GB2534542A - Dynamic functions implicit in the variation of normal forces at an interface exhibiting isotropic friction - Google Patents

Abstract

An inertial propulsion arrangement comprises a trolley 1 whose weight is supported by supports such as wheels 4, 5, 12, 13. At least one of the wheels 4, 5 is oscillated horizontally along its rolling direction, with direction reversal at the ends of the oscillation. The arrangement is such that the load acting on the wheel 4, 5 is different at the two reversal points. It is proposed that the impulse is proportional to the normal force acting on the wheel, and that this gives rise to propulsion.

Description

DYNAMIC FUNCTIONS IMPLICIT IN THE VARIATION OF THE NORMAL

FORCE AT AN INTERFACE EXHIBITING ISOTROPIC FRICTION

JR DREWE HS SUSSMAN BAR COCKETT

The present invention is a method of inducing a pulsatory impulse on a dynamic configuration while it is exposed to an external gravitational field.

Fig. 1 illustrates one embodiment of the present invention. It consists of a horizontal platform, 1, attached to four wheels in separate frames. A rectangular runway, 2, supports this trolley, which rests on a horizontal roller track, 3. The two front wheels, 4, 5, oscillate about a fixed point on platform 1, usually at a predetermined frequency fo, and under conditions where the phase difference, A4, remains constant. Each wheel then rebounds from an elastic block, at its oscillatory limit (Fig.2). Hence, the normal force acting on each wheel varies as a periodic function of time.

The rebound of the two front wheels is simultaneous. At the time these wheels reverse their velocity, their normal forces are unequal. It is demonstrable that a linear impulse is induced at the rebound, propelling the trolley forwards.

Mathematically, the description of this phenomenon must incorporate an impulsive reaction acting directly on the source of the external field. It is remote from any part of the dynamic configuration, and its supporting frame. It is the absence of a detectable reaction which exemplifies the novelty of this invention.

The measured propulsive impulse is associated with a rebound of the normal forces acting on the oscillating wheels. It derives from the proportion of the total weight of the assembly which acts upon each wheel at the runway interface.

Implicitly, such an impulse arises from an interaction between the oscillatory configuration and the external gravitational field. There are various oscillatory geometries developing a sustained impulse, where the reaction occurs on the body generating the external gravitational field.

Generally the applications of this invention rely on the reaction acting away from the runway along which the trolley accelerates.

Consider a trolley one or more of the wheels of which oscillate about a fixed point along a line g, lying perpendicular to its rotational axis. All the trolley's weight acts through its wheels, on a horizontal runway, at right angles to a uniform external gravitational field, of intensity g. The distribution of the normal force acting at the runway interface of a moving wheel, is a function of time. Let the wheel oscillate about a fixed point on that line, S. Over the interval (At) where the wheel reverses its velocity, the external field induces a dynamic interaction. It is manifested during the velocity change, and is directly proportional to the normal force.

Referring to Fig. la, the trolley platform 1 is a rectangle, ABCD. It is rigid, and constructed of sheet steel of uniform density and thickness. Its four wheels are supported by a horizontal runway, 2, which, in turn, rests on a roller track, 3. Let the runway, 2, exhibit an isotropic coefficient, p, of dynamic friction. The x'-axis of a Cartesian local reference frame, 8', bisects platform 1 transversely. Its mid-point denotes the origin 0'. The z-axis bisects it longitudinally, and the y'-axis is perpendicular to 01. The Earth's external gravitational field, of constant intensity g, lies along the y'-axis.

A set of eight identical roller bearings, 6, are in mounts screwed to the cylindrical rim, 9, of each front wheel, 4, 5. They rotate freely about an inner cylinder, 10, which is fixed to a frame 8. The rollers are constructed from perspex, with a chrome-steel jacket. Each mount incorporates a helical spring, to push its roller onto cylinder, 10. Two parallel channels, 11, of equal depth, are cut into the underside of platform 1. Three parallel rows of identical, equally-spaced, small wheels, 7, are fitted to the top of each wheel frame 8, thereby enabling it to slide smoothly back and forth along its path. Referring to Fig. 2a, each front wheel is displaced along a separate straight line, Si or;2, lying parallel to the z-axis, and equidistant from it. At a given time, t, one wheel is displaced by Az' from its midpoint z-coordinate, z'," and the other by -Az'. Each of the rotational axes, 131,132, has a z'-displacement limit of ±Az",', from 4. These two axes coincide at 4, at an x'-coordinate of x' = Fig. lc shows schematically the external features of the oscillatory mechanism, 16. A rack, 14, is attached to each front wheel frame, 8. Each rack meshes with a toothed wheel, 15, with its rotational axis fixed to platform 1. The mechanism, 16, acts to displace simultaneously the two front-wheel frames, 8, in opposite directions, at a constant speed, Uz', along a path gl or c2. Fig. 2b illustrates this geometry. A front wheel axis transcribes an angle of 20,z about;pt. Throughout their oscillations, these two front wheels remain out-of-phase by 2'. The oscillatory mechanism serves to displace the axes, p,, P2, of each front wheel, at a preset constant speed, Uzs, along a track c. On reaching its z'-displacement limit, either zu,' or -z",', a front-wheel frame, 8, then rebounds from a rubber block, 17, 18, fixed to the lower surface of platform 1. A weight, 19, is screwed underneath oscillator 16. It serves to ensure the trolley's centre of gravity lies on the z'-axis.

A pair of rear wheels, 12, also rotates axially in roller bearings inserted into frames, 13. Each frame is fixed to the underside of platform 1, whereupon the rear wheels have a common rotational axis, y. Note that the rotational axes of the four wheels arc at the same y'-coordinate.

Immediately before a frame, 8, impinges on one of the rubber blocks, the oscillatory mechanism, 16, disengages from it. After rebounding, the mechanism again locks onto each wheel frame. Its i-velocity, Uzi, is thereby reversed.

Referring to Fig. la, a xenon stroboscope, 20, and two cameras, 21, 22, are in fixed positions relative to the trolley's rest frame, S. They enable the position of a mark 23, inscribed on the edge of platform 1, to be determined at a series of equal time intervals, Atz, the stroboscopic pulse interval. A linear scale, 24, is fixed to the roller-track frame, 3.

An electromagnetic clamp, 25, secures the trolley platform, 1, and runway 2, to the steel frame of track 3. Once the electric current through clamp 25 is switched off, the trolley and runway are then detached, simultaneously.

The i-coordinate at the centre of lines gl and c2, is designated by z."i. The oscillator, 16, can be preset for frequency f", and z-displacements, Zcoj = Z.(2). However, the x'-distance between GI and c2 is constant in this specific configuration. Let c denote the trolley assembly's centre of gravity. It lies on the zi-axis, and its distance from z'pi is denoted by 4 The position of c is displaced by putting slotted weights, 26, on the five rods, 27, which screw into the upper surface of platform 1, at right angles to it. *1 Referring to Fig. lb, the axis of each wheel, 7, is mounted on two helical springs enabling the frame 8 to be displaced smoothly along its indented path.

Implicitly the z'-velocity of each wheel axis, p, in frame S is: Uzi = 2; fo, (1) provided the frequency fo is constant, and where the displacement; is positive when the z'-coordinate is increasing.

Initially, platform 1 and runway 2 are stationary with respect to the roller-track assembly, 3. Frame S has its x-, y-and z-axes as shown in Fig. 1 a. By definition, it is at rest with respect to the surface of the Earth. The trolley platform and runway are locked to the roller track by a clamp, 25, whereupon oscillator 16 is switched on. Each front wheel oscillates about the midpoint of its path, gl, g2, as depicted in Fig. 2.

On detaching clamp 25, it is observed that, while the oscillation continues: (i) A z-impulse APzo-,2) is imparted to the trolley at intervals of T/2, where T = 1/f," ; (ii) Each impulse is of equal magnitude and is associated with the simultaneous rebound of the two front wheels, 6, 7, at the rubber blocks, 17, 18; (iii) The trolley accelerates along the runway, 2, supporting it on the track, 3; (iv) During the inter-rebound interval, a trolley moving along the runway, experiences retardation; (v) There is a negligible recoil on the runway as each impulse APz072) is absorbed by the trolley; A capital X, Y, or Z, specifies a distance along the respective axis. The lower case denotes a coordinate.

(vi) By altering the value of one of the preset variables (fa, ZED, the impulse APB (T/2) can be compared for different trolleys. The stroboscopic photographs demonstrate that APB (T2)is directly proportional to the difference in the normal forces acting on the rebounding front wheels, and (vii) The analysis also indicates that this impulse AP./r2) is directly proportional to the z-velocity change AU: of the rotational axes, 13,, Pt, when they rebound from the rubber blocks, 17, 18.

Let the normal forces on the front wheels, 4, 5, be denoted by Al and A2, respectively. A convention is imposed whereby a wheel with a positive i-velocity UzI, moving towards rubber block 17, and rebounding from it, exerts a normal force Al. The second wheel has a i-velocity of-U) towards block 18, exerting a normal force of A2 during the rebound. By definition, therefore, over each inter-rebound interval, At decreases with time, while A2 increases.

Fig. 3 illustrates schematically a cross-section through the trolley platform, 1, with the i-coordinates of the normal forces, Ali A2 and A3, superposed on it. Let the trolley be in equilibrium, where its weight is entirely supported by the runway, 2. Under these conditions, the normal forces produce moments about an x'-axis, passing through the centre of gravity, c. The sum opposing moments are necessarily of equal magnitude, so that: API +. rata = 0; IMP) AlZis + A2 Z2' ; ALZ,' = A2 Z2' ; Tiler; 2A3Zyr. where

The normal force on each rear wheel is A3, the i-distance between e and the centre of the rotational axis 131 is Z1', that for p, is 4, and the z'-distance between c and the centre of the rotational axis 7 is 4'.

On putting mag as the total weight of the intact trolley, Eq.'s (2), (3) and (4), require that: Z2 Z21 mag -2 A2 -A2 -A2 -= 0. (5) 4' z11 Once the values of Z1s, Z2 and 4' are known: A2 can be determined from Eq. (5), and Al = A2 Z21 Z1'. An expression for the rebound z-impulse AP.(T72) in terms of At and A2, adopts the form: 2 (A2(m) -At (m)) AP. g

where A1(m) and A2(m) represent the normal forces, At and A2, respectively, at the instant they impinge on the rubber blocks, 17, 18, at their respective displacement limits of Zioz; and Z2(m), as shown in Fig. 2b. *2 *2 Under conditions where 1..J; varies over the inter-rebound path, other interactions occur. They require Eq. (6) to be modified by introducing second-order terms, which act independently of the rebound.

Alternatively, the values of Ai(n) and A2(m) can be measured directly. One method entails suspending the horizontal trolley by four identical helical springs, each holding a wheel in a sling.

In the absence of friction, the z-velocity of the trolley is given by: APz (T/2) (7) where B is, the total number of rebounds (an integer) at the oscillatory limit, during a series of continuous oscillations.

Over each period, T, the sum z-impulse arising from friction during the oscillation of each front wheel on the runway, is necessarily zero, given that V(z)a 0. Let T = 442 -te, whereupon: 41+2 Pio z = I g A. dt; (17(=)a= (8) to = 0, (9) where g is the coefficient of dynamic friction.

Instead of a straight line, 5, the midpoint of each front-wheel axis, can execute a circular path, of radius r,', about The channels, li, guiding each wheel frame are then curved. Each axis ft then transcribes an angle 241," subtending x9i, before reversing its velocity U'. A vertical rod, the axis a of which passes through oscillates the front wheels. It emerges from the oscillator, 16, and screws onto the midpoint of a horizontal strut holding a front-wheel frames, 8, at each end.

Note that z,' = zg' + rcrsin 4)., and Z2 = Zz' -rpl sin 4m, in Eq. (5).

In such a configuration, Eq. (6) is modified by putting: Uzi = U 'cos 4", , (10a) where U = 2n fornly, (10b) = 4m in radians, and the angular velocity of a, = f d$ /dt.

Consider the type of system shown in Fig. 1 a. Let each front-wheel axis oscillate, through an angle 2$m, about a point x9, in a path of radius r.1. Assume that the axis speed, U', is constant, and the oscillations of 133 and (32 remain out-of-phase by 2+m. The dotted line in Fig. 2b illustrates this geometry.

Let the mass of the intact trolley be, ma = 7.35 kg, with = 30°, = 5Hz, r: = 0.080 m, 4 = 0.215 m, Zy = 0.323 m, and a stroboscopic pulse interval of 0.050 s. Referring to Fig. 1, runway 2 is a rectangular sheet of polycarbonate, of uniform thickness, with a total mass of m" = 1.140 kg. The maximum errors in the preset quantities, listed above, are. ± 0.010 kg for mass, ± 0.05Hz for oscillatory frequency, ± 0.001 m for a distance, ± 0.03° for angle 4), and ± 0.0005s for the stroboscopic interval. The i-distances are: Z.' + = Z-r', (Fig 1c).

Table 1 records the measured z-velocity, V(z)a, of the trolley from rest with respect to time, t. The clamp, 25, is disengaged after the oscillatory frequency, fa, is constant. Referring to Fig. 1, V(.),, is defined in the rest frame, S. Graph 1 displays the stroboscopic results, comparing them with V(.),, (dotted line), as calculated from the sum impulse and trolley mass ma, (Eq. 7). Friction between the wheels and runway, and at the wheel bearings, ensures that the measured z-velocity is lower than that predicted by Eq. (9). (1, 2)

TABLE 1

Velocity V(z)a / m s-1 0.00 0.15 0.31 0.46 0.61 0.77 0.93 1.06 1.20 1.36 1.49 1.66 (± 0.01 ms') Time t / s 0.01 s) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100 Table 2 relates the ti-distance, Z.', between the centre of gravity, e, and the oscillatory focus)4', to the z-velocity, V(z)a, of the trolley, from rest, after a continuous series of rebounds over 1 s. The preset quantities are as stated for Table 1, except that f. = 3 Hz, = ± 21.0°, and 4' is varied by altering the positions of the slotted weights, 26. Note, however, that ZT' = 0.5380 m in each example.

Graph 2 shows V(.* as a function of Z.' at t = 1 s, with time on the abscissa. The dotted line represents Vwco as predicted by Eq. (7) from the sum impulse at t = 1 s.

TABLE 2

Velocity V(z) a / m CI 0.713 0.430 0.294 0.236 0.150 0.090 (±0.005ms') M (±0.001111) 0.120 0.155 0.215 0.253 0.305 0.355 Table 3 records the observed z-velocity VR.)," of the trolley after a time interval of At = ls, from rest, as a function of f,". The oscillatory limit Om = ± 21.0°. Other 'preset quantities are as stated for the results in Table L

TABLE 3

Velocity V(a) a / m 0.132 0.290 0.515 0.842 1.215 1.642 2.115 (±0.005 m ri) / Hz (±0.05 Hz) 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Graph 3 plots V(.)," at At = ls from rest, as a function of f. , with time on the abscissa. The dotted line denotes a calculated value of Vcaz, (Eq. 7).

Each result was obtained by measuring the average value of Vwc, from fifteen trials along each direction (± z) of the runway, 2. The roller track, 3, is horizontal within 0.1°, and runway 2 exhibits isotropicity.

Usually the electric circuit to clamp, 25, is set to disengage it immediately each wheel axis, 13, subtends an angle ± On, at x9'.

The above results are by way of example. Further experiments record V(z)a as a function of ch",, 1.9', and m". Eq. (6) is derived a posteriori. It predicts.the z-impulse, APzu/2), imparted to a given trolley at each rebound of its oscillating front wheels.

An oscillatory rebound is observed to produce an impulse on the trolley only when the wheels; (i) reverse their velocity, and (ii) experience a normal force. It is the rebounding of the normal force at the wheel interface which is fundamental, and underlies the phenomenon. The primary concern, however, is to account for the absence of any detectable reaction to the rebound interactions.

The runway 2 does not recoil as the trolley accelerates along it. On the contrary, the runway develops a small positive impulse along the z-axis. *3 A study of friction and drag acting on the trolley as it is propelled along the runway, is the subject of a separate report. (3) However, the results in Table 1, above, demonstrate that the observed value of Ywc, is lower than that calculated from Eq. (7). A uniform retarding force exists when At < 0.6 s. Subsequently, when the elapsed time At > 0.6 s, the external forces serve to produce a non-uniform rate of change of V(i)a.

Coulombic friction between the wheels and runway accounts for the result when At < 0.6 s. (4) It also explains the small positive velocity of the runway. At higher velocities, two other processes intervene. Both are non-linear functions of its z-velocity. First, the oscillatory motion of the trolley introduces a sliding of the wheels on the runway. Second, friction at the wheel bearings increases in proportion to m",2, where cob, is a wheel's angular velocity about its rotational axis, p. The coefficient IA of dynamic friction is then no longer constant. (5) Mathematically, the rebound impulse, AP. 072), of Eq. (6), is predictable in terms of the properties of the normal force, A, acting at the interface of each wheel. Referring to Fig. 3, when the complete trolley is supported on a runway, the force of gravity, mug, acting on the trolley is distributed over its four wheels. The normal forces, A1, A2, acting on the two front wheels, are determined from Eq.'s (3b) and (5). When the y-forces are in equilibrium on the trolley, a normal reaction is present, thereby counteracting the normal force on each wheel. The sum reaction over all four wheels is -mag.

*3 By definition of frame S (Fig. la), a positive z-impulse acts in the same direction as V(z)a.

Mass and inertia are regarded as separate properties intrinsic in any material body. However, it appears that the determinative property governing the rebound impulse, is inertia, rather than mass.

Suppose a normal force mag, exists as an interaction of an external gravitational field, of intensity g, with the mass of a trolley. A normal force acts on each of its wheels, provided the trolley's weight is supported by the runway. At a given time, t, normal force and normal reaction then exist at a wheel interface. They are equal in magnitude and in opposite directions. Hypothetically, a normal force, mag, is then associated with inertia, X. Let the direction of the gravitational field intensity, g, be constant. It follows that the normal reaction, -mag, must incorporate negative inertia, -x. While a wheel travels at a given velocity U', along the runway, equal positive and negative inertia exist simultaneously.

Hypothetically, momentum, P, is then associated with a normal force, and the reaction to it, at each moving interface supporting the trolley on the runway. It is defined in terms of xU ' and -xU respectively. While the velocity is constant over the inter-rebound interval, there is a zero momentum change. However, as it rebounds from a rubber block, each wheel comes to rest, momentarily. An impulse is thereby generated as the wheel reverses its momentum: the positive and negative momentum changes are then equal and opposite. They exist separately, and resolve spontaneously in opposite directions.

While the wheels oscillate, the normal forces, AI, A2, are unequal. The force Al, rebounding nearer the centre of gravity, e, is greater than A2, which rebounds at a greater distance from it (Eq. 3b).

Both front wheels reverse their velocities simultaneously. Over any oscillatory half period, T/2, each generates a positive z-impulse.*4 It arises from the momentum change of the positive inertia associated with each normal force, Al, A2. These impulses are unequal, and act in opposite directions. The process is represented by Eq. (6). A corresponding negative z-impulse is produced, simultaneously, as the negative inertia rebounds.*5 The positive z-impulse, generated over each half period, is associated with the trolley weight, and acts on it. However, its negative counterpart arises from the reaction, -mag, counteracting the trolley's weight. It is generated at the runway and roller track, by the external gravitational field.

Physically, normal force is produced by an external gravitational field. Inertia, is regarded as a property of gravitation, as expressed by Mach's principle. (6, 7, 8) Once a normal reaction, -AP1072), exists, it is accompanied by negative inertia. Any change in the normal reaction produces a negative z-impulse.

Hilbert's equations have been extended to describe universal gravitation in terms whereby inertia is manifested by normal force. (9, 10, 11) *4 Over a complete period T, the two wheels then generate equal and opposite x-impulses from the positive-inertia rebounds. The corresponding negative-inertia x-impulses also cancel.

*5 Note that the rebound of inertia at a normal-force interface necessarily produces a positive impulse. Its-surn, for the two front interfaces, is a z-component propelling the trolley forwards. Simultaneously, the rebounding reaction induces a negative impulse away from the trolley and runway.

A series of experiments enables this hypothesis to be tested. One test entails clamping the trolley to runway 2, while it is supported by the roller track, 3. On activating the oscillator, 16, it is observed that a pulsatory impulse is imparted to the intact assembly, comprising a trolley with the runway locked onto it. It develops a z-velocity according to Eq. (7), but with ma+ mp substituted for ma.

Clearly, the absence of a detectable impulsive reaction, -APz cum has profound implications. It is necessary to provide a theoretical basis for the propulsion of the trolley, in terms of the existing schemes of conservation. A pump truck (Rollatruc, L 2000) can be used to demonstrate the phenomenon. An operator need only stand on the truck's loading platform, and oscillate the steering mechanism attached to its two front wheels. When the floor is smooth and horizontal, the truck accelerates forwards.

An unladen truck has a mass of 63 kg. Let the operator have a mass of 70 kg. Such an assembly can, in turn, be supported on a runway, fitted with wheels, able to move freely along the workshop floor. As soon as the pump truck accelerates, it is evident that there is no recoil on the runway.

In principle, this method of propulsion has a number of applications. Most depend on the reaction to the driving impulse acting remotely, away from the trolley and runway. It appears that this property is intrinsically gravitational.

Hilbert's theory requires that positive and negative inertia are associated with the normal force on a wheel supported on a runway. Once a wheel is moving, positive and negative momentum coexist at the supporting interface. However, at each rebound, the resulting changes in momentum then produce impulses of equal magnitude in opposite directions.*6 At the present time, the impulsive reaction to the rebound impulse propelling the trolley, remains to be detected.

A set of formulae embody the mathematical scheme emerging from Hilbert's equations. They have enabled an oscillatory system to be constructed which develops only a y '-impulse. (3) Additionally, it is explicit that the reaction acts through the external gravitational field.

Referring to Fig. 1 a, let the trolley accelerate, from rest, along a roller track, 3, which is stationary with respect to an external gravitational field. Under such conditions, it is demonstrable that the gain in kinetic energy of the trolley derives entirely from the power fed to the oscillator, 16.

Dynamically, the rebound impulse imparted to the trolley by the oscillatory system is determined by the kinematics of the inertia.. Suppose the intact trolley develops a z-velocity, V(z)a, along the runway. There is both a normal force and a normal reaction at each wheel interface with the runway. It is the velocity of the inertia associated with these forces, which then determines the impulse and kinetic-energy changes at each rebound.

*6 J. P. Terletslcy (1962-3) published (e.g: in Journal de Physique) a mathematical exposition of positive and negative proper mass. Physically, his scheme requires a negative impulse to interact in this way on rebounding from matter of the opposite sign.

Implicitly, the impulse developed by the trolley frame, together with the kinetic energy transferred to it, at each oscillatory rebound, are geometrical functions of the velocity of the normal force within the frame it is produced. In practice, this velocity, usually, is V(z)a. It can, however, be a term where V(z),, is superposed upon the runway's velocity along the roller track, when trolley and track are coupled. *7 Whatever the practical details, it is fundamental significance that the properties of the oscillatory mechanism are defined with respect to the velocity of the normal force on each wheel at its rebound.

Let the power input *V,' to the oscillator, 16, at a given time, t" during the time interval, At" be: 43,1= V I, where V is the p.d., and I the current (A). These variables are monitored continuously, using a voltmeter and ammeter in the circuit from the battery B, as depicted in Fig. lc. Over an interval of time, At" the sum energy input is then: = .c (1),' dt, (11a) 1111e, V4).2 (1 1 b) + , where 41 is a constant representing the energy dissipation (over At,), arising from friction on all four wheels, air resistance and thermally, as a result of the rebound. Note that 141 is governed by the trolley specification, its preset oscillatory functions, and the frictional coefficients.

Suppose, however, the roller track has a uniform velocity V(./T. There is then a fundamental relation between the total energy input to the oscillator (over At,), which imparts a z-velocity, V(z)a, to the trolley relative to that track, and the roller-track velocity V(z)r. Let the velocity of that trolley with respect to the track, be: 3/40. = 0; (12a) Vux, = 0; (12b) V(z)a > 0. (12c) Note, however, that that Eq. (11) represents the trolley's velocity, V(z)a, after it accelerates from rest (t,= 0), during a time interval At" with respect to track 3, in response to a series of internal rebound impulses, 0 APzcra), (Eq. 7).

Suppose that, during this process, the roller-track frame already has a uniform z-velocity in S, i.e: with respect to the external gravitational field of intensity, g, so that: V(zyr > 0. (13a) In practice this can be achieved by installing the track frame in a lorry, and driving it at a uniform velocity along the runway of a disused aerodrome.

*7 Referrring to Fig. la, frame S' is defined as being at rest with respect to the trolley platform, I. It is the frame where acceleration is detectable internally as a result of the rebound of the moving interfaces. The oscillator's power input, CD:, is measured in S'.

The trolley's z-velocity, Vwci, is measured relative to the track. At a given time, t" it is determined by: (i) the total z-impulse, 0 APz(v2), imparted to the trolley over the time interval At" and (ii) the sum opposing z-impulses from wheel friction and drag. Significantly, it can be shown that after the trolley has absorbed a given internal impulse 0 APz cm), V(z)a is entirely independent of any pre-existing z-velocity, Vwf, of the roller-track, 3. *8 However, the trolley develops additional kinetic energy, AE(K)s, throughout this interval, At" with respect to the reference frame, S, so that: ma 07(z)a± V(402 InaV(z)i 2 -InaV(z)(t2 AE(K)s (13b) Once a trolley starts to accelerate from rest along a roller track which already has a uniform z-velocity, V(zyr, in S, its increase in kinetic energy exceeds that supplied by the oscillatory mechanism. At a given time, t" the trolley's total kinetic energy in S, is: Eoos = /2 Mall(z)a± * (eft)2 * In order to attain this state, it must receive: (i) an external energy input of: MaV(z)r 2, to bring the trolley from rest, in S, to the track's velocity, V(z)T, and (ii) an internal (oscillatory) input, of: t /2 rn"V(,)"2 + V, increasing its z-velocity, from rest, relative to this track.

Note that three expressions, Eq.'s (11a), (I lb) and (13b), are predicated upon Eq. (6). It then follows that: (i) the z-momentum imparted to the trolley is equal at each rebound, and (ii) the speed U4, (Eq. 10b), remains constant.

However, the oscillator must supply a greater energy, over an interval, At" if the increment, AV(z)a, in the trolley's z-velocity is to be constant at each rebound As the trolley's z-velocity increases from rest, the rate of change of its kinetic energy, with respect to the stationary track, is necessarily: AE(e)a = /2 rna(V(z)a(0+1)2 -V(z)a02) ; CV(z)T = 0) , (13c) At As,' ; = const.) , (13d) At where V(z)cco+i) is its velocity at the 0 + I rebound, and V(z)a(e) at 0. Note that AS," is measured over At" which is the interval between successive rebounds.

It is implicit that the oscillatory power output should then increase with time. In practice, it cannot do so indefinitely. Many oscillators supply a preset, constant, power, (IN', to the front wheels. The angular velocity co about the oscillatory focus, z,", then diminishes with time, while the period, T, increases.

*8 The normal force necessarily originates in the frame supporting the trolley's weight. Electrical power is fed to the reciprocating mechanism, 16, oscillating the two front wheels. When these wheels rebound, energy is transferred to the trolley. However, the oscillating normal forces are defined in terms of the external gravitational field, with which they interact. In addition to the power fed to the oscillator from the battery on the trolley platform, 1, there are conditions where the external field also contributes to the oscillatory power input. Specifidally, such conditions arise when: (i) there is no reaction to the rebound impulse on either the runway, 2, or roller-track, 3, and (ii) V(z)r # 0.

Suppose the kinetic energy supplied to the trolley to increase Vwto is constant at each oscillatory rebound. Eq. (6) is then no longer applicable. Over a series of consecutive rebounds, the impulse, AP, (r/2) , diminishes with each term in the series. There is a smaller change, AV(4., in trolley velocity, at each rebound when the oscillatory mechanism generates a kinetic-energy pulse of fixed magnitude.

Under such conditions, each velocity change, of AV(z)c, = V(4(0,1)-Vws, is one of a series, starting at 0 = 0, where successive terms decrease progressively, so that: A2 V(z)a < 0; (F = const.) . (13e) Ate When a trolley incorporating an oscillator with such characteristics, accelerates from rest along a runway, it can be demonstrated that: K a = ince(vcs (944) 2 -V(z)a 82), (13f) where K represents the proportion of the total energy supplied to the oscillator during each interval of T/2, which is converted into kinetic energy of the trolley. As V(z)a increases, x and is also increase. Moreover, a smaller velocity change, AV(z)a, arises from a given AE"*" An investigation of the trolley's dynamics is necessarily limited by such considerations. Inevitably, once a critical time has elapsed a trolley's average rate of acceleration diminishes progressively at each rebound. Eventually it attains a terminal velocity. Each z-impulse arising internally during the interval, At" is then cancelled by the opposing external impulses, arising from wheel friction and air resistance. Over this time, all the oscillatory energy, 9,', is entirely dissipated thermally, whereupon x = 0.

According to Hilbert, there are conditions where the reaction impulse is directed through the external gravitational field. Implicitly, neither the runway, nor the roller track, exhibit any counter impulse, -0 APzerm. Under such conditions, the trolley derives additional kinetic energy, AE(K)s, (Eq. 13c), from this external field. The classical schemes of conservation, however, remain valid.

These phenomena have been investigated. (3) Hilbert's scheme imposes a constraint on the geometry of the gravitational radiation field. Any absorption of energy by the reaction impulse is prohibited in certain configurations. They include the type shown in Fig. 1. It is, moreover, explicit that the impulsive reaction, -0APi(T,2), acts at the centre of the Earth.

Crucially, the z-momentum thereby transferred to the trolley at each oscillatory rebound, is entirely independent of the roller-track's velocity, V(z)r. The trolley's velocity attributable to a series of oscillatory rebounds, is a function of the kinetic energy it has absorbed. It is demonstrable that this depends on preset oscillatory quantities (f0, 4m, r", etc.), governing the magnitude and velocity of the normal force in the reference frame where it originates.

Under conditions where the roller track has a z-velocity, Vw-r, there is an influx of energy directly from the external gravitational field to the trolley (Eq. 13b).*9 Gravitation is the determinative property, in relation to both the rebound impulse itself, and the energy changes induced by the oscillatory mechanism. Hypothetically, there are configurations with a reaction, -0 AP, cr/2), which is directed through the gravitational radiation field. Theory predicts, however, that the oscillatory z-impulse on the trolley shown in Fig. 1, produces a reaction which does not absorb energy from the oscillator.

Hilbert describes Banach space in Euclidean terms, with a norm induced by an inner product: = I x, xi. These geometries, as developed by Hilbert, interpret electromagnetism and gravitation in purely geometrical terms. They have been extended to derive a set of functions associating gravitational interactions with wave propagation, and satisfying the hyperbolic second-order differential equations of the Cauchy problem.

Inertia is incorporated in these functions, and expressed as a property of gravitation (Mach's principle). Within this dynamic scheme, positive and negative inertia accompany any gravitational normal force in equilibrium at an interface perpendicular to the field intensity, g. Such a scheme then predicts the rebound impulse of Eq. (6).

*9 Strictly, the analysis requires all extemal gravitational fields to be taken into account. An exposition of the origin of inertia in accord with Mach's principle emphasizes the role of the matter in the Universe (8). Mach stated: "The inertia of a body must increase when ponderable masses are piled up in its neighbourhood", and "A body in an otherwise empty Universe should have no inertial mass".

REFERENCES

1 Feeny B, Guran A, Hinrichs N and Popp K (1998), A Historical Review on Dry Friction and Stick-Slip Phenomena, App. Mech. Rev., 51, 321 -341.

2 (de) Coulomb CA (1785), Theorie des machines simples, Memoirs de mathematique et de physique de l'academie royale, 161 -342.

3 Drewe JR, Sussman HS, and Cockett BAR (2014), Dynamic Functions Implicit in a Variation of the Normal Force at an Interface Rotating with Isotropic Coulomb Friction, UK Patent Specification Pending 4 Goyal S (1989), Planar Sliding of a Rigid Body with Dry Friction: Limit Surfaces and Dynamics of Motion, PhD thesis, Cornell University Department of Mechanical Engineering Trinkle JC, Pang JS, Sudarsky S, and Lo G (1997), Dynamic Multi-Rigid-Body Contact Problems with Coulomb Friction, Zeitschriftfur Angewandte Mathematik and Mechanik, 77(4): 267 -279.

6 Raine DJ (1975), Mach's Principle in General Relativity, Royal Astronomical Society, 171, 507.

7 Sciama DW (1953), On the Origin of Inertia, Royal Astronomical Society, 113, 34.

8 Assis AKT (1999), Relational Mechanics, Apeiron, Montreal.

9 Hilbert D (1899), Grundlagen der Geometric: Festschrift ztu. Einweihung des Gottinger Gauss-Weber Denkmals, BG Teubner, Leipzig.

Klein F (1917), "Zu Hilbert's erster Note uber die Grundlagen der Physik", Nachrichen der Konigliches Gesellschaft der Wissenschaften zu Gottingen. Mathematisch-Physikalische Klasse, 469-82. Reprinted, with a commentary: Klein F (1921), Gesammelte Abhandlungen, Bd. I., Berlin: J Springer, 553-567.

11 Hilbert D (1922), "Neubegrundung der Mathematik: Erste Mitteilung", Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universitat, 1,157-77.

Claims (13)

1. I * * CLAIMS * . * *** * I An apparatus of total mass m, incorporating a platform supported by two or more wheels, floats, skis, gas jets, or an electromagnetic suspension, or any combination thereof, through which its total weight, mg, is directed and interacts either directly on the ground, or on a frame external to it, for example a ship or an aircraft in flight, under conditions where one or more of these weight-bearing supports is repeatedly displaced back and forth relative to a given point (usually the centre of gravity of the intact assembly), along a path which is perpendicular to the gravitational acceleration g, or has a component perpendicular to it, so that the proportion of the total weight acting through each support then varies as a function of the relative distances from that point, with the result that the assembly experiences a linear impulse perpendicular to g at each reversal of the path of displacement.
2. 2 An apparatus as in claim 1 where the total weight, mg, of the entire assembly is counteracted by a reaction -mg, acting entirely or partially at right angles to the displacement path(s) of the moving supports, so that the impulse arising from the rebound of a support (i.e: at the time its velocity is reversed) is greater when the rebound occurs at the position of the greater reaction force, i.e: the position where the rebound is nearer to the centre of gravity of the assembly.
3. 3 A method of propulsion entailing a body or assembly of connected bodies, of total weight mg being supported relative to the surface of the Earth by two or more structures which are displaced (usually against frictional forces) along any path which is perpendicular to g, or has a component perpendicular to it, so the reversal of the direction of displacement generates an impulse perpendicular to g, and where the magnitude of this impulse is proportional to the normal reaction on a support when its direction of motion is reversed. *

   * *
4. 4 An apparatus constructed according to claims I and 2, and operating as described in claim 3, where the displacement of the weight supports is achieved by oscillating an axle attached to one or more support by * incorporating a mechanical or electrical oscillator in the assembly, or, by a person resting on the platform and * * * displacing the axle back and forth. * *
5. An apparatus operating in accord with claim 3, where the weight acting through any one support varies * "" 41. according to its position relative to the other supports, so that when one or more of them experiences a ** * 4 ^ displacement back and forth, cyclically, relative to a given point on the platform, an overall propulsive * impulse acts on the intact configuration in a specific direction, perpendicular to g, during each complete oscillatory cycle, with the impulsive reaction acting on the Earth as the source of the gravitational field.
6. 6 An apparatus as in claims, 1, 2, 4 and 5, which is confined inside a frame, such as a trolley moving along the ground, or a ship or an aircraft, and which may be fixed to it, thereby producing an impulse in a given direction during each successive oscillatory cycle and serving to accelerate the frame, or to produce a force acting to oppose any forces acting extemally on it.
7. 7 An apparatus as in claims 1, 2, 4, 5 and 6, where the point on the platform with respect to which displacement of the weight-supports is measured, can either be fixed on the platform, or can vary its position in relation to it.
8. 8 An apparatus as in claims 1, 2, 4, 5, 6 and 7, where the centre of gravity of the assembly in relation to the platform is either fixed, or varies because of changes to the position of bodies incorporated in it, or because of additional bodies being attached to it.
9. 9 An apparatus as in claims 1, 2, 4, 5, 6, 7 and 8, which uses the Earth's gravitational field to generate a propulsive impulse the reaction to which acts on the Earth.
10. An apparatus as in claims 1, 2, 4, 5, 6, 7, 8, and 9, which uses the Earth's gravitational field to generate an increasing velocity in the absence of frictional forces acting against the displacement of the moving weight supports.
11. 11 An apparatus as described in claim 10, which is inserted in a frame, such as a trolley a ship or an aircraft which is already moving at a given velocity, perpendicular to g and relative to the Earth, so the sum kinetic energy of this entire system is increased over each rebound cycle of the weight supports, with a correspondingly diminished kinetic energy of the Earth as a result of the reaction impulse.
12. 12 A method as described in claim 3 of using the Earth's gravitational field to increase the kinetic energy of the apparatus described in claims 1, 2, 4, 5, 6, 7, 8, 9, 10 and 11.
13. 13 An apparatus, as described in claims 1, 2, 4, 5, 6, 7, 8, 9, 10 and 1, producing a propulsive impulse by having one or more of its weight supports displaced along a given path, where the weight is different at each (of the two) velocity reversal points, producing an overall impulse, perpendicular to g, over the entire cycle, substantially as shown in the drawings.* ' * * * * * * * * * * * * * * * * * * * ** ** * *f**