Experimental and Analytical Investigation of Inertial Propulsion Mechanisms and Motion Simulation of Rigid-Multi-Body Mechanical Systems

Methodologies are developed for dynamic analysis of mechanical systems with emphasis on inertial propulsion systems. This work adopted the Lagrangian methodology. Lagrangian methodology is the most efficient classical computational technique, which we call Equations of Motion Code (EOMC). The EOMC is applied to several simple dynamic mechanical systems for easier understanding of the method and to aid other investigators in developing equations of motion of any dynamic system. In addition, it is applied to a rigid multibody system, such as Thomson IPS [Thomson 1986]. Furthermore, a simple symbolic algorithm is developed using Maple software, which can be used to convert any nonlinear n-order ordinary differential equation (ODE) systems into 1 st_order ODE system in ready format to be used in Matlab software. A side issue, but equally important, we have started corresponding with the U.S. Patent office to persuade them that patent applications, claiming gross linear motion based on inertial propulsion systems should be automatically rejected. The precedent is rejection of patent applications involving perpetual motion machines.


ACKNOWLEDGMENTS
Professor Philip Datseris, with unlimited appreciation and respect, I would like to let you know that I will miss you. When you are tired and rest on the leather chair in your office, please ask yourself 'why was Mohammed so good to me!!' Thank you dear doctor forever. I would like to thank Jim Byrnes on his electronic circuitry, and his humble, supportive and friendly personality. Jim, thank you so much.
The assistance and inspiration I received from Donna and, Kathy is greatly acknowledged. Thank you, you make our department lovely.
I extend great gratitude to the faculty at URI with whom I have interacted throughout the years.
To Mom, and Dad, no words reward your love, care, and prayers for me.
To my lovely wife and children: Ahmed, Shrief, and Mustafa, I greatly appreciate your patience, love, and support during the years of my study toward this Ph.D.

PREFACE
Since the beginning of the last century and until the present time, almost all the patented devices in the inertia propulsion field were built with the belief that centrifugal forces can cause "Gross Motion" (the system center of mass is in unidirectional motion). Chapter 1 reviews some of the previous work since 1912.
Over fifty patents have been granted to inventors in the inertial propulsion field, but unfortunately, this technology has not yet been exploited for commercial use . This was one of the initial reasons of this study. Chapter 1 includes experimental work, measurement techniques, and devices, developed to investigate the existence of the inertia forces due to rotating bodies. In addition, chapter 1 includes the experimental technique, which is used to define the phase angle between the angular position of a body, and the corresponding position of its inertial force. Where, the position of a body is the line passing by both of the center of rotation and the center mas. s of this body. It concludes that the rotational arm and the · angular displacement are the two parameters influencing both the magnitude and direction of the dynamic system's linear motion respectively. However, this conclusion is modified in Chapter 2.
In the design field, dynamic simulations are required to answer many questions with regard to motion and joint forces between components of a Furthermore, examples of the code, which is used to simulate the motion are also provided. This code is numerically solved (integrated) with Runge-Kutta method in Matlab software.
In Chapter 3, we simulate and analyze in detail one of the most promising new inertial propulsion systems (IPS), which is Thomson IPS ].
The computational method used in this work is systematic and reliable, once the four parameters (T, V, F, Q) of the mechanical system are defined, where, T, and V, are the kinetic and the potential energies of the system respectively, F is the dissipative forces (i.e Rayleigh function) of the system, and Q represents external forces on the system. Furthermore, this method enables the investigators to develop the equations of motion symbolically. It is a good tool to verify the correctness of commercial dynamic analysis software. The only drawback of this computational method (versus the virtual commercial dynamic analysis software) is that one must provide the correct four parameters T, V, F, Q as they are explained in Chapters 2, and 3.

Horizontal configuration with slip table, top view
Simulations showed that these mechanisms could deliver] vibratory motion only.
Since the beginning of the last century, almost all the patented devices in the inertia propulsion field were built with the belief that centrifugal forces can cause linear motion. Therefore, one of the goals of this work is to define the dynamic term( s ), which can cause unidirectional linear motion in rotating devices.
In order to accomplish this, three main tasks will be completed: The first task is to design a simple mass-rotating device that is shown in Figure 4 in order to investigate the generated forces associated with the rotating unbalanced masses from a novel viewpoint to determine if these forces can be utilized for propulsion or unidirectional linear motion. Also, this device is used to accomplish two other major tasks.
The second task is composed of two steps. The first step is to generate and to quantify forces generated by eccentric rotation of masses. The second step is to determine the exact direction of these generated forces as function of the angular position of these masses with respect to the testing machine's loading train axis.
The third task is to test the interactions and effects of these forces on the device's motion.  Table. It is composed of an electric motor 4, motor shaft 9, two couplings 6, two rotating arms (radius of rotation) 10 with one mass on each, and the supporting structure. Supporting structure is composed of two longitudinal beams 2, four transverse brackets 7, motor's mounting plate 3, and the base plate 1.

l. Specifications of the Rotating Device
To investigate the exerted forces due to the rotation of masses, the 'eccentric mass device' shown in Figure 4 is designed to provide rotational motion of eccentric masses. This device consists of one motor, two couplings, two rotating shafts, two adjustable arms, two weights, four transverse brackets with ball bearings each, two longitudinal brackets, and the base plate. Upon running the electric motor, a rotational motion is transferred to the two rotating shafts through the two couplings. The two rotating shafts in tum drive the two masses, where the center gravity of these masses have radius of rotation 'r' from the axis of rotation of the rotating shaft. To determine the exact position of the generated forces, a sensor-system shown in Figure 6 is mounted on the device shown in Figure 4. This system is composed of two infrared sensors, a disc with a slot, and an electronic circuit (see Figure   22). The slotted disc is mounted on the free end of one of the two rotating shafts where the slot was aligned with both masses. The two infrared sensors are mounted at 90° from each other on the transverse bracket where the slotted disc is free to run across the sensors' infrared light beam. One of the two sensors is positioned parallel to the axis of the testing machine's loading train. This position permits the sensor to send its signal when the rotating masses are parallel to the axis of this loading train. However, each sensor is in a U shape where the light beam can travel from the emitter side to the receiver side. The receiver converts the light beam into voltage signal. This voltage signal will be permitted only when the slot in the slotted disc reaches the infrared beam. Furthermore, the voltage electronic signals coming out of the two sensors are sent to the electronic circuit. The electronic circuit in tum amplifies and shapes the signal respectively (see III.3.2). The output signals are displayed on and recorded through one channel with an oscilloscope (Tektronix TDS 340A). The oscilloscope's adding channel is adjusted to show and record two signals. The first signal is from the light sensors and the second signal is from a load transducer.
The rotation of the eccentric masses will cause a repeated dynamic loading on each of the rotating shafts. Therefore, a reasonable service life of the rotating shaft, which could carry the effects of these eccentric masses, is a crucial matter.
It is felt that this issue should be addressed to insure that a system which has eccentric masses can be designed for infinite life under expected cyclic loading.
Therefore, this work will provide design procedures of the structure, which supports the shafts and the eccentric rotating masses.  Therefore, upon defining a specific sinusoidal loading and during the early stage of designing this type of machinery, the prediction of the service life will be always close to the actual service life.
Design procedures of the rotating device follow four major steps. The first step is to choose the proper preventive fatigue criteria model. The second step is to estimate the maximum possible dynamic load, which a lf4 inch diameter motor shaft can carry. The third step is to select a motor. The fourth step is to design a frame structure. This structure should be able to support both of the dead weights of the motor and the eccentric masses, and the life load of the exerted dynamic load due to the mass rotation. Furthermore, deflection should be maintained below 0.001 inch. The importance of the deflection minimization is not only to assure a proper mechanical running operation but also to assure that the rotation plane of the eccentric load is approximately parallel to the loading train axis of the testing machine.

Previous Work
A number of rotating devices will be investigated to determine if claims of unidirectional motion can be proven. These devices have been patented. In addition, other researchers in the field have investigated the motion of these patented devices. Some of these patents are listed below and in the reference section. Laskowitz invented a centrifugal variable thrust mechanism wherein the principle of centrifugal force is utilized and which is capable of developing a thrust, as he claimed. However, the centrifugal or propelling driving force will be entirely independent of the medium upon or through which the vehicle travels. . Figure 9-The vehicle where a drive means is enclosed therein at the oscillating frequency of the component, which is also the motor frequency, enforced vibration frequency. Thus, the vibrating system has two principal directions of vibration. In this invention, each spring section has its own resonance frequency. Figure 11 shows that if the enforced frequency is below or close to resonance frequency, very large spring deflection results. Whereas, if the enforced frequency is above and far removed from the resonance frequency, very small or nearly zero deflection occurs. Therefore, the short sections 24a and 24a are designed to operate at the points indicated in Figure 11. It shows that the motor frequency, enforced frequency, is close to the resonance frequency of spring 24a, and too large compared to the resonance frequency of spring 24b.

Movemsnf
However, considerable force is imposed upon the vehicle structure by the vibration of spring section 24a. The vertically pulsating force in direction 3 Oa produced by spring 24a, is the propulsive force. Vehicle weight and friction force between the vehicle bottom surface and the ground overcome the force in other direction during the weight's rotation.
I. 2.2.3. Cuffs Device (U. S. Patent No. 3,968,700) Cuff believed that the shown device is able to produce motion in the shown direction 13 in Figure 12. The invention converts the centrifugal forces produced by the rotating masses 8a, 8b, .. , 8h into a propulsive force acting in one direction which is perpendicular to the plane of rotating masses. Propulsive force is due to the continuously variation of the radius of gyration of each mass during its cycle of revolution. Cuffs invention produces an unbalanced centrifugal force by varying the radius of rotation of rotating masses at predetermined moments in their cycle of revolution by means of an eccentrically disposed circular member.
The path of rotation of the rotating masses contains certain predetermined sectors in which each rotating mass attains a maximum radial distance and then, after 180° more of rotation, a minimum radial of distance.    Figure 14 shows that each wheel carnes a pair of gearwheels 26, and 27, which rotate around the axis of the wheel with the wheel and support eccentrically a pair of planet masses 33, and 34. The masses are arranged such that their distance from the axis of rotation of the wheel increases and decreases under control of the gearwheels. At a position immediately prior to the maximum distance of the planet from the axis, electromagnetic devices 3 7, and 3 8 restrain outward movement of the planet mass so that when released the planet mass provides whip-like action inducing a resultant force in a direction at a right angles to the plane containing the axes of the wheel, according to Thomson. For example, Figure 14 shows four positions of mass 34 indicated respectively at "A", which is the immediate position prior to the maximum distance through "D".
Where D is the position of mass 34 at the maximum spacing distance. It should be noted that the positions "B" and "C" are inhibited inwardly of their normal positions so that the center of mass of the planet 34 in the positions "B" and "C" are no longer on the radius joining the rotation axis and the pivot axis 25. In addition, the inhibiting means preferably is arranged on the body for rotation therewith and uses electromagnetic forces to-restrain the movement of the planet mass. In addition the positioning of the electromagnetic restraining device is such that the planet mass is released immediately prior to its position of maximum spacing from the first axis so that it provides a whip-like action while traveling at its maximum velocity. I. 3.

Inertia Force Measurement and Its Effects on the Device's Motion
In this section, the dynamic inertia force will be measured experimentally.
This measure is accomplished with the shown layout In Figure 16. It shows the device with the two eccentric rotating masses attached to an Instron machine. In addition, the effects of this dynamic force on the device motion will be experimentally tested with the layout shown in Figure 25. Testing procedures involve three steps, which are: 22 1 _ Mounting the rotating mechanism on the automated testing machine as shown in Figure 16. Zeroing the system of the Instron machine so that the weights of the loading train and the rotating mechanism are ignored when the data acquisition records the load cell output signals to a file.

2-
Writing a simple 'dynamic load program' to utilize the data acquisition of the testing machine to collect data from the load cell in a file. Instantaneously, this data also will be sent through one of the external channels located in the automated testing machine to an oscilloscope.
3-The oscilloscope is prepared to add and record data signals from the testing machine and light sensors instantaneously. The oscilloscope adding signal feature channel is activated to add the two signals. The save and record procedures are followed to record them in a numerical file.

Measuring the Centrifugal Force (CF)
This section investigates the measurement of CF in two different conditions. The first condition is when the point of application of the CF is stationary and the second when the point of application of the CF is permitted to move.

I.3.2.l Condition# 1: Measuring the Centrifugal Force -No Linear Motion 23
In this test the device is attached to the lnstron machine, and therefore the device does not move, no linear motion. .E 0 (.) . The two eccentric masses are machined of four thin discs with a centered hole in each. The weight of each set of discs is obtained with a sensitive digital scale and is found to be about 0.15 lb. Each set of discs is mounted on an arm with a distance of 6.5 inches between the geometry centers of the discs to the geometry centerline of the motor shaft. The weight of each arm is found to be about 0.125 lb.
The load cell in the automated testing machine measures the magnitudes or the components of forces acting in or parallel to the centerline of the loading train axis (see Figure 16). Therefore, The load cell will sense and read the maximum magnitude of the exerted forces by the rotating masses when the angle between the rotating arms and the loading train axis is zero or a multiple of 180°.
The testing machine is programmed to run in displacement mode and the displacement magnitude is programmed to be very small. This will provide no linear motion of the device during the dynamic test. The output file is programmed to obtain load signals versus the time increments every 2.0 milliseconds. Figure 17 shows that the dynamic load, Fm, fluctuate between '-9' and '+ 1 O' lb with 5.5 cycles per second or 330.0 rpm.
The Measured Force, Fm, output results are then, as follows: The Estimated Force, Fe, is calculated using the well-known centrifugal force equation, as follows: (c) a motor which rotates with 5.5 rps (330 rpm) Figure 19 shows the maximum WM's output force of 9.3 lb is almost equal to the experimentally measured force of 9 .5 lbs. when using an lnstron testing machine. This small error of 2% is possibly due to speed variation during testing. Therefore, WM is sufficiently accurate. 10 -8 .c 6 :::::..
Time (sec) Figure 19 -Exerted dynamic force along the X-axis; measurement is made only when the center of gravity of eccentric mass is in the horizontal plane.  The rotating device is now permitted to have linear motion along the Y-axis.
The goal of this simulation is to find out the effects of the rotating device on the movable machine. This is done by measuring the forces at the joint, see pin joint in Figure 18. For different rotational speeds, the model in Figure 18 is used to find out the corresponding exerted dynamic force on the pin joint as a function of different weight ratios of the eccentric weight to the carriage weight. Figure 20 shows the relationship plot of the ratio of the eccentric weight to the carriage weight versus the normalized measured applied force along the Y-axis, along which the carriage is free to move.
The output result in Figure 20 shows that the force at the joint pin decreases as the ratio of the eccentric weight to the carriage weight decreases and vice versa.
This can be also interpreted as follows; as the carriage weight increases then the required force to move it from a stationary mode to a motion mode will be higher until it reaches its maximum which is the stationary CF.
It should be mentioned that eccentric weight is normally very small compared to the carriage weight. Then in Figure 20, a machine based on the principle of unbalanced rotating masses should operate at weight ratios magnitudes close to zero and take almost full advantage of CF. It is also important to note that a the load at the pin joint has a sinusoidal profile, see Figure 17. In other words, in these systems, the force at the pin joint fully reverses and its magnitude corresponds to the weight ratio, eccentric weight to carriage weight, as shown in Figure 20.

I.3.3 Identifying the Angular position of the Exerted Inertia Forces
The angular position of the maximum exerted inertia force, centrifugal force, with respect to the initial angular position of the eccentric mass is determined by using the sensors shown in Figures 6, and 16. Figure 21 shows the schematic diagram of the device components with respect to the Instron machine.
It shows that the arms of the rotating eccentric masses and one of the two light sensors are placed in line with the loading train while the second sensor is placed with 90° from the first sensor.
The first signal is from the load cell in the testing machine, and the second signal is from the infrared sensors. These signals are sent to an oscilloscope as shown in Figure 22. However, the test preparation to achieve these two signals is composed of the following three steps: Step # 1 : Automated Testing Machine Signal Preparation.

Load Cell Loading Train
Rotating Arm Eccentric Mass Figure 21 -Schematic diagram showing th~ layout of the slotted disk, light sensors, and the eccentric mass with respect to the machine loading train centerline at the initial condition. The eccentric mass, the slotted disc and the arm rotate in the arrow direction; the load cell, the loading train centerline, and the two sensors are stationary. Figure 22 -The Oscilloscope receives the Two Signals from the Loading Cell and the Light Sensors. The screen shows the sum of the Two Signals. 1.3.3.1 Step #1: Automated Testing Machine Signal Preparation The automated testing machine is programmed to perform as follows: 1. Its movable head moves under the displacement control with 0.1 in/hr.

2.
Its data acquisition collects one sample every 2.0 millisecond. 3. The sample is composed of the outputs of the load cell and the computer registrar clock. 4. One of its analog output channel (Yl or number 3) is activated to receive the load cell output signal. 5. After attaching the rotating device in the automated testing machine, the load cell reads the weight of the attached device. Therefore, before running the test to measure only the force due to the rotating eccentric mass, the load cell is zeroed.
I. 3.3.2 Step# 2: The Infrared Sensor's Signal preparation As mentioned earlier that the two U-shape, see Figure,  Multivibrator is used to shape the digital signal. The original shape of the sensors' output signal which is converted to Analog signal is a step function with a period of time equal to the elapsed period of time which required to pass the gab in the disc through the sensors. However, the Multivibrator is operated in a Monostable mode where the step signal width shape can be decreased to a spike shape, see Figure 22, to represent the start at which the gap reaches the light sensors. (   I. 3.3.3 Step #3: The Oscilloscope Preparation The TekTronix TDS 340A oscilloscope has two input channels, one math channel, and two reference channels waveform. The two oscilloscope's input channels are connected to the two output digital signals which are coming from the testing machine, and the output signals of the infrared sensors. The math channel is used to add the two incoming signals. One of the reference channels is 5 Motion Due to Inertia Forces I. 3. 36 The rotating eccentric mass device is mounted on a carriage with two screws.
The carriage has free rolling motion on a slide rail as shown in Figure 25. The rail's horizontal elevation is adjusted to be even by utilizing a bubble level scale as shown in the right hand side of the figure. The motor in the rotating device rotates the eccentric masses, which in tum induce the linear motion of the carriage along the rail. Three tests are performed. The first test involves motion of the carriage as a function of two different angular velocities of the rotating masses. It is observed that increasing the angular velocity of the masses is not coupled with any increase in the magnitude of the displacement of the system.
The second test involves motion of the carriage as function of the radius of rotation of the eccentric mass. It is observed that increasing the radius of rotation is coupled with an increase in the magnitude of the system displacement.
As expected, in the previous two cases, system motion after one full rotation of the eccentric mass, is such that the carriage moves to the right direction with certain displacement then moyes back to the left with exactly the same displacement. Therefore, there is no net linear motion since at the end of each cycle the carriage returns to its initial position. Figure 25 -The rotating device with its unbalanced masses is mounted on a carriage with rolling wheels. The wheels are free to rotate along the right and left directions on the rail. To the right hand side a bubble level scale is placed to show the even horizontal elevation of the rail.
The third test involves the same mechanism but the carriage is restrained by a physical stop and cannot move in one direction. It is observed that the carriage reaches a certain amount of net linear · displacement in the opposite direction of the physical stop and after a period of time only the oscillating motion remains. Figure 26 shows this motion in a qualitative fashion since no measurements are made in this test. However, this experiment is video taped.
The net linear motion is due to the impact force between the carriage and the stop and it vanishes due to friction. The oscillation motion, which the system maintains is due to the rotation of the unbalanced masses.

Displacement
Figure 26 -Schematic plot shows the resultant motion of the device shown in Figure 25 when a physical stop is used to prevent motion of the carriage in xdirection. The resultant motion is oscillating with certain unidirectional linear displacement but after a certain amount of time only oscillating motion remains J. 4

. Results and Conclusions
I. 4. 1. Results: 1. The eccentric rotational masses generate centrifugal force effects at the pin joint and the carriage. These forces act outward and are collinear with the radius of rotation.

2.
The centrifugal force is a function of the weight ratio of the rotating eccentric masses to the weight of the carriage. 3. The eccentric rotating mass exerts a fully reversed loading profile at the pin joint, which attaches the arm of eccentric mass to the carriage. 4. In the dynamic system, the magnitude of the force at the pin joint is determined as function of the weight ratio, eccentric weight to the carriage weight. 5. The rotational motion of the eccentric masses causes oscillating motion. 6. An increase of the angular velocity of the rotating mass does not affect the magnitude of the motion but increases the frequency. 7. The magnitude of the linear displacement has a proportional relationship to the radius of rotation.

4 .2. Conclusions:
These results show that the magnitude of motion is dependent on two factors.
The first factor is the radius of rotation at which the eccentric masses are located and the second factor is the angular position of the arm that holds the eccentric masses, which changes the direction of the linear motion every half cycle.
Therefore, additional work will investigate the possibility of manipulating the two factors to achieve unidirectional linear displacement.
In addition, an in-depth study of the patents in this field will be done to choose the most promising Rotating Inertial device for further improvements.
Specifically, the aim is to invert a geared system such that eccentric masses do not make full rotations in the plane of motion as shown in the Figure 27.    The developed procedures will also greatly aid in teaching advanced courses in dynamics and enhance knowledge and experience of students.
Dissipative forces include all forces where energy is dissipated from the system when motion takes place, i.e frictional, viscous, and proportional forces, see [Walls 1967].
In general, the Lagrangian technique is extensively used to develop the equations of motion, especially for multiple-degree-of freedom systems with distributed masses [Saeed, 2001, Fu 19 87, Featherstone 1983, Shahinpoor 1988]. The Lagrangian method is based on energy terms only and, thus, is easy to use it and results in the least number of equations of motion. The Lagrangian method is chosen for its simplicity instead of matrix methods, see [Vibet 1994].
Therefore, the developed symbolic EOMC based Lagrangian mechanics is demonstrated on two textbook examples and the invention work of Thomson ]. The first example is particle motion in a uniform gravitational field expressed in spherical coordinates, see example 6-3 ] which is used to reduce the n-order equations of motion to 1st order, and rewriting the symbolic 1st order equations of motion is provided in the Appendices.
Finally, the third example simulates the dynamics of the model shown in Figure   35 . This example is chosen because the mechanical system it represents is the building block for many IPS.
However, the simulation results of the third example show clearly that the Gross motion of a mechanical system depends on the motion of the system's center mass. This fact will be used in a subsequent paper to investigate the motion capabilities of inertial propulsion systems, especially Thomson's Mechanism, see ].

• 3 Example 1: A Body in a Uniform Gravitational Environment.
z x Figure 28 -Body P is located at distance r in the spherical coordinate system r, e, and cp. The unit vector triad represents the instantaneous motion direction of Pin this coordinate system.
However, these steps are discussed in detail in the example 2. Figure 29 -The system incll.ldes two masses ml, and m2, two springs with stiffness Kl, and K2, and two dashpots with damping coefficients A, and B. Its motion is in a horizontal frictionless plan

II.4 Example 2: One Dimensional Lumbed-Parameter Mechanical System
The example shown in Figure  At this stage, an investigator must define the T, V, F, and Q expressions in step 1, while the other steps are automated systematic procedures. II.4.1 Step 1: Developing the expression of the four variables T, V, F, and Q.
Lagrangian (Lag) is a function of T and V, Lag= T -V, where, T, and V are the system kinetic energy and the non-dissipative potential energy of a system respectively.
The existence of V is due to the masses being exposed to gravitational, magnetic, spring effects, etc.

Case# 2
Defining kinetic energy and other key parameters in this case ( Figure 31) requires a little care. By inspection in Figure 31, assume that ml is displaced in Therefore, the four variables can be defined as follows: the damping coefficients as shown in Figures 30, and 31. II.4.2 Step 2: Developing the equations of motion Figure 30 shows that the general coordinates are X, and Y. Therefore, for each case, the EOMC is used twice to develop two equations of motion. This is done by substituting for the q in the EOMC with X, and Y respectively. For example, to develop the equation of motion 6 in X-coordinate, which is shown below, the EOMC becomes as follows,

D_Lag_X=-D_F_dX + Q_X),diff);
Again to develop the equation of motion in Y-coordinate, all what one has to do is to substitute Y for q in the EOMC. This provides the second equation of motion (7) in the Y-coordinate as shown below, EQ2:::::.
Note that all the procedure in step 2 should be repeated again for case 2 to develop the corresponding equations of motion as those of equations 6, and 7 for case 1. 11.4.3 Step 3: Rewriting the 2°d order equations of motion for Matlab simulation The mathematical system, which is composed of equations 6, and 7, must be solved numerically to simulate the motion of the mechanism shown in Figure   30. In order to simulate this system, which is composed of two coupled 2nd order nonlinear, non-homogeneous deferential · equations 6, and 7 with Matlab software, this 2nd order system is automatically converted to 1st order system. Also, the variables in coordinates X, Y and its first derivatives must be assigned as y(#), i.e y( 1 ), y (2), etc. This is to make the equations of motion in Maple software ready to be sent to Matlab. Therefore, an example of the Transform Symbolic Code (TSC) is provided and explained in the Appendix ·A. However, the TSC develops the following two equations 8, and 9 from equations 6, and 7 respectively which shall be solved in Matlab numerically.
B y( 4) + K2 y( 2) -Kl y ( I ) pp4 := Y3 == lnl (9) II. 4.4 Step 4: Simulation #1 Some arbitrary system parameters are assumed for the two cases to obtain simulation results. These parameters are assumed as follows: Equations of motion 8, and 9 ·are coded in m-file as shown in Appendix B.
Simulations results of cases 1, and 2 are shown in Figures 32, and 33 respectively. Again, to simulate the motion of the mechanism shown in Figure 35, the following steps should be followed.
1) Development of T, V, F, and Q expressions to form the Lag expression 2) Development of equations of motion utilizing the EOMC (see example 1) 3) Converting the n-order ODE of motion system to a first order ODE utilizing code similar to TSC, see Appindex A. 4) Simulation. II.5.1 Step 1 Where, X is the displacement of gear #2 in the X-coordinate, and r is the gear (9)] + [2 r 9" cos (9)] i 11.5.2 Step 2: Developing the equations of motion Figure 35 shows that the general coordinates are X, and 9. Therefore, in Maple software the EOMC is used twice by subsisting for the q with X, and 9 respectively.

X-Coordinate Equation:
EQl:=  (11) II. 5.3 Step 3: Rewriting the 2nd order equations of motion for Matlab simulation Again, simulation of the differential equations (Equations 10, and 11) in Matlab can be done after the following two steps are performed: The first step is separation of the highest derivations (i.e x··, 8.). The second step, is writing the equations in an acceptable form for Matlab (see Appendix A). First Step: Separation of the highest derivatives The first step includes three commands to be written in Maple software.
The first command substitutes Y3, and Y4 for X .. , and 8 .. (which are the second derivatives with respect to time of 8, and X generalize9 coordinates) respectively.
The second command pulls out each equation from the result of the previous command. The third command solves the outputs (EQ 1, and EQ2) of the previous command (second command) using Cramer's rule for the Y3, and Y4.  ~rsil(y(l))M _ 4 _1 r(4co~y(l))nu 2 +41 co~y(l)))y (3)
In Matlab file, paste the previous two equations and substitute dy (3), and dy (4) for Y3, and Y 4 respectively, see Appendix D.
The previous two Equations 10, and 11 are coded in a Matlab file (see Appendix D). This file is used to provide the simulation of the motion of the Epicyclical Gear System's model shown in Figure 35. The output simulation results are shown in Figures 36, and 37 for generalized coordinates 8, and X respectively. Figure 37 shows that the carriage is reciprocating between 0 to 0.6 m. Figure 38 shows the projection (AB * cos (8)), which represents the relative motion displacement of gear #2 in the X-coordinate with respect to the c.m of gear #1. Figure 39 shows X2, the absolute position of gear #2 in X-coordinate.
Where, m 1 , and m 2 are equal mass of gears #1, and #2 respectively. In addition, X 1 , and X 2 are the absolute motion displacements in X-coordinate of gears #1, and #2 which are shown in Figures 37, and, 39 respectively. Therefore, the previous rule can be simplified to become: (m 1 *X1 + m2*X2) I (m1 + m2) = (X1 + X2) I (2). Figure 40 shows that the motion of the c.m. of the whole system is stationary although the components of the mechanical system are in motion. ll.5.4 Step 4

II.6 Conclusions:
In this work equations of motion are automatically developed in a symbolic form. using the EOMC symbolic code. Furthermore, a simple transforming code TSC has been developed, which converts any non-linear n-order ODE system to a 1st ODE system. EOMC, and TSC together can be used as a tool to develop the equations of motion of any dynamic system in a format which can be numerically simulated in Matlab. Solver ode45 in Matlab is used, to integrate the equations of motion. This solver, which is based on the single step Runge-Kutta integration method, provides accurate simulation outputs .
Therefore, it is concluded that EOMC, TSC, and ode45 together is a powerful and accurate tool to simulate motion of any dynamic system.
The third example show_ s that 'Gross Motion' is not possible for a simple mechanical device, which is the building block of many IPS. These results have been verified using WM.
Note, that the integration time step in working model and Matlab can be adjusted to provide output integration at th~ same period step using 'the integration time step' and the 'MaxStep' option in WM and Matlab respectively. 11.7 Future Work: We will work with the U.S.Patent office to persuade them to reject patent applications, claiming gross linear motion based on Inertial Propulsion Systems.

Abstract:
In this work, Thomson inertial propulsion system (IPS)  is modeled as a rigid multibody system. The motion of this model is numerically and virtually simulated with Matlab and WM respectively.

Claim 2:
Furthermore, Thomson [1], page 5 claimed that the magnet causes the eccentric mass 33 to have a whip-like effect, which in turn provides Thomson IPS with unidirectional gross motion.
However, to investigate the prev10us two claims, two submodels of Thomson's device are examined.

I I
The objective of the model shown in Figure 43 is to investigate the first claim that the eccentric mass 33 nearly positions itself outward during 360° rotational motion of gear 26. Figure 43, assumes a massless disc 26 is driven with a motor.
The motor provides torque M to disc 26 at its center B. Disc 33 is mounted on disc 26 with hinge C.
This model shows that the two generalized coordinates 81, and 82 are required to describe the system's motion. However, to simulate the motion, the four steps procedures which are explained in [2] shall be applied again as follows: III. 3.1 Step# 1: Developing the expression of the four variable T, V, F, and Q Gear 26 and the eccentric mass 33 both rotate in a horizontal plane, Figure   43. Therefore, potential energy V is zero. Also, friction between disc 26 and eccentric mass 33, is assumed negligible. Therefore, no energy dissipation exists and therefore, F is equal zero. Q is equal to M (motor torque) at coordinate 8 1 , because the motor is located at the center of Disc 26, Figure 43.
111. 3.4 Step# 4: Simulations Figures 44,and 45 show the simulation of motion of disc 26, and the eccentric mass 33, respectively. Results appear to be identical but they are not.
The plot in Figure 46 is calculated by subtracting the angular position (81) of disc 26 ( Figure 44) from the corresponding one (8 2 ) of the eccentric mass 33 ( Figure   45). Figure 46 shows that the ·angular motion of the eccentric mass 33 w.r.t joint C looks like a damped harmonic motion during the first 10 seconds of the simulation. This apparent damping is because the system is driven with a constant torque (not constant velocity). However, Figure 46 shows that disc 33 has a substantial harmonic motion amplitude of ± 1.25 rad with respect to line BC, Figure 43, where the system starts from rest. This amplitude decreases as the motion continues but the angular magnitude of the reciprocating motion of disc 33 with respect to BC is about ± 70° after more than 10 full revolutions of disc 26, which is significant amplitude.
It is concluded that the eccentric mass 33 cannot be considered "nearly outward" as Thomson claims in his patent ] and as Valone claims .

Jil.4 Submodel #2
Carriage Motion 30 YI y Frictionles Slider Figure 4 7 -shows the same model shown in Figure 42, but with an additional assumption that the three discs 30, 26, and 33 are similar. The Y axis of the inertial frame is not shown in this Figure. 92 The objective of the model shown in Figure 4 7 is to investigate the second claim by Thomson [1,p.5] that the magnets 37, and 38 shown in Figure  This model shows that only three generalized coordinates, 8 1 , 8 2 , and X are required to describe its motion. The rotation of disc 26 around disc 30 is related to the rotational motion of the arm AB and is defined as 2* 8 1 , see Figure 47.
Therefore, there is no need for an additional generalized coordinate. The fourstep procedures, which are explained in [ Almesallmy et al.] shall be applied again as shown below to develop the equations of motion. III.4.1 Step# 1: Developing the expression of the four variable T, V, F, and Q By investigating Thomson model, which is shown in Figure 47, one can realize that it is composed of two submodels. The first submode} is the third example in [Almesallmy et al.], where the motio_ n was investigated for discs# 1, and# 2. The two discs 30, and 26 in Thomson are the same as discs# 1, and# 2 respectively. The second submodel is the previous one shown in Figure 43, where the motion of the eccentric mass 3 3 was investigated.
Therefore, to simulate Thomson motion, the two submodels are added together. This is done by adding the variables T, V, F and Q of the two submodels to develop the equations of motion for Thomson model as shown below: Where, {T}subl = Kinetic energy of submode} 1, which is composed of the kinetic energy of discs 30, and 26, see ]. T 0 ' = kinetic energy of disc 33 due to its small oscillations about hinge C and for simplicity can be ignored.
The previous added terms in T expression, express the kinetic energy of the three bodies 30, 26, and 33 respectively, see Figure 47. The first term represents the kinetic energy of disc 30. The second and third terms represent the kinetic energy of disc 26. The fourth, and fifth terms represent the kinetic energy of disc 33 which is calculated at the c.m of the disc.
In spite of the magnetic force between the eccentric mass 33 and the magnet pole 37 can be expressed directly in Lagrange's equations, but the magnet effect is introduced to Lagrangian as ICs to simulate Thomson's claim.
Thomson's claim is as follows: " Figure 48 shows that the magnet pole 37 holds the eccentric mass 33 for a period of ~ of revolution of disc 26 or hinge C.
The onset the magnet release the eccentric mass 33 till the next release the whiplike motion of mass 33 provides the mechanism with a certain magnitude of linear displacement in X-coordinate." The first stage of the simulation is solving the previous three equations of motion for the ICs of set A shown in Table 1 in order to develop the new I Cs of set B shown in Table 2 where its c.m is placed at the origin of XY inertial frame. Gear 30 is fixed on the carriage and translates with it, A is the center of rotation of arm AB, which in tum rotates gear 26 around gear 30 (as indicated with arrow 2*8 1 ), and the eccentric mass 3 3 is free to rotate around hinge C.
Solving the previous equations of motion for the IC 5 of set A provides the motion simulation shown in Figures 50, 51, and 52. Figure 50 shows the motion of link AB in 8 1 -coordinate takes 9.4 sec to complete five full revolutions. Figures 51 and 52 show that the linear and the angular velocities of disc 30 and arm AB are zero m/sec, and -8.5 rad/sec respectively, at 9.4 seconds.
The I Cs of set B is chosen at that instant at which arm AB completes five full rotations in 8 1 -coordinate. Therefore, the IC 8 of set B can be written as follows: Note, that the initial angular position of the eccentric mass 33 (In_ 8 2 ) is assumed negative 90° with respect to the positive X-coordinate, to consider Thomson claim that 'the magnet holds back the eccentric mass for a period of 1 A revolution,' see        Figure 54 shows that at the onset that arm AB completes one full revolution, disc 30 returns back to its initial position. In other words, the simulation for the ICs set A predicts that Thomson IPS will have no gross motion. Therefore, it is concluded that the magnet poles 37, and 38 and the claim of the whip-like motion effect of the mass 33 will provide no gross motion for Thomson mechanism.
However, using the model shown in Figure Figure 56 shows the mechanism moves in circular motion in X-Y plane.

Discussion and Conclusions:
In this work EOMC and ode 45 solver in Matlab is a good tool for developing equations of motion of complex mechanical systems.
In the Thomson's mechanism, the eccentric mass 33 cannot assume a "mostly" outward direction when magnets are not present. It is proven that the eccentric disc 33 will maintain angular motion amplitude of about ± 1.25 rad.
Furthermore, the Thomson device cannot provide gross motion, as the only contribution the magnet poles 3 7, and 3 8 provide is an increase in the frequency of the harmonic motion.
In summary, the conclusion of this work are as follows: 1-Maple code of eight steps code (EOMC) can be used to develop the equations of motion of any dynamic system.
2-WM is a good tool to show the motion pattern of any 2-D mechanism.
3-No motion can be achieved without an external force. Note that P3, P4, PP3, and PP4 are dummy symbols, so once equations 6, and 7 are transformed to 8, and 9 past the right hand side of PP3, and PP4 in Matlab after replacing Y3, and Y4 with dy (3), and dy (4) respectively as shown in Appendix B.