Thereby: Force, average = mass * Velocity, gain / Time, duration

Transcription

1 Proofs Next two pictures, Picture #6 and #7 presents the Proof of the natural depletion of kinetic energy in a rotational to straight line coupled motion pertaining to the principles set out in picture #2 and #3 applying to the separation of unequal inertial masses by the work of the potential energy contained in an inertial mass moment of a flywheel-rotor. Picture #6 presents the formula for a mutual separation-acceleration between a straight line displacement motion of a rotor- flywheel separating from the inertial mass of a device. Picture #7 presents the stopping- de-acceleration of the straight line displacement rotor- flywheel motion. There are two separating motion and two stopping motion of the rotor- flywheel rotational to straight line displacement coupled motion: Page -59-

2 Important to note in picture #6 are the inverse quare root out of three sums wich indicates a compound feedback system. Next presentation is picture #7, the calculation for the rotor-flywheel angular velocity progression pertaining to the rotational to straight line displacement coupled motion progressing from a straight line velocity Vf from a rotational velocity ωb to a momentary stop condition after a 90º turn increasingly progressing onto ωc. The straight line velocity of the isolated system, during this type of motion, is opposing the velocity +Vd due to the straight line displacement energy conserving collision type motion of the flywheel-rotor on proven on page 70 with vectors and is also applicable if additional energy is induced during this type of motion. Important to note in picture #6 we have a compound feedback system while in the next picture #7 we have a singular feedback system: Page -60-

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4 MATHEMATICAL FOOTPRINT SUMMARY UNIFORM VERSUS NON-UNIFORM MASS MOTION FOR STRAIGHT LINE DISPLACEMENT Postulation: The secant line connecting the end-points of a curve represents the average slope of that curve and is the proven average slope of the curve. Thereby: Force, average = mass * Velocity, gain / Time, duration No matter how the mass got there, only the velocity gain and time duration determines the average force. Speed, average = Distance, displacement / Time, duration No matter how the mass got there only the distance displacement and the time duration determines the average Speed. Therefore: The first two formulas combined: Force, average = mass * Velocity, gain * Velocity, average / Distance, displacement Finally: The above formula converted from force to work Work, performed = mass * Velocity, gain * Velocity, average = Kinetic, energy, consumed And: Work, performance = mass * Velocity², gain / 2 = Kinetic, potential, energy Ref: Proven by experiment: Gaspard Coriolis Because real usable: impulse² =mass * 2 * work. performed THEREFORE IMPULSE = mass / (2 * Velocity, gain * Velocity, average) MATH FOOTPRINT FOR CALCULATING THE UNIFORM ROTATIONAL TO STRAIGHT LINE COUPLED MOTION The Continuous Centripetal Force for Rotational motion is: Page -62-

5 Force, continuos, centripetal = mass, orbital, motion * Acceleration, centripetal Acceleration, continuous, centripetal =Velocity², tangential / Radius, crank ω = Angular, velocity Velocity, tangential = ω * Radius, orbit, motion Acceleration, continuous, centripetal = (ω * Radius, orbit)² / Radius, crank Acceleration, centripetal =ω² * Radius, orbit Therefore average Force for ¼ turn: Force, average, ¼, turn = mass, flywheel * Radius, orbit * ω² * 2 / π Furthermore by multiplying by time duration we get the impulse: Time, duration,¼, turn = π / 2 * ω Impulse, average,¼, turn = mass, straight line, displacement* Radius, orbit * ω MATH FOOTPRINT FOR CALCULATING THE NON- UNIFORM ROTATIONAL TO STRAIGHT LINE COUPLED MOTION Rotor Angular Velocity at 0º, the start of straight line motion = ω,a Rotor Angular Velocity at 90º, the end of straight line motion = ω,b ω,a > ω,b Average (Mean Value) Angular velocity for the ¼ Rotor turn: ω mean, value = ( ½(ω,a + ω,b )) Squaring the mean angular velocity we get:( ½(ω,a + ω,b ))² Therefore the Average force for non uniform Rotational to Straight line Coupled motion for a drive phase (the drive phase will be presented later) is: Force, average,¼ turn, non, uniform = = mass, straight line, displacement * Radius *( ½(ω,a + ω,b ))² *2 /π Therefore by multiplying Force, average, ¼turn, non, uniform * Time, duration, ( ½(ω,a + ω,b )) turn; we get Impulse: Page -63-

6 Time, duration,¼, turn = π / (2( ½(ω,a + ω,b ))) Impulse, average,¼ turn = mass, straight line, displacement * Radius, crank * ½(ω,a+ω,b) Average angular velocity for rest of the ¾ rotor turn is a ω,b progression rising up by the multiplying ω,b with constant C2. Then slowing back down by multiplying ω,bc2 with constant C1 because of the kinetic energy flow into and out of the straight line mass motion. This principle is presented in the next picture #8: Page -64-

7 The picture #8 reveals that the drive phase impulse having a larger mechanical energy potential is ( ½(ω,a + ω,b )) average progression and is opposed by one Idle phase impulse having a (½(ω,b +ω,bc2) average progression. The ω,bc2 accounts for the kinetic energy flow from the stopping of the straight line displacing mass during the Idle Phase presented in picture#6. Thereby, the three impulses progressing for each 1/4 turn after the drive phase must be alternately subtracted and added to arrive at the exact resultant internal self contained impulse. The sum of impulses during the idle phase: impulse, idle=mass *radius*(-(½(ω,b+ω,b C2))-(½(ω,b+ω,b C2*C1)+(½(ω,b+ω,b C2²*C1)) Refined to: Impulse, idle=mass *radius*(½ω,b(-1-c2-1-c2c1+1+c2²c1)) Drive phase and Idle Phase will be defined later. The idle phase impulses from 90º to 360º can be algebraically solved to the mathematically exact: Impulse,idle=mass*radius*1/2ω,b(-1-C2-C2C1+C2²C1) C3=(-1-C2-C2C1+C2²C1) The Net Impulse is:impulse, net= mass, straight line, flywheel, displace * Radius, crank * ½(ω,a+ ω,b(1+c3)) The Self contained internal impulse is divided by the total cycle time duration to arrive at the NET Internal self contained motivating force because the total cycle time is diluting the generated impulse. Accordingly the total Cycle Time is: ¼cycle time t1=π /(ω,a + ω,b)), plus one ¼cycle time t2=π /(ω,b + ω,b * C2)) plus t3=π /(ω,b * C2+ ω,b * C2*C1)). If no new mechanical energy is induced during t4 then we add: time t4=π /(ω,b*c2*c1+ ω,b* C2*C1*C2). If new mechanical energy is induced then ω,b * C2*C1 is boosted up to ω,a then we add: t4=π /(ω,b*c2*c1+ω,a) Thereby: The effective internal motivating force, the Internal Propulsion Formula is: Force, internal, selfcontained=mass, flywheel * Radius, orbit*½(ω,a+ ω,b(1+c3))/(t1+t2+t3+t4) Thereby the self contained Inertial Propulsion motivating energy is: E=mass, flywheel*radius², orbit*½(ω,a+ ω,b(1+c3))/(t1+t2+t3+t4) Page -65-

8 In the next Diagram, the final proof of the presented math footprint for a 1/4 rotor rotation is presented. It is important to note the avalanche fashion of the nonuniform force progressing from an exponentially large value to zero. The initial exponential instantaneous force magnitude and the near straight line progression down to zero pertains to the non- uniform motion and has its root cause in the mean value theorem combined with the centripetal force having a defined value for every displacement position of the angular motion, a quadratic function of the angular speed for every angular position. This is proven with the basic calculus theorem of the secant line being always the average slope of the sinusoidal curve having its mean value always at 47º, no matter of the degree of concavity. This is proven with the congruence of the average force line cross-sectioning with the force slope line always at the vertical 47º line, independent of the degree of concavity. The force footprint above the mean value line, up to the 47º vertical line, has the identical force footprint as the force below the mean value line from 47º to 90º, therefore the mean value line in respect to displacement is valid. Accordingly: It is proven, ½ the difference of the angular velocities multiplied by mass* radius* is always representing the mean value of the impulse for a 1/4 turn rotation. The relationship between the harmonious Newtonian and non-uniform forces is also depicted with a force vector phase shift triangle graph used in electrodynamics phase shift Physics, it further proves the congruence of inertial mass motion with electrodynamics: Page -66-

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11 PROOF REALITY CHECK Mean value straight line motion: Impulse = /mass 2 * Velocity, gain * Velocity, average Page 55: Harmonious mass motion; Velocity, gain, straight line, displacement = ω,a* radius = 6.28 meter/second, blue line; Velocity, Average = distance / time = 1 meter / 0.25 seconds = 4.0 meter / second Straight line displacement motion for uniform speed. 1N=0.102Kgf Impulse=0.102*1 * /(2 * 6.28 * 4.0)=0.723Kgf, seconds Non Uniform Motion compared; Velocity, gain, straight line, displacement = ω,a * radius = 6.28 meter/second, blue line; Velocity, Average = distance / time = 1 meter / seconds = 4.86 meter/sec. Straight line motion non uniform angular speed: 1N=0.102Kgf Impulse=0.102* 1 * / (2 * 6.28 * 4.86) =0.797Kgf seconds The difference between and is kgf seconds impulse magnitude. This proofs the reality that a shorter cycle time will generate a larger impulse and proves the presented Physics. The difference presented in figures page 55 is the difference between the mean value averaging calculation applying to straight line displacement inertial mass motion and the rotational to straight line coupled inertial mass motion applying the centripetal force calculation = Tangential², velocity / Radius, crank. Accordingly, the plot on page 55 proofs that the force averaging applying to a ¼ rotational turn coupled to a straight line displacement motion using a 2/π = multiplier to the maximum force at the beginning of the straight line motion relating to the average angular speed at 45º of the ¼ turn has a high degree of accuracy. Page -69-

12 THE FUNCTIONAL ELEMENTS OF THE INERTIA DRIVE The described combined effort inertial drive has seven main functional elements: 1. A pair of flywheels, for providing a rotational inertial reluctance backrest to produce a reaction less rotational force impulse. The pairs of flywheels move in alternating straight line reciprocal motion in direction of vehicular travel, and have parallel axial orientation, opposing rotations, equal peak straight line motion velocities, equal straight line stroke length and differential straight line reciprocal motion cycle times. Each flywheel has complete freewheeling freedom of rotation in relation to the propulsion device. 2. An impact rotor in axial alignment with each flywheel, for the purpose of accumulation and storing temporarily rotational kinetic energy to be used for the propulsion of the device. 3. A motive kinetic force generator in form of a motor-generator engaging with each flywheel and reciprocal with each impact rotor. The motor-generator has the purpose of energizing the impact rotor with rotational kinetic energy, while using the flywheel as a inertial backrest. The flywheel, the impact rotor and the motor-generator are assembled into an integral assembly. 4. A transmission for converting the rotation of the impact rotor into reciprocation straight line motion of the flywheel assemblies. The transmission therefore can be called a rotational-to-reciprocation transmission. For the purpose of mathematical simplicity, a complimentary cam and cam followers are used as a rotational-to-reciprocating transmission for all the following propulsion discussions, because of the simplicity of the straight line rise and fall of the straight line velocity in relation with the rotation of the impact rotor. This type of motion is also referred to as a saw-tooth motion. 5. A pair of straight line guides for guiding the flywheel assemblies in a straight line reciprocal motion. 6. A reciprocal touch friction break, for removing excess flywheel angular velocity reciprocally between the two flywheels. The touch friction break, because of Page -70-

13 reciprocal operation, therefore does not interfere with the propulsion of the device and it represents the simplest form of the device. 7. A supporting frame, for the purpose of supporting items 1 to 7. DESCRIPTION OF THE COMBINED EFFORT PROPULSION CYCLE The combined rotational and straight line motion inertial propulsion is accomplished with a four phase process. Each phase is a quarter turn of the impact rotor. The impact rotor rotation is used as a measure of reference because of the workable characteristic of the angular motion position as a reference and because of the variable character of the cycle time duration. The impact rotor direction of rotation is counter clockwise, the rotation the flywheel is clockwise. 1. Accumulation Phase: Accomplishes the accumulation of rotational kinetic energy into the impact rotor by mutual rotational reciprocal inertial exertion against the reluctance of a flywheel, by the motive force of the motor-generator. The accumulation phase thereby increases the angular velocity of the impact rotor. The utilization of the motor-generator is 1/8 of the total capacity of the motor, because of the ¼ turn of the drive phase and the reciprocal exertion between the flywheel and the impact rotor, which distributes thereby kinetic energy into both, the impact rotor and the flywheel. Thereby 1/8 of nominal rated power. 2. Drive Phase: Release of the rotational kinetic energy, accumulated in the impact rotor, into the straight line inertial kinetic energy of the flywheel and into the straight line inertial kinetic energy of the propulsion device, by mutual reciprocal separation. 3. Rotor Break Phase: Removal of excess (unused) rotational kinetic energy from the impact rotor to accomplish the angular velocity ω,b. The impact rotor break phase is an on demand function, which depends on the relative resistance of the device to motion and occurs during the end of the drive phase. The break phase is a complex vector force Page -71-

14 de-acceleration. The intensity of the break phase, in combination with the drive phase, also determines the overall gain/variance of the angular velocity of the impact rotor. 4.Idle phase: When no new energy is induced into the impact rotor, the stored rotational kinetic energy of the impact rotor will alternately flow into the straight line kinetic energy of the flywheel assembly and back into the impact rotor through the motion of the rotational-to-reciprocating transmission. The straight line reciprocating motion of the flywheel assemblies and the alternating acceleration and de-acceleration of the impact rotor is thereby an alternating feedback loop. The straight line reciprocating flywheel motion has an equal peak velocity and the straight line acceleration and de-acceleration forces of the two flywheel assemblies are in reciprocal equilibrium. Therefore no motion or vibration forces act onto the device. The idle motion frequency is preferably the maximum allowable motion frequency of the employed mechanical design. The accumulation phase and drive phase represents each one quarter turn of the impact rotor, for a total of one half cycle. The second half of the cycle are idle phases. continued next page Page -72-

15 THE MATHEMATICAL AND PHYSICS PRINCIPLES OF THE PROPULSION CYCLE The Accumulation Phase: During the accumulation phase, rotational kinetic energy is accumulated into the impact rotor, by mutual reciprocal exerting against the free wheeling rotational reluctance of the flywheel. The action of the accumulation of the rotational kinetic energy is reaction less, in relation to the device mass, due to the reciprocal action and the declining slope of the straight line velocity of the flywheel assembly, in Page -73-

16 comparison to the distance moved by the flywheel assembly and the new induced rotational kinetic energy was developed against the free wheeling reluctance of the flywheel. The declining slope is illustrated by the Cartesian coordinate in the drawing and has the effect of removing the straight line kinetic energy from the flywheel assembly mass, even when new straight line kinetic energy is introduced, and feeding the kinetic energy into the impact rotor, thereby reducing the straight line kinetic flywheel energy to zero. Such a negative slope of the velocity is the characteristic of rotational-to-reciprocating transmissions, like the crank and connecting rod, the scotch joke, the rotational cam and cam follower, to mention a few. The negative slope of the mentioned rotational-to-reciprocating transmissions is occurring during the approach to the point of no straight line motion, also referred as the approach to the top dead center or the bottom dead center of the crank shaft. It is called the dead center because no straight line motion of the piston occurs at that point. The declining slope characteristic of the crank shaft and connecting rod has the effect that the straight line moving cylinder mass, having a kinetic energy and being attached to the end of the connecting rod, is being converted to rotational kinetic energy of the rotor mass centered on the crank, at the point of top dead center. Therefore, the rotational to straight line coupled motion of the before mentioned rotational-to-reciprocating transmissions employs a negative feedback loop, feeding and depleting one form of kinetic energy, straight line or rotational, into another form of kinetic energy, rotational or straight line. The conservation of kinetic energy applies also to any new kinetic energy introduced into the system at any point of the motion rotation. And it can be further concluded that the action of the accumulation phase has no net kinetic effect on the device mass due to the equal action and reaction of the all the straight line forces at play and the fact that all straight line kinetic energy has been feed into the impact rotor at the end of the accumulation phase. Thereby one can conclude that the accumulation phase is complying with, and is working with, the principle of conservation of kinetic energy and the conservation of momentum. Ref. Kurt Gieck Engineering Formulas P.10; epi-eng.com. The next Vector plot further explains the Drive Phase Force distribution with Vectors. Vector A is equal to Vector B because the tangential acceleration Vector C is equal the Flywheel acceleration Vector D. Therefore a net Zero Force. Page -74-

17 The Accumulation Vector plot: Page -75-

18 The Drive Phase: The drive phase is accomplished by releasing the accumulated rotational kinetic energy of the impact rotor into the device mass, like a spinning yoyo releases its rotational kinetic energy into very fast acting straight line motion. The drive phase is a reciprocal impulse by mutual reciprocal separation, distributing the accumulated rotational kinetic energy of the impact rotor into straight line kinetic energy of the flywheel and straight line kinetic energy of the device mass respectively. The drive phase can thereby considered to be a distribution of kinetic energy, conserving the Page -76-

19 kinetic energy of the impact rotor into the kinetic energy of the device and into the flywheel assembly respectively. The distribution ratio is the reverse ratio of the device mass to the flywheel assembly mass, which will be mathematically developed in the math section. While one flywheel assembly is operating the drive phase with a high impact rotor energy level, the second flywheel assembly is operating the idle phase with an idle impact rotor energy level. This energy differential causes a balance beam weight scale tip in favor of the flywheel operating the drive phase, thereby investing partially the impact rotor stored kinetic energy into the device, motivating the device forward. The amount of kinetic energy invested into the device, is then discounted from the amount of kinetic energy necessary to be absorbed by the impact rotor break phase, to accomplish the regular peak straight line flywheel assembly velocity. While the accumulation phase employs equal reciprocal forces. The drive phase, in comparison, only has extending forces acting straight line against the device and the flywheel assembly. Thereby, causing an impulse by separation with the kinetic energy of the impact rotor as the motivating energy. Therefore, the action of the forces are causing the reaction less propulsion of the device. The distribution of rotational kinetic energy, made available in the impact rotors by the accumulation phase, is distributed according the ratio of masses of the device and flywheels. From the indicated formula for Kinetic Energy the following plot of the Velocities is generated. Next is picture #9: Page -77-

20 The above picture #9 diagram illustrates the force foot-print and the reversal of force foot-print between the idle phase and drive phase. During the idle phase the highest intensity of force is at the end of the idle phase. During the drive phase the highest intensity of force is at the beginning of the drive phase with a much larger force foot-print due to the much larger acceleration because force = mass * velocity², change / 2 * distance, change. The higher initial force foot-print and power foot-print is causing a force imbalance which is motivating the device forward. The motivation of the device starts a dynamic process which depletes the larger drive phase impact rotor kinetic energy faster then the depletion of the idle phase. In conclusion: Only the flywheel assembly with the higher impact rotor initial kinetic energy is distributing kinetic energy into the device in direction of vehicular travel, Page -78-

21 which is the impact rotor delivering the drive phase. The drive phase starts a dynamic process of impact rotor accelerated kinetic energy depletion and resultant straight line kinetic energy gain of the device, due to the initial force imbalance between the drive phase and the idle phase. The drive phase is complying with the principle of conservation of energy and the conservation of momentum because the kinetic energy of the drive phase impact rotor is distributed into the device and into the flywheel assembly. The distribution ratio is the reversed ratio of the device mass to the flywheel assembly mass. Continued next page: Page -79-

22 The Impact Rotor Break Phase: The impact rotor break phase removes excess kinetic energy from the impact rotor before the flywheel reaches the regular peak straight line velocity during the drive phase. It can be therefore assumed to be part of the drive phase action. Page -80-

23 Only additional break de-acceleration vector forces are added with complex vector math to the declining straight line centripetal forces because all forces are reciprocal rotational forces between the impact rotor and the flywheel. This means the resultant average complex vector impulse is larger then the regular repeating peak straight line flywheel velocity times the flywheel mass. The break action is activated by reversing the polarity of the motor-generator, whereby the motor-generator acts as a generator which removes rotational kinetic energy from the impact rotor before it reaches the straight line mass of the flywheel assembly. Thereby one can conclude that the break phase works with and complies with the principle of conservation of energy and the conservation of momentum. The break phase break energy flow is the most intense during the stall of the vehicular motion due to gravity or obstruction, subsequently, the average complex vector force addition has he highest magnitude. With a further analysis of the break action, it can be rationalized that the break action makes the subsequent following idle phase to have the same regular Idle impulse intensity. Thereby, the drive phase has a large impulse by separation force foot-print at the beginning with the same subsequent idle phase impulse intensity at the end. Thereby the averages Drive Phase impulse is larger then the Idle Phase because of the differential in time duration. Continued next page: Page -81-

24 The Idle Phase: The motion of flywheel 1 and flywheel 2 are in an opposing alternating motion and thereby all acceleration and de-acceleration forces are canceling to a sum of ZERO as long as no NEW kinetic energy is induced into the system, as evident in the diagram above. The next two graph illustrates the dynamic process of the propulsion cycle considering the shifting of the reference frame using the principle of negative feedback. Page -82-

25 The gain of the device velocity is feed back reducing the flywheel effective internal self-contained force producing the reduced resulting force. The feedback logic describes the inclusion of gravitational pull into the reference frame formula thereby determining the maximum incline climbing ability. Page -83-

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