ............................................................... 71
66. Adams Simulation Phase1 Mass 1, 2, and 3 for ωC ............................................................... 72
67. Adams Simulation Phase1 Mass 1, 2, and 3 for ωD ............................................................... 72
68. Adams Simulation Phase1 Mass 1, 2, and 3 for ωD ............................................................... 73
69. Adams Simulation Phase1 Mass 1, 2, and 3 for ωF................................................................ 73
70. Adams Simulation Phase2 Mass 1 and 2 for ωB .................................................................... 74
71. Adams Simulation Phase2 Mass 1 and 2 for ωC .................................................................... 75
72. Adams Simulation Phase2 Mass 1 and 2 for ωE .................................................................... 76
73. Adams Simulation Phase2 Mass 1 and 2 for ωF..................................................................... 76
xiii
NOMENCLATURE
𝑎
Scalar acceleration
𝑎
Acceleration vector
𝑎𝑑
Acceleration of the disk relative to the world
𝑎𝑝
Acceleration vector of projectile relative to the
world
𝑐𝑣
Viscous damping
𝐹
Scalar force
𝐹
Acceleration vector
𝐹𝑚𝑎𝑠𝑠
Force due to the rotating mass (sub-notation
indicates which one)
𝐼
Moment of inertia
𝑚
Total mass of projectile
𝑚𝑑
Mass of rotating weight (sub-notation indicates
which one)
𝑚𝑝
Mass of projectile minus moving parts
𝑚𝑡
Mass of translating Weight
𝑝
Point of rotation of the mass (sub-notation
indicates which one)
𝑟
Radius of rotating masses’ arm relative to the
projectile (sub-notation indicates which one)
𝑟
Radius vector of rotating masses’ arm relative
to the projectile (sub-notation indicates which
one)
𝑟𝑝
Radius vector of the projectile relative to the
world’s origin
𝑅
Radius of the projectile
xiv
NOMENCLATURE (continued)
rpm
Rotations per minute
𝑣
Scalar velocity
𝑣
Velocity vector
𝛼
Scalar angular acceleration
𝛼
Angular acceleration vector
𝛼𝑑
Vector angular acceleration of the mass
relative to the world
𝛼𝑝
Vector angular acceleration of the projectile
relative to the world
𝜏
Scalar torque
𝜏
Torque vector
𝜔
Scalar angular Velocity
𝜔
Vector angular velocity
All units are presumed to be in the metric system unless noted.
xv
CHAPTER 1
INTRODUCTION
1.1
Background and motivation
From medieval times with trebuchets destroying castles to launching modern satellite into
orbit, the study of ballistics has been conducted to improve the flight course and end results.
Early studies of control were done by means of changing the initial physical conditions. As
technology developed, more study and research has been conducted in active control of the
projectile during flight.
The direct idea for control of the flight of a simple bullet in the lateral direction emerges
from personal experience of watching bullets err from the desired target due to the influence of
the wind. This idea of a single internal rotating mass is explored by Frost and Costello. Several
other researchers have explored different means of controlling and calculating trajectory [1].
Figure 1 Projectile Sketch of Traditional and Concept Idea
1
Figure 2 Personal Motivation for Study
1.2
Literature Review
Control of a missile can be accomplished externally by fins and engines in flight.
Internally, a projectile can be controlled by translating and rotating masses. The idea of a
translating mass is that a weight is contained in a linear cavity within the projectile. The mass
then slides within the cavity, creating a change of the center of gravity and conversely the angle
of attack. Another version of the translating mass concept is attaching the mass to a rod of
constant length which then rotates about the major axis of the projectile [2]. With the use of
several of these rods, the center of gravity can be changed, and thus also changing the angle of
attack.
The concept of a translating mass was explored by Rogers and Costello and Byrne,
Sturgis, and Robinett. According to Rogers and Costello, the trajectory changes due to a moving
mass from the dynamic coupling between the two bodies [3]. The equation that was found for
the force acting from the translating mass in Equation (1-1) was converted from the frequency
domain to the time domain [3]. When the mass was oscillated for a finned missile, the missile
deviated in the lateral direction as shown in Figure 3.
2
𝐹𝐼𝑛𝑝𝑢𝑡 =
𝑚 𝑝 𝑚𝑡
− 𝑚𝑡 𝑎 + 𝑐𝑣 − 2𝑚𝑡 𝜔𝑛 𝑣 + 𝑚𝑡 𝜔𝑛2
𝑚
(1-1)
Figure 3 Cross Range verses Time of a Missile with a Translating Mass Perpendicular to
the Major Axis [3]
The mass in this parametric study was offset from the center of gravity in a slot
perpendicular to the major axis and oscillated. As was expected, an increase was found in the
swerve as the mass moved away from the center of gravity, as shown in Figure 5 [3]. It was
found as the slot moved away from the center of gravity, the control authority increased
proportionally. If the slot is perpendicular to the major axis, as shown in Figure 6, the control of
the missile is increased during the flight [4]. This is due to fact that the stability of the missile
during launch can be increase, therefore reducing launch perturbations. Yet, when the missile is
in flight, the stability can be reduced, giving greater control of the path of the missile. This
effect is visible in Figure 7 as the missile with translating mass reaches the target, as opposed to
a conventional missile just shy of reaching the target.
3
Figure 4 Missile with Translating Mass Perpendicular to the Major Axis [3]
Figure 5 Projectile Lateral Movement versus Offset of a Projectile with a Translating Mass
Perpendicular to the Major Axis [3]
4
Figure 6 Projectile with Translating Mass Parallel to the Major Axis [4]
Figure 7 Deviation of a Missile with a Translation Mass along the Major Axis [4]
Changing the center of gravity by slowly rotating weights affixed about the major axis
changed the angle of attack and conversely the flight path, was studied by Byrne. He found that
while a minimum of two weights could be used, three or more weights require less drastic
5
movement [2]. The base equation for the resultant center of mass in the polar coordinates was
derived to Equation (1-2) and converted to Cartesian coordinate in Equation (1-3).
𝑅𝑒𝑞 𝑒 𝑖𝜃𝑒𝑞 =
𝑅𝑒𝑞
𝑅 𝑖𝜃
𝑒 1 + 𝑒 𝑖𝜃2 + 𝑒 𝑖𝜃3
3
cos 𝜃𝑒𝑞
𝑅 cos 𝜃1 + cos 𝜃2 + cos 𝜃3
=
sin 𝜃𝑒𝑞
3 sin 𝜃1 + sin 𝜃2 + sin 𝜃3
(1-2)
(1-3)
The application of the translating mass is explored for possible use for reentry from orbit;
the result of this system is shown Figure 8. In these figures, as the vehicle descended, the
internal moving masses changed the direction of flight and directed the vehicle to the desired
point of impact.
Figure 8 Reentry Vehicle Trajectory with Three Slowly Rotating Masses [2]
The notion of a single internal rotating disk was considered by Frost and Costello in
several of their studies [1,5]. They found that, as the offset of the mass from the center of gravity
increased, so also did the control. The equation of motion for the disk was found in Equation (1-
6
5) and for the projectile body in Equation (1-6). Combining these equations together gives
Equation (1-6) which is the equation of motion for both the projectile and the rotating disk.
𝑚𝑑 𝑎𝑑 = 𝐹𝑅 + 𝑊𝑑
(1-4)
𝑚𝑝 𝑎𝑃 = −𝐹𝑅 + 𝑊𝑃 + 𝐹𝐴
(1-5)
𝑚𝑝 𝑎𝑃 + 𝑚𝐷 𝑎𝑑 = 𝑊𝑑𝐷 + 𝑊𝑃 + 𝐹𝐴
(1-6)
From these equations, a control system for the projectile can be developed. The
movement in the lateral direction from this study is shown in Figure 9 with the two body systems
containing the rotating disk. This concept of a single rotating disk is expanded in this thesis in a
parametric study of three rotating disks and the effect of the rotation on the lateral movement of
the projectile.
Figure 9 Cross Range Offset verses Range of a Projectile with a Single Internal Rotating
Disk [1]
7
1.3
Objectives
The objective of this thesis is to conduct a parametric study on the concept of controlling a
projectile by the controlled spinning of one to three internal masses. The masses are mounted
atop mass-less rods. These rods are then pinned on the opposite end to the main body of the
projectile, equally distanced from each other and the major axis of the projectile. In the
parametric study, the mass, starting orientation, and angular rotation of the disks are
systematically changed, and the results are captured. The results are later examined, compared,
and contrasted with each other. From these results, recommendations for possible further studies
and improvement are given.
1.3.1 Objective of Mathematical Model of Projectiles
One of the sub-objectives of this study is to create a mathematical model of the different
projectiles examined and model their actions. The computer program MATLAB will be used for
the modeling. From this mathematical model, the results will be compared with the dynamics
from the simulation program. The final equation of motion will be composed of a seven degrees
of freedom--four for the projectile and one for each rotating disk. Controllable inputs into the
model are initial linear and angular velocity and acceleration of the projectile and disk. After the
projectile is in flight, control will be established by the changing rotational velocity of the disks,
which will in turn change the course of flight. Other force beyond direct control and acting upon
the projectile during flight beyond direct control will be drag and gravity.
8
1.3.2 Objective of Computer Dynamic Models of Projectiles
The concept for control of a projectile will be modeled on computer dynamic simulation
software. The software package of choice is MSC Adams because it is readily available. Two
models were created--one of a simple projectile and the other of the concept model. The outputs
from the simulation of the Adams’ models were saved to a spread sheet. This data was then
compared with other results of the simulations models and conditions.
9
CHAPTER 2
SIMULATION MODELS
2.1
Physical Projectile Properties
The two models analyzed are projectiles 20 mm in diameter. The shape of the projectile is
a simple, cone shape for the head attached to a cylinder. Figure 10 shows the projectile with
axes sketched; the major axis is equivalent to the x-axis in this figure and in all subsequent
references. The mass of the model considered is between 130 gm and 400 gm depending on
conditions. The results from the following study, however, can be sub sequentially scaled up for
larger models because equations of motion are linear with respect to masses, rod lengths for the
rotating mass, and linear velocity. The non-linear part of the equation of motion is the angular
velocity of the mass, which has an exponential effect, and the angle of the mass rod which
changes trigonometrically.
Figure 10 Sketch of Projectile with Axis
10
2.2
Solid Core Projectile Model
The traditional model is a projectile of a solid core with no moving parts. It is assumed the
greatest force acting upon it during it flight is gravity and the path will parabolic due to the
acceleration. The environment considered is a simple world of six degrees of freedom. Many
studies have been conducted on predicting the flight path of a projectile and simplifying the
equations to linear forms [5,6].
Figure 11 Sketch of Solid Core Projectile
2.2.1 Solid Core Projectile Dynamics
The bases of the mathematic model is Newton’s first law, 𝐹 = 𝑚𝑎. The components of
𝐹 in this case are gravity and force induced from flight, the largest of which, in this case, is drag.
For the simple projectile, 𝐹 is equal to the sum of the drag forces, which is given in equation ,
and gravity, which is acting in the direction given in Equation (2-2) where: 𝜌 is the density of the
air; 𝐶𝑑 is the drag coefficient; 𝐴 is the cross sectional area; and 𝜃 is the angle of the major axis
relative to direction of gravity.
1
𝐹𝑑𝑟𝑎𝑔 = − 𝜌 𝑣 2 𝐶𝑑 𝐴𝑣
2
11
(2-1)
𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑦
0
= sin 𝜃 𝑔
cos 𝜃 𝑔
(2-2)
Combining these forces in the equation of motion, Equation (2-3) is derived and gives
the basic parabolic arc for a projectile.
𝐹𝑑𝑟𝑎𝑔 + 𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑦 = 𝑚𝑎
(2-3)
2.2.2 Solid Core Projectile Adams’ Model
A model was created in Adams with similar physical properties to the more complex
model described. The construction of the model was a cylinder with a cone mated to the top.
The model is initially at point zero with the bottom of the cylinder located thus and the major
axis passing through the point. Because all forces are acting through the center of gravity, torque
will not be induced or need to be accounted for. The gravity force vector was angled to simulate
a 15-degree tilt and initial velocity was set to 800 m/s. The flight of the projectile perfomed as
expected with the projectile traveling in a parobalic arc, as shown in Figure 13.
Figure 12 Adams’ Model of Solid Core Projectile
12
Figure 13 Solid Core Projectile Flight of Elevation versus Distance Traveled
2.3
Concept Projectile Model
The concept of the new model, as shown in Figure 14, was to incorporate three weights
that could be rotated about a point within the projectile and cause a deviation in the flight path,
specifically in the lateral direction. The rotation points are the tips of an equilateral triangle
centered about the major axis of the projectile, as shown in Figure 15. Letting 𝑟 be the
maximum length of the rotation mass such that 2𝑟 is one of the legs of the equilateral triangle.
Then the position of the rotation points are (0,
2 3
3
𝑟), (−𝑟, −
3
3
𝑟), and (𝑟, −
3
3
𝑟) respectively
with the origin located in the center of the triangle. The minimum radius for the project, 𝑅, in
terms of 𝑟 is shown in Equation (2-4).
𝑅=
2 3
+ 1 𝑟 ≈ 2.155𝑟
3
13
(2-4)
Figure 14 Sketch of Concept Projectile
Figure 15 Rotation Point Diagram
The idea is when the masses are rotated the gyroscopic force generated will cause the
projectile to deviate. By the use of three masses, the direction of the deflection can be controlled
in the lateral direction.
2.3.1 Concept Projectile Dynamics
The equation for motion of the projectile was derived from equations from several studies
and modified for this study [1,7,6]. Combining Equations (2-1) and (2-2) with Equation (2-5) for
each mass gives Equation (2-6), which gives the overall force equation for the system. Thus, the
overall encompassing equations for force is
14
𝐹𝑚𝑎𝑠𝑠 = 𝑚𝑑 (𝑎𝑝 + 𝛼𝑝 × 𝑟𝑃 + 𝜔𝑝 × 𝜔𝑝 × 𝑟𝑃 + 𝛼𝑑 × 𝑟𝐶𝐷 + 𝜔𝑑 × 𝜔𝑑 × 𝑟 )
𝑚𝑝 𝑎𝑝 + 𝜔𝑝 × 𝑣𝑝 +
𝑛
𝑖=1(𝑚𝑑 𝑖 (𝑎𝑝
+ 𝛼𝑝 × 𝑟𝑃𝐶,𝑖 + 𝜔𝑝 × 𝜔𝑝 × 𝑟𝑝,𝑖 + 𝛼𝑑,𝑖 × 𝑟𝑖 +
𝑛
𝑖=1 𝑚𝑑 𝑖
𝜔𝑑,𝑖 × 𝜔𝑑′𝑖 × 𝑟𝑖 ) = 𝑚𝑝 +
1
∗ 𝑔 − 8 ∗ 𝐶𝑑 ∗ 𝜌 ∗ 𝑣𝑝
𝑖
2
(2-5)
(2-6)
𝜋𝑑 2 𝑣
Torque is created by the offset of the rotation point from the major axis, with the
resulting force from the normal acceleration upon that point as given in Equation (2-5). Using
𝜏 = 𝐼𝛼, the angular acceleration of the projectile can be found, and conversely the angular
velocity of the project can be found[8]. This rotation of the projectile changes the angle of the
rod of the rotating disk and the location of the point of rotation with respect to the ground and
subsequently the direction of force that is created by the disk on the projectile.
3
𝜏=
(2-7)
𝑝 × 𝐹𝑚𝑎𝑠𝑠 𝑖
𝑖=1
The polar moment of inertia about the major axis was estimated by treating the main body of the
projectile like a solid cylinder and the masses as points, giving Equation (2-8). With the moment
of inertia and torque, the angular velocity of the projectile was found along with the subsequent
changes in the location of rotation points and arms.
1
𝐼 = 𝑚𝑝 𝑅 2 +
2
3
𝑚𝑑 𝑟𝑖 + 𝑝𝑖
2
(2-8)
𝑖=1
2.3.2 Concept Projectile Computer Simulation
Using the equations discussed above, a simulation program was created in MATLAB to
find velocity and position, both linear and angular, of the projectile and the disk. This was
primarily accomplished by breaking Equation (2-6) and (2-5) in to sub-equations and solving for
15
the parts for ease and cleanness of code. Using Euler method and the given equations, the
instantaneous acceleration for that point was found, and the corresponding velocity and change
in position could be solved. This process was continued iteratively for a time step and continued
on to a predetermined termination point. The error for the Euler’s method when the time step
was halved varied depending what direction was examined. Error for the x and y axis (forward
and vertical) was between 0.002 and 0.004 depending on what conditions were examined. The
maximum found error for the z axis, the lateral deviation direction, had a magnitude of 1.1 ×
10−6 and thus negligible when compared to the deviation. The negligibility holds for the other
directions of travel as well.
The structure of the program was loops within loops. For each major loop, one of the
initial starting conditions was changed. These variables were the disk mass, initial phase, and
initial angular velocity. This structure of the codes is evident in the copy of the code in
Appendix A. The results of position and velocity of the disks and projectile would then be stored
in a file for later recall to be analyzed either manually or by a sub program.
2.3.3 Concept Projectile Dynamic Computer Model
The model in Adams, as pictured in Figure 16, was created using parametric references.
The model was constructed with three rotating masses represented as cylinders connected to
mass-less rods which rotated about a point, as shown in Figure 17. The three rotation points
were located at the peaks of an equilateral triangle with the center of the triangle coinciding with
the major axis of the projectile as seen in Figure 15. The parameters for design were segment
lengths, masses of each part, and the initial starting angular and linear velocities.
16
Figure 16 Adams’ Concept Model
Figure 17 Rotation Masses in Adams’ Concept Model
By the use of parameters, it was possible for the model to be quickly changed to simulate the
scenario changes. The changes made to the model were the variations of initial angular
velocities, mass ratios, and phases of the weights’ positions to analyze the effect of the change
down-range in the lateral direction.
17
CHAPTER 3
CONCEPT MODEL PARAMETRIC STUDY
The model was subjected to a series of parametric changes to determine what parameters
induced the greatest deviations of the projectile in the lateral directions. The concept is if a
deviation can be induced to change the flight path from the spinning weight; then conversely, the
same force can be used to correct for an external random force upon the projectile. Therefore,
the deviation of the path of the projectile in the lateral direction was observed compared.
The parametric study change occurred in the order presented in Figure 18. First, the phase
or initial position of the mass was set according to the parameter being examined. The mass was
then either changed to one of three possible weights or remained the same while the angular
velocity possibilities were cycled. The angular velocity of the masses were changed to reflect a
given scenario represented by ωN, where N is a letter between A and F designating what
conditions are occurring, as seen in the following tables. The results from these changes are
recorded in spreadsheet form. The results are plotted and then compared.
18
Phase
Mass
ωN
•Change Initial Position of the Swinging Weights
•Phase 1 – All masses point upwards
•Phase 2 – All masses are evenly balanced about the major axis
•Change or Keep the Mass Settings
•Mass weights 10 gr, 50 gr, or 100 gr
•Change the Angular Velocity of the Masses
•The angular velocity of the mass is set to a predetermined value based on the
preceeding chart
•Run Simulation on Designated Program
Simulation •Record Results for Analysis
Results
Figure 18 Parametric Study Flow Chart
3.1
Parametric ω Change Study
As stated, the angular velocities of the masses were changed to one of six conditions
designated by a letter.
In Table 1, the order of what masses are rotated for a given conditions is
shown. The first case, 𝜔A, all the masses are held at a standstill and there is no deviation lateral
of the projectile. In 𝜔B, the top mass is rotated at a constant velocity relative to the projectile.
This rotation causes the projectile to both rotate and move because of the induced force. In cases
𝜔C and 𝜔D, the other two masses are spun sequentially. In the next case, 𝜔E, the two lower
masses are spun at half the velocity of the upper mass and in the opposite direction. In 𝜔F, mass
3 is at half the angular velocity of 𝜔B and in the opposite direction.
19
Table 1 ω Conditions and Diagrams
3.2
ωA
ωB
ωC
ωD
ωE
ωF
Parametric Weight Change Study
The weight of the masses was changed a total of four times, as shown in Table 2, 2, and
Table 4, with the other parameters changing for simulations to be compared. The mass of the
non moving part of the projectile was held at 100 grams equally distributed about the center of
mass. The lowest ratio of weight of the rotating masses to the overall weight is 0.23 and the
largest is 0.75.
20
Table 2 Mass Conditions 1
𝜔Condition
𝜔1 (rpm)
𝜔2 (rpm)
𝜔3 (rpm)
𝜔A
0
0
0
𝜔B
167
0
0
𝜔C
167
167
0
𝜔D
167
167
167
𝜔E
167
-83.3
-83.3
𝜔F
0
0
-83.3
𝜃1 = 𝜃2 = 𝜃3 = 90°
𝑚𝑑,1 = 𝑚𝑑,2 = 𝑚𝑑,3 = 10 𝑔𝑟
Table 3 Mass Conditions 2
𝜔Condition
𝜔1 (rpm)
𝜔2 (rpm)
𝜔3 (rpm)
𝜔A
0
0
0
𝜔B
167
0
0
𝜔C
167
167
0
𝜔D
167
167
167
𝜔E
167
-83.3
-83.3
𝜔F
0
0
-83.3
𝜃1 = 𝜃2 = 𝜃3 = 90°
𝑚𝑑,1 = 𝑚𝑑,2 = 𝑚𝑑,3 = 50 𝑔𝑟
Table 4 Mass Conditions 3
𝜔Condition
𝜔1 (rpm)
𝜔2 (rpm)
𝜔3 (rpm)
𝜔A
0
0
0
𝜔B
167
0
0
𝜔C
167
167
0
𝜔D
167
167
167
21
𝜔E
167
-83.3
-83.3
𝜔F
0
0
-83.3
𝜃1 = 𝜃2 = 𝜃3 = 90°
𝑚𝑑,1 = 𝑚𝑑,2 = 𝑚𝑑,3 = 100 𝑔𝑟
3.2.1 Parametric Weight Change Computer Simulation
A MATLAB model was created to verify the characteristic actions of the Adams’ model.
The form of the program was loops embedded within each other to change the conditions
sequentially. While simulations did not directly correlate completely with the Adams the
overall results were similar.
The rotation masses were set to 0.010 kg and spun with angular velocity as specified in
Table 2. The results of this simulation are shown in Figure 19 with each color line representing a
different omega condition as laid out in the table. The greatest deviation for this case occurred
for ωD condition, magnified in Figure 20, with a total deflection of 39 mm.
22
0.01
0.005
-0.005
-0.01
0
0.119
0.238
0.357
0.476
0.595
0.714
0.833
0.952
1.071
1.19
1.309
1.428
1.547
1.666
1.785
1.904
2.023
2.142
2.261
2.38
2.499
2.618
2.737
2.856
2.975
3.094
Lateral Deviation (m)
0
ωA
ωB
-0.015
ωC
-0.02
ωD
-0.025
ωE
-0.03
ωF
-0.035
-0.04
-0.045
Time (sec)
Figure 19 Computer Simulation Mass Condition 1 and All ω Conditions
-0.005
0
0.119
0.238
0.357
0.476
0.595
0.714
0.833
0.952
1.071
1.19
1.309
1.428
1.547
1.666
1.785
1.904
2.023
2.142
2.261
2.38
2.499
2.618
2.737
2.856
2.975
3.094
0
Lateral Deviation (m)
-0.01
-0.015
-0.02
ωD
-0.025
-0.03
-0.035
-0.04
-0.045
Time (sec)
Figure 20 Computer Simulation Mass Condition 1 and ωD Condition
The extra weight, according to the MATLAB model, caused the projectile to deflect more
in Weight Condition 2 than in Weight Condition 1 as expected. Also, according to the computer
23
model, the greatest deflection occurred when all the masses rotated in the same direction with the
same angular velocity. The results for all these simulations are shown in Figure 21 with each ω
condition, as specified in Table 3. The greatest deviation shown in Figure 22 of 101 mm; this
deviation was the greatest deviation out of all the masses cases examined for this phase.
0.04
0
-0.02
0
0.119
0.238
0.357
0.476
0.595
0.714
0.833
0.952
1.071
1.19
1.309
1.428
1.547
1.666
1.785
1.904
2.023
2.142
2.261
2.38
2.499
2.618
2.737
2.856
2.975
3.094
Lateral Deviation (m)
0.02
ωA
ωB
ωC
-0.04
ωD
-0.06
ωE
-0.08
ωF
-0.1
-0.12
Time (sec)
Figure 21 Computer Simulation Mass Condition 2 and All ω Conditions
24
Lateral Deviation (m)
-0.02
0
0.119
0.238
0.357
0.476
0.595
0.714
0.833
0.952
1.071
1.19
1.309
1.428
1.547
1.666
1.785
1.904
2.023
2.142
2.261
2.38
2.499
2.618
2.737
2.856
2.975
3.094
0
-0.04
-0.06
ωD
-0.08
-0.1
-0.12
Time (sec)
Figure 22 Computer Simulation Mass Condition 2 and ωD Condition
In Weight Condition 3, the greatest deviation for most cases of the weight conditions
occurred in the MATLAB when all the angular velocities are the same, similar to the other cases,
as seen in Figure 23. The maximum deviation for this case, and all the computer simulation
cases, was 82 mm and this is magnified in Figure 24 when the ωD condition was being
simulated. This is in contradiction to the computer dynamic simulations, which will be
examined in a later section.
25
Lateral Deviation (m)
-0.14
-0.02
-0.14
0
0.119
0.238
0.357
0.476
0.595
0.714
0.833
0.952
1.071
1.19
1.309
1.428
1.547
1.666
1.785
1.904
2.023
2.142
2.261
2.38
2.499
2.618
2.737
2.856
2.975
3.094
-0.02
0
0.119
0.238
0.357
0.476
0.595
0.714
0.833
0.952
1.071
1.19
1.309
1.428
1.547
1.666
1.785
1.904
2.023
2.142
2.261
2.38
2.499
2.618
2.737
2.856
2.975
3.094
Lateral Deviation (m)
0.04
0.02
0
-0.04
-0.06
-0.08
-0.1
-0.12
Time (sec)
Figure 24 Computer Simulation Mass Condition 3 and ωD Condition
26
ωA
ωB
ωC
ωD
-0.08
ωE
-0.1
ωF
-0.12
Time (sec)
Figure 23 Computer Simulation Mass Condition 3 and All ω Conditions
0
-0.04
-0.06
ωD
3.2.2 Parametric Weight Change Adams’ Model Simulation
The masses were changed as specified in Table 2,2, and 3. The simulation was
conducted on Adams with each successive simulation utilizing a different change in the angular
velocities as specified. The greatest deviation occurred in Mass Condition 3 when all the initial
angular velocities, ωD, were the same. This is what was expected--that the greatest deviation
would occur when the masses made up a greater proportion of the overall mass of the system.
In mass condition 1, all the masses were set to 10 gm, composing of 13% of the overall
weight of the projectile, as laid out in Table 2. When an angular rotation was introduced, again
as in Table 2, the missile trajectory changed coming out of the plane of travel as seen in Figure
25. The greatest deviation occurred for ωD and this action is exemplified in Figure 26 with a
total deviation of 38 mm.
20.00
0.00
-10.00
ωA
0.00
0.12
0.24
0.36
0.48
0.60
0.71
0.83
0.95
1.07
1.19
1.31
1.43
1.55
1.67
1.79
1.90
2.02
2.14
2.26
2.38
2.50
2.62
2.74
2.86
2.98
3.09
Lateral Deviation (mm)
10.00
ωB
ωC
-20.00
ωD
ωE
-30.00
ωF
-40.00
-50.00
Time (sec)
Figure 25 Adams’ Simulation Mass Condition 1 and All ωConditions
27
-5
0.00
0.12
0.25
0.37
0.50
0.62
0.74
0.87
0.99
1.12
1.24
1.36
1.49
1.61
1.74
1.86
1.98
2.11
2.23
2.36
2.48
2.60
2.73
2.85
2.98
3.10
0
Lateral Deviation (mm)
-10
-15
-20
ωD
-25
-30
-35
-40
-45
Time (sec)
Figure 26 Adams’ Simulated Deviation for Phase 1 Condition, Mass 1 Conditions for ωD
The weight was increased and subjected to the same simulation as the first condition.
The increased weight increased the lateral distance the projectile moved. The extent of the
lateral movement is seen in Figure 27, with the greatest deviation, as seen before in ωD, as
shown in Figure 28, is 27 mm.
28
30.00
25.00
ωA
15.00
ωB
10.00
ωC
5.00
ωD
ωE
0.00
-5.00
0.00
0.12
0.24
0.36
0.48
0.60
0.71
0.83
0.95
1.07
1.19
1.31
1.43
1.55
1.67
1.79
1.90
2.02
2.14
2.26
2.38
2.50
2.62
2.74
2.86
2.98
3.09
Lateral Deviation (mm)
20.00
ωF
-10.00
-15.00
Time (sec)
Figure 27 Adams’ Simulation Mass Condition 2 and All ωConditions
30.00
Lateral Deviation (mm)
25.00
20.00
15.00
ωD
10.00
5.00
-5.00
0.00
0.12
0.24
0.36
0.48
0.60
0.71
0.83
0.95
1.07
1.19
1.31
1.43
1.55
1.67
1.79
1.90
2.02
2.14
2.26
2.38
2.50
2.62
2.74
2.86
2.98
3.09
0.00
Time (Sec)
Figure 28 Adams’ Simulated Deviation for Phase 1 Condition Mass 2 Conditions for ωD
As with the other Adams’ simulations, the greatest deviation for Weight Condition 3 was
in the ωD condition which is visible in Figure 29 and exemplified in Figure 30 with a value of 49
29
mm. While in Mass Condition 1 the greatest deviation had a negative slope, the projectile
deviation Mass Condition 2 and 3 had a positive trend in the slope for ωD.
60
40
ωA
30
ωB
ωC
20
ωD
10
ωE
0
-10
-20
ωF
0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00
1.11
1.22
1.33
1.44
1.55
1.67
1.78
1.89
2.00
2.11
2.22
2.33
2.44
2.55
2.66
2.78
2.89
3.00
3.11
Lateral Deviation (mm)
50
Time (sec)
Figure 29 Adams’ Simulation for Mass Condition 3 and All ωConditions
60
Lateral Deviation (mm)
50
40
30
ωD
20
10
-10
0.00
0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1.00
1.11
1.22
1.33
1.44
1.55
1.67
1.78
1.89
2.00
2.11
2.22
2.33
2.44
2.55
2.66
2.78
2.89
3.00
3.11
0
Time (sec)
Figure 30 Adams’ Simulated Deviation for Phase 1, Condition Mass 3 Conditions for ωD
30
3.3
Parametric Phase Change Study
Phase in this study is specified as the location of the rotating masses with respect minor
axes. The initial phase of the mass was changed by rotating the mass from the starting position,
as shown in Figure 31, to an evenly-balance position about the major axis, as shown in Figure
32. The position of the masses corresponds to Table 5 and Table 6 respectfully. Each of the
phase changes was subjected to similar parametric studies as to weight changes. It was found
that if all the initial angular velocities of the mass were the same for all three masses, as stated
for ωD, the movement of the projectile would be cyclic with the mean being a negated
movement as revealed in Figure 34 and Figure 36.
Figure 31 All Masses 90 Degrees as Specified in Phase Condition 1
Figure 32 Masses in Equilateral Triangle as Specified in Phase Condition 2
31
Table 5 ω and Phase Condition 1
𝜔Condition
𝜔1 (rpm)
𝜔2 (rpm)
𝜔3 (rpm)
𝜔A
0
0
0
𝜔B
167
0
0
𝜔C
167
167
0
𝜔D
167
167
167
𝜔E
167
-83.3
-83.3
𝜔F
0
0
-83.3
𝜃1 = 𝜃2 = 𝜃3 = 90°
𝑚𝑑,1 = 𝑚𝑑,2 = 𝑚𝑑,3 = 10𝑚𝑔
32
Table 6 ω and Phase Condition 2
𝜔Condition
𝜔1 (rpm)
𝜔2 (rpm)
𝜔3 (rpm)
𝜔A
0
0
0
𝜔B
167
0
0
𝜔C
167
167
0
𝜔D
167
167
167
𝜔E
167
-83.3
-83.3
𝜔F
0
0
-83.3
𝜃1 = 90, 𝜃2 = 210, 𝜃3 = 330°
𝑚𝑑,1 = 𝑚𝑑,2 = 𝑚𝑑,3 = 10𝑚𝑔
3.3.1 Parametric Phase Change Computer Simulation
Using the same computer simulation program on MATLAB, the initial starting phase of
the masses wqas changed according to the Table 5 and Table 6. The results from Condition 1, as
specified in Table 5, were exactly the same, produced the results as shown in Figure 19.
The results from Phase Condition 2, as specified in Table 6, are shown in Figure 33.
The greatest deviation occurred for 𝜔E, shown by the light blue line in Figure 33 with a total
deviation of 19 mm. In the case of ωD, the gyroscopic force was all but negated within the
system, and the projectile path was along the center line with no deviation laterally, as shown in
Figure 34.
33
Lateral Deviation (m)
-0.1
-0.5
0.00
0.12
0.23
0.35
0.46
0.58
0.69
0.81
0.92
1.03
1.15
1.26
1.38
1.49
1.61
1.72
1.84
1.95
2.07
2.18
2.30
2.41
2.53
2.64
2.76
2.87
2.99
3.10
Lateral Deviation (m)
-0.005
-0.025
0
0.119
0.238
0.357
0.476
0.595
0.714
0.833
0.952
1.071
1.19
1.309
1.428
1.547
1.666
1.785
1.904
2.023
2.142
2.261
2.38
2.499
2.618
2.737
2.856
2.975
3.094
0.005
0
-0.01
-0.015
0
-0.2
-0.3
-0.4
Time (sec)
Figure 34 Computer Simulation of Mass Condition 1, Phase Condition 2 for ωD
34
ωA
ωB
ωC
ωD
ωE
ωF
-0.02
Time (sec)
Figure 33 Computer Simulation of Mass Condition 1 Phase Condition 2
0.5
0.4
0.3
0.2
0.1
ωD
3.3.2 Parametric Phase Change Adams’ Model
The conditions for Phase Condition 1 are the same as in Table 2 and responded in the
same manner, with the results shown in Figure 19.
According to the Adams’ model, when the phase is changed according to Phase
Condition 2, the deflection of the model behaves as predicted, as shown in Figure 35. The
greatest deflection occurs in ωE condition with a value of 21 mm. But when all the angular
velocities are the same, as in ωD, the projectile does not have any deflection as shown in Figure
36. This action is on par with the computer simulation as discussed.
5.00
0.00
0.12
0.23
0.35
0.46
0.58
0.69
0.81
0.92
1.04
1.15
1.27
1.38
1.50
1.61
1.73
1.84
1.96
2.07
2.19
2.30
2.42
2.53
2.65
2.76
2.88
2.99
3.11
Lateral Deviation (mm)
0.00
-5.00
ωA
ωB
ωC
-10.00
ωD
ωE
-15.00
ωF
-20.00
-25.00
Time (sec)
Figure 35 Adams’ Simulation of Mass Condition 1 Phase Condition 2
35
Lateral Deviation (mm)
-5
-20
0.00
0.12
0.23
0.35
0.46
0.58
0.69
0.81
0.92
1.04
1.15
1.27
1.38
1.50
1.61
1.73
1.84
1.96
2.07
2.19
2.30
2.42
2.53
2.65
2.76
2.88
2.99
3.11
20
15
10
5
0
-10
-15
Time (sec)
Figure 36 Adams’ Simulation of Mass Condition 1 Phase Condition 2 for ωD
36
ωD
CHAPTER 4
RESULTS AND COMPARISONS OF PARAMETRIC STUDY
The selected results of the deviation were over overlaid with each other for comparison.
The effect of the different masses and starting phase along with programs, MATLAB and
Adams, used to calculate the deviation were compared and commented on.
4.1
Comparison of Reaction to Parametric Mass Change
The case for which the greatest deviation occurred varied from mass case and what
program was used to calculate the distance traveled as will be shown in the proceeding sections.
Phase 1, according to Table 5, was used for all initial phases for the Mass Conditions changes.
4.1.1 Computer Simulation Comparison of Reaction to Mass Change
According to MATLAB, Mass Condition 2 had the greatest overall deviation for the Case
ωD, as seen in Figure 23; this trend continued in all the other cases looked at where ωD had the
greatest deviation. In case of ωB, ωC, and ωD the trend of the projectile was to have a negative
slope regardless of what mass was being used. This can be seen in Figure 37, Figure 57 in the
appendices, and in
Figure 38 for each case respectively. In case of ωF the slope of the trajectory was
positive, as seen in Figure 40, because of the one mass rotating in an opposite direction when
compared to ωB. In ωE, the masses swinging in the opposite direction of the upper mass had a
net canceling effect, and the projectile merely alternated the direction of traveling, as shown in
Figure 39, with a mean deviation of zero.
37
-0.01
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
0.00
Lateral Deviation (m)
-0.02
-0.03
-0.04
Mass1 ωB
-0.05
Mass2 ωB
Mass3 ωB
-0.06
-0.07
-0.08
-0.09
Time (sec)
Figure 37 MATLAB Mass Comparison of ωB
0.00
0.00.10.20.40.50.60.70.91.01.11.21.41.51.61.71.92.02.12.22.42.52.62.72.93.03.1
Lateral Deviation (m)
-0.02
-0.04
Mass1 ωD
-0.06
Mass2 ωD
-0.08
Mass3 ωD
-0.10
-0.12
-0.14
Time (sec)
Figure 38 MATLAB Mass Comparison of ωD
38
0.00
0.00
0.00
0.00
Mass1 ωE
0.00
0.00
0.0
0.1
0.2
0.4
0.5
0.6
0.7
0.9
1.0
1.1
1.2
1.4
1.5
1.6
1.7
1.9
2.0
2.1
2.2
2.4
2.5
2.6
2.7
2.9
3.0
3.1
Lateral Deviation (m)
0.00
Mass2 ωE
Mass3 ωE
0.00
0.00
0.00
0.00
Time (sec)
Figure 39 MATLAB Mass Comparison of ωE
0.03
Lateral Deviation (m)
0.02
0.02
Mass1 ωF
0.01
Mass2 ωF
Mass3 ωF
0.01
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
0.00
Time (sec)
Figure 40 MATLAB Mass Comparison of ωF
4.1.2 Adams’ Simulation Comparison of Reaction to Mass Change
The result from the Adams’ simulation was somewhat chaotic in the projectile’s manner
of traveled. For all masses cases the largest overall deviation occurred for ωD, as seen in Figure
39
41, Figure 57 in the appendices, and Figure 42. The greatest overall deviation occurred for Mass
Condition 3 with the masses rotating as specified for condition ωD with a total of 49 mm of
deflection, more readily observed in Figure 30.
Intriguing were the other cases as the projectile traveled during flight. For ωB, as seen in
Figure 41, Mass Conditions 2 and 3 moved slightly off the 0 line and then traveled nearly
parallel in the horizontal direction. While Mass condition 1 in ωF had a positive slope as in
Figure 44 but Mass Conditions 2 and 3 had an initial positive slope, reversed course passing back
over the 0 line, and continued with a shallow negative slope. The same is true for ωE in Figure
43. Overall, it appeared that the projectile was tossed about with the larger mass and did not
exhibit fully cyclic functions.
10.00
8.00
6.00
2.00
Mass1 ωB
0.00
Mass2 ωB
-2.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
Lateral Deviation (mm)
4.00
-4.00
-6.00
-8.00
Time (sec)
Figure 41 Adams Mass Comparison of ωB
40
Mass3 ωB
60.00
50.00
40.00
Lateral Deviation (mm)
30.00
20.00
Mass1 ωD
0.00
Mass2 ωD
-10.00
0.0
0.1
0.2
0.4
0.5
0.6
0.7
0.8
1.0
1.1
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.0
2.1
2.3
2.4
2.5
2.6
2.7
2.9
3.0
3.1
10.00
Mass3 ωD
-20.00
-30.00
-40.00
-50.00
Time (sec)
Figure 42 Adams Mass Comparison of ωD
15.00
5.00
0.00
Mass1 ωE
0.0
0.1
0.2
0.4
0.5
0.6
0.7
0.8
1.0
1.1
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.0
2.1
2.3
2.4
2.5
2.6
2.7
2.9
3.0
3.1
Lateral Deviation (mm)
10.00
-5.00
Mass2 ωE
Mass3 ωE
-10.00
-15.00
-20.00
Time (sec)
Figure 43 Adams Mass Comparison of ωE
41
10.00
0.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
Lateral Deviation (mm)
5.00
-5.00
Mass1 ωF
Mass2 ωF
Mass3 ωF
-10.00
-15.00
-20.00
Time (sec)
Figure 44 Adams Mass Comparison of ωF
4.1.3 Comparison of Adams Model with the Computer Simulation for Parametric Masses
Changes
The results of the computer simulation compared to the results gathered from Adams
express similar trends for about half of the cases examined. For the Mass 1 case seen in Figure
45, the ωB condition the MATLAB and Adam trend is to deviate in the opposite direction, with
the MATLAB slope being more severe. The only condition for which the projectile has the same
trajectory is for ωF, where both simulations show the projectile having positive slopes which
nearly overlap. For ωE, the path differed, with the Adams’ simulation sloping downward while
the computer simulations held to the zero line.
42
15.00
5.00
MATLAB Mass1 ωB
MATLAB Mass1 ωE
0.00
0.0
0.1
0.3
0.4
0.6
0.7
0.8
1.0
1.1
1.3
1.4
1.5
1.7
1.8
2.0
2.1
2.2
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (mm)
10.00
ADAMS Mass1 ωB
ADAMS Mass1 ωE
-5.00
-10.00
-15.00
Time (sec)
Figure 45 Adams verse MATLAB Mass1 of ωB and ωE
For Mass Case 2, as shown in Figure 46, the results were slightly different from the
results in Mass Case 1. The trajectories of the projectile move closely together for the case of
ωB with both trajectories sloping in opposite directions. For ωE, the path of the projectile was
even closer with the computer simulation path holding to the zero line while the Adams model
moves slightly away with an average slope of -0.002. Unlike Mass Case 2, the ωF path deviates
largely from each other, nearly the same as the case of ωB with the computer simulations
deviating away quickly with a positive slope, while the Adams’ model cleaves to the zero line.
43
10.00
5.00
-5.00
0.0
0.1
0.3
0.4
0.6
0.7
0.8
1.0
1.1
1.3
1.4
1.5
1.7
1.8
2.0
2.1
2.2
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (mm)
0.00
MATLAB Mass2 ωB
-10.00
MATLAB Mass2 ωE
-15.00
ADAMS Mass2 ωB
-20.00
ADAMS Mass2 ωE
-25.00
-30.00
-35.00
-40.00
Time (sec)
Figure 46 Adams verse MATLAB Mass2 of ωB and ωE
In the Mass Condition 3, the behavior was identical to Mass Condition 2 with all of the
projectile paths deviating in the same manner to each other, only with greater magnitude.
44
20.00
0.00
0.0
0.1
0.3
0.4
0.6
0.7
0.8
1.0
1.1
1.3
1.4
1.5
1.7
1.8
2.0
2.1
2.2
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (mm)
10.00
-10.00
MATLAB Mass3 ωB
MATLAB Mass3 ωE
ADAMS Mass3 ωB
-20.00
ADAMS Mass3 ωE
-30.00
-40.00
-50.00
Time (sec)
Figure 47 Adams verse MATLAB Mass3 of ωB and ωE
The results of ωE in all the mass cases closely resembled each other for the most part, as
visible in Figure 48. The actions of the paths are very similar with only a slight phase shift
between where the peaks of the curve of the paths occur. The biggest difference is that the
computer simulation holds tightly to the zero axis line while the Adams’ model has a slight slope
in all cases.
45
15.00
5.00
MATLAB Mass1 ωE
MATLAB Mass2 ωE
0.00
0.0
0.1
0.3
0.4
0.6
0.7
0.8
1.0
1.1
1.3
1.4
1.5
1.7
1.8
2.0
2.1
2.2
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (mm)
10.00
MATLAB Mass3 ωE
ADAMS Mass1 ωE
-5.00
ADAMS Mass2 ωE
-10.00
ADAMS Mass3 ωE
-15.00
-20.00
Time (sec)
Figure 48 Adams verse MATLAB Mass 1, 2, and 3 of ωE
4.2
Comparison of Reaction to Parametric Phase Change
The two cases of initial phase that were examined were when the rod of the masses were
parallel to each other as shown in Figure 31. The other case was when all masses were equally
spaced apart with the angle between the rods being 120 degrees, as shown in Figure 32. The
case of Phase 1 among the different simulations has already been compared above. In all cases
when Phase 1 is compared to Phase 2 for case ωB, the deviation of the projectile is identical as
would be expected because it is the center mass which is in a vertical position for both cases.
4.2.1 Computer Simulation Comparison of Reaction to Phase Change
In the MATLAB simulation, the difference in the deviation of the projectile depending on
phases can be seen in Figure 49. The greatest difference in the action of the projectile took place
46
for ωD and ωE, which is amplified in Figure 50. The projectile path behaved as expected in that,
for Phase2, the projectile stayed along the axis for ωD and deviated for ωE. But for Phase1 it
behaved in the opposite manner--deviating for ωD and traveling along the axis for ωE.
0.01
0.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
Lateral Deviation (m)
0.01
-0.01
Phase1 Mass1 ωB
Phase1 Mass1 ωE
Phase1 Mass1 ωF
Phase2 Mass1 ωB
-0.01
Phase2 Mass1 ωE
-0.02
Phase2 Mass1 ωF
-0.02
-0.03
Time (sec)
Figure 49 MATLAB Phase 1 and 2 Comparison Mass1 of ωB and ωE
47
0.01
Lateral Deviation (m)
-0.01
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
0.00
-0.01
-0.02
Phase1 Mass1 ωD
-0.02
Phase1 Mass1 ωE
-0.03
Phase2 Mass1 ωD
Phase2 Mass1 ωE
-0.03
-0.04
-0.04
-0.05
Time (sec)
Figure 50 MATLAB Phase 1 and 2 Comparison Mass1 of ωD and ωE
4.2.2 Adams’ Simulation Comparison of Reaction to Phase Change
The difference in phase made similar changes to the Adams’ model as it did for the
computer simulations, as observed in Figure 51. In the case of ωE and ωF, the path of the
projectile in Phase 1 was opposite of the path for Phase 2. Also, the paths for ωB were similar
because on the top mass was spinning and the difference in the path can be attributed to the
slightly different moments of inertia for the projectile due the position of the other masses.
Figure 52 shows the path of the projectile for ωD and ωE for both phases. The difference in the
two phases is as expected, with the path staying close to the axis for ωD in Phase 2 and ωE for
Phase 1 while deviating for ωE and ωD for Phase 1 and 2 respectively.
48
15.00
5.00
Phase1 Mass1 ωB
0.00
-5.00
0.0
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
1.5
1.6
1.7
1.9
2.0
2.1
2.3
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (mm)
10.00
Phase1 Mass1 ωE
Phase1 Mass1 ωF
Phase2 Mass1 ωB
-10.00
Phase2 Mass1 ωE
-15.00
Phase2 Mass1 ωF
-20.00
-25.00
Time (sec)
Figure 51 Adams’ Phase 1 and 2 Comparison Mass 1 of ωB, ωE, and ωF
20.00
0.00
0.0
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
1.5
1.6
1.7
1.9
2.0
2.1
2.3
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (mm)
10.00
-10.00
Phase1 Mass1 ωD
Phase1 Mass1 ωE
-20.00
Phase2 Mass1 ωD
Phase2 Mass1 ωE
-30.00
-40.00
-50.00
Time (sec)
Figure 52 Adams’ Phase 1 and 2 Comparison Mass 1 of ωD and ωF
49
4.2.3 Comparison of Adams’ Model with Computer Simulation for Parametric Phase
Change
When the simulations are compared for difference for phase change, the results from each
program generally mirror each other, especially in Phase1; this can be seen in Figure 53. In
Figure 53, the paths of the projectile for ωB, ωD, ωE, and ωF lie side by side or overlap each
other for both programs. The path for both programs’ case of ωD lies on the axis, covering each
other.
5.00
Lateral Deviation (mm)
0.00
0.00.10.30.40.60.70.91.01.21.31.51.61.81.92.02.22.32.52.62.82.93.1
MATLAB Phase2 Mass1 ωB
MATLAB Phase2 Mass1 ωD
-5.00
MATLAB Phase2 Mass1 ωE
MATLAB Phase2 Mass1 ωF
-10.00
ADAMS Phase2 Mass1 ωB
-15.00
ADAMS Phase2 Mass1 ωD
ADAMS Phase2 Mass1 ωE
-20.00
-25.00
ADAMS Phase2 Mass1 ωF
Time (sec)
Figure 53 Computer Simulation versus Adams’ Phase 2, Mass 1 of ωB, ωE, and ωF
50
CHAPTER 5
CONCLUSION
The projectile deviated from line of travel when it was subjected to an internally gyroscopic
force. Because the deviation of the projectile can be controlled, the flight path could be
corrected to counteract an undesired movement in the lateral direction. The deviation varied
depending on the initial conditions, with the greatest deviation just over 100 mm in Mass
Condition 2, Phase 1, for ωD, as seen in Figure 22 for the computer simulation. And the greatest
deviation for the Adams’ simulation occurred in Mass Condition 3, Phase 1, for ωD as visible in
Figure 30. Yet, considering the diameter of the projectile being considered, 20 mm, the
deviation was just over 5 times the diameter. With a continuous slope over a range of 2500
meters, the implementation of this system is probably not worth the effort or the expense. Also
with the given slope of the deviation, it renders nearly impractical for arcing the projectile
around an object in the flight path or correcting to external deviating forces.
Lateral Deviation (mm)
140
120
100
MATLAB Phase 1 Mass 1
80
MATLAB Phase 1 Mass 2
60
MATLAB Phase 1 Mass 3
40
Adams Phase 1 Mass 1
20
Adams Phase 1 Mass 2
0
Adams Phase 1 Mass 3
A
B
C
D
E
F
ω Conditions
Figure 54 Maximum Deviation Comparison for all Mass Conditions, all ω Conditions, for
Phase 1, both Adams' and MATLAB
51
Lateral Deviation (mm)
45
40
35
30
25
MATLAB Phase 1 Mass 1
20
MATLAB Phase 2 Mass 1
15
Adams Phase 1 Mass 1
10
Adams Phase 2 Mass 1
5
0
A
B
C
D
E
F
ω Conditions
Figure 55 Maximum Deviation Comparison for Mass 1, all ω Conditions, for Phase 1 and
2, both Adams' and MATLAB
5.1
Improvements
To improve this system, one could devise a control system to do ‘real time’ correction for
minor deviations. More research in the effect of the initial phase of the mass could be examined
or orienting the position of the rotation axis to not parallel to the major axis of the projectile.
Moving the orientation of the axis would have a similar effect, but initial speculation would
anticipate a more chaotic flight path.
5.2
Areas for Further Research
The speculative development and production cost of design for one-use projectiles would
most likely adversely outweigh any improved accuracy adversely. The funds and efforts could
be put to better use in designing a more robust adaptive targeting system to consider initial
conditions before launching the solid core round at the target. For far larger applications such as
reentry vehicles the idea of rotating internal members may have use by adding stability or control
to the system. This concept may pose merit, especially if already existing systems such as robot
52
arms could be implemented and if it would merely be adding a control system to existing
onboard computers.
Overall, the movement of the projectile behaved as expected with greatest deviation
occurring for the case of ωD for the case of Phase 1 yet this movement was negated when the
initial phase was changed to Phase 2 for the MATLAB model and the Adams’ model. Also, for
ωE for Phase 1, the lateral deviation was minimal, as was anticipated. Overall, this research was
productive and may have possible application for other flight vehicles other than small simple
projectiles.
53
REFERENCES
54
REFERENCES
[1] G. Frost and M. Costello, "Stability and Control of a Projectile with an Internal Rotating
Disk," in AIAA Atmospheric Flight Mechanics Conference and Exhibit, Austin, TX, 2003.
[2] R. H. Byrne, R. D. Robinett, and B. R. Sturgis, "Moving Mass Trim Control System Design,"
in AIAA, Guidance, Navigation and Control Conference, San Diego, Jul. 1996, pp. 1-9.
[3] J. Rogers and M. Costello, "Flight Dynamics and Control Authority of a Projectile Equipped
with a Controllable Internal Translating Mass," in AIAA Atmospheric Flight Mechanics
Conference and Exhibit, Hilton Head, SC, 2007, pp. 1-24.
[4] J. Rogers and M. Costello, "A Variable Stability Projectile Using an Internal Moving Mass,"
in AIAA Atmospheric Flight Mechanics Conference and Exhibit, Honolulu, HI, 2008.
[5] G. Frost and M. Costello, "Linear Theory of a Rotating Internal Part Projectile Configuration
in Atmospheric Flight," Journal of Guidance, Control, and Dynamics, vol. 27, no. 5, pp. 898905, Sep. 2004.
[6] D. Ollerenshaw and M. Costello, "Simplified Projectile Swerve Solution for General Control
Inputs," Journal of Guidance, Control, and Dynamics, vol. 31, no. 5, pp. 1259-1265, Sep.
2008.
[7] L. C. Hainz III and M. Costello, "Modified Projectile Linear Theory for Rapid Trajectory
Prediction," Journal of Guidance, Control, and Dynamics, vol. 28, no. 5, pp. 1006-1014, Sep.
2005.
[8] F. P. Beer, E. R. Johnston, and W. E. Clausen Jr., Vector Mechanics for Engineers Dynamics,
7th ed. New York, NY: McGrawHill, 2004.
55
APPENDICES
56
APPENDIX A
COMPUTER SIMULATION CODE
Main Program Code
%Christopher Umstead
%Thesis Program
%Fall 2009
%prepping program and clearing all constants
clc; clear;
%constant variables
r=0.003; mp=0.100; g=-9.81;
%projectile body geometry and some initial conditions
loc=[0, 2*sqrt(3)*r/3, 0;0, -sqrt(3)*r/3, -r; 0, sqrt(3)*r/3, r]';
Othetai=0; Owi=0;
R=(2*sqrt(3)/3+1)*r;
cd=0.03;
%acceleration of the masses
alpha=[0;0;0];
%time and time step
t0=0; ti=0.001; tf=3.2;
%initial conditions
aangle=15; %angle of attack
vi=[800; 0; 0]; %initial linear velocity
ai=[0; 0; 0]; %initial acceleration
%%note counting numbers i will be for conditions changes and j will be used for
%%time tracking
%%Omega%%
%different conditions for omega in deg/sec
omegaC=-1*[0, 0, 0; 1000, 0, 0; 1000, 1000, 0; 1000, 1000, 1000; 1000, -500, -500; 0, 0,-500]';
omegaCsize=size(omegaC);
%%%%mass changing%%%%
massC=[0.01, 0.05, 0.1];
phase=[90; 90; 90];
i=1;
57
APPENDIX A (continued)
massCsize=size(massC);
while i<=massCsize(1,2)
%setting variables
%naming file for output
namea=['MassCon' int2str(i)];
m=ones(3,1)*massC(1,i);
p=[0;0;0];
%%omega change%%
ii=1;
while ii<=omegaCsize(1,2)
%naming file for output
nameb=[namea 'OmegaCon' int2str(ii)];
%setting initial condition for parametric changes
w=omegaC(:,ii);
v=vi;
theta=phase;
Otheta=Othetai;
Ow=Owi;
%calculating moment of inertia
ID=0;
for ilj=1:3
ID=ID+m(ilj,1)*norm(r*[0; sin(theta(ilj,1)*pi/180); cos(theta(ilj,1)*pi/180)]+loc(:,ilj))^2;
end
I=(1/2)*mp*R^2+ID;
%priming the loop
t=t0; j=1;
while t(j)<=tf
%calculating the instantaneous acceleration and forces of the
%masses by calling sub program
for mj=1:3
fdisk(:,mj)=acel(m(mj,1), w(mj,1), r, theta(mj,j)+Otheta(1,j), alpha(mj,1));
end
%Drag
Fdrag=-(1/8)*pi*R^2*cd*norm(v(:,j))*v(:,j);
%calculating the instantaneous acceleration of the projectile due
58
APPENDIX A (continued)
%to the masses
a=zeros(3,1);
a(1,1)=g*sin(aangle*pi/180);
a(2,1)=g*cos(aangle*pi/180);
a=a(:,1)+(fdisk*ones(3,1)+Fdrag)/(sum(m)+mp);
%calculating the angular acceleration of the projectile
tau=0;
for lj=1:3
A=[1, 0, 0; 0, cos(Otheta(j)*pi/180), -sin(Otheta(j)*pi/180); 0,
sin(Otheta(j)*pi/180), cos(Otheta(j)*pi/180)]*loc(:,lj);
B=fdisk(:,lj);
tau=tau+cross(A,B);
end
Oalpha=tau/I;
%progressing counter
j=j+1;
%calculating angle of rotation for the projectile
Ow(j)=Ow(j-1)+sum(Oalpha)*ti;
Otheta(j)=Otheta(j-1)+Ow(j)*ti;
%calculating velocity and position
v(:,j)=v(:,j-1)+a*ti;
p(:,j)=p(:,j-1)+v(:,j)*ti;
theta(:,j)=theta(:,j-1)+w*ti;
%calculating new moment of inertia for the projectile due to
%the movement of the masses
ID=0;
for ilj=1:3
ID=ID+m(ilj,1)*norm(r*[0; sin(theta(ilj,j)*pi/180);
cos(theta(ilj,j)*pi/180)]+loc(:,ilj))^2;
end
I=(1/2)*mp*R^2+ID;
t(j)=t(j-1)+ti;
end
%saving selected data to file with predetermined name
59
APPENDIX A (continued)
save(nameb, 'v', 'p', 'theta', 'Ow', 'Otheta', 't');
%progressing the counter
ii=ii+1;
end
%%End Omega Change%%
i=i+1;
end
%%%Phase Change%%%
%resetting and setting initial conditions
m=[0.01; 0.01; 0.01];
phaseC(:,1)=[90; 90; 90];
phaseC(:,2)=[90; 210; 330];
i=1;
phaseCsize=size(phaseC);
while i<=phaseCsize(1,2)
%setting variables
namea=['PhaseCon' int2str(i)];
theta=phaseC(:,i)
p=[0;0;0];
%%omega change%%
ii=1;
while ii<=omegaCsize(1,2)
nameb=[namea 'OmegaCon' int2str(ii)];
w=omegaC(:,ii);
v=vi;
Otheta=Othetai;
Ow=Owi;
ID=0;
for ilj=1:3
ID=ID+m(ilj,1)*norm(r*[0; sin(theta(ilj,1)*pi/180);
cos(theta(ilj,1)*pi/180)]+loc(:,ilj))^2;
end
I=(1/2)*mp*R^2+ID;
%priming the loop
t=t0; j=1;
60
APPENDIX A (continued)
while t(j)<=tf
%calculating the instantaneous acceleration and forces of the
%masses by calling a sub program
for mj=1:3
fdisk(:,mj)=acel(m(mj,1), w(mj,1), r, theta(mj,j)+Otheta(1,j), alpha(mj,1));
end
%Drag
Fdrag=-(1/8)*pi*R^2*cd*norm(v(:,j))*v(:,j);
%calculating the instantaneous acceleration of the projectile due
%to the masses
a=zeros(3,1);
a(1,1)=g*sin(aangle*pi/180);
a(2,1)=g*cos(aangle*pi/180);
a=a(:,1)+(fdisk*ones(3,1)+Fdrag)/(sum(m)+mp);
%calculating the angular acceleration of the projectile
tau=0;
for lj=1:3
A=[1, 0, 0; 0, cos(Otheta(j)*pi/180), - sin(Otheta(j)*pi/180); 0,
sin(Otheta(j)*pi/180), cos(Otheta(j)*pi/180)]*loc(:,lj);
B=fdisk(:,lj);
tau=tau+cross(A,B);
end
Oalpha=tau/I;
%progressing counter
j=j+1;
%calculating angle of rotation for the projectile
Ow(j)=Ow(j-1)+sum(Oalpha)*ti;
Otheta(j)=Otheta(j-1)+Ow(j)*ti;
%calculating velocity and position
v(:,j)=v(:,j-1)+a*ti;
p(:,j)=p(:,j-1)+v(:,j)*ti;
theta(:,j)=theta(:,j-1)+w*ti;
%calculating new moment of inertia for the projectile due to
%the movement of the masses
61
APPENDIX A (continued)
ID=0;
for ilj=1:3
ID=ID+m(ilj,1)*norm(r*[0; sin(theta(ilj,j)*pi/180);
cos(theta(ilj,j)*pi/180)]+loc(:,ilj))^2;
end
I=(1/2)*mp*R^2+ID;
t(j)=t(j-1)+ti;
end
save(nameb, 'v', 'p', 'theta', 'Ow', 'Otheta', 't');
%progressing the counter
ii=ii+1;
end
%%End Omega Change%%
%progressing the counter
i=i+1;
end
%plotting graphs by calling the sub program
graphgenerator;
Subprogram Code A
%Christopher Umstead
%Thesis Sub Program
%Fall 09
%sub program to calculate the instantaneous force due to rotation of the
%masses
function f=acel(m, w, r, theta, alpha)
%force due to normal acceleration of the rotating disk (omega x omega x r)
f1=m*cross(w*[1;0;0]*pi/180,cross(w*[1;0;0]*pi/180,r*[0; sin(theta*pi/180);
cos(theta*pi/180)]));
%force due to the tangential acceleration of the rotating disk (alpha x r)
f2=-m*cross((alpha*pi/180)*[1;0;0],r*[0; sin(theta*pi/180);cos(theta*pi/180)]);
%summing up and returning the two forces
62
APPENDIX A (continued)
f=f1+f2;
Subprogram Code B
%Christopher Umstead
%Thesis sub program
%Fall 2009
%Note, a large portion of this code was found online a couple of years ago
%and has been heavily modified for the selected process. The original source of the
%has been lost
%Setting the directory to load file from
figdirectory = 'C:\Users\Umstead\Desktop\Thesis\ThesisMatLab\CleanProgram';
%change the directory name to match the desired location.
%setting path name to direct the saving of the figures
fullpath = sprintf('%s/*.mat',figdirectory);
d = dir(fullpath);
length_d = length(d);
%counting number of files in the directory
if(length_d == 0)
disp('couldnt read the directory details\n');
disp('check if your files are in correct directory\n');
end
%creating file to save figures to
f1=fullfile(figdirectory,'Graphs');
if (exist(f1) == 0)
mkdir (f1);
end
%setting counters for the loop
startfig = 1;
endfig = length_d;
%loop to open the files and create the graphs
for i = startfig:endfig
fname = d(i).name;
load(fname)
63
APPENDIX A (continued)
%creating and saving graph of deviation vs time
figure;plot(t,p(3,:));title(fname);ylabel('Lateral Devation (m)');...
xlabel('time (sec)');
fname_output = sprintf('%s/%s.jpg',[figdirectory '\Graphs'],fname);
saveas(gcf,fname_output,'jpg');
pause(0.05);
close all
%Creating and saving graph of elevation vs distance
figure;plot(p(1,:),p(3,:));title([fname ' Elivation vs Distance']);ylabel('Lateral Devation (m)');...
xlabel('Down Range (m)');
fname_output = sprintf('%s/%s.jpg',[figdirectory '\Graphs'],[fname '2']);
saveas(gcf,[fname_output],'jpg');
pause(0.05);
close all
%creating and saving angular orientation vs Time
figure;plot(t,Otheta(1,:));title([fname ' Angular Orentation vs Time']);ylabel('Angle (deg)');...
xlabel('Time (sec)');
fname_output = sprintf('%s/%s.jpg',[figdirectory '\Graphs'],[fname '3']);
saveas(gcf,[fname_output],'jpg');
pause(0.05);
close all
end
64
APPENDIX B
SELECTED COMPUTER SIMULATION GRAPHS

Computer Simulation for Phase 1, Masses 1, 2, and 3, for ωA is not shown because it
would merely be a line along the axis for all three cases
-0.01
0.0
0.1
0.2
0.4
0.5
0.6
0.7
0.9
1.0
1.1
1.2
1.4
1.5
1.6
1.7
1.9
2.0
2.1
2.2
2.4
2.5
2.6
2.7
2.9
3.0
3.1
0.00
Lateral Deviation (m)
-0.01
-0.02
Mass1 ωB
-0.02
Mass2 ωB
-0.03
Mass3 ωB
-0.03
-0.04
-0.04
-0.05
Time (sec)
Figure 56 Computer Simulation Phase1 Mass 1, 2, and 3 for ωB
65
APPENDIX B (continued)
-0.01
0.0
0.1
0.2
0.3
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.3
1.4
1.5
1.6
1.7
1.8
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.8
2.9
3.0
3.1
0.00
Lateral Deviation (m)
-0.02
-0.03
-0.04
Mass1 ωC
-0.05
Mass2 ωC
Mass3 ωC
-0.06
-0.07
-0.08
-0.09
Time (sec)
Figure 57 Computer Simulation Phase1 Mass 1, 2, and 3 for ωC
0.00
0.00.10.20.40.50.60.70.91.01.11.21.41.51.61.71.92.02.12.22.42.52.62.72.93.03.1
Lateral Deviation (m)
-0.02
-0.04
Mass1 ωD
-0.06
Mass2 ωD
-0.08
Mass3 ωD
-0.10
-0.12
-0.14
Time (sec)
Figure 58 Computer Simulation Phase1 Mass 1, 2, and 3 for ωD
66
APPENDIX B (continued)
0.00
0.00
0.00
0.00
Mass1 ωE
0.00
0.0
0.1
0.2
0.3
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.3
1.4
1.5
1.6
1.7
1.8
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.8
2.9
3.0
3.1
0.00
Mass2 ωE
Mass3 ωE
0.00
0.00
0.00
0.00
Time (sec)
Figure 59 Computer Simulation Phase1 Mass 1, 2, and 3 for ωE
0.03
0.02
0.02
Mass1 ωF
0.01
Mass2 ωF
Mass3 ωF
0.01
0.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
Lateral Deviation (m)
Lateral Deviation (m)
0.00
Time (sec)
Figure 60 Computer Simulation Phase1 Mass 1, 2, and 3 for ωF
67
APPENDIX B (continued)

Computer Simulation for Phase 2, Masses 1 and 2 for ωA is not shown because it would
merely be a line along the axis for both cases.
-0.01
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
0.00
Lateral Deviation (m)
-0.01
-0.02
Phase2 Mass1 ωB
-0.02
Phase2 Mass2 ωB
-0.03
-0.03
-0.04
-0.04
Time (sec)
Figure 61 Computer Simulation Phase2 Mass 1 and 2 for ωB
68
APPENDIX B (continued)
0.00
Lateral Deviation (m)
0.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
0.00
0.00
-0.01
Phase2 Mass1 ωC
-0.01
Phase2 Mass2 ωC
-0.01
-0.01
-0.01
-0.02
Time (sec)
Figure 62 Computer Simulation Phase2 Mass 1 and 2 for ωC

Computer Simulation for Phase 2, Masses 1 and 2 for ωD is not shown because it would
merely be a line along the axis for both cases.
69
APPENDIX B (continued)
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
0.00
Lateral Deviation (m)
-0.01
-0.02
Phase2 Mass1 ωE
-0.03
Phase2 Mass2 ωE
-0.04
-0.05
-0.06
Time (sec)
Figure 63 Computer Simulation Phase2 Mass 1 and 2 for ωE
0.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
0.00
Lateral Deviation (m)
0.00
0.00
0.00
Phase2 Mass1 ωF
-0.01
Phase2 Mass2 ωF
-0.01
-0.01
-0.01
-0.01
-0.01
Time (sec)
Figure 64 Computer Simulation Phase2 Mass 1 and 2 for ωF
70
APPENDIX C
SELECTED Adams SIMULATION GRAPHS

Adams Simulation for Phase 1, Masses 1, 2, and 3, for ωA is not shown because it would
merely be a line along the axis for all three cases.
10.00
8.00
6.00
2.00
Mass1 ωB
0.00
Mass2 ωB
-2.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
Lateral Deviation (mm)
4.00
-4.00
-6.00
-8.00
Time (sec)
Figure 65 Adams Simulation Phase1 Mass 1, 2, and 3 for ωB
71
Mass3 ωB
APENDIX C (continued)
40.00
30.00
10.00
Mass1 ωC
Mass2 ωC
0.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
Lateral Deviation (mm)
20.00
Mass3 ωc
-10.00
-20.00
-30.00
Time (sec)
Figure 66 Adams Simulation Phase1 Mass 1, 2, and 3 for ωC
60.00
50.00
40.00
Lateral Deviation (mm)
30.00
20.00
Mass1 ωD
0.00
Mass2 ωD
-10.00
0.0
0.1
0.2
0.4
0.5
0.6
0.7
0.8
1.0
1.1
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.0
2.1
2.3
2.4
2.5
2.6
2.7
2.9
3.0
3.1
10.00
-20.00
-30.00
-40.00
-50.00
Time (sec)
Figure 67 Adams Simulation Phase1 Mass 1, 2, and 3 for ωD
72
Mass3 ωD
APENDIX C (continued)
15.00
5.00
0.00
Mass1 ωE
0.0
0.1
0.2
0.4
0.5
0.6
0.7
0.8
1.0
1.1
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.0
2.1
2.3
2.4
2.5
2.6
2.7
2.9
3.0
3.1
Lateral Deviation (mm)
10.00
-5.00
Mass2 ωE
Mass3 ωE
-10.00
-15.00
-20.00
Time (sec)
Figure 68 Adams Simulation Phase1 Mass 1, 2, and 3 for ωD
10.00
0.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
Lateral Deviation (mm)
5.00
-5.00
Mass2 ωF
Mass3 ωF
-10.00
-15.00
-20.00
Mass1 ωF
Time (sec)
Figure 69 Adams Simulation Phase1 Mass 1, 2, and 3 for ωF
73
APENDIX C (continued)

Adams Simulation for Phase 2, Masses 1 and 2, for ωA is not shown because it would
merely be a line along the axis for both cases.
15.00
5.00
0.00
0.0
0.1
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
1.3
1.4
1.5
1.7
1.8
1.9
2.1
2.2
2.3
2.5
2.6
2.7
2.8
3.0
3.1
Lateral Deviation (m)
10.00
Phase2 Mass2 ωB
-5.00
-10.00
-15.00
-20.00
Phase2 Mass1 ωB
Time (sec)
Figure 70 Adams Simulation Phase2 Mass 1 and 2 for ωB
74
APENDIX C (continued)
8.00
6.00
2.00
Phase2 Mass1 ωC
0.00
-2.00
0.0
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
1.5
1.6
1.7
1.9
2.0
2.1
2.3
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (m)
4.00
Phase2 Mass2 ωC
-4.00
-6.00
-8.00
Time (sec)
Figure 71 Adams Simulation Phase2 Mass 1 and 2 for ωC

Adams Simulation for Phase 2, Masses 1 and 2, for ωD is not shown because it would
merely be a line along the axis for both cases.
75
APENDIX C (continued)
25.00
20.00
10.00
5.00
Phase2 Mass1 ωE
0.00
-5.00
0.0
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
1.5
1.6
1.7
1.9
2.0
2.1
2.3
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (m)
15.00
Phase2 Mass2 ωE
-10.00
-15.00
-20.00
-25.00
Time (sec)
Figure 72 Adams Simulation Phase2 Mass 1 and 2 for ωE
5.00
4.00
2.00
1.00
Phase2 Mass1 ωF
0.00
-1.00
0.0
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
1.5
1.6
1.7
1.9
2.0
2.1
2.3
2.4
2.5
2.7
2.8
2.9
3.1
Lateral Deviation (m)
3.00
-2.00
-3.00
-4.00
-5.00
Time (sec)
Figure 73 Adams Simulation Phase2 Mass 1 and 2 for ωF
76
Phase2 Mass2 ωF