The "Eccentric-Fulcrum Cylinders Activated by Relative Transmission of Opposed Revolution for Mechanical Efficiency Converging on 1" described in this private-patent invention report allows for the elevation of indices of efficiency in many applications of classical mechanics. Transportation of payloads, reversible presses, winches, sky-trams, cranes, elevators, launchers and catapults are examples in which relation of motor forces acting on a useful, resistant payload impose conditions that are critical to the operation. In situations in which it is desirable to have practical values to safeguard energetic costs and environmental risks, target index close to 100% appear impossible.

In the systems of pulleys and sheaves normally used, the mechanical tackles carry heavy burdens while die motor forces are applied in a reduced manner. Conversely, launchers require a projectile to run along a course when strong force acts upon the least point of dislocation. The theory predicts linear proportional behavior between the forces of action and payload as a function of their reduction forces. In practice, we see that die efficiency decays rapidly when compared to these expectations, primarily for large payloads that require significant relations between the useful work-energy and the work-motor used.

Other solutions involve using gears or "endless" equipment, as they are called.

Attrition in the fixed elements is frequendy calculated in a cumulative series. Hydraulic systems achieve higher levels of efficiency, but their telescopic design limits their operation to smaller dislocations. In these often-used examples, response curves of the efficiency graphs demonstrate exponentially declining behavior.

New solutions deserve important conceptual exploration: to consider constructive variations that prevent loss by connecting sequences of components; to configure arrangements that block progression of dissipative overload. Directions resulting from forces of rotational or deformative forces, which are responsible for the dispersion of work-energy, will have geometric support straight lines purposefully low to their point of rotation of torque.

Rigidity of flexible elements and mobile couplings between fixed elements will offer solutions in multiple parallel arrangements of economic architecture. Security at points of direct or structural support will have considerations for dynamic equilibrium independent of dieir increase for the purpose of mere redundancy. Higher payload tensions will have fewer repercussions.

The intent of the figures is to clarify, and here is presented their correlation to the text:

Fig. 1 : Comparison of the mechanical efficiency as a factor of reduction. Above in mounting for a tackle conceived as a single unit, the principal object of this patent for invention. Below, we see a known example of the response curve associated widi conventional simple tackles.

Fig. 2: The principal object described in this invention-patent report, the mobile module activated will be called cylinder 6 of catenary eccentric torque. To the fixed module we will consider drum 7 as conventional.

Fig. 3: Opposed rollers in course b. Constructive variant provided by a ring with 2 filets for each channel of transitive entry 11.

Fig. 4: Geometric profile in a transversal view of cylinder 6 showing the initially points, straight lines, angles, and arrangements of tension under conditions absent of fields of external force.

Fig. 5: Reinforcement of the general concepts of the previous figure including the effects of a restricted gravitational field. There may be observed a collector/base/emittcr arrangement that may suffer permutations in respect to the forces of neutral/payload/phase, satisfying die imperative application of die project. Also notice the useful possibility of divergence as a factor for die conservation of energy in lateral dislocations.

Fig. 6: Reinforcing the concepts of the previous figure for convergent tensions, detailing the notable and directional angles of the course. There may be seen the useful possibility of a polarity field as a factor in the conservation of energy of vertical dislocation.

Fig. 7: Presentation of the basic module in some constructive variants of fabrication, mounting, or installation.

Fig. 8: Preliminary arrangement for elevator in which "W" // to the "£' for fixed drum 7.

Fig. 9: Here, we have constructive models for the matricial class-2 tackles in parallel cylinder and drum arrangements. There may be seen die useful possibility of the polarity field as a factor in the conservation of energy in vertical dislocations.

Fig. 10: This option alternates "W w" orthogonals and aggregate cylinders forming a single unit, both in "g' verticality. Notice the identity "mc=nd". There may be seen the useful possibility of the polarity field as a factor in the conservation of energy in vertical dislocations. Fig. 11: Constructive variation in which pulleys are substituted for toothed wheels and flexible elements for hook slots. The base varies in inclination for vertical, horizontal, and intermediate transports. Suggestion for winches, inclined ramps, or bridge-tackles.

Fig. 12: Utility model serving as transitive tackle of Vz reduction q* suggested for industrial applications, civil construction, or home use. Fig. 13: Utility model for motorized or manual elevator in pairs "w-" transverse cylinder drum. By the functional alternation of C/E for E'/C, the machine operates as a projectile launcher. The vertical launch represented could be redirected to work on an inclined ramp.

Fig. 14: Launcher with dual action for long course. Fig. 15: Constructive model with scalar pulley 4 interchangeable for varied reductions.

Options for internal or external bearings are shown middle-bottom of this detail for the tubular-axel alternative. The construction of rings can be dispensed with if we configure the flexible elements of base 10 with an inclination of λ° identical to the step of the helix. Suggestion for industrial production serving as the handle of buckets in wells, work-sites, or manual, simple-reduction hoisting. Fig. 16: Reversible press.

Figs. 17 to 28: Comparative graphs. Odd-numbered figures show matricial tackles varying every two lines by a value of r. The even-numbered figures are modified for known models. The reduction factors are changed for each line compared. According to the illustrations in figure 2, the module that is the object of this patent of invention consists of principle axel 1 to which we apply a quantity g of reverse pulleys 2 of radius r. Provided are fixation anchors 8 to which we connect the elements of base 10 in the same quantity. There are also inserted a quantity m of scalar pulleys 4 of radius R=q.r and of lifter 5 totaling £ bearings of width x and radius r°. The pulleys are all stopped and move in concert with axel 1 around bearings 3. In variations in construction, such as in figure 15, the bearings can be internal, in which case principle axel 1 will be hollow or tubular and r° is fixed to lifter 5. Thus, r° will either be considered as having a hollow or solid radius.

For an active module, lifters 5 are coupled to transport car 16. Each reverse pulley 2 constitutes a channel to collect elements of base 10 such as cables or flexible steel, hemp, or nylon belts, chains, and similar items. From the start, we have constructed g channels for the catenaries of base 12 that are tangent osculating lines of the same. Rigid rails are shown as constructive variants in figure 11. The remaining free extremities will hang from a suspension beam capable to support the weight of the system that occupies it. Thus, for each channel g

pulled, we will have a corresponding fixed base-point 9. Assured that the assembly at rest is in a state of equilibrium and maintained at a distance b from the base-points 9, this will become the maximum path allowed for transportation of payload. Stressors 13 equalize the tensions on base T individually. — We can incorporate another similar module in which the lifters 5 are installed directly on the support beam. We reiterate that the fixed modules coupled to the base will be called drum 7 and considered as conventional, but different to the catenary eccentric cylinder 6 attached to the car 16 for purposes of novelty to clarity in the text. The reverse pulleys 2 of drum 7 will be encased in n number and will have a radius of r' in value arbitrated freely an independently of r. The inverted transitive elements 11 of scalar pulleys 4 starting from cylinder 6 will have their free extremities coupled to anchorages 8 of each of the n channels that will be considered fixed points of phase 15, osculating lines tangent to the inert perimeter of r ¹ of the respective drums 7.

Also the scalar pulleys 4, in quantity u, can be considered one single piece in which u=l for each drum 7 that is installed in the collection of d drums. They will have a radius of R'=q'.r' where q ¹ is a value arbitrated freely and independently of q. The revolution of remote element 17 for conventional transmission of straps, knuckles, cogwheels, or cables will activate the entire system by the motor force P applied.

Back to the cylinder 6, calculating the number of turns necessary to revolve a sheaf of g elements of base 10 around reverse pulleys 2 and along the entire path b, we begin to move in the opposite direction a new series of transitive elements 11 in each one of the m scalar pulleys 4 in an equal number of turns. It is thus that when the elements of base 10 are wound, the transitive elements 11 are in the maximum condition of stretching and individual tension t. As a consequence, scalar pulleys 4 are empty whenever the payload occupies the end of course b and are full whenever the car is in the initial starting position.

Figure 3 clarifies this mounting in an alternative construction for the pulleys. Scalar pulleys 4 especially exhibit double-filet rings, and each entrance serves as a channel for the transitive cables ml and m2 parallel to the rollers. The spindles of reverse pulleys 2 have single channels but can have multiple filets. The alternated inclination of rings gl and g2 impede the longitudinal translation of car 16, but if desirable, they can be parallel without causing harm.

We see that cylinder 6 constitutes an activated differential assembly that combines rotational-axial movement over axel 1, longitudinal translation along b and translated relative to the catenary points of base 12 of the elements of base 10 in the case that they have an inclination of λ° in respect to that of the straight line of course "b". It is important to note that they serve as a multiple fulcrum for dynamic locomotion configuring steps in helix or exceptionally cyclic. Elements 10 are contained in a single plane to be called the base grade. Seen in a transverse section, the results of the tensions T°=Tcos λ° and P=g.T°. Thus, P is the result of forces on the base grade being a distant orthogonal of r at the central axis of axial rotation "w". Figure 3 shows T°=P/2. We say that λ° is the constant base angle the director cosign of which is that of the tensions T-P. We also have to see that t°=t.cos σ°, p=m.t°, and σ° is the constant angle of the phase: its director cosign is that of the tensions t-p. Taking fig. 4 as the transversal cut of cylinder 6, we see that the straight line of the course b is tangent to r. As such, a vector ^=OB is simultaneously perpendicular to w and to b where B serves as a multiple fulcrum in the catenary arrangement of base 12 for tension of base P.

Analogically, f is a straight tangent to R and serves to support tension of pulse or phase p. It represents a grade plane for transitive elements 11 and forms in F, that is to say the catenary point of phase 14, a right angle with a vector R=OF.

When p is active force, F will be called an emitter E. For forces of retention, we consider F a collector C. Mobile point B, catenary of base 12, provides eccentric torques. The Central Point of Symmetry O, it will receive attributes in opposition to that of the phase. This ..serves as a base fulcrum. For example, reverse pulleys 2 at the n fixed points F' of phase 15 serve as collectors or emitters in counterpart alternation to remote elements 17 of scalar pulley 4 whenever drums 7, taken as conventional, provide coccentric centered torques. Center O of cylinder 6 serves as a hybrid reference for rotations w and translations s*. The distances for the torques generated on B will be called effective radii R*, r*, and r°* and, in figure 4, are forming rectangles represented by the lines drawn. We have thus constituted a trio of force of base P/B, of phase p/F, and of payload

Q*/O. The point N*=μ in the figure signals the contact chosen for the rotational wear surface, and here we see that it deals with an bearing inside of an axis that, for its part, is hollow or tubular.

The angles formed by the directions of b/f at the point of concourse S are oriented by the vectoral tensions pP taken as a universal referential due to the known fact that | τ | ≤π and I α I = π - I τ I for definition of the angles between the vectors. From there, p.P=pP.cos X = - pP.cos α. We conclude COST— -cosα. As such, (-π ≤ T ≤ π), and we will call x the angle of tensions or, more formally: subordinate angle of tangency.

Considering the angle opposite to base p=BF of δBOF, we see that ( r = BO) + (R = OF) = p corresponding to the vectoral difference R-r. This evaluation leads us to notice that a vector taken from p — -r + R seen from O would have α opposite of p. As we saw for τ in S, we see here in O between catenary points B and F that π- 1 α | = | τ | . The same restrictions taken for τ apply, and j α | ≤ π. We will call α the catenary angle, or more formally: subordinate radial angle. Defining γ as the angle of bracket of the tensions on the axis of the pulleys of center O, we see by comparison with fig. 6 that γ ± | τ | — π. The expression is equivalent to γ ± (π- 1 Ci I)= π. Thus, γ = 2π- | α | or γ = | α | . We also see γ varying between 0 and 2π. From cos. | α | =cos.α and cos.γ = cos. | α | for both the hypotheses of γ, we deduce cos γ = cos α. If γ > π, then γ ≠ | α | because | α | ≤ π. If γ>π and γ ≠ | α | , then ( γ = 2π- | α | ) > π and I α I <π are simultaneously true. If γ ≤ π and γ φ | α | , then 2π- 1 α | ≤π and | α | > π is false.

If γ ≤ π and γ= | α | , then | α | ≤π is true. We conclude that γ is equal to | α | only if γ

≤ π. If γ<π, we say that there is class-2 divergence. When γ=π, we say that there is a convergent polarization. If γ≠ | α | , then γ=2π- 1 α | and γ>π. We say that there is convergence if π<γ<3 ^ι /2π. If γe (3 ViK, 2π), we have a class-1 divergence. Finally, if γ— 2π, we have divergent polarity. Polarities approaching γ guarantee the full parallelism of straight lines b// f//s* are aligned with O with an effective course obtained from compound referentials without transversal detours to the internal force field P-p-Q*-

We except α=0 algebraically implying in γ=0 or γ=2π. The null value of γ characterizes the disconnection of bracket tension with the pulleys, and there is no physical sense for an undefined α. In practical considerations, we see that if α=0, then γ=2π, and we have divergent polarization. When γ='λπ or γ=3Yzπ, we have ortho-polarization or that is to say that the straight line f is orthogonal to course b. This means that we will have the action of y) made from the same referential O starting from car 16. In this case, the elements of transmission will be conventional, and such a system lacks novelty for the technical state. The three trivial points are γ= {0, ViTi, V/zπ).

The five significant domains are γe {(0, π), π, (π, 3 ¹ An), (V/zπ, 2π), 2π} respective to the conditions of divergence=2, convergent polarity, convergence, divergence=l, and divergent polarity. While the angles of tension τ and the catenaries α vary between -π and π, including the value null, that of bracket γ = (0 ≤ π + τ < 2π) varies from 0 to 2π. Inasmuch as γ ≥ 0 is always true, we know that | γ | =γ can be measured with absolute trigonometric accuracy starting from r, independent of its argument γ with reference to the horizontal. From this, we can state that bracket γ is the subordinative angle of α and τ. Moreover in fig. 4, we see the similarity between triangles, δ/r = Sz/SB = cos.τ and cosτ=cos | τ| =cos(π- | α |)= -cosα=-cosγ. From the illustration, R*=R+δ leads to R*=r.q + r.cos.τ where R*/r= (q+cosτ) — (q-cosα) = (q-cos γ).

The mechanical advantage between class- 1 differential types, or additive=class2, is determined by comparing | τ| and | α | with π.Vz, and speaks to the calculation for brace- lever Bz of the moment of the effective torque. In particular, in figs. 4 and 5, we see (| α | >π. ^ι /2> | x|), and we can guarantee that (cos.τ) >0>(cos.α). In practice, this means that R*>R and the binaries PQ*+Q*p constitute a lever based on a class-2 fulcrum on which payload Q* is interposed between point B = (catenary of base 12) and point F = (catenary of phase 14). If on the contrary we see (| τ| >%.Vi> | α|), then (cos.τ)<0<(cos.α), and we conclude R*<R. This is the case in fig. 8, indicating | α | <Ti. ^λ h< | τ | . Thus, the binary pP+PQ* constitutes a class-1 lever: the payload is found at O, the pulse/F on the other extremity, and between the two is fulcrum/B of reverse pulley 2.

When S is convergent, we see a bracket-angle greater than flat line π and less than3.!/2.π. Moving on to fig. 6, we note that for γ that the third trigonometric quadrant is the only region offering conditions under which S is a real (i.e., non-imaginary) point in the triple concourse between p, P, and the normalized median "N-O". The definition around the convergence obtained by the inclusion of γ in the open interval of the diird quadrant complements considerations for the effective trajectories s* when the direction of approximation or repulsion F toward F' on the straight line f involves compound referentials. If γ φ {(π, 3 'λπ) }, then S is not convergent, and F moves away from F' except for y—π in the convergent polarization. We are aware of the possibility that γ>π is no longer convergent if it pertains to the interval [3.'/2.Jt, 2π]. Polarization guarantees the parallel alignment b/f/s*; convergence guarantees foci F-S-F'. The next example seeks to clarify:

If the binomial pP is parallel with superior motorization, then | τ | =0< | α | , cos.τ=l, and R*=r (q+1) is the maximum for the leverage of class-2 torque. The convergence exists in the polarity because | Ot | =π ^r =γ, and even though S is undetermined, we may assume that it belongs to the medial for O. F approaches F'.

If p and P are anti-parallel with the inferior motorization, τ=π> | α | , cos.τ=-l, and there will be polarization by divergence. S is undetermined, and b is interposed between the medial NO and the straight line of phase f. In both cases, the effective course of O will be parallel to the straight line g of the direction of the field because sin λ will be null and constant in time. We see γ=π and γ— 2π polarized in the two cases. The trajectory b does not suffer horizontal detours, and s*≡b as well as s* over O makes it parallel at distance r. The course of phase f*, this indeed alters the direction of p. In the second case with the dislocation, O maintains directions equal to that of P, b, and s*, but nevertheless, F moves away from F' because S is divergent.

It is relevant to note that in the conventional mobile sheaves, the possible increase in the angle of the aperture between the cables represents a waste of useful potential. With the

trajectory resulting from the vertical bisection of τ, causing it to vary, we have increasing motor traction amplified by {cos. ( ¹ A)T) tending to be a very high value. In one module of cylinder 6, we have movement allowed in the direction of a course b unmistakably linearly pre- established even though angle λ, subordinate to the normal level of base NB, may vary over some inertial referential. This property has to be reconverted into useful work of dislocation. We may imagine a constructive model, say a "virtual ramp", the trajectory of which is an inclined curve composed of referentials. The clear build-up of tensions in the opposite direction, since R ≠ r, will result in an interesting solution of a model for overhead trams if seen as another example. Of the habitual pulleys, the univalent results extended to almost 180° will not have another result except to expend energy or expose the mobile item to risk of accident. For this reason, we can consider that the eccentric fulcrum models in B, called cylinders 6, have a directional quality that characterizes them.

The angle λ* in fig. 4 is the subordinate normal of the base, and σ* the subordinate normal of the phase when "I" is a right angle support referential with subordinate direction of payload ι, imposed and oblique to force-field^. For its part, b forms with i the angle ψ oriented by forces P and J2* encounter each other at the point of concourse I.

Note that | τ j = |λ| + | σ | = | λ* | + | σ*| , firstly because G is a force field that, when imposed, falls across O and S in parallel. The second equality is true because τ is outside of the triangle of side IS. | β | + | λ | + | ι | = (Y2)π in which β is seen via BOI or BIFat in which we conclude by the equality BIO=λ+ι.= λ*. We say that β is the subordinate angle of payload. We remember that l is the normal or imposed subordinate of payload, and Fat is the force of resistance per rotation and, as such, is normal from i through N. The vector Fat was transposed from N* to I merely to facilitate graphic representation and improve understanding of β/I = β/O in the figure.

The subordinate radial α is more easily seen from referential O in translation and leads to the expressions maximum δ for maximum α. We can consider as a more direct way to refer to the increment of effective ray R*. Inasmuch as α is equal or replementary to bracket- angle γ, we see cos.α— cos.γ: It is useful, as such, to reaffirm that R* = r {( q + cos.τ) - ( q - cos.α) — ( q - cos.γ)}. The reduced rays are: r*=r sin.β and r ⁰ *^ (r° - r cos.β). The effective reason of torque in B will be q*— R*/r* therefore q*= (coseq.β)(q-cos.γ). The effective distance r°*< r° will reduce the torque of rotoidal resistance. Sin.β equals cos.λ* because β+λ* = (V-;)π and thus r*= r.cos.λ*. In apposition to the cosign angles diat compose each expression, we may formalize denominations for r°*, r*, and R* as being the effective ray of normal wear, the effective ray of payload, and the effective ray of phase. The imposed angle l, of small rotation when we fix bearings 3 to the platform of car 16, availing ourselves of articulated lifers 5, can be redirected to straight support i of the normal wear in N*.

To have equilibrium in the imposed trinomial, s must intercept I, a fact not shown in fig. 4 because until now we have dealt with a very broad representation.

Fig. 5 suggests the final state of σ' and p' for the total natural pendulation between payload and divergent tensions already imposed on the gravitational field.

The control of polarized course to evaluate horizontal detours and the phase for convergences /divergence is visually listed by the comparison of distance AD with respect to (R+r) or analyzing the trigonometric interval of γ that is a whole multiple of the level for polarization and the third Quadrant for convergences. Regarding that which we previously stated, if we see (F)=E, we deduce that the payload will ascend and increase in its gravitational

potential Q.δh, h being the distance from O to the base in N. The balanced angle of traction is τ'= -λ ¹ -σ', remembering that | τ' | = | λ' | + | σ' | . Curve s* is composed of rectilinear translations b on referential O and circular translation centered on A in the space of fixed reference and, in the most general cases of divergence, will not be a straight line. It is traced from point O obtained from the movement over the transversal plane to w. As a result, the compliment s* will always be greater than or equal to that of b except in polarizations. Comparing fig. 4 with fig. 5, it may be noted that the rotation of γ causes R* to increase. The unbalanced angles were removed from the scene. In ψ', we see clear agreement between points I, S, and g, previously represented by I and S separately, unbalanced in the unrestricted field.

Fig. 6 seeks to reinforce in a convergent state of equilibrium the concept for a bracket- angle γ. Reviewing that it is the sum of the flat-line π added to τ, or, that is: γ = π ± | τ | — π+τ. In this configuration for b being greater than h, we will have a convergence impeding unplanned lateral translations, confining us to the utilization of motor potential by demand of the vertical courses. It is demonstrated that | σ| = | λ| + | δ | , and then we also see that | τ | = | σ | + | λ| = 2 | λ| + |δ | where γ=π±(| σ | + | λ|)=π±(2 | λ| + | δ |). We will consider the angle of inclination of b compared to the horizontal: ξ = (Vz)Ii- 1 λ | . Developing for the expressions, we arrive at the identity ξ = Vz (γ+ | δ |). This is the variable angle that P inscribes with the horizontal in the most general cases and serves as the directional angle for AB as well as for ray r constantly normal to curve b of cylinder 6 in "parallel" translation from its point O to B. There are methods to determine effective course s* in the studies for the constructive models. Let us touch on the purpose of this report into not extending considerations that require such specific calculations.

It is worth considering the trigonometric angle of clear rolling of b around r not seen in the transverse section: because b^ b°+r.δθ, where θ = θ°+2+κπ+ γ, we make θ° = τ°. The "°" indices indicate the initial values of the course and angular phase starting from time t° of measurement We conclude: δθ =2κπ + γ= (2κ+ l)π+τ, and we will accept that γ is the plane angle θ with variation δθ = (δb)/r. Derived in relation to time, we find ( w.r ) = Vb as the linear velocity of approximation A-B of course b. Considering δv the number of rotations around axel 1 with center O, we must find

2πδv = δθ. For an observer from a fixed referential, 2πδv — (2κ+ l)π+τ, and we recalculate δv= ( τ / 2π ) + Vz + K.

It is equally important that an observer placed in car 16 in movement will clearly be able to determine position, velocity, and other necessary variables starting from referential O in a longitudinal section as we see in fig. 3. By direct count of filets transported from one to the other pulley rolling in the opposite direction, we would know the number of rotations there are on the axis of the cylinder..., it would remain that the transversal angles could be calculated. For this observer, the identity δv = ( ¹ A ) (γ/π)+ K allows him to obtain the subordinate angles α and τ starting from the subordinate γ comparing it to the horizontal. To calculate the linear and angular velocities, he would verify the indications of his wristwatch.

Fig. 6 adds views for δ, defining it on the terrestrial horizon as a coefficient angle. It orients the straight support for the base p opposed to σ of the triangle BOF. Knowing p and δ, or that is to say p, we can consider an orthonormal base of the complex numbers in O where p = p ( cosδ + i.signδ ) and r - r (cosλ + ϊ.signλ ) are represented. Thus, the conjugate \p] of/) would have the argument -δ and would apply to the divergent configuration for cylinder 6.

In another example, the parallel condition would bring the λ arg. of r to the value of null; the δ arg. of p would also be null; we find a complex vector R = /) - r = p - r as pure real and R — p - r. Given the complex vectors p and r, all the construction of cylinders 6 will be perfectly defined in inertial space including the calculations for R^ q, H(t) as a time/height function, and other values will be implied!

Fig. 7 gives us two models for drums 7 and two varieties for cylinders 6. Fig. 8 is an elevator application in which the cylinders are circularly disposed around a vertical drum. Even though only two cylinders 6 have been represented, complementary designs could be configured in overview, polygonals including multiple straight axial supports for w possessing on the middle point of each side a fixation center R+r' distant from an overview taken with W as the center of these polygons. Especially in figure 8, we have null λs and as such β= Vϊlt. -l. We can make 1=0, and from there φ = l/[ q'.(q - cos.γ)]. Seeing acute angle α=γ, we conclude R*<R and have a class-1 tackle, neither polarized nor convergent.

We can suggest transitive elements 11 of inflexible construction for, remembering the friction of rigid couplings, we convert said assembly into another class-2 polarized and convergent assembly and thus take optimal advantage of the potential. This is only possible in cases in which w and W pertain to the intersecting planes for solution of non-trivial coincidence. Drum 7 would be a screw of c=n threads, α=π, the longitudinal translation W of F' would be parallel to the polarized vertical course b//s* and coincident with F. We make an exception for the maintenance defined for F' to be a fixed point in phase 15 because of the transverse section of W, we have a horizontal plane over which the orthogonal projection of F' remains unchanged on the perimeter r' of center W revolving around a vertical axle. Pulleys 4 would be made in the form of gears of inclination σ° that for maximum return would be the arc (π/ 4 - φ.'λ) where tg φ=μ of the material coefficient of friction. F°=[c.p=c.m.t.cosσ°] tg(σ°+φ) where for ι=0=λ=σ,

P=FVq'- (l/q^[cmQ.secσVcni(qcosλ+cosσ)].cosσ°.tg(σ°+φ), then we will see an elevator of reduction φ= f°/Q = [l/q'(q+l)|. tg(σ°+<p) capable to innovate the state of technology because such traction cables as those of the suspension could be totally eliminated when we also substitute the flexible elements of the mobile base (10) with rigid tracks as we see in the example in fig. 11.

For tackles, these concepts allow for various convenient arrangements, and some models in variant constructions are shown in figures 9 and 10. Presented up to the final page of this report the results of measurements involving practical formulae of real forces, we may evaluate efficiencies in comparative indices. In the graphics were considered friction on the remote elements 17 that limit the definitive goal to something below 100%. The theoretical force f° on remote element 17 will be Q° / q' ( q - cos.γ ) where Q° is the projection of Q* over b, q'=R'/r', and q=R/r. The weight is Q equaling Mg= Q*.cos(l). As Q°=Q*.sin.β=Q.sec(ι).sin.β, the final reduction of the tackle will be: f°/Q = φ = [ sec(ι). sin.β]/[q'.( q- cos.γ) ], where very generally 1=0 and Q*=Q- According to the assemblies, the payload is suspended by a total of g.c elements of base 10 with m.c transitive elements 11. Each parcel of this sum represents respectively one base-station and another phase-station. It is interesting to note that taking g cc m.q, the tensions of base T and of transit t could be statistically equalized, which will allow for the use of identical cables for all of the tie-offs. For dynamic regimes of constant velocity, tensions T become reduced to T _d in the class-2 assemblies.

All we have to do is increase the value of g to reduce the tension T of each element of base 10. This allows for lower coefficients of rigidity starting from thinner cables, for

example. If we choose a higher quantity m, we can reduce tensions t in the transitive grade, enriching our rate of return.

As we see in 6g. 3, p = (g T cos.λ°) and p= (m t cos σ°) result in P/p = (g cosλ° / m cosσ°) (T/ 1). For equilibrium of torque in O of fig. 4, Pr equals pR therefore P— q.p. The weight is divided among c(P.cos.λ+ p.cos.σ) normalized elements of suspense tension. Resolving Q=cp(qcosλ+cosσ) and the other affirmatives, we find t=Qsecσ°/cm(qcosλ+cosσ) and T = qQsecλ°/cg(qcosλ+cosσ). In normal applications, we might want, for example, identical cables for all of the pulleys, and then we must equalize T with t. All that we need is a construction in which g equals (mq cosσ° secλ°), and we choose an immediately superior integer value for g.

Another factor for lost useful work is friction of the axle-bearing type. One physical solution could be achieved in our project when we augment the number £ of bearings 3 or we may separate them by the width of contact x. This done, we can reduce the ray r° without changing rolling surface of N*, conserving the specific pressure of the material. Finally, we acquire better- advantage relation r°/ q.r as was desired. Solutions in architecture could be thought of: in fig. 12, we see a model for £=0 and therefore without bearings!

The prototype of fig. 13 achieves a double utility and can convert from a tackle to a catapult launcher. For such, all we need to do is rotate point F one half turn, releasing the cored in R' below. Transforming R' into a collector, we use Q as an emitter of action-force in O. The 4 fixed points of base 9 support the catenary of base 12 in B, while the lifters of cylinder 6 keep the descending along with car 16. The result is that the extremity F2 will serve as a catapult launcher for smaller bodies.

Some applications can restrict the length of axle 1; others may require exaggerated payload capacity. These obstacles will be overcome if we assemble a quantity c of mobile cylinders 6 to the car forming the collection. This is demonstrated by lower fig. 10 with two cylinders. In the same way, we can choose greater quantities d of fixed drum 7, amplifying the phase-station represented in the figure by d=l of the single cylinder.

The need to overcome greater distances for course b is resolved simply and directly by increasing the value of r as well as elongating the pulleys. This is a good solution for, as in the case of fig. 13, achieve a reduction in dispersion due to the rigidity of the roller, and as in that of fig. 11, through rolling friction of its geared reverse pulleys 2 in connection with the rigid tracks of base 10.

Increasing the payload provision, we choose adequate values for €.x, g, m, c, and d, bearing in mind that m.c = n.d and therefore n= m.c / d is a function of this variable. Thus, reducing the diameter of the cables and the r° of the bearings, there is used smaller specific physical constants of rigidity or friction for the same materials.

Increasing the reductions by way of an increased q, the type-m resistances are reduced by the inverse of the increase of R.

Increasing the ray of reverse pulley r= | r\. = OB, all of the dispersions diminish — those of type g, m, or £.x — because the new proof r° / ( q.r ) amplifies the torque-relation and because the rigidities are inversely proportional to r and R inasmuch as R=r.q. It is interesting to employ the properties of linear Algebra when a specific element of the assembly acquires vectoral representation for purpose of determining the parameters. The assembly of fig. 9 could be described as [c(g,m,l); d(n,u,/)]= [1(3,2,4); 1(2,1 ,2)] for w//w. Also for perpendiculars Ww:

Q= £°/φ= 2{ |T ^o .(4,0,0)]+[t ^o (0,l,0)] }= 2 {2κ*.x.r°(0,0,4) } where K* is the constant of specific pressure for the bearing. Note that each vector can represent a module, assembly, or configured tensions, and each equation a condition of equilibrium for the stations.

Organizing these variables, we can easily resolve the most complicated calculations. For this reason, the tackles were called matricials.

It is most important to note that the transitive elements 11 constitute a property that is exclusive to them: they transmit, starting from fixed points F ¹ of phase 15, the tangential velocity W.r ¹ for the action of drum 7 at point F'. The opposite extremity F of cylinder 6 does not have equivalence of type w.R. To this, we add the velocity v _b =w.r belonging to the translation assuming that the base- and phase-stations are vertical as in the examples given in figs. 9 and 10. The resulting effective for velocity at point F is w.(R+r), which is a hybrid of a double referential in the movement over the last fixed referential, in our case that of the earth. Even though we can equalize W=w=ω, we see that, in this condition, r' should equal R+r, or that is to say: we will have r'=r.(q+l). This is possible in construction but violates the initial value defined for r' as being an arbitrary independence in regards to r. Conventional transmissions via belts, gears, or coupling joints present identical tangential velocities due to the straight nature of their rays and inverse to the angular variations of each one of its axles. Therefore their rays of simultaneously identical tangential and angular velocities should also be identical. Of this principle applied to our transitive module, if V=Wr' =wR and (D=W=W are simultaneously true, then V/ω =r'=R or that is to say r'=q.r, and we arrive at a contradiction inasmuch as {r,r',q} φ 0, and of the as described above, we see r' to be in the form of r'= r (q+1) safely different from q.r. We conclude that transitive elements 11 are not conventional. In common tackles in which the cables consecutively move through the roller- assemblies, vectoral variables such as velocity or force necessarily change the attributes of the module and direction. Transitive cylinders 6, which are catenary eccentric, are the carriers of the acclivities of the transitive elements 11 of base 10. They always guarantee what we desire, that all of the vectoral bulk is conserved in either direction and intensity. Assembling this proof, we offer practical results in which graphic comparisons of the responses are shown in fig. 1 for a single transitive module of cylinder 6 and just below that for known simple tackles with motor action over a single fixed pulley. From figures 17 to 28 for tackles with a stationary base and a parallel phase for drum 7, we accept λ=σ=l=0 and α=π, therefore β= ( ¹ A) π. We made q'=q+l, therefore φ= f°/Q - 1 / (q+1) ² , to which we give the name "quadratic matricial tackle".

We made small changes in the value of r for each pair shown. We blocked reductions up to 1/100 for the known systems for reasons of practical order. We limited reductions to 1/400 in the matricial tackles. From the exposition, for greater reduction, the index of efficiency is the convex and convergent profile to the left for a maximum value equal to one, that is to say: mechanical efficiency of 100%.