TIP-PATH AIRFOIL THRUST PRODUCTION IN ROTARY-WING AIRCRAFT

TECHNICAL FIELD

The present invention is generally directed to moving thrust to the tip-path of a rotary-wing bladed system of a helicopter (or any propeller) assembly where energy required to rotate the blade is reduced by the force multiplied by the rotor blade moment arm (length of blade span). Morphing airfoil surfaces electrically with piezoelectric wafers improves the production of rotational relative wind.

BACKGROUND OF THE INVENTION

This invention teaches that the magnitude of torque depends on three quantities: the force applied, the length of the lever arm connecting the axis to the point of force application, and the angle between the force vector and the lever arm. The equation for the magnitude of a torque, arising from a perpendicular force: torque is multiplied, because it equals the distance to the center of rotation from the wing tip multiplied by the force applied to the wing tip-path. Torque transfer through a central axis hub is eliminated.

Yet a further drawback of the prior art is that a rotary airfoil blade of a propeller typically is mounted to a plane where air is crossing the bottom and top of a shaped airfoil wing to decrease the air density on top and increase the air density below the wing. This invention teaches that in milliseconds, piezoelectric wafers mounted onto propeller airfoil blades morph from a symmetrical airfoil into a nonsymmetrical airfoil (chambered), to increase air density for more lift below the airfoil during high speed propeller rotation.

To be considered economically feasible, an aircraft must apply available sources of energy in the most efficient way. In prior art, aircraft do not have common flexible of morphing designs that provide the ability to fly aircraft together into one single aircraft that combines the resources of all or some of the individual aircrafts functions into one single aircraft. An example is FIGURE 22 where a rotating tube ring aircraft is provided with helix air controls that enable it to fly up to the ends of another similar aircraft's frontend to backend making a longer more efficient tube to fly together. In addition, several smaller diameter aircraft are provided with smaller of larger diameters, so individual aircraft unit assembles concentrically, fitting within each other, some are larger in diameter, and others are smaller in diameter to fit together concentrically at approximately the same length. These round tubes that have road tire material added to the outside diameter, can land on terrestrial ground and travel as a tire to deliver goods or services to customers, across roads, water, or soil, controlled by gyro-motor devices to balance and propel a tire in desired directions.

Piezoelectric wafer morphing airfoil surfaces, FIG 19 helices air lift optimization to control a rotating tube flying, and increased efficiency from tip-path increased torque technologies is taught to physically connect many aircraft together into one aircraft to travel more efficiently as one aircraft using all the capabilities of each individual aircraft, integrating electric power, fuel sources, morphing aerodynamically on surfaces, and spinning rotating blades together or individually. Laser light and other sensors can be combined to fly aircraft into each other. Helix airduct control is one of the key features of tip-path propulsion, reaching the greatest distance from the center axis for aircraft control. Pentazolate rings are applied as fuel in this inventions airborne aircraft: A carbon- free inorganic-metal complex [Ζη(Η2θ)4(Ν5)2] 4H2O can be synthesized by the ion metathesis of [Na(H20)(N5)] 2H20 solution with Zn(N0 ₃ )2 6H2O from any energy source on earth or airborne solar energy, because it is primarily produced from air and water requiring just electric input (wind power, solar, hydroelectric, or any other power source). Nitrogen fuel source are taught in this invention that do not emit any pollution but increase lift and propulsion most effectively along the helices air pathways. Just enough water can be added to an aircraft helix in FIG 19 to increase lift during takeoff minimizing the use of conventional fuels for propulsion.

In accordance with the present invention, the thrust is placed on the tips of rotors to optimize force by multiplying force by the moment arm. Torque transfer through a central hub axis will not occur, and any power sources for thrust will be easier to manage, since the thrust physically moves from the body of an aircraft location out to the most optimized location onto the tip-path.

Further, in some embodiments piezoelectric wafers mounted onto propeller airfoil blades morph the airfoils to optimize force and minimize drag by ultrasonically vibrating leading edges of the morphing airfoil shape, so wind never contacts some surfaces, wind just contacts ultrasonically compressed smooth air, keeping surfaces clean, reducing drag.

A roller chain in prior art is a way of transmitting mechanical power from one place to another through linear links arranged in or extending along a straight or nearly straight line "linear movement" that are assembled around on sprockets providing a chain drive. This invention teaches that a "triangular-chain link" replaces the past linear links assembled around on sprockets by providing a chain drive that extends a mechanical structure, an additional moment arm, at a greater radial distance from the center of sprocket's conventional linear chain drive links outer surface. The conventional past chain roller is still functioning in the same way, but with the addition of a triangular-chain link (FIG 42) an additional motion translational geometry is added to reciprocate rods of a piston during the movement from straight to curves around the sprocket, a motion translational function, to compress and expand piston cylinder assemblies sequentially in a series, stretch and compress piezoelectric material to drive the chain assembly or the forces from energy, like wind or water, that can force movement of the chain assembly to generate electricity by stretching and compressing materials like piezoelectric material. In an example, Stirling engines would have much more time to be cooled and heated when assembled between a series of triangular-chain roller links.

DESCRIPTION OF THE RELATED ART

According TO U.S. FAA.

The rotation of rotor blades as they turn about the mast produces rotational relative wind (tip-path plane). The term rotational refers to the method of producing relative wind.

Rotational relative wind flows opposite the physical flightpath of the airfoil, striking the blade at 90° to the leading edge and parallel to the plane of rotation; and it is constantly changing in direction during rotation. Rotational relative wind velocity is highest at blade tips, decreasing uniformly to zero at the axis of rotation (center of the mast). This invention teaches how to improve rotational relative wind in the tip-path plane. SUMMARY OF THE INVENTION

Accordingly, one of the objects of the present invention is to provide an improved structure and system to optimize airborne aircraft acted upon by four aerodynamic forces; thrust, drag, lift, and weight. Understanding how these forces work and knowing how to control them with the movement of thrust to the tip-path plane and adding to flight controls by adding piezoelectric wafers mounted onto airfoils to morph shapes are essential to future flight.

This invention teaches that thrust of the aircraft can be moved to the tips of rotary- wings and in milliseconds, piezoelectric wafers mounted onto propeller and wing airfoil blades morph from a symmetrical airfoil into a nonsymmetrical airfoil (chambered), to increase air density for more lift below the airfoil during high speed propeller rotation, improving efficiency and lowering the cost of fuel.

ELECTRIC MOTOR BACKGROUND:

An electric motor is an electrical machine that converts electrical energy into mechanical energy. The reverse of this is the conversion of mechanical energy into electrical energy and is done by an electric generator, which has much in common with a motor.

Most electric motors operate through the interaction between an electric motor's magnetic field and winding currents to generate force. In certain applications, such as in

regenerative braking with traction motors in the transportation industry, electric motors can also be used in reverse as generators to convert mechanical energy into electric power.

Found in applications as diverse as industrial fans, blowers and pumps, machine tools, household appliances, power tools, and disk drives, electric motors can be powered by direct current (DC) sources, such as from batteries, motor vehicles or rectifiers, or by alternating current (AC) sources, such as from the power grid, inverters or generators. Small motors may be found in electric watches. General-purpose motors with highly standardized dimensions and characteristics provide convenient mechanical power for industrial use. The largest of electric motors are used for ship propulsion, pipeline compression and pumped-storage applications with ratings reaching 100 megawatts. Electric motors may be classified by electric power source type, internal construction, application, type of motion output, and so on.

Electric motors are used to produce linear or rotary force (torque), and should be distinguished from devices such as magnetic solenoids and loudspeakers that convert electricity into motion but do not generate usable mechanical powers, which are respectively referred to as actuators and transducers.

Major categories:

Electric motors operate on three different physical principles: magnetic, electrostatic and piezoelectric. By far the most common is magnetic. In magnetic motors, magnetic fields are formed in both the rotor and the stator. The product between these two fields gives rise to a force, and thus a torque on the motor shaft. One, or both, of these fields must be made to change with the rotation of the motor. This is done by switching the poles on and off at the right time, or varying the strength of the pole.

The main types are DC motors and AC motors, the former increasingly being displaced by the latter. AC electric motors are either asynchronous or synchronous.

Once started, a synchronous motor requires synchronism with the moving magnetic field's synchronous speed for all normal torque conditions.

In synchronous machines, the magnetic field must be provided by means other than induction such as from separately excited windings or permanent magnets.

A fractional horsepower (FHP) motor either has a rating below about 1 horsepower (0.746 kW), or is manufactured with a standard frame size smaller than a standard 1 HP motor. Many household and industrial motors are in the fractional horsepower class.

Motor construction:

Electric motor rotor and stator Rotor

Main article: Rotor (electric)

In an electric motor, the moving part is the rotor, which turns the shaft to deliver the mechanical power. The rotor usually has conductors laid into it that carry currents, which interact with the magnetic field of the stator to generate the forces that turn the shaft. However, some rotors carry permanent magnets, and the stator holds the conductors.

Bearings:

The rotor is supported by bearings, which allow the rotor to turn on its axis. The bearings are in turn supported by the motor housing. The motor shaft extends through the bearings to the outside of the motor, where the load is applied. Because the forces of the load are exerted beyond the outermost bearing, the load is said to be overhung.

Stator:

The stator is the stationary part of the motor's electromagnetic circuit and usually consists of either windings or permanent magnets. The stator core is made up of many thin metal sheets, called laminations. Laminations are used to reduce energy losses that would result if a solid core were used.

Air gap: The distance between the rotor and stator is called the air gap. The air gap has important effects, and is generally as small as possible, as a large gap has a strong negative effect on the performance of an electric motor. It is the main source of the low power factor at which motors operate. The air gap increases the magnetizing current needed. For this reason, the air gap should be minimal. Very small gaps may pose mechanical problems in addition to noise and losses.

Salient-pole rotor

Windings:

Windings are wires that are laid in coils, usually wrapped around a laminated soft iron magnetic core so as to form magnetic poles when energized with current.

Electric machines come in two basic magnet field pole configurations: salient-pole machine and nonsalient-pole machine. In the salient-pole machine the pole's magnetic field is produced by a winding wound around the pole below the pole face. In the nonsalient-pole, or distributed field, or round-rotor, machine, the winding is distributed in pole face slots. A shaded- pole motor has a winding around part of the pole that delays the phase of the magnetic field for that pole.

Some motors have conductors that consist of thicker metal, such as bars or sheets of metal, usually copper, although sometimes aluminum is used. These are usually powered by electromagnetic induction.

Commutator (electric):

A toy's small DC motor with its commutator

A commutator is a mechanism used to switch the input of most DC machines and certain AC machines consisting of slip ring segments insulated from each other and from the electric motor's shaft. The motor's armature current is supplied through the stationary brushes in contact with the revolving commutator, which causes required current reversal and applies power to the machine in an optimal manner as the rotor rotates from pole to pole. In absence of such current reversal, the motor would brake to a stop. In light of significant advances in the past few decades due to improved technologies in electronic controller, sensorless control, induction motor, and permanent magnet motor fields, electromechanically commutated motors are increasingly being displaced by externally commutated induction and permanent-magnet motors.

With these and other objects in view that will more readily appear as the nature of the invention is better understood, the invention consists in the novel process and construction, combination and arrangement of parts hereinafter more fully illustrated, described and claimed, with reference being made to the accompanying drawings in which: BRIEF DESCRIPTION OF THE DRAWINGS

FIGURE (FIG.) 1 is an isometric view of an three-rotor blade system assembly; FIGURE 2 is a side view of FIGURE 1 and 3;

FIGURE 3 is an elevated top view of FIGURES 1 and 2;

FIGURE 4 is a data table of PL112 - PL140 PICMA® Bender

All-Ceramic Bending Actuators with High Displacement;

FIGURE 5 illustrates a PL112.10 - PL140.10. L, LF, W, TH, see data table;

FIGURE 6 PICMA® Bender bending actuators have differential control;

FIGURE 7 Displacement of the PICMA® bending actuators: Clamped on one side

(top) and on both sides (bottom);

FIGURE 8 illustrates a symmetrical airfoil and nonsymmetrical airfoil (cambered) airfoil;

FIGURE 9 is an elevated top view of a four-rotor blade system assembly with a 90- degree power strip providing power to all four blades during 360-degree of rotation;

FIGURE 10 illustrates a cycloid curve shaped sprocket rotated 90-degrees, 42.19- degrees, and 0-degrees to provide the area of the cycloid sprocket, which is equal when divided along the line going through the centerpoint of rotation;

FIGURE 11 illustrates the cycloid sprocket of FIGURE 10 coordinate points generated when rotating a circle along a straight line the length of the circle's circumference;

FIGURE 12 A and B illustrates the cycloid sprocket in FIGURE 11 rotated 90- degrees;

FIGURE 13 illustrates an isometric view of a circle-arc stator roller chain type assembly chain links stator;

FIGURE 14 illustrates roller chain motor stator fastened along the circumference of circle in a circle-arc of about 90-degrees of rotation requiring four motor poles arrayed as radius lengths from center at 90-degrees, a circle-arc of about 120-degrees of rotation requiring three motor poles arrayed as radius lengths from center at 120-degrees, and a circle-arc of about 180-degrees of rotation requiring two motor poles arrayed as radius lengths from center at 180-degrees;

FIGURE 15 illustrates a right-handed helix (cos t, sin t, t) from Θ = 0 to 4π with arrowheads showing direction of increasing t, and the height of one complete helix turn, measured parallel to the axis of the helix; FIGURES 17 and 18 illustrates two options to fasten the edges of surface curved snap fastener edges together around different shapes.

FIGURES 18 and 17 illustrates two options to fasten the edges of surface curved snap fastener edges together around different shapes;

FIGURE 19 illustrates a perpendicular view of letter "t" 136 optical center C (also known as camera projection center) 130 of the image 134 on the projected plane;

FIGURE 20 illustrates flexible helix video display camera dimensional observation for software programming an image from the math;

FIGURE 21 illustrates a fuselage of an aircraft with airduct helix pathways to optimize lift of any aircraft, which could include any shape;

FIGURE 22 illustrates a fuselage of an aircraft in FIGURE 21 with airduct helix pathways that have been isolated into airfoils around the fuselage, each scaled concentrically from the central axis of FIG 21 to optimize lift of any aircraft, which could include any shape;

FIGURE 23 illustrates a side perspective view of FIGURE 22 helix fuselage scaled around one-point origin, which forms a more aerodynamic fuselage than concentric scaling radially rather than radially and along the axis of the fuselage;

FIGURE 24 illustrates a top view of FIGURE 23 helix fuselage unrolled around helix intersecting mirrored sides;

FIGURE 25 illustrates an end view of FIGURE 24 helix fuselage unrolled around helix intersecting mirrored sides of FIGURE 23;

FIGURE 26 illustrates a perspective view of FIGURES 23, 34, 25, and 26 providing a view of helices and data centerline at rear of fuselage;

FIGURE 27 illustrates a perspective view of FIGURES 23, 34, 25, 26, and 27 providing a view of rear wings and tail of aircraft unrolled from top helices airfoil and data centerline at rear of fuselage;

FIGURE 28 illustrates a perspective view of the fuselage in FIGURE 21 provided a separated shell of the fuselage shape partially elevated into a fixed location to convert fuselage into a glider eliminating the need for power to land safely;

FIGURE 29 illustrates a perspective view of the fuselage in FIGURE 28 provided a hinge to pivot the split fuselage, converting it into a glider with more surface a perspective view of the aircraft in FIGURE 34 and resistance to falling;

FIGURE 30 illustrates a perspective view of the fuselage in FIGURE 29, before the fuselage to opened around the hinge; FIGURE 31 illustrates a perspective view of the fuselage in FIGURE 30 with diagonal air pathways between the fuselage two halves, which are separated enough to direct top of fuselage air to the bottom for more lift, like the helix in FIGURE 21;

FIGURE 32 illustrates a perspective view of the fuselage in FIGURE 30 with the bottom fuselage ends rotated to the top providing a glider function to land with minimum or no power;

FIGURE 33 illustrates a hexagonal fuselage with all the elements of this inventions Tip-Path elements;

FIGURE 34 illustrates a perspective view of the fuselage in FIGURE 30 with rotating wings attached to the nose of the fuselage;

FIGURE 35 illustrates a perspective view of the aircraft in FIGURE 34 with wings rotated into a rotary wing helicopter type lift or forward thrust configuration, which provides a tip- path thrust system with forces and aerodynamics around the same circular path;

FIGURE 36 illustrates a perspective view of the aircraft in FIGURES 34 and 35 with additional rotary wings to add to the capacity of the aircraft, including a fuselage that rotates with each independent rotary wing assemble, providing additional X wing four rotary wings on back to stand the aircraft on the backs of the wings during lading or takeoff;

FIGURE 37 illustrates a perspective view of the aircraft in FIGURE 36 with the front two sets of rotary wings position to rotate in opposite directions;

FIGURE 38 illustrates a side view of a wing aerodynamic profile with a hinge at the front of the wing to pivot the two halves of the wings around the hinge;

FIGURE 39 illustrates a perspective view of the aircraft wing in FIGURE 37 pivoted around hinge 180-degree angle from a 90-degree angle from the original wing positions, which provides a maximum gliding function for aircraft in FIGURES 34 and 37;

FIGURE 40 illustrates a perspective view of the aircraft in FIGURE 38 providing wings that slide open rather than hinge;

FIGURE 41 illustrates a perspective view of a roller chain and sprocket chain drive as a way of transmitting mechanical power from one place to another, which a triangular

modification provides an additional motion translational geometry to reciprocate rods, a motion translational function, to compress and expand piston cylinder assembly, stretch and compress piezoelectric material to drive the chain assembly or energy that moves the chain assembly can generate electricity; FIGURE 42 illustrates an expanded assembly to configure piston and cylinders, piezoelectric material insertion, and other energy conversion systems, like Stirling engines expanded to a 1.41 -centimeter distance between the two rocker arms in FIGURE 43;

FIGURE 43 illustrates a perspective view of FIGURE 42 expanded to a maximum 1.41 -centimeter distance between the two rocker arms;

FIGURE 44 illustrates a perspective view of FIGURE 42 closed to a 0.00-centimeter distance between the two rocker arms;

FIGURE 45 illustrates a perspective view of FIGURE 42 reduced to a 0.39- centimeter distance between the two rocker arms;

FIGURE 46 illustrates a perspective view of FIGURE 42 reduced to a 1.06- centimeter distance between the two rocker arms;

FIGURE 47 illustrates a side view of a roller chain and sprocket chain drive in

FIGURE 41;

FIGURE 48 illustrates a roller chain triangular-link 370 with an axle inserted at a triangular distance from pairs of elements 80 and 79 that contact the sprockets 301 at a distance that will fit the double sprocket in FIGURE 49. FIGURE 49 illustrates a side view of a roller chain and sprocket chain drive in FIGURE 41, 47, 50, 52, and 53. FIGURE 50 illustrates a roller chain triangular-link 370 with an axle inserted at a distance that will fit the double sprocket in FIGURE 49, with greater detail in FIGURE 53 illustration of perspective view.

FIGURE 51 illustrates a roller chain triangular-link with an axle inserted at a distance that will fit the double sprocket in FIGURE 49;

FIGURE 52 illustrates a roller chain triangular-link with an axle inserted at a distance that will fit the double sprocket in FIGURE 49;

FIGURE 53 illustrates a perspective view of FIGURE 52 to observe the double sprocket assembly and triangular-link connection, which provides a double sided triangular-link for bending the chain in reverse to increase the compression and expansion angles for a given geometry and sizes;

FIGURE 54 illustrates a perspective view of a roller chain and sprocket chain drive as a way of transmitting mechanical power from one place to another, which a triangular modification provides an additional motion translational geometry to reciprocate rods, a motion translational function, and a modification of the second link in a chain drive provides a rotation function when transitioning from straight chains to curving around the sprockets and back to straight pathways; FIGURE 55 illustrates a perspective top view of FIGURES 54 and 56 to observe the angle of 26.1 -degrees that a design provides;

FIGURE 56 illustrates a side view of FIGURE 55 to view the chain drive pathway relative to the sprocket that rotates the links configured around second link;

FIGURE 57 illustrates an end and side view of an all-ceramic bending circular actuator with high displacement;

FIGURE 58 illustrates an end and side view of an all-ceramic bending circular actuator with high displacement.

FIGURE 59 illustrates an aircraft wing aerodynamic, provided with a chain roller and sprocket to move the wing surface around the aircraft wing, which may be used to position the aircraft on a terrestrial surface too;

FIGURE 60 illustrates a ball joint that has the freedom to rotate around its axis 360- degrees, which differentiates it from a chain link, and a 25-degree angle is possible in this view providing the ability to curve around a chain roller sprocket designed to receive the ball shape;

FIGURE 61 illustrates an end view of FIGURE 62 which is a propulsion ball in the airduct on the surface of an aircraft or migrating to the surface of an engineering mobility system;

FIGURE 62 illustrates a perspective view of FIGURE 61 for viewing the propulsion ball entering the firing tube and a second propulsion ball is exiting the firing tube into the open air propulsion chamber to move air aerodynamically on the surface of an aircraft;

FIGURE 63 illustrates a perspective view of a roller chain and sprocket chain drive in FIGURE 41 stacked three wide in the axial direction of rotation;

FIGURE 64 illustrates a perspective close up view of a roller chain and sprocket chain drive in FIGURE 63 stacked three wide in the axial direction of rotation;

In FIGURE 65 illustrates a hinge 450 for compressing and stretching piezoelectric or reciprocating components at and radial distance that material strength will tolerate.

DESCRIPTION OF THE PREFERRED EMB ODEVIENT

Referring now to the drawings, particularly FIGURE 1 is an elevated isometric view of a three-rotor blade system assembly 1 comprised of helicopter rotor airfoil blades 2, 3, and 4 fastened to rotor blade spoked mast hub 5 and cap 6 to provide a central mast for a 360-degree rotational direction 11 for assembly 1 to rotate around axis 16. The rotation of rotor airfoil blades 2, 3, and 4, as they turn about the mast hub 5, produces rotational relative wind (tip-path plane). The term rotational refers to the method of producing relative wind. Rotational relative wind flows opposite the physical flightpath of the airfoil, striking the blade at 90° to the leading edge and parallel to the plane of rotation illustrated in the side view of FIGURE 2 and elevated top view in FIGURE 3; and it is constantly changing in direction during rotation. Rotational relative wind velocity is highest at blade tips 20, decreasing uniformly to zero at the axis of rotation (center of the hub mast) 16.

This invention teaches that in prior art torque from an engine (or electric motors, magnetic drive, jets, or turbines) transferring forces through the hub mast 5 is eliminated by moving the thrust production to the airfoil blade tips 20 of three rotor bladed (two-bladed, or any number of blades can be added) system assembly 1 by spinning propellers rotor system assemblies 12, 13, and 14, comprised of helicopter three rotor (propeller) blades 7, 8, and 9 connected to electric motor hub systems 17, that rotates blades 360-degrees in rotational direction 10. Thrust is moved to the tip 20 of rotor bladed system assembly 1 where energy required to rotate the blade is reduced by the force times the rotor blade moment arm, length of blade span 18. Hub 5 provides electrical connections to electric motor hub 17, from the body of the aircraft with the power source. Propeller assemblies 12, 13, and 14, could be dimensionally the same as assembly 1 blades 7, 8, and 9, scaled down to size. Any number of thrust sources can be placed along the length of blade span 18 and then can be rotated in any direction to control airflow, even to stop the rotation of assembly 1 to just hover in the air.

Angle of attack on the leading edge of the airfoil blades 2, 3, and 4 can be increased or decreased by rotating blades around rotating coupling joint 19 in FIGURE 1.

The rotational velocity of the rotor blade 1 is lowest closer to the hub 5 and increases outward towards the tip 20 of the blade during rotation, providing an aircraft that is more efficient than a central axis driven system: engines, electric motors, jets, or turbines forcing rotation through a central mast, like assembly 1, which transferred torque through hub 5 in the plane of rotational direction 11.

Generating Angular Momentum: If the line of action of the force 15 is directed through the rotational axis 16 or rotary-wing assembly 1 axis of rotation 16, then assembly 1 one will not spin around 11. The force 15 must cause a torque, or moment, which means it must be applied some distance 18 from the axis of rotation 16 and have a line of action force 15 which does not go through the axis of rotation 16. The larger the force 15 or the farther the force 15 is from the axis of rotation 16, the larger the torque. The larger the torque, the greater the angular momentum. Torque, moment, or moment of force is rotational force 11. A torque can be thought of as a twist 11 to an object 1. Mathematically, torque is defined as the cross product of the vector by which the force's application point 20 is offset relative to the fixed suspension point 16 (distance vector 18) and the force vector 15, which tends to produce rotational motion 11.

Generally, torque is a measure of the turning force 11, 15 on an object such as a rotary-wing assembly 1. Pushing or pulling the rotary-wing 1 produces a torque (turning force 11) that rotates the rotary-wing assembly 1.

The symbol for torque is typically τ, the lowercase Greek letter tau. When it is called moment of force, it is commonly denoted by M.

The magnitude of torque depends on three quantities: the force applied 15, the length of the lever arm 18 connecting the axis 16 to the point of force application 20, 15, and the angle between the force vector and the lever arm. In symbols:

Where

T is the torque vector and τ is the magnitude of the torque,

r is the position vector (a vector from the origin of the coordinate

system defined to the point where the force is applied)

F is the force vector,

x denotes the cross product,

Θ is the angle between the force vector and the lever arm vector.

The SI unit for torque is the newton metre (N-m).

The term torque is used for the closely related "resultant moment of a couple".

Torque is defined mathematically as the rate of change of angular momentum of an object. The definition of torque states that one or both; of the angular velocity or the moment of inertia of an object are changing. Moment is the general term used for the tendency of one or more applied forces to rotate an object about an axis, but not necessarily to change the angular momentum of the object (the concept which is called torque in physics). A rotational force 15 applied to a mast axis 16 of assembly 1 causing acceleration, such as a rotary-wing assembly 1 accelerating from rest, results in a moment called a torque 11.

Definition and relation to angular momentum: A force 15 applied at a right angle to a lever 2, 3, and 4 multiplied by its distance 18 from the lever's fulcrum (the length of the lever arm 18) is its torque 11. A force of three newtons applied two meters (metres) from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. The direction of the torque can be determined by using the right-hand grip rule: if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque.

More generally, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product:

τ — r X F

Moment arm formula: A very useful special case, often given as the definition of torque in fields other than physics, is as follows:

T— (moment arm) (fores)

The construction of the "moment arm" is illustrated in the Figures 1, 2, and 3, along with the vectors r 18 and F 15 mentioned above. The problem with this definition is that it does not give the direction of the torque but only the magnitude, and hence it is difficult to use in three- dimensional cases. If the force is perpendicular to the displacement vector r, the moment arm will be equal to the distance to the center 18, and torque will be a maximum for the given force 15.

r = (-distance to center }{ [force)

For example, if a person places a force of 10 N at the terminal end of a wrench that is 0.5 m long (or a force of 10 N exactly 0.5 m from the twist point of a wrench of any length), the torque will be 5 N.m - assuming that the person moves the wrench by applying force in the plane of movement and perpendicular to the wrench.

Machine torque: Torque is part of the basic specification of an engine: the power output of an engine is expressed as its torque multiplied by its rotational speed of the axis. Internal- combustion engines produce useful torque only over a limited range of rotational speeds (typically from around 1,000-6,000 rpm for a small car). The varying torque output over that range can be measured with a dynamometer and shown as a torque curve.

Steam engines and electric motors tend to produce maximum torque close to zero rpm, with the torque diminishing as rotational speed rises (due to increasing friction and other constraints). Reciprocating steam engines can start heavy loads from zero RPM without a clutch.

This invention teaches that Capstone power generation solutions help to improve operations by putting the end-user in control of their energy costs and fuel selection. By integrating an aero-based turbine engine, a magnetic generator, advanced power electronics, with patented air bearing technology, Capstone microturbines provide distributed energy to this invention's aircraft. Capstone aero-based turbine with high speed air bearings provides reliability and a thrust force source for the tip of a rotary wing to drive a propeller, electric motor, wired to an electric motor from the center 16 of the aircraft to just provide the electricity, or be distributed into the wing tip with the intake and exhaust thrusting the aircraft wing. If a Capstone turbine is installed in the wing tip, a fuel supply can be provided through the central hub 5 (or in a large wing), as it rotates, it can capture fuel for transport through the rotor blades 2, 3, and 4. Propellers 12, 13, and 14 would be replaced by Capstone turbines on the tips 20 of the rotor blades 2, 3, and 4. In FIGURE 3, a round airfoil structure can be provided with an inside diameter of ring 27 where rotor bladed airfoils are attached to the ring 27, adding a structure to house Capstone turbines, connect a helium or hot air balloon, house batteries, and connect to larger aircraft airfoil wings as a subsystem. Several layers of rotary-wings could be stacked onto www.LevX.com magnetic levitating bearing rings. The ring 27 could also provide a magnetic drive to force the tip 20 with a magnetic coupling provided to couple around the ring 27, providing force 15. A two-rotor bladed airfoil in a rotary- wing aircraft could change the function of the rotor blades into conventional airplane wings, then return back into rotary-wings.

Capstone Microturbines operate on a variety of fuels, including natural gas, associated gas, biogas, LPG/propane, and liquid fuels (diesel, kerosene, and aviation fuel). In resource recovery applications, microturbines burn waste gases that would otherwise be flared or released directly into the atmosphere. Capstone microturbines: Capstone Turbine Corporation (CPST), 21211 Nordhoff Street, Chatsworth, CA 91311, www.capstoneturbine.com .

Capstone microturbines feature low maintenance air bearing technology, the lowest emissions of any non-catalyzed gas combustion engine, and digital power conversion to stand as the optimal power generation solution.

The ability to operate on a wide variety of gaseous and liquid fuels makes Capstone microturbines stand out as a robust source of clean power.

Capstone microturbines can be installed individually or in a "multi-pack"

configuration and function as thrust on rotary-wings. Capstone microturbines are compact, quiet and lightweight for integration into the tips of rotary wings to provide reliable energy. In FIGURES 1 and 2, an aircraft, like a conventional helicopter system can be attached to hub 5.

In FIGURE 3 a cycloid curve 26 is illustrated on the leading edge, providing a wind air particle path that would traverse a segment of an inverted cycloidal arch in the same amount of time, regardless of its starting point on curve 26. As the speed of rotation increases wing pressure on the rotor blade 2, this invention teaches a cycloid curved rotor bladed airfoil inner leading-edge curve 26 (whole airfoil surface wind entry and exit could be a cycloid curve), first capturing air, wind air particle would traverse a segment of an inverted cycloidal arch 26 illustrated, in the same amount of time, regardless of its starting point on curve 26. Controlling air dynamics by providing a synchronized wind departure cycloid curve increases control, reducing drag and turbulence of wind at the compressed air particle level. Every wind path on the aircraft would be a design optimization by applying cycloid shapes, which could be optimized by adding piezoelectric wafers to reshape the aircraft. Micro propellers of drones or human air taxis could be comprised of piezoelectric wafers shaped into turbines, fans, and propellers.

FIGURE 8 illustrates a symmetrical airfoil and nonsymmetrical airfoil (cambered) airfoil.

Airfoil Types

Symmetrical Airfoil

The symmetrical airfoil 31 is distinguished by having identical upper and lower surfaces. The mean camber line and chord line are the same on a symmetrical airfoil, and it produces no lift at zero angle of attack (AO A). Most light helicopters incorporate symmetrical airfoils in the main rotor blades. The upper and lower curvatures are the same on a symmetrical airfoil and vary on a nonsymmetrical airfoil.

Nonsymmetrical Airfoil (Cambered):

The nonsymmetrical airfoil 30 has different upper and lower surfaces, with a greater curvature of the airfoil above the chord line than below. The mean camber line and chord line are different. The nonsymmetrical airfoil design can produce useful lift at zero AO A. A nonsymmetrical design has advantages and disadvantages. The advantages are more lift production at a given AOA than a symmetrical design, an improved lift-to-drag ratio, and better stall characteristics. The disadvantages are center of pressure travel of up to 20 percent of the chord line (creating undesirable torque on the airfoil structure) and greater production costs.

Blade Twist

Because of lift differential due to variations of rotational relative wind values along the blade, the blade should be designed with a twist to alleviate internal blade stress and distribute the lifting force more evenly along the blade. Blade twist provides higher pitch angles at the root where velocity is low and lower pitch angles nearer the tip where velocity is higher. This increases the induced air velocity and blade loading near the inboard section of the blade in FIGURE 1.

Rotary wing blades 2, 3, and 4 in this invention can be spinning around the blades individual axis relative to the blade not the blade assembly 1. A central rotary blade hub axis 16 can have a stationary gear fixed relative to the fuselage, so when all the blades rotate around the central hub 5 axis 16, each blade is rotating around its own axis to provide more lift with near the same power source, air frame, and blade assembly. Each blade could be mounted to independent electric motors to spin the blades independent of each other. In this invention, blades can morph

aerodynamically by piezoelectric wafers movement, so spinning blades around the blades individual axis can be optimizes by the changing the shape of the blade, including providing hollow blade air vessels to fill and release air during spinning the blades themselves. Blade assembly 1 can also rotate all the blades together at the same time blades 2, 3, and 4 are rotating independently.

FIGURE 6 PICMA® illustrates Bender bending actuators have differential control to reshape the aero dynamics of airfoil surfaces. PICMA® Bender All-Ceramic Bending Actuators with High Displacement: PI Ceramic GmbH, Lindenstrasse, 07589 Lederhose Germany

www.piceramic.com .

In FIGURE 3 and 8 a rotary airfoil blade 1 of a propeller typically is mounted to a plane where air is crossing a shaped airfoil wing to decrease the air density on top of area 23 and increase the air density below bottom area 22, where the wing is dependent upon the wing airfoil shape. In FIGURE 8 this invention teaches that in milliseconds, piezoelectric wafers mounted onto propeller airfoil blades 2, 3, and 4 of FIGURE 3 are morphed from a symmetrical airfoil 31 into a nonsymmetrical airfoil (chambered) 30, to increase air density for more lift during high speed propeller rotation. FIGURE 3 illustrates an upper half of a circle 27 area 23 divided by dashed line 21 where airfoil shape 31 (FIG. 8) is provided to capture or decrease wind by morphing

piezoelectric wafers. FIGURE 3 illustrates a lower half of a circle 27 area 22, divided by dashed line 21, where airfoil shape 30 (FIG. 8) is provided by morphing piezoelectric wafers to produce more rotational relative wind (tip-path plane) for more lift and travel speed. When airfoil shape 31 (FIG. 8) is applied to capture wind, it is released where higher density air is desired to lift or travel.

FIGURES 4 to 7 provide the data for selecting all-ceramic bending actuators with high displacement.

FIGURE 4 is a data table of PL112 - PL140 PICMA® Bender

All-Ceramic Bending Actuators with High Displacement;

FIGURE 5 illustrates a PL112.10 - PL140.10. L, LF, W, TH, see data table;

FIGURE 6 PICMA® illustrates Bender bending actuators have differential control to reshape the aero dynamics of airfoil surfaces; and

FIGURE 7 Displacement of the PICMA® bending actuators: Clamped on one side (top) and on both sides (bottom). PL112 - PL140 PICMA® Bender All-Ceramic Bending Actuators with High

Displacement

Displacement to 2 mm: Fast response in the ms range, nanometer resolution, low operating voltage, and operating temperature up to 150 °C.

PICMA® multilayer bender elements with high reliability.

Operating voltage 0 to 60 V. Bidirectional displacement, bimorph design. Ceramic insulation, polymer free. UHV compatible to 10-9 hPa, no outgassing, high bakeout temperature. Reliable even under extreme conditions. Applied in laser technology, sensor technology, automation tasks, pneumatic valves.

Resonant frequency: Measured at 1 Vpp, clamped on one side with remaining length LF, unloaded. Standard connections: Solderable contacts (PLlxx.10) or PTFE-insulated stranded wires, 100 mm, AWG 32, 0 0.49 mm (PLlxx. l l).

Recommended mounting: Epoxy resin adhesive.

All specifications depend on actual clamping conditions and mechanical load applied. Custom designs or different specifications on request.

Design Information:

PL112.10

PICMA® multilayer piezo bending actuator, 200 μπι travel range, 18 mm x 9.60 mm x 0.67 mm.

PL122.10

PICMA® multilayer piezo bending actuator, 620 μπι travel range, 25 mm x 9.60 mm x 0.67 mm.

PL127.10

PICMA® multilayer piezo bending actuator, 900 μπι travel range, 31 mm x 9.60 mm x 0.67 mm.

PL128.10

PICMA® multilayer piezo bending actuator, 900 μπι travel range, 36 mm x 6.15 mm x 0.67 mm.

PL140.10

PICMA® multilayer piezo bending actuator, 2000 μπι travel range, 45 mm x 11.00 mm x 0.55 mm.

With PTFE-insulated wire leads, 100 mm, AWG 32 (0 0.49 mm) PL1 PICMA® multilayer piezo bending actuator, 200 μιη travel range, 18 mm x 9.60 mm x 0.67 mm, stranded wires.

PL122.11

PICMA® multilayer piezo bending actuator, 620 μιη travel range, 25 mm x 9.60 mm x 0.67 mm, stranded wires.

PL127.11

PICMA® multilayer piezo bending actuator, 900 μιη travel range, 31 mm x 9.60 mm x 0.67 mm, stranded wires.

PL128.11

PICMA® multilayer piezo bending actuator, 900 μιη travel range, 36 mm x 6.15 mm x 0.67 mm, stranded wires.

PL140.11

PICMA® multilayer piezo bending actuator, 2000 μιη travel range, 45 mm x 11.00 mm x 0.55 mm, stranded wires.

Hollow propellers can be opened by opening piezoelectric wafers to fill propeller with air on top of a wing, then piezoelectric wafer can be shut on one side, opened on the other to release captured air under a wing or structures configured to lift an aircraft. Wafers can be wired with nano wires made of graphene that are very dense and flexible, so vibrating wafer at ultrasonic speeds to not damage the wire windings.

According to the United States Department of Energy VEHICLE TECHNOLOGIES

OFFICE https://www.energy.gov/eere/vehicles/electric-motors-researc h-and-development :

To reach the EV Everywhere Grand Challenge goal, the Vehicle Technologies Office (VTO) is supporting research and development (R&D) to improve motors in hybrid and plug-in electric vehicles, with a particular focus on reducing the use of rare earth materials currently used for permanent magnet-based motors.

In an electric drive system, an electric motor converts the stored electrical energy in a battery to mechanical energy. Electric motors consist of a rotor (the moving part of the motor) and a stator (the stationary part of the motor). A permanent magnet motor includes a rotor containing a series of magnets and a current-carrying stator (typically taking the form of an iron ring), separated by an air gap. There are three types of electric motors that can be used in hybrid or plug-in electric vehicle traction drive systems. This invention teaches that the current-carrying stator is reduced in size to match the number of "poles" on the rotor (Example Given: four blades of a helicopter would have a 90-degree Circle-arc (arc-power strip stator)), which would provide 360-degrees of force as each of the tip-path ends of four electrically wired blade poles pass within the 90-degree arc-power strip stator. Permanent magnets can be reduced in size, weight and power capacity by mounting the magnet on the end of piezoelectric wafer that move the magnet back and forth up to an ultrasonic speed, providing more eddy currents increasing the electromagnetic currents across an airgap between a stator and rotor.

Internal permanent magnet (IPM) motors have high power density and maintain high efficiency over a high percentage of their operating range. Almost all hybrid and plug-in electric vehicles use rare earth permanent magnets in their traction motors. Because of the high costs of magnets and rotor fabrication, these motors are relatively expensive. Other challenges to using IPM motors include the limited availability and high cost of rare earth magnetic materials. Despite the challenges, the automotive industry anticipates continuing to use IPM motors in the majority of electric drive vehicles over the next decade.

Induction motors have high starting torque and offer high reliability. However, their power density and overall efficiency are lower than that of IPM motors. They are widely available and common in various industries today, including some production vehicles. Because this motor technology is mature, it is unlikely research could achieve additional improvements in efficiency, cost, weight, and volume for competitive future electric vehicles.

Switched reluctance motors offer a lower cost option that can be easy to manufacture. They also have a rugged structure that can tolerate high temperatures and speeds. However, they produce more noise and vibration than comparable motor designs, which is a major challenge for use in vehicles. Also, switched reluctance motors are less efficient than other motor types, and require additional sensors and complex motor controllers that increase the overall cost of the electric drive system.

VTO Electric Motor R&D

VTO's primary goal is to decrease electric motors' cost, volume, and weight while maintaining or increasing performance, efficiency, and reliability. To meet the 2022 cost targets described in the EV Everywhere Blueprint, research must reduce the cost of the motor by 50%.

To achieve these goals, VTO and its partners are examining many research avenues: The Beyond Rare Earth Magnets (BREM) R&D project led by Ames Laboratory is investigating lower-cost permanent magnets and magnetic materials. This effort is closely coordinated with the Critical Materials Institute also led by Ames Laboratory. Oak Ridge National Laboratory and industry projects are pursuing reduced rare-earth magnet motors, non-permanent magnet motor designs, and innovative motor materials and designs.

The National Renewable Energy Laboratory is focusing research on improving electric motor thermal management, performance and reliability.

FIGURE 9 is an elevated top view of a four-rotor blade 42 system assembly with a

90-degree arc-power strip stator 41 providing electric power to all four blades 42 during full 360- degrees of rotation by electromagnetically forcing the magnetic tip 40 along the arc-power strip stator 41 aligned close enough to rotor tip-path tip 40 to transfer electromagnetic force across air gap between assembly stator 41 and tip 40 of blade 42. This invention teaches that a full 360- degrees of force can be applied to each rotor blade 42 pole tip 40 by providing an electric arc-power strip 40 shaped to match the path of the rotor pole blade 42 tip 40 and each blade rotating next through the path of arc-power strip stator 41. 90-degree arc-power strip arc 41 can be mounted to the body of the aircraft, like a helicopter or drone. FIGURE 3 illustrates a three-blade assembly, so an 120-degree arc-power strip stator would be mounted in the arc between any two of the triangles illustrated next to the blades 2, 3, and 4 tips. Two-blade rotors would have an 180-degree arc power strip, so any design and number of blades can be provided the corresponding arc-power strip stator length. Example given: A six-bladed rotary wing would have 60-degree of arc-power strip stator. Electromagnetic arc-power strips can be designed with any rotating electric motor technology, including any controls and sensors to keep the power transfer from arc-power strip 41 to blade 42 electric motor element tip 40. Tip 40 can be a permanent magnet, electric coil, or any other element that will electromagnetically transfer rotational motion to the rotor assembly. In addition, tip 40 could be shaped into a "jet" energy receptacle to capture high speed air or exhaust fumes jetting out to blow or vacuum the rotor blade tip 40 to rotate all four of the blades 42 arrayed around a center point of rotation. High speed rotating elements at the tip (or anywhere along the radial distance to the tip) of the blades can be electromagnetic elements that add to the energy for forcing rotation or generating electricity.

Arc-power strip divisions and positioning: In yet a further embodiment of this invention, teaches that in a three-blade assembly in FIGURE 3, an 120-degree power strip is taught to be mounted in the arc between any two of the three triangles illustrated next to the blade tips, which 120-degree arc could further be divided up into three 40-degree arc-power strip stators arrayed into separate positions to provide one of three tips of the blades to each engage the tip of a corresponding 40-degree arc-power strip stator. A full 360-degrees of force is provided by FIGURE 3 as blade 2 arc-power strip stator forces the blade the first 40-degrees, blade 3, now rotated 40- degrees from its original position is the "continuous" force for the second 40-degree arc-power strip stator, then blade 4 now rotated 80-degrees from its original position would provide the last (third) 40-degrees of force from the last 40-degree arc-power strip stator totaling 120-degrees of forced motion, first phase of three. In second phase, blade 4 rotated 120-degrees into the original position of blade 2 starting point, which provides the same sequences as blade 2, 3, and 4 were forced through in the first phase. In the third phase, blade 3 rotated 120-degrees into the original position of blade 2 starting point, which provides the same sequences as blade 2, 3, and 4 were forced through in the first phase. 360-degrees of continuous force from three 40-degree arc-power strip stators teaches that further breakdown of any rotor with any number of blades can be positioned to provide 360-degrees of rotation forces. Arc-power strip stators can be placed anywhere along the axis in a perpendicular position relative to the axis, and in addition can be moved too, if they add force to the rotation at the correct time to force rotation around that axis. A belt or chain arc-power strip, full of permanent magnets or electric coils (any power source) can be rotated into the form of an arc-power strip, where the motion of the arc-power strip is driven into motion with another power source of any shape or size. Engineered hybrid devices could have arc-power strips provide half the power of rotation, while some other force, like water, wind, or steam provides the balance of 360-degree force. Capacitors could be charged remotely, then moved into the cycle.

This invention teaches that piezoelectric All-Ceramic Bending Actuators with High Displacement in FIGURES 4 through 7 are provided with permanent magnets or a wired

electromagnetic coil to drive a motion translation device, like arc-power strip stators taught in this invention, or adding force to any other motion translational device: tip-path of turbine, turboprop, turbo fans, jets, and morphing objects, like robots. Diving rotational devices with piezoelectric wafers at the tip-path of the structures increases the efficiency and the devices can reach ultrasonic speed staying clean during operation in addition to higher electromagnetic force, because of the ultrasonic speed capability magnetic field density increases. Many paths can be "shaped" to increase wafer efficiency and most important the active waver at the tip-path end in motion can have a corresponding stationary wafer in its path to optimize the sensing and motion from the same wired wafer. Circuits can be applied to the wafer too for sensing and driving the electromagnetic forces or physical force from the wafer alone.

Wind power blades can be equipped with electromagnetic (permanent magnets or wired coils) tips to generate electricity from wind energy with a full 360-degrees, or arc-power strips corresponding to the number of blades (blades are poles, like a pole motor wired or containing permanent magnets). Automotive, bicycle, or other motion translational devices can provide equally spaces electric components inside or outside a tire, wheel rim, or axil to adapt this arc-power strip to an application. An innertube could be a wired arc-power component to upgrade a car and bike tire, or the tire can be made into an arc-power strip with corresponding components on the frame of the engineered motion machine.

FIGURES 10 through 12 illustrate a cycloid sprocket. In the case of single-speed bicycles and hub gears, the chain length must match the distance between crank sprocket 40 and rear hub 60 and the sizes of the front chain ring 46 and rear sprocket 63. These bikes usually have some mechanism for small adjustments such as horizontal dropouts, track ends, or an eccentric mechanism in the rear hub or the bottom bracket. In extreme cases, a chain half-link may be necessary. In FIGURES 11 and 12:

All modern bicycle chains in use today are of the "roller chain" design.

(In commercial designs: Do = Sprocket diameter. Dp = Pitch diameter)

Sprocket 40 is a 26 tooth sprocket along cycloid curve 46.

Sprocket 60 is a 12 tooth sprocket along circle circumference 63.

Sprockets can be designed in any diameter, chain length, and number of sprockets. Belts can have teeth, functioning like sprockets, and belts made with spaced permanent magnets along the belt would equal a sprocket when the sprocket is driven by the magnets. Spiral around cylinders can be divided up and on curved surfaces, like circles parallel arc-power strips can be configures along the axis length, yet the arc-power strip would change in radial distance and still be aligned and in length to continue a 360-degree force.

FIGURE 10 illustrates a cycloid curve shaped sprocket 40 rotated 90-degrees, 42.19- degrees, and 0-degrees to provide one half the area of the cycloid sprocket, which is equal when divided along the line 44 going through the centerpoint of rotation. Area = 942.465709 square centimeters for each half at any angle:

90-degree angle of rotation 41 around centerpoint 45 on measurement line 44 of cycloid sprocket 40.

42.19-degrees angle of rotation 42 around centerpoint 45 on measurement line 44 of cycloid sprocket 40.

0-degree (also 360-degrees) angle of rotation 43 around centerpoint 45 on

measurement line 44 of cycloid sprocket 40. Cycloid curve circumference line 46 is related to the pitch diameter in a circle, but this invention teaches a cycloid curve mirrored around line 58 into a sprocket 40 has more leverage, providing force multiplied by the moment arm half of line 58 leveraging around centerpoint 45.

FIGURE 11 illustrates cycloid assembly 50, cycloid sprocket 40, and circular sprocket 60 are chained together, like a bicycle roller sprocket.

A cycloid curve is provided by rotating a circle 59 illustrated in five positions, 51, 52, 52, 54, and 55 rotated 360-degrees along line 58. Line 58 and the circumference of the circles 59 are equal (example: when circle 59 Diameter is 20 centimeter (cm), Area is 314.1592653589793 square cm, Circumference is 62.83185307179586 cm, which is also the length of cycloid 40 line 58. Area of cycloid sprocket 40 is 1,884.931418 square centimeters within curve 46 mirrored around line 58). Point 57 on the circumference of circle 59 is marked on 51, 52, 53, 54, and 55 to illustrate the cycloid path-line drawn, cycloid curve 46, which is mirrored around line 58 to provide the sprocket 40 cycloid geometric dimensions. A center hole 48 and two holes 49 are cut out to mount the sprocket 40 onto a bike or machine, like a piston engine or compressor. As a human peddles a bike cycloid sprocket 40 connected by a belt or chain 64 moving in direction 61 and 62 of wheel sprocket 60 mounted to a wheel with a tire, circular sprocket 60 is rotated with more torque than just a circular sprocket that human peddles are mounted to, because of the equation for the magnitude of a torque, arising from a perpendicular force: torque is multiplied by the greatest distance from the center of rotation of the cycloid sprocket, endpoints of line 58 (point 57 at circular position 51 and position 55), multiplied by the force applied to each position. On most bicycles, with multi-speeds, peddles can be reversed to optimize the greatest torque position relative to feet pressing the peddles, so the cycloid optimized leverage can be positioned by the bicyclist. On fixed bicycles, combustion engines, compressors, and other machines, the cycloid sprocket can be mounted to keep the piston compressed for a longer period of time during combustion, providing more efficient combustion and less pollution by completely burning the fuel air mixture under high pressure for longer time. Air intake piston positions, at bottom of their cycle, will also have a longer period of time for air intake, increasing air, increases efficiency of an engine. Cycloid sprocket 40 can be positioned on machines, like piston engines, to obtain the desired efficiency. The load against the engine with a cycloid sprocket mounted to the crankshaft with pressure the changes in engine piston speed. In FIGURE 10 sprocket 40 at 0-degree angle horizontal position 43, the bottom half of cycloid curve 46 would guide two equal sized ball bearings down to bottom center of the curve at the same time no matter where the ball bearings are placed on the curve. This physical function of a cycloid curve provides many benefits in optimizing motion along with forces applied against the curve. No other curve behaves the same, but an oval sprocket and other near cycloid shapes that provide the leverage can be applied in this invention for more torque, but without the benefit of the cycloid curve above-mentioned physics.

In FIG 11 circular sprocket 60 can be mounted inside cycloid sprocket 40 rotating within sprockets 40 surrounded by an outer housing structure, so the inner circle 53 can rotate all the way around cycloid curve 46 and 64. Center line 58 can provide the reference points to separate the two halves 46 and 64 enough to fit circle 53 around each end if line 58. Circle 50 with sprockets or just smooth roller bearing outer diameter can be attached to any mechanical motion device, like floats under a boat or floating docks to provide a

synchronized motion between several identical cycloid assembles with inner roller 50 whenever the circle 50 is rolled against the inner surfaces of cycloid curve 46 or 64 when the reference line 58 is floating horizontally on water fluids. Building frames, boat floats, and other motion translational devices can have several moving structure where circle 50 rotates within cycloid curve 46 or 64 placed horizontal to synchronize motion back to where circle 50 is in FIG 11, after it is displaced. A boat would stay level in waves and a house would return to a level house, if shaken back and forth on top of a common cycloid curve 46 or 64. Any motion translational device like robots would return to a rested uniform state without ower when applying cycloid curves.

Cycloid curve 46 coordinate point positioning is illustrated in FIGURE 11 as circletated along line 58, which is equal to the circles circumference length:

At a 0-degree angle, the starting point of circle 59 position 51 radial circumference point 57 is located on fixed line 58, which is then rotated in direction arrow 56 around the centerpoint of circle 59 in position 51 traveling along line 58 to provide a cycloid curve 46 through the following additional coordinate points:

At a 90-degree angle of rotation, point 57 on the circle 59, in position 52, provides the second cycloid curve 46 coordinate point.

At an 180-degree angle of rotation point 57 on the circle 59, in position 53, provides the third cycloid curve 46 coordinate point.

At a 270-degree angle of rotation, point 57 on the circle 59, in position 54, provides the forth cycloid curve coordinate point. At a 360-degrees angle of rotation, point 57 on the circle 59, in position 55, provides the fifth, last cycloid curve 46 coordinate point.

At every moment of circle 59 360-degrees of rotation, cycloid curve 46 is drawn from one point 57 on the circle 59.

In FIGURE 11, independent of the sprocket, the geometric cycloid path of circle 59, a turbofan, turboprop, and propellers (including round balls and cylinder shapes) can be cycled on both sides of line 58 to power an air or water craft. In power generation applications, the motion on one side of line 58 can be covered protecting the motion from power generation resistant forces, while the opposite side of line 58 provides the energy capture phase of the cycle, where forces of wind or water move the objects for power generation. Circle 59 would cycle around both ends of line 58 to continuously cycle over time. Circle 59 would be linked to line 58, providing the full surface of the circle as a wind capture device with the torque generation from the tip-path of circle 59 power motion translational device. A floating ball with a frictional extended surface (water paddles) that is captured under the near-water surface can be blown across the water, rotating at twice the torque, over the full diameter of the ball, generating electricity at more capacity than just a wind power blade above the water, on the ocean for example. A ball with water capture paddles could travel inside a floating tube, covering the ball from the wind forces and water interference (or the ball could be rotated onto a non-paddle friction side of the ball, floating like a boat or completely lifted off the water), on one side of line 58, providing the opposite side of line 58 with maximum wind water capture forces to generate power when floating with surface paddle resistance. One of the endpoints of line 58 could be "fixed" (anchored relative to forces), providing rotation in alignment with the maximum wind or water currents. Electric energy capture can be provided by arc-power strips composed of permanent magnets, electric winding, coils, or any electric power capture technology, even conventional full circle generators.

FIGURE 12 illustrates cycloid sprocket 40, a description NOT related to rotating sprocket 40, is rotated 90-degrees, aligning datum line 58 around to line 44 in FIGURE 1. This invention teaches that the chain or belt 64 connecting cycloid sprocket 40 to circular sprocket 60 can be the same length no matter what the angle of rotation is relative to sprocket 40 and 60. For adjustments, line 58 divides the cycloid sprocket 40 into two equal halves, which can be spaced equally along line 58 to adjust chain or belt length to be "exactly" equal without any tolerance on a chain, belt, or gear drive under motion pressure. FIGURE 10 illustrates that the cycloid curve 46 area is divided along the centerpoint 45 by dividing line 44, providing the same symmetrical equal area above and below the line 44 no matter what the rotation angle. The belt or chain length mounted around the cycloid sprocket 40 and circular sprocket 60 will not change, because the cycloid area does not change in FIGURE 10 as the cycloid is rotated around a centerpoint 45.

FIGURE 13 illustrates an isometric view of a circle-arc stator roller chain type assembly 70 (chain links stator 70), comprised of bolt fastener 71 inserted through the pin (tube) 80, which provides the assembly of outer plate 75, roller bearing bushing tube 78, roller bearing tube 79, inner plate fastener 76, and outer plate fastener 75 to complete roller chain 70 assembly. View 70b is provided to illustrate stacking chain link 70 on top of one another to provide a helix when rotating around line 77 to align with the circumference of a circle specified. These helix 70b can be left or right-handed helix.

Pin tube 80 is pressure fit into hole 81 of outer plate fastener 75. Ends of bushing tube 78 outside diameter fits within the pressure fit hole 82 of inner plate fastener 76. Outside diameter of pressure fit fastener pin 80 fits inside diameter 83 of roller bearing bushing tube 78. Roller bearing tube 79 inside diameter 84 assembles over roller bearing bushing tube 78 to provide a roller bearing clamped between plates 75 and 76. 3D Printing surfaces can be specified on any or all the components: plate 75 and 76, bearing bushing 78, and roller bearing 79. Any additional components can be added to the roller chain link stator 70 to adapt any electronic components designed as a motor or generator. A high-speed motor can be substituted for roller bearing 79.

FIGURE 14 illustrates roller chain motor stator 70 fastened along the circumference of circle 84 in a circle-arc of about 90-degrees of rotation requiring four motor poles arrayed as radius lengths from center at 90-degrees. Rotation of the circle-arc stator roller chain type assembly 70 in FIGURE 13 links around the centerline 77 of bolt 71 (screw, bolt, or dowel can be applied), which provides any length and curve of a circle-arc by adding or subtracting the number of circle- arc stator roller chain links 70 to optimize and adapt manufacturing the stator into structures, like automotive, aerospace, and generators. Exploded view in FIG 13 of all components assembled along axis centerline 77 illustrates all components fastened around center axis of bolt 71 comprised of plain unthreaded shank (called the grip length) to make a strong dowel 73 to fasten a circle-arc stator to any structure designed to match the specifications. Bolt 71 is a form of threaded fastener, external male thread 74. Hexagonal head 72 (any other type head) of bolt 71 is turned with a spanner or wrench to fasten roller chain circle-arc motor stators any curves and length can be provided. FIG 14 also illustrates a linear straight assembly that can be the wired rotating pole of the motor. The wire windings of the motor can be assembled on any of the components of chain link stator 70, including winding around the roller bearing tube 70, or replacing the roller bearing with a winding, permanent magnet, or other elements. A motor circle-Arc stator can be slightly longer that the circle-arc in order to phase in the electromagnetic field between the independent poles. This chain link stator can be rotated around more than one pole rotating rotor, like a bicycle spoke chain in sprockets. The chain link electric motor stator can also phase from straight linear motors to any curve of a motor (generator) or drive, providing mechanical links, but also providing the electric motor function in motion in almost any configuration (e.g. a bicycle chain that also is an electric motor relative to the sprocket acting as motor poles, stator and poles of the electric chain sprocket motor would be in motion).

In FIGURES 13, 14, and 41 through 56 Chain roller drive materials in pins 80, bushings 78, and rollers 79 are electrically insulated materials, sandwiched between the electrically conductive plates 75 and 76 (FIG 42 and FIG 44 plate 376, and FIG 54 plate 375) to provide plate- circuits in stationary stators or rotating sprockets. These electrically conductive and electrically insulating materials in the chain roller assembly provide electric circuits for circle-arc stators in FIGURE 14, motors, generators, or sensors: All the small chain link plates 75 and 76 can be assembled into roller chains with all the electrically insulating pins 80, bushings 78, and rollers 79, sandwiched between the plates 75 and 76, providing two Plate-Circuits in electrically independent contact with two sprockets faces illustrated in FIGURES 41 through 56. In FIGURES 41 through 56 two Plate-Circuits electrically contact sprockets 301 and 302 that have one side of sprocket face electrically conductive and the opposite surface electrically insulating to provide electrical connections for all the electrically conductive chain-link plate-circuits traveling on one side of each sprocket, so one sprocket would be the positive electric charge and the other sprocket the negative charge (electric source or ground). In another embodiment of this invention each face of one sprocket could be electrically conductive with a layer of insulating material between to complete an electric circuit along independent plate circuits traveling in contact with each sprocket face.

Sprockets could five layers: conductive, insulating, conductive, insulating and conductive, so that the two outer surfaces are circuits with an independent circuit in the middle that only connects to pins, bushings, and rollers sandwiched between the conductive plates. Each individual sprocket can be a circuit, so each chain link is an independently connected circuit component. Once the chain roller is configured into an electric circuit motors, generators, and sensors can be applied within the design. This invention teaches that any of the roller chain components can become completed circuits to provide functions that prior art did not provide. FIGURE 41 illustrates a perspective view of a roller chain triangular-chain links 303 and 304 assembled around on sprockets 301 and 302 providing chain drive 300 as a way of transmitting mechanical power from one place to another, which a triangular modification to link 303 in FIG 42 provides an additional motion translational geometry to reciprocate rods 321 and 322 of a piston 323 and cylinder 324 in FIGURES 43 through 46, a motion translational function, to compress and expand piston cylinder assembly, stretch and compress piezoelectric material in FIGURES 57 and 58 to drive the chain assembly or the forces from energy, like wind or water that moves the chain assembly can generate electricity by stretching and compressing materials like piezoelectric material.

FIGURE 42 illustrates a perspective view of an expanded triangular-chain link assembly 303 to configure piston and cylinders 321, 322, 323, and 324 provided in assembly 320 (replacing a roller bearing tube 79, piezoelectric material insertion, and other energy conversion systems, like Stirling engines, expanded to a 1.41 -centimeter distance between the two rocker arms in FIGURE 43. Triangular chain link assembly 303 is and expanded view along three axis lines 377 comprised of the pin (tube) 80, which provides the assembly of outer plate 375 (replacing plate 75 in FIG 13), roller bearing bushing tube 78, roller bearing tube 79, inner plate fastener 76 (in FIG 43), and outer plate fastener 375 to complete roller chain 303 assembled around in FIGURES 41 and 47 spaced and linked by 311, 312, 313, and 314 in corresponding FIGURES 43, 44, 45, and 46. A piston is a component of reciprocating engines, reciprocating pumps, gas compressors and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-tight by piston rings. In an engine, its purpose is to transfer force from expanding gas in the cylinder to the crankshaft via a piston rod and/or connecting rod. In a pump, the function is reversed, and force is transferred from the crankshaft to the piston for compressing or ejecting the fluid in the cylinder. In some engines, the piston also acts as a valve by covering and uncovering ports in the cylinder. Large slow-speed Diesel engines may require additional support for the side forces on the piston. These engines typically use crosshead pistons. The main piston has a large piston rod extending downwards from the piston to what is effectively a second smaller- diameter piston. The main piston is responsible for gas sealing and carries the piston rings. The smaller piston is purely a mechanical guide. It runs within a small cylinder as a trunk guide and carries the gudgeon pin. Lubrication of the crosshead has advantages over the trunk piston as its lubricating oil is not subject to the heat of combustion: the oil is not contaminated by combustion soot particles, it does not break down owing to the heat and a thinner, less viscous oil may be used. The friction of both piston and crosshead may be only half of that for a trunk piston. Because of the additional weight of these pistons, they are not used for high-speed engines.

A connecting rod is a shaft which connects a piston to a crank or crankshaft in a reciprocating engine. Together with the crank, it forms a simple mechanism that converts reciprocating motion into rotating motion. A connecting rod may also convert rotating motion into reciprocating motion, its original use. Earlier mechanisms, such as the chain, could only impart pulling motion. Being rigid, a connecting rod may transmit either push or pull, allowing the rod to rotate the crank through both halves of a revolution. In a few two-stroke engines the connecting rod is only required to push. Today, the connecting rod is best known through its use in internal combustion piston engines, such as automobile engines. These are of a distinctly different design from earlier forms of connecting rod used in steam engines and steam locomotives. Radial engines typically have a master rod for one cylinder and multiple slave rods for all the other cylinders in the same bank. Fork and blade rods: The usual solution for high-performance aero-engines is a "forked" connecting rod. One rod is split in two at the big end and the other is thinned to fit into this fork.

A Stirling engine is a heat engine that operates by cyclic compression and expansion of air or other gas (the working fluid) at different temperatures, such that there is a net conversion of heat energy to mechanical work. More specifically, the Stirling engine is a closed-cycle regenerative heat engine with a permanently gaseous working fluid. Closed-cycle, in this context, means a thermodynamic system in which the working fluid is permanently contained within the system, and regenerative describes the use of a specific type of internal heat exchanger and thermal store, known as the regenerator. Strictly speaking, the inclusion of the regenerator is what differentiates a Stirling engine from other closed cycle hot air engines. Stirling engines have a high efficiency compared to internal combustion engines, [4] being able to reach 50% efficiency. They are also capable of quiet operation and can use almost any heat source. The heat energy source is generated external to the Stirling engine rather than by internal combustion as with the Otto cycle or Diesel cycle engines. Because the Stirling engine is compatible with alternative and renewable energy sources it could become increasingly significant as the price of conventional fuels rises, and also in light of concerns such as depletion of oil supplies and climate change. This type of engine is currently generating interest as the core component of micro combined heat and power (CHP) units, in which it is more efficient and safer than a comparable steam engine. [5][6] However, it has a low power-to-weight ratio,[4] rendering it more suitable for use in static installations where space and weight are not at a premium. Like the steam engine, the Stirling engine is traditionally classified as an external combustion engine, as all heat transfers to and from the working fluid take place through a solid boundary (heat exchanger) thus isolating the combustion process and any contaminants it may produce from the working parts of the engine. This contrasts with an internal combustion engine where heat input is by combustion of a fuel within the body of the working fluid. Most of the many possible implementations of the Stirling engine fall into the category of reciprocating piston engine. As a consequence of closed cycle operation, the heat driving a Stirling engine must be transmitted from a heat source to the working fluid by heat exchangers and finally to a heat sink. A Stirling engine system has at least one heat source, one heat sink and heat exchangers. Some types may combine or dispense with some of these.

The heat source may be provided by the combustion of a fuel and, since the combustion products do not mix with the working fluid and hence do not come into contact with the internal parts of the engine, a Stirling engine can run on fuels that would damage other engines types' internals, such as landfill gas, which may contain siloxane that could deposit abrasive silicon dioxide in conventional engines.

Cooler / cold side heat exchanger

In small, low power engines this may simply consist of the walls of the cold space(s), but where larger powers are required a cooler using a liquid like water is needed to transfer sufficient heat.

Heat sink

The larger the temperature difference between the hot and cold sections of a Stirling engine, the greater the engine's efficiency. The heat sink is typically the environment the engine operates in, at ambient temperature. In the case of medium to high power engines, a radiator is required to transfer the heat from the engine to the ambient air. Marine engines have the advantage of using cool ambient sea, lake, or river water, which is typically cooler than ambient air. In the case of combined heat and power systems, the engine's cooling water is used directly or indirectly for heating purposes, raising efficiency.

Alternatively, heat may be supplied at ambient temperature and the heat sink maintained at a lower temperature by such means as cryogenic fluid (see Liquid nitrogen economy) or iced water.

Displacer The displacer is a special-purpose piston, used in Beta and Gamma type Stirling engines, to move the working gas back and forth between the hot and cold heat exchangers.

Depending on the type of engine design, the displacer may or may not be sealed to the cylinder, i.e. it may be a loose fit within the cylinder, allowing the working gas to pass around it as it moves to occupy the part of the cylinder beyond.

Configurations

There are three major types of Stirling engines, that are distinguished by the way they move the air between the hot and cold areas:

The alpha configuration has two power pistons, one in a hot cylinder, one in a cold cylinder, and the gas is driven between the two by the pistons; it is typically in a V-formation with the pistons joined at the same point on a crankshaft.

The beta configuration has a single cylinder with a hot end and a cold end, containing a power piston and a 'displacer' that drives the gas between the hot and cold ends. It is typically used with a rhombic drive to achieve the phase difference between the displacer and power pistons, but they can be joined 90 degrees out of phase on a crankshaft.

The gamma configuration has two cylinders: one containing a displacer, with a hot and a cold end, and one for the power piston; they are joined to form a single space with the same pressure in both cylinders; the pistons are typically in parallel and joined 90 degrees out of phase on a crankshaft.

Other suitable heat sources include concentrated solar energy, geothermal energy, nuclear energy, waste heat and bioenergy. If solar power is used as a heat source, regular solar mirrors and solar dishes may be utilised. The use of Fresnel lenses and mirrors has also been advocated, for example in planetary surface exploration. [39] Solar powered Stirling engines are increasingly popular as they offer an environmentally sound option for producing power while some designs are economically attractive in development project

FIGURE 43 illustrates a perspective view of rod assembly 320 in FIGURE 42 connected to triangular-chain link 303 with rods expanded to a maximum 1.41 -centimeter distance between the two rods 321 and 322 (rocker arms), which is position 311 in FIGURES 41 and 47. FIGURE 44 illustrates a perspective view of rod assembly 320 in FIGURE 42 connected to triangular-chain link 303 with rods expanded to a maximum 0.00-centimeter distance between the two rods 321 and 322 (rocker arms), which is position 312 in FIGURES 41 and 47.

FIGURE 45 illustrates a perspective view of rod assembly 320 in FIGURE 42 connected to triangular-chain link 303 with rods expanded to a 0.39-centimeter distance between the two rods 321 and 322 (rocker arms), which is position 313 in FIGURES 41 and 47.

FIGURE 46 illustrates a perspective view of rod assembly 320 in FIGURE 42 connected to triangular-chain link 303 with rods reduced to a 1.06-centimeter distance between the two rods 321 and 322 (rocker arms), which is position 311 in FIGURES 41 and 47.

FIGURE 47 illustrates a side view of a roller chain and sprocket chain drive in

FIGURE 41, providing a more precise view of the 180-degree angles around the sprockets 301 and 302, which moves the spacing between triangular tip of 303 with rods 321 and 322 attached. When the triangular-chain link 303 in FIGURE 47 is observed in position 313, a spacing of 0.39- centimeters is measured, because as link 303 transitions around circular sprocket 301, aligning one pin on the start of 180-degree line, the angle starts to widen between any two of links 303. In

FIGURE 43 maximum spacing is measured on 311 in FIG 41 and 47 under angle of 153.29-degrees, then position of 314 provides a reduced distance of 1.06-centimeters as the triangular-chain link passed the 180-degree of travel around sprocket 301. In FIGURE 44 position 312 is minimum distant spacing is measured on 312 in FIG 41 and 47 after passing by 180-degrees, then position of 312 provides a reduced distance of 0.00-centimeters as the triangular-chain link passed the 180- degree of travel around sprocket 301. As an example: the measurement of 25-centimeters and the radial distance and thickness of the sprockets 301 and 302 are all related to the triangular-chain link 303 and 304 configuration: angles of separation, size of the rods or pistons, length of the chain for cooling or heating energy elements, matching the size of an application (railroad engine versus micro-robot), and integration into engineering mobility (automotive, aerospace, oceanography, mining,..).

FIGURE 63 illustrates a perspective view of a roller chain 444 and sprocket 301 and 302 chain drive in FIGURE 41 with triangular-chain link assembly 303 in FIGURE 42 stacked three wide with four plates 375 stacked too in the axial direction of rotation. FIGURE 64 illustrates a perspective close-up view of a roller chain 444 and sprocket 302 chain drive in FIGURE 63 stacked three wide in the axial direction of rotation. Three separate reciprocating elements 321 and 322 can be removed or added from the multi-wide chain, including providing any function: one could compress air, one could be a closed Stirling piston cylinder, one could be a piezoelectric material, or other mechanical or electric function like reciprocating a permanent magnet within a coil to to generate electricity. Pumping fluids or any valving system can be provided. In FIGURE 65 illustrates a hinged motion for compressing and stretching piezoelectric or reciprocating

components at and radial distance that material strength will tolerate,

FIGURE 48 illustrates a roller chain triangular-link with an axle inserted at a distance that will fit the double sprocket in FIGURE 49.

FIGURE 49 illustrates a side view of a roller chain and sprocket chain drive in FIGURE 41, 47, 50, 52, and 53.

FIGURE 50 illustrates a roller chain triangular-link 370 with an axle inserted at a distance that will fit the double sprocket 301 and 301 spaced along their axis of rotation in FIGURE 53. FIGURE 51 illustrates a roller chain triangular-link with an axle inserted at a distance that will fit the double sprocket in FIGURE 49, which provides more expansion and compression cycles on a chain length with less time expanded and more time compressed. FIGURE 52 illustrates a roller chain triangular-link 370 with an axle 331 inserted at a distance that will fit the double sprocket 301 and 302 in FIGURES 49 and 54. FIGURE 53 illustrates a perspective view of FIGURE 52 to observe the double sprocket assembly 301 and 302 and triangular-link connection 370, which provides a double sided triangular-link for bending the chain in reverse to increase the compression and expansion angles for a given geometry and sizes. The expansion of rods in FIGURE 43 are limited to 1.41 -centimeters with current triangle distance, but by curving the chain roller pathway 330 around the pair of sprockets 301 and 302 the expansion distance is increased to 2.82- centimeters.

FIGURE 54 illustrates a perspective view of a roller chain and sprocket chain drive LINK 370 and 376 as a way of transmitting mechanical power from one place to another, which a triangular modification triangular-chain link 370 provides an additional motion translational geometry to reciprocate rods, a motion translational function, and a modification of the second link 376, 335, and 336 in a triangular-chain link 376 from the "side" of the roller chain along the X axis, which provides a rotation function viewed in FIGURES 55 and 56 when transitioning from straight chain pathways on 330 to 330 curving around the sprockets and back to straight pathways on 330. A pair of triangular-chain links 370 are on the straight path of chain link, then as the second pair of triangular-chain links 370 curve around the sprocket 301, arm 333 rotates blade 341 at a 26.1-degree angle on top of curved sprocket 301 relative to blade 341 position on straight chain link pathway 330 in FUGURES 55 and 56. FIGURES 57 illustrates a piezoelectric high displacement cylinder actuator and 58 illustrates an end and side view of an all-ceramic bending circular disk actuator 250 with high displacement too by fastener central hole 251, location viewed from top and side. This high displacement cylinder or disk can be installed between connecting rods 321 and 322 in FIGURES 43, 44, 45, and 46 to generate power when forces bend and then expand the piezoelectric materials, or electricity applied to the piezoelectric material can expand and contract rods 321 and 322 as they curve around sprocket 301 or 302 as illustrated in position 313, 31 1, 314, and then 312 providing forces for torque in the tip-path of sprockets 301 and 302 in the most efficient manner possible, at the greatest distance from the center of rotation. Each period of expansion and contraction of piezoelectric elements has almost 180-degrees of open position then -almost the same time period in the closed compressed position along the straight length of 25-centimeters. In FUGURES 41, 47, and 49 the 180-degrees of sprocket contact with the chain links and the straight chain drive links are each 25-centimeters, equal number of chain-links at equal distances, which in these FIGURES is five sets of triangular-chain links 370. One link 370 expands about 180-degrees around sprockets 301 and 302 for 25-centimeters, and then compresses twice along the 25-centimeters of straight chain drive illustrated. No other prior art provides such a long-time period to heat, cool, clean, or stabilize components after a moment of energy exchange during relative motion of a reciprocating pair of connecting rods in motion. No other prior art has placed the energy moment (thrust, force, compression, or expansion) onto the tip-path, furthest distance from the CenterPoint of rotation, which provides the greatest torque when each of the twenty triangular-chain links 303 reciprocate twice around sprockets 301 and 302 with fixed distances during compression and expansion about 23 -centimeters each transition period.

FIGURE 15 illustrates a helix 100. FIGURE 16 applies the helix in FIGURE 15. The right-handed helix (cos t, sin t, t) from Θ = 0 to 4π with arrowheads 103 showing direction of increasing t . The pitch of a helix is the height of one complete helix turn, measured parallel to the axis 101 of the helix 100. A curve 1 1 1 is called a general helix or cylindrical helix 1 1 1, if its tangent makes a constant angle with a fixed line 101 in space. A curve is a general helix, if the ratio of curvature to torsion is constant.

FIGURE 16 illustrates a helix 100. The right-handed helix (cos t, sin t, t) from Θ = 0 to 4π with arrowheads 103 showing direction of increasing t .

The pitch of a helix is the height of one complete helix turn, measured parallel to the axis 101 of the helix 100. A double helix consists of two (typically congruent) helices with the same axis, differing by a translation along the axis.

A conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis. An example is the Corkscrew roller coaster at Cedar Point amusement park.

A circular helix, (i.e. one with constant radius) has constant band curvature and constant torsion.

A curve 111 is called a general helix or cylindrical helix 111 if its tangent makes a constant angle with a fixed line 101 in space. A curve is a general helix if and only if the ratio of curvature to torsion is constant.

A curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space. It can be constructed by applying a transformation to the moving frame of a general helix.

Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as tendril perversions.

Mathematical description

A helix composed of sinusoidal x a dy components

In mathematics, a helix is a curve in 3-dimensional space. The

following parametrisation in Cartesian coordinates defines a particular helix, Perhaps the simplest equations for one is

x( cos(t),

yit) sin(t),

z(t) t.

As the parameter t increases, the point {x{i)y{i),z{i)) traces a right-handed helix of pitch 2π (or slope 1) and radius 1 about the z-axis, in a right-handed coordinate system.

In cylindrical coordinates (r, Θ, h), the same helix is parametrised by: r(t) = l,

0(t) = t,

(t) = t.

A circular helix of radius a and slope bla (or pitch 2 b) is described by the following parametrisation: x(t) = a\cos(t),

y{t) = a\sin(t),

z(t) = bt.

Another way of mathematically constructing a helix is to plot the complex -valued function e ^xl as a function of the real number x (see Euler's formula). The value of x and the real and imaginary parts of the function value give this plot three real dimensions.

Except for rotations, translations, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, the simplest being to negate any one of the x, y or z components.

Arc length, curvature and torsion

Examples

FIGURE 16 illustrates a helix assembly 200 with a helical circuit fastener line 1 12 to electrically connect the two edges 1 12 and 1 13 of surface 99 wrapped around a three-dimensional cylinder 1 10 with top end 1 15 and bottom end 1 14. Surface 99 has a clearance distance 1 16

(contacting surfaces too) between the backing and cylinder 1 10 rounded surfaces illustrated as circular ends 1 14 and 1 15.

Applying a flexible video display 99 in the configuration of surface 99 provides complete flexible-display 99 covering cylinder 1 10, which provides the helix 100, central axis 101, helix 1 1 1, arrows 103, and starting point 102 in FIG 15 specification and geometry, where helix 1 1 1 and its arrows are also illustrated in FIG 16 as a centerline in flexible-display 99 with outer edges 1 12 and bottom edge 1 13. Helix 1 1 1, in FIG 16, is wrapped three full turns around axis 101 : starting at 0-degree angle 1 1 1, 102, 1, then first 360-degrees of rotation, second 720-degrees of rotation, and third 1580-degrees of rotation to cover cylinder 1 10 with a flexible display. Circuit line 1 12 is a circuit that senses the location of flexible-display 99, providing data to control what is displayed: one image of a "t" in FIGURE 19 can be displayed across the top edges 1 12 and bottom edges of flexible display 99, or can provided separated display data of any image. Human anatomy is not uniform like the cylinder 1 10 and human muscle compression and contraction movement will compress and stretch the flexible display requiring motion joints (fasteners that provide flexible- display movement relative to edges 1 12 and 1 13 and measure flexible-display 99 motion) that provide data to the flexible-display circuit, so measurements of the display location relative to edges 112 and 113 is required to provide one image across line 112 over the full length of the helix flexible-display 99.

FIGURES 17 and 18 illustrates two options to fasten the edges 93 of surface curved snap fastener 92 edges 93 together around different shapes, like cylinders, cones, circles, and flat surfaces. Further, this invention teaches that the fasteners of edges 93 and 94 of surface 92 can be any shape that will securely fasten the edges together. In addition, this invention teaches that the fastener elements provide an electric circuit when fastened together. Examples given: A cylinder 91 can be pressed into a flexible fastener 92.

This invention teaches bendable, rollable, and curving video display screens

"flexible-displays" have been disclosed by companies like Sony, Samsung, Sharp, and LG, but adapting flexible-displays to wrap around humans, robots, and aerospace drone machinery in motion is not provided. Helix 99 in FIG 17 is a flexible-display 99.

In FIGURES 17 and 18 potentiometers are illustrated, which is just a resistor with a sliding contact that can act as a voltage divider or a variable resistor. A resistor's 90a resistance usually depends directly on its length 96 in FIG 17 or diameter of 120 in FIG 18. Longer the length 96 intercepted within the potentiometer 90a, larger the resistance. A sliding contact called Wiper 92 slides over the resistor rod 91 and varies its resistance in a circuit according to the positioning distance 96 at which it touches the resistor 91. Any sensor can be applied to collect location data for software programs and circuits.

In FIG 18 two identical flexible-displays 124 and 125 (sections of helix flexible- display 99 in FIG 16) are assembled around a rotating circular potentiometer with a common center 120 and two points 121 and 122 to vary the resistance of the potentiometer relative to the motion around point 120. Both flexible-displays 124 and 125 in view 120a are at 90-degrees relative to points 121 and 122 and the full display is rotated 90-degrees relative to each other with the letters AZ displayed only on flexible-display 125. Flexible-display 124 is on top of flexible display 125. Both flexible-displays 124 and 125 in view 120b are at 45-degrees angle relative to points 121 and 122 and the full display is rotated 45-degrees relative to each other with the letter A of AZ partially displayed on 124 completing the letter AZ partially covered on flexible display 125. Flexible- display 124 is on top of flexible display 125. Both flexible-displays 124 and 125 in view 120c are at 0-degrees angle relative to points 121 and 122 (could be lightbulbs, micro LED, or any relative location sensors) meeting at the same point and the full display is rotated 0-degrees relative to each other with the letter A of AZ fully displayed on flexible-display 124and 125 positioning overlapping area 126, completing the letter AZ partially covered on flexible display 125. Flexible-display 124 is on top of flexible display 125 with the full A of AZ displayed on 124, 125 and their overlapped area 126. A volume knob works on the principle of a potentiometer only, but this invention teaches a potentiometer is only one method of aligning an image in the same location as the flexible-display moves relative to a helix or segments display 124 and 125 of a helix flexible-display 99. In FIG 17 rod 91 can be attached to the top edge 112 of flexible-display helix 99 or sleeve 92 can be attached to the bottom edge 112 of flexible-display helix 99, to provide the same complete video image by converting all relative motion of edges 112 and 113 into a single image across dividing lines, overlapped flexible-display material, and other motion sensors. FIG 17 is a sliding potentiometer and FIG 18 is a rotating potentiometer, both providing the location of a flexible-display, so images can remain complete relative to motion of the flexible displays or adjoining edges. Any sensors configure under the full magnetic spectrum or GPS satellite systems can be applied or linked to smartphones. Pinpoints of light can document location between related materials, like a

potentiometer, and a touch sensitive flexible display can provide the data of "bumps" around the display that it can display. A programmed set of bumps (protrusion points) can be laid out under the display to locate it, or recordable 3d elements can be added to the flexible-display providing data about where the display is relative the overlapping, jointing, motion, and any other changes that keep the flexible-display image defined by the programmer/user.

Other fasteners below that can be circuits measuring the location of flexible displays relative to movement of the flexible display. Nano graphene circuits and other nano-circuits can measure motion at a nano-scale, so motion of flexible-displays can be measured with extreme accuracy. Management of the energy required to display a flexible-display, like helix 99, can be managed with sensors in the observer's possession.

A snap fastener (also called press stud, popper, snap or tich) is a pair of interlocking discs, made out of a metal or plastic, commonly used in place of buttons to fasten clothing and for similar purposes. A circular lip under one-disc fits into a groove on the top of the other, holding them fast until a certain amount of force is applied. Different types of snaps can be attached to fabric or leather by riveting with a punch and die set specific to the type of rivet snaps used (striking the punch with a hammer to splay the tail), sewing, or plying with special snap pliers.

A retaining ring is a fastener that holds components or assemblies onto a shaft or in a housing/bore when installed in a groove. Once installed, the exposed portion acts as a shoulder which retains the specific component or assembly. Circlips are a type of retaining ring.

Self-locking retaining rings may be installed in applications where there is no groove. Retaining rings are typically made from carbon steel, stainless steel or beryllium copper and may feature a variety of finishes for corrosion protection depending on the type of environment in which they are used

A Terry clip (or Terry's clip) is a spring metal clip used to hold a cylindrical object. Snap fasteners (also named poppers and press studs) and the hook and loop fastener are a few less common zipper alternatives.

A zipper, zip, fly, or zip fastener, formerly known as a clasp locker, is a commonly used device for binding the edges of an opening of fabric or other flexible material, like on a garment or a bag. It is used in clothing (e.g., jackets and jeans), luggage and other bags, sporting goods, camping gear (e.g. tents and sleeping bags), and other items. Zippers come in all different sizes, shapes, and colors. Whitcomb L. Judson was an American inventor from Chicago who invented and constructed a workable zipper. The method, still in use today, is based on interlocking teeth. Initially, it was called the "hookless fastener" and was later redesigned to become more reliable.

The bulk of a zipper/zip consists of two rows of protruding teeth, which may be made to interdigitate, linking the rows, carrying from tens to hundreds of specially shaped metal or plastic teeth. These teeth can be either individual or shaped from a continuous coil and are also referred to as elements. The slider, operated by hand, moves along the rows of teeth. Inside the slider is a Y-shaped channel that meshes together or separates the opposing rows of teeth, depending on the direction of the slider's movement. The word Zipper is onomatopoetic, because it was named for the sound the device makes when used, a high-pitched zip.

FIGURE 19 illustrates length of the circumference of flexible-display dl rolled open flat. Length d2 is equal to one half of the circumference 115. cl . Axial length d3 of cylinder 110 unrolled to visualize the video image of a "t", circumference cl of the cylinder 110, vertical length il of the "t" 136, horizontal circle-arc length i2 of horizontal element of a "t" on a flexible display 200 in FIGURE 15. Width i3 and i4 (can vary relative to the observation distance from the center axis 101) is the width of vertical il and horizontal i2 length of a "t" on flexible display 200 in FIGURE 15. Width i4 of the horizontal length of "t" rolled out flat on plane 110, length i6 of the actual "t", length i8 on axis 101 of flexible display 200 in FIGURE 15 of the actual "t" as it appears to an observer camera point 130, C, which could be from any angle 131 providing software programed to control the image pixels of the "t" appearing as a flat "t" 134 no mater which angle 13 lthat "t" was observed from, so the length of the horizontal elementl37 of the "t" wrapped around a cylinder in FIGURE 15 is longer the length d2 of half of the cylinder's circumference cl to make it appear as a "t" dimension length 132. i6 on the flat plane 110 . This invention teaches that overlapped flexible video displays can be measure by optical scanning for a 3d image, measure by sensors, or fasteners locations can be measured to determine the programed image the display pixels illuminated for an observer. Bluetooth, WiFi, wired, or other communication inks can provide any shape with a programmed image and observers can visualize in any configuration. A touch sensitive flexible video display can have the sensors of the touch display defined to coordinates and three- dimensional geometry for the programmed display of the modified image to display the real-life image to an observer. In touch sensitive displays a known protrusion that a touch sensitive display can record, provides the programable data of where the element is located relative to the rest of the display that defines the software programed image. Flexible displays can be loose when worn by people or motion translational devices, so motion sensors, triangulation measurements, and any measuring device that measures the required points of the three-dimensional geometry of the application the flexible display covers, and then senses the location of the observer or camera to display the image desired. Several images can be displayed and divided into observation windows for several observers.

FIGURE 19 illustrates a perpendicular view of letter "t" 136 optical center C (also known as camera projection center) 130 of the image 134 on plane 110, which is the cylinder 110 in FIG 15 with length dl equal to the circumference of cl on cylinder edge 115 in FIG 15 unrolled providing one half-length d2 of the circumference cl, of edge 115 marked by arrows on the plane of 112 on 110. The distance of the image plane 112 on 110 from C, 130 is the focal length f from the arrow on top of 130 to the intersection of the "t" center lines 132 and 134 on the plane 110's surface.

The line from the camera center C, 130 perpendicular to the image of plane 110 is called the principal axis 138 or optical axis 138 of the camera C, 130.

The plane 140 parallel to the image plane 110 containing the optical center C, 130 is called the principal plane 140 or focal plane 140 of the camera C, 130. The relationship between the 3-D coordinates of scene points 136 and 137, and the coordinates 134 and 133 of its projection onto the image plane 112 on plane 110 is described by the central or perspective projection of C, 130.

FIGURE 19 illustrates Pin-hole camera geometry of C, 130. The left figure illustrates the projection of the point M on the image plane 110 by drawing the line 138 through the camera center C and the point 130 to be projected. The right figure illustrates the same situation in the XZ plane, showing the similar triangles used to compute the position of the projected point m in the image plane 140 from image plane 112 with length d2 on fully rolled out plane 110 with length dl . 3. The camera projection matrix

The pin-hole camera C is described by the following:

If the world and image points are represented by homogeneous vectors, then perspective projection can be expressed in terms of matrix multiplication as

fxfyz=f0000f000010xyzl (Equation 3)

The matrix describing the mapping is called the camera projection matrix P.

Equation (3) can be written simply as:

zm=PM (Equation 4) where M=x,y,z,lT are the homogeneous coordinates of the 3-D point and m=fx/z,fy/z, IT are the homogeneous coordinates of the image point.

The projection matrix P in Eq. (3) represents the simplest possible case, as it only contains information about the focal distance f.

3.1.1. General camera: bottom up approach

The above formulation assumes a special choice of world coordinate system and image coordinate system. It can be generalized by introducing suitable changes of the coordinates systems.

Changing coordinates in space is equivalent to multiplying the matrix P to the right by a 4x4 matrix:

G=Rt01 (Equation 5)

G is composed by a rotation matrix R and a translation vector t. It describes the position and orientation of the camera with respect to an external (world) coordinate system. It depends on six parameters, called extrinsic parameters.

The rows of R are unit vectors that, together with the optical center, define the camera reference frame, expressed in world coordinates.

Changing coordinates in the image plane is equivalent to multiplying the matrix P to the left by a 3 x3 matrix:

K=f/sxf/sxcot9ox0f/syoy001 (Equation 6)

K is the camera calibration matrix; it encodes the transformation in the image plane from the so-called normalized camera coordinates to pixel coordinates.

It depends on the so-called intrinsic parameters:

focal distance f (in mm),

principal point (or image center) coordinates ox,oy (in pixel),

width (sx) and height (sy) of the pixel footprint on the camera photosensor (in mm), angle Θ between the axes (usually π/2).

The ratio sy/sx is the aspect ratio (usually close to 1).

Thus, the camera matrix, in general, is the product of three matrices:

P=K[I|0]G=K[R|t] (Equation 7)

In general, the projection equation writes:

(Equation 8)

where ζ is the distance of M from the focal plane of the camera (this will be shown after), and m=u,v,lT.

Note that, except for a very special choice of the world reference frame, this "depth" does not coincide with the third coordinate of M.

3.1.2. Projection center

If P describes a camera, also λΡ for any 0≠ GR describes the same camera, since these give the same image point for each scene point.

In this case we can also write:

m-PM (Equation 9)

where ^ means "equal up to a scale factor."

In general, the camera projection matrix is a 3 x4 full -rank matrix and, being homogeneous, it has 11 degrees of freedom.

Using QR factorization, it can be shown that any 3 x4 full rank matrix P can be factorized as:

P= K[R|t], (Equation 10)

(λ is recovered from K3,3=l).

— 3.2. Camera Anatomy

—3.2.1 Projection Center

The camera projection center C is the only point for which the projection is not defined, i.e. :

PC=PC~1=0 (Equation 11)

where C~ is a 3-D vector containing the Cartesian (non-homogeneous) coordinates of the optical center.

After solving for C~ we obtain:

C~=-P1 :3-1P4 (Equation 12)

where the matrix P is represented by the block form: P=[P1 :3|P4] (the subscript denotes a range of columns). 3.2.2. Depth of a point

We observe that:

Cm=PM=PM-PC=PM-C=Pl :3M~-C~. (Equation 13)

In particular, plugging Eq. (10), the third component of this equation is C= r3TM~-C~

where r3T is the third row of the rotation matrix R, which correspond to the vector of the principal axis.

If λ=1, ζ is the projection of the vector M~-C~ onto the principal axis, i.e., the depth of M.

3.2.3. Optical ray

The projection can be geometrically modelled by a ray through the optical center and the point in space that is being projected onto the image plane (see Fig. 2).

The optical ray of an image point m is the locus of points in space that projects onto m.

It can be described as a parametric line passing through the camera projection center

C and a special point (at infinity) that projects onto m:

Μ=-Ρ1 :3-1Ρ41+ζΡ1 :3-1ιηΟ,ζε (Equation 14)

If λ=1 the parameter ζ in Eq. (14) represent the depth of the point M.

Knowing the intrinsic parameters is equivalent to being able to trace the optical ray of any image point (with P=[K|0]).

3.2.4. Image of the absolute conic

We will prove now that the image of the absolute conic depends on the intrinsic parameters only (it is unaffected by camera position and orientation).

The points in the plane at infinity have the form M=M~T,0T, hence

m-K[R|t](M~T,0)T=KRM~. (Equation 15)

The image of points on the plane at infinity does not depend on camera position (it is unaffected by camera translation). The absolute conic (which is in the plane at infinity) has equation M~TM~=0, therefore its projection has equation:

mTK-TK-lm=0. (Equation 16)

The conic co=KKT-l is the image of the absolute conic.

Its knowledge allows one to measure metrical properties, such as the angle between two rays.

Figure 3. Angle Θ between two rays. Indeed, let us consider a camera P=[K|0]. Then the angle Θ between the ray's trough

Ml and Ml is:

cos0=M~lTM~2M~lM~2=mlTrom2mlTromlm2Trom2 (Equation 17) (it follows easily from m=lzKM~.)

3.3. Camera calibration (or resection)

A number of point correspondences mi→Mi is given, and we are required to find a camera matrix P such that

mi— PMifor all i. (Equation 18)

The equation can be rewritten in terms of the cross product as

miPMi=0. (Equation 19)

This form will enable a simple a simple linear solution for P to be derived. Using the properties of the Kronecker product (®) and the vec operator (Magnus and Neudecker, 1999), we derive:

mixPMi=0<^[mi]xPMi=0<^vec([mi]xPMi)=0<^<^(MiT®[ mi]x)vecP=0

These are three equations in the 12 unknown of vecP. However, only two of them are linearly independent: Indeed, the rank of MiT®mix is two because it is the Kronecker product of a rank-1 matrix by a rank-2 matrix. Therefore, from a set of n point correspondences one obtains a 2xnx 12 coefficient matrix A by stacking up two equations for each correspondence. In general A will have rank 11 (provided that the points are not all coplanar) and the solution is the 1- dimensional right null-space of A. The projection matrix P is computed by solving the resulting linear system of equations, for n>6.

If the data are not exact (noise is usually present) the rank of A will be 12 and a least- squares solution for vecP is computed as the singular vector corresponding to the smallest singular value of A. This is called the Direct Linear Transform (DLT) algorithm according to (Hartley and Zisserman,2003), which can be programmed to provide any image on any flexible display.

This invention teaches that any image on the flexible-display around any shape can be software programmed to provide the appearance of a flat "plane" image by programing the flexible display to illuminate pixels within the "projected" image from a camera observation point plane to all the curves and overlapping flexible display edges, and joining edges, including during motion. A sensor could be worn on the bar between the eyeglass lenses, a neckless, or other sensors to replicate a camera observation point onto a flexible display of any shape by applying the Direct Linear Transform (DLT) algorithm. 3D printing, printed documents or colored clothing, painting, posters, billboard imaging, lighting, LED sourced images (colored LED elements), retroreflecting materials, and architecture designs can all apply this imaging technology on any shape to provide a flat plane image appearance to the observer.

A charged particle in a uniform magnetic field following a helical path FIGURE 21 illustrates how to increase the climb gradient of a fuselage assembly 150 travelling in direction of centerline arrow 161 through atmospheric airstream relative resistant path 149, which provides double helix aerodynamic air paths consisting of two (typically congruent) helices 147 and 148 with the same axis 161, differing by translation pathways 147 and 148 along the centerline axis 161. Helix 147 has the same mirrored dimension as helix 148, which this invention teaches is a mirrored helix around intersection points 152 and 158 on an aircraft fuselage 160 with center-axis 161 providing more lift 163 for an increase climb gradient (during long term flight altitude too), after exhausting through each bottom intersecting compression points 153, 155, and 157. An airduct can be constructed around the fuselage 160 surface in the path of each helix; helix 147 is in a clockwise rotation relative to moving in the direction of centerline arrow 161 from back- tail end 159 to front-nose 151, and helix 148 is in a counterclockwise rotation around center-axis 161 moving from the back-tail end 159 on fuselage 160 up to the front nose end 151. A mirrored double helix of equal helices intersects at 180-degree intervals starting at point 152, followed by 153, 154, 155, 156, 157, 158, and ending up at 159, which this invention teaches is an optimized pathway for air currents during flight of an aircraft providing more climb gradient lift forces 163 by directing air on top of the aircraft from helices intersections 152, 154, 156, and 158 to the corresponding compression zones underneath the aircraft, where the helices intersect 153, 155, and 157. Helices are designed into air ducts to capture air to be directed under the aircraft for lift 163 arrows following each bottom intersecting air compression points 153, 155, and 157. Valving or reshaping of the ducts can be provided to modify the location force vectors 162 and 164 of the air flow around the whole fuselage 160. Directing air from the top of the aircraft to the bottom of the aircraft with helices of the least air turbulence shape (length, duct diameter, coating surfaces, and other standard best practices) reduces the density of air above the aircraft by accelerating air from above to below the aircraft sooner than prior art, increases density during compression of air under the center of the fuselage within the intersecting points of the helices geometry 153, 155, and 157. Dividing air in atmospheric resistance path 149 by capturing air 149 from the top of the aircraft at intersecting air-dividing points 152, 154, 156, and 158, air is moved to the bottom air compressing intersections 153, 155, and 157, under the aircraft, providing more lift 163 illustrated under each bottom intersection 153, 155, and 157. Ducting the air down under the aircraft in the most optimized pathways, helices 147 and 148, provides a much higher acceleration of air, reducing the air density above the aircraft, reducing the resistance force of arrow 149 representing atmospheric air pressure, to more lift 163 from under the aircraft surfaces pressured by compressed air 163. The outer surface can increase in diameter segmented into a stepped fuselage surface along each helix line between helix intersecting points to multiply the amount of air the surfaces of the fuselage captures by stepping the fuselage surface to greater diameters while moving from front nose 151 to the tail-end 159. This invention teaches several concentric spaced fuselage's surfaces relative to centerline 161 capture more air for increased lift 163.

FIGURE 22 illustrates a fuselage 160 of an aircraft in FIGURE 21 with airduct helix pathways that have been isolated into airfoils, 162, 164, and 166 around the fuselage centerline 161, each airfoil is scaled concentrically from the central axis of FIG 21 to optimize lift of any aircraft, which could include any layered out shape, like the tubes. These construction lines of scaled tubes off the surface of fuselage 160 are in line with the helixes 147 and 148 segments with arrows pointing down, example given, point 152 to point 153 on helix 147 and 148 (hidden lines), point 154 to point 155 along helix 147 and 148 (hidden lines), and point 156 to point 157 along helix 147 and 148 (hidden lines).

FIGURE 23 illustrates a side perspective view of FIGURE 22 helix fuselage scaled around one-point of origin 158, which forms a more aerodynamic fuselage than concentric scaling radially rather than radially and along the axis 166 of the fuselage 160. Helix lines from point 156 to point 157 along helix 147 and 148 (hidden lines) are copied back to the scaling origin point 158, providing tube 166, after scaling concentrically, not including scaling along the center line 161. Every concentric tube in the end view of the tubes originates from scaling 166 from point 158 as the origin, 2 ^nd point was point 157, scaling to third point 152 to construct tube 166d, scaling another tube 166c from as the second point 157 to 153, and then scaling another tube 166b from as the second point 157 to point 155. All the tubes were scaled from tube 166 origins that were just scaled concentrically from fuselage 160 in FIGURE 21.

FIGURE 24 illustrates a top view of FIGURE 23 helix fuselage unrolled around helix intersecting mirrored sides around points 152 and 158 in FIGURE 21. Wings 166 on top, then 166b, 166c, and 166d are unrolled from the fuselage illustrated in FIGURE 23. FIGURE 25 illustrates an end view of FIGURE 24 helix fuselage unrolled around helix intersecting mirrored sides of FIGURES 23 and 27. FIGURE 26 illustrates a perspective view of the fuselage in

FIGURES 23, 24, 25, and 27 unrolled flat, providing a view of helices 150 and data centerline at rear of fuselage when rolled up into a tube in FIGURE 23 and 27. FIGURE 27 illustrates a perspective view of FIGURES 23, 44, 25, and 26 providing a view of rear wings 166a and tail 166b of aircraft unrolled from top helices airfoil 166 three dimensionally along data centerline from points 158 to points 156 at the rear on top of fuselage. Three to four extra layers of tube 166 is provided to form tail 166c and 166d, and then a second set of three layers of tube 166 form wing 166a and 166b. these unrolled wings have geometry that stops the unrolling to the aerodynamic destination tail or wing, including wing modifying infrastructure. Ribs can unfold with flexible material under the wings that tightens when the wing aerodynamic location is reached. Tail 166c and 166d can connect by the same method as the wings, but a fastening structure is recommended. The wing and tail can have many configurations relative to speed, propulsion source, and stacking aircraft for transportation when being transported.

FIGURE 31 fuselage 160 has airducts 171 moving air from the top of fuselage 160 to the bottom through the center of the fuselage. These ducts could be exhaust from energy supplies or added to the fuselage in FIGURE 21 to optimize the control and redirecting of airflow that is leftover from helix 147 and 148. FUGRE 31 diagonal ducts could also be functioning alone as a method to redirect air below a fuselage with other structures, like wings, engines, electric fans or propellers.

Water vapor is a lighter gas than other gaseous components of air at the same temperature, so humid air will tend to rise by natural convection. This is a mechanism behind thunderstorms and other weather phenomena. Relative humidity is often mentioned in weather forecasts and reports, as it is an indicator of the likelihood of precipitation, dew, or fog. In hot summer weather, it also increases the apparent temperature to humans (and other animals) by hindering the evaporation of perspiration from the skin as the relative humidity rises. This effect is calculated as the heat index or humidex.

This invention teaches that regulating the redirected cycling air between the top and bottom of the fuselage can add to the propulsion and fueling economy by managing moisture, temperature, and heat source in air flowing within the helices air ducts constructed to compress air below the aircraft.

Air with higher humidity is less dense, providing lower lift and lower propeller or turbine efficiency (this is somewhat offset by decreased drag), so this invention teaches how to regulate the less dense air to the top of the aircraft and the dryer air with greater density to the bottom of the aircraft. In prior art Pilots must take humidity into account when calculating takeoff distances, because high humidity will require longer runways and will decrease the climb gradient. Density altitude is the altitude relative to the standard atmosphere conditions

(International Standard Atmosphere) at which the air density would be equal to the indicated air density at the place of observation, or, in other words, the height when measured in terms of the density of the air rather than the distance from the ground. "Density Altitude" is the pressure altitude adjusted for non-standard temperature.

An increase in temperature, and, to a much lesser degree, humidity, will cause an increase in density altitude. Thus, in hot and humid conditions, the density altitude at a measured location, may be significantly higher than the true altitude.

A useful rule of thumb is that the maximum absolute humidity doubles for every 20 °F or 10 °C increase in temperature. Thus, the relative humidity will drop by a factor of 2 for each 20 °F or 10 °C increase in temperature, assuming conservation of absolute moisture. For example, in the range of normal temperatures, air at 68 °F or 20 °C and 50% relative humidity will become saturated if cooled to 50 °F or 10 °C, its dew point, and 41 °F or 5 °C air at 80% relative humidity warmed to 68 °F or 20 °C will have a relative humidity of only 29% and feel dry. By comparison, thermal comfort standard ASHRAE 55 requires systems designed to control humidity to maintain a dew point of 16.8 °C (62.2 °F) though no lower humidity limit is established.

A device used to measure humidity is called a hygrometer; one used to regulate it is called a humidistat, or sometimes hygrostat. (These are analogous to a thermometer and thermostat for temperature, respectively.)

Relative humidity (RH) is the ratio of the partial pressure of water vapor to the equilibrium vapor pressure of water at a given temperature. Relative humidity depends on temperature and the pressure of the system of interest. It requires less water vapor to attain high relative humidity at low temperatures; more water vapor is required to attain high relative humidity in warm or hot air. The relative humidity of an air-water mixture is defined as the ratio of the partial pressure of water vapor in the mixture to the equilibrium vapor pressure of water over a flat surface of pure water at a given temperature:

Relative humidity is normally expressed as a percentage; a higher percentage means that the air-water mixture is more humid. Airliners operate with low internal relative humidity, often under 10%, especially on long flights. The low humidity is a consequence of drawing in the very cold air with a low absolute humidity, which is found at airliner cruising altitudes. Subsequent warming of this air lowers its relative humidity, increasing air density. Humidifiers are not employed to raise it to comfortable mid-range levels because the volume of water required to be carried on board can be a significant weight penalty. As airliners descend from colder altitudes into warmer air (perhaps even flying through clouds a few thousand feet above the ground), the ambient relative humidity can increase dramatically. Some of this moist air is usually drawn into the aircraft helices airducts and into other non-pressurized areas of the aircraft and condenses on the cold aircraft skin. Liquid water can usually be seen running along the aircraft skin, both on the inside and outside of the cabin. Drastic changes in relative humidity inside the vehicle, components must be qualified to operate in those environments. The recommended environmental qualifications for most commercial aircraft components is listed in RTCA DO-160.

Cold humid air can promote the formation of ice, which is a danger to aircraft as it affects the wing profile and increases weight. Helix airducts have a further danger of ice forming inside the airducts. Aviation weather reports (METARs) therefore include an indication of relative humidity, usually in the form of the dew point.

Pentazolate rings for fuel: A carbon-free inorganic-metal complex

[Zn(H ₂ 0) ₄ (N ₅ ) ₂ ] 4H ₂ 0 can be synthesized by the ion metathesis of [Na(H ₂ 0)(N ₅ )] 2H ₂ 0 solution with Zn(N0 ₃ ) ₂ 6H ₂ 0. The axial position is formed by two nitrogen atoms (Nl) from two pentazolate rings (cyclo-Ns ^~ ) and the equatorial plane is formed by four oxygen atoms (01) from four coordinated water molecules. Thermal control of [Zn(H ₂ 0)4(N ₅ ) ₂ ] 4H ₂ 0 water stabilizes cyclo- N ₅ ^~ , dehydration does not cause immediate decomposition of the anion. However, cyclo- N ₅ ^~ decomposed into N ₃ ^~ and N ₂ gas at temperature of 107.9 °C (onset). Based on its chemical compatibility and stability, the complex exhibits promising potential as a modern environmentally- friendly energetic material for increasing density of air below the aircraft and decreasing air density on top of the aircraft, including the fact that heating air provides more relative humidity in air, reducing air density when ducted on top of the aircraft for lift through helix 147 and 148. This invention teaches regulating the dehydration by thermal control of [Zn(H ₂ 0)4(N ₅ ) ₂ ] -4H ₂ 0; water stabilizes cyclo-Ns ^~ , so dehydration causes immediate decomposition of the anion, providing cyclo- N ₅ ^~ decomposed into N ₃ ^~ and N ₂ gas at 107.9 °C (onset), which is an energetic material denser than air with moisture and a force to propel the aircraft in any direction crafted by design. Above- mentioned piezoelectric wafer can morph the airducts aerodynamic properties and can open and close airducts. Piezoelectric wafer can also ultrasonically decompose [Zn(H ₂ 0)4(N ₅ ) ₂ ] 4H ₂ 0 into energy within any airduct, in any direction of airflow for optimizing the forces of lift and propulsion to move the aircraft through air up or down through the atmosphere. This invention teaches piezoelectric wafers configured with some magnetic fields pulsing provide an electric energy to spark [Zn(H ₂ 0)4(N ₅ ) ₂ ] 4H ₂ 0 into decomposition, releasing energy. Other standard fuels can be included as heat sources, which also can be managed by piezoelectric wafers. FIGURE 33 illustrates a hexagonal fuselage provided with helix ducting provided in FIGURE 21. Hexagonal fuselage components can update existing aircraft that are round cylindrical shapes. Hexagonal fuselage can also be a flying tube in the helix direct illustrated without a standard fuselage but propelled and controlled by the FIGURE 21 helix airducts molded (or 3D printed) into the curved Hexagons that are offset from the CenterPoint to one of the six hexagon points, piezoelectric wafers morphing airfoils, and the application of rotor blade elements on the inside diameter of the tube for Tip-Path propulsion, if desired.

FIGURE 1 through 4 rotary blades can be hollow, providing a rotating air vessel to capture, compress, then release air to optimize control through airducts shaped for capturing air, all controlled by valving with optional morphing piezoelectric wafers controlled at fractions of a second, controlled by the revolutions per minute. Hollow bladed rotary wings are provided optional small high-speed air fans (turbofan, propeller, or other air capture devices) near the blades tip-path that can capture more air than just the outside surface of a rotary blade airfoil and then release air adding to propulsion and noise reduction at the Tip-Path of propulsion. A cylindrical ring can be part of the rotary blades structure, but another turbine or propeller can be ported from inside the ring to outside the ring to manage the air wasted exiting the tip-path of a rotary wing, again capture and direct the air inside the tip-path ring, but port the air to an air path optimized shape, or through another outer propulsion device that can redirect inner air inside the inner ring diameter or from the hollow rotary blade to the outside of the ring diameter, or work independently to control Gyroscopic Precession the spinning main rotor of a helicopter acts like a gyroscope moving the blades in and out of the tip-path.

FIGURE 19 illustrates a round cylindrical fuselage, but the concept of redirecting air from the top of an aircraft around to down under the aircraft into compression zones, could be applied to any shape and aerodynamic element in air, water, or space. The helix can be spread over the length of the fuselage into just an 180-degree of each helix around a fuselage length, up to many complete 360-degree turns around a fuselage as illustrated in fuselage 150 of FIGURE 19. In addition, optimizing the airducts around wings attached to the fuselage can modify a perfect helix, including shaping the curves illustrated in FIGURE 21 into a cycloidal curve that moves all air down to the compression exhaust port during the same period of time during travel, because duplicated specifications applied to several cycloid curves can move objects to the same point starting at different distances along the curves. In addition, airduct intake, exhaust, and air pathways can have turbines, fans, and any propulsion installed to add to control of the aircraft. This invention teaches that aircraft surfaces can be coated with superhydrophobic surfaces that provide the Cassie-Baxter state when water droplets become very mobile and quickly slides off the aircraft surface by repelling water. The mobility of the water droplets on the aircraft surfaces has another effect by water zooming around the surface of the aircraft rather than sticking, the droplets of water collect small particles of dust, hoovering them up. This cleaning mechanism of these superhydrophobic surfaces is called the lotus effect, which is an option to control the humidity of high speed air more accurately through the helices pathway to direct higher or lower density air through the proper airducts.

Superhydrophobic surfaces have been synthesized and studied in labs for decades, but it is only recently that commercial versions have been produced. Now there are quite a few coming on to the market (e.g. neverwet.com), and they are impressive - when water is poured on to these surfaces it behaves like mercury and bounces off providing the ability to control humidity for lift or the direction of travel desired. A simple helix paint pattern of stripes from top to bottom of an aerodynamic surface improves lift, including adhesive shaped materials applied to a surface for more efficient aerodynamic control. Hot air balloons, dirigibles, and other drones can be improved by managing elements described in this invention.

Reverse rotation of airflow can remove water, increasing the density of air, which can be added to mange water, removing water for increasing density of air below the aircraft, while increasing density of air by adding the water to the airstream up on top of the aircraft to reduce air density. Managing water is taught in this invention by reverse rotating the water for its separation from air as it flows around the aircraft.

According to United States FAA:

The rotation of rotor blades as they turn about the mast produces rotational relative wind (tip-path plane). The term rotational refers to the method of producing relative wind.

Rotational relative wind flows opposite the physical flightpath of the airfoil, striking the blade at 90° to the leading edge and parallel to the plane of rotation; and it is constantly changing in direction during rotation. Rotational relative wind velocity is highest at blade tips, decreasing uniformly to zero at the axis of rotation (center of the mast).

Rotor Blade and Hub Definitions.

Hub— on the mast is the center point and attaching point for the root of the blade.

Tip— the farthest outboard section of the rotor blade.

Root— the inner end of the blade and is the point that attaches to the hub.

Twist— the change in blade incidence from the root to the outer blade The angular position of the main rotor blades is measured from the helicopter's longitudinal axis, which is usually the nose position and the blade. The radial position of a segment of the blade is the distance from the hub as a fraction of the total distance

Coriolis Effect (Law of Conservation of Angular Momentum) The Coriolis Effect is also referred to as the law of conservation of angular momentum. It states that the value of angular momentum of a rotating body does not change unless an external force is applied. In other words, a rotating body continues to rotate with the same rotational velocity until some external force is applied to change the speed of rotation. Angular momentum is the moment of inertia (mass times distance from the center of rotation squared) multiplied by the speed of rotation.

Changes in angular velocity, known as angular acceleration and deceleration, take place as the mass of a rotating body is moved closer to or farther away from the axis of rotation. The speed of the rotating mass varies proportionately with the square of the radius.

An excellent example of this principle in action is a figure skater performing a spin on ice skates. The skater begins rotation on one foot, with the other leg and both arms extended. The rotation of the skater's body is relatively slow. When a skater draws both arms and one leg inward, the moment of inertia (mass times radius squared) becomes much smaller and the body is rotating almost faster than the eye can follow. Because the angular momentum must, by law of nature, remain the same (no external force applied), the angular velocity must increase.

The rotor blade rotating about the rotor hub possesses angular momentum. As the rotor begins to cone due to G-loading maneuvers, the diameter or the rotor disk shrinks. Due to conservation of angular momentum, the blades continue to travel the same speed even though the blade tips have a shorter distance to travel due to reduced disk diameter. The action results in an increase in rotor rpm which causes a slight increase in lift. Most pilots arrest this increase of rpm with an increase in collective pitch. This increase in blade rpm lift is somewhat negated by the slightly smaller disk area as the blades cone upward.

Gyroscopic Precession The spinning main rotor of a helicopter acts like a gyroscope.

Water propulsion can apply this invention's airfoil technologies underwater in the marine industry, because air and water are fluid. This invention teaches how to improve propulsion in fluid; water or air.

In prior art of parafoils, on October 1, 1964, Domina C. Jalbert applied for a patent for his "Multi-Cell Wing" named "Parafoil" (also known as a "ram-air" wing), which was a new parachute design. His ideas were registered as a U.S. patent on November 15, 1966. However, in 1964 Lowell Farrand had already flown a motorized version called "The Irish Flyer" by Nicolaides. Farrand was the first person to put an engine on a ram-air inflated parachute wing, starting the evolution of the powered parachute with the Irish Flyer. This wing evolved into today's modern powered parachute canopies, which include rectangular, elliptical, semi -elliptical, and hybrid wings.

The United States (U.S.) government had a number of test programs that used the square parachute as a means to glide spacecraft back to earth or glide payloads dropped out of airplanes to a specific location.

Two-place powered parachutes have years of testing, development, and evolution. Training exemptions to Title 14 of the Code of Federal Regulations (14 CFR) part 103, Ultralight Vehicles, permitted individuals to give instruction in two-place ultralight vehicles, instead of being restricted to vehicles intended for single occupants. The Federal Aviation Administration (FAA) allowed ultralight vehicle pilots to train in two-place ultralights until January 31, 2008. After this date, the ultralight vehicle training exemption expires, and only N-numbered aircraft may be used in two-place PPC instruction and flight.

Powered Parachute Terms

Different terms have been used throughout the powered parachute community. The terms standardized throughout this book are as follows:

• Powered Parachute - The complete aircraft.

• Cart - The engine and seats, attached by a structure to wheels; may also be referred to as the fuselage, cockpit, chaise, or airframe.

· Wing - Typically a ram-air inflated and pressurized wing including lines that attach to the cart. The wing is not in position to fly until the aircraft is in motion; when not inflated, referred to as a parachute or chute.

Introduction to the Powered Parachute

The powered parachute is a category of aircraft that flies in a manner unique among light-sport aircraft. Three significant differences separate the PPC from other types of light sport aircraft (LSA):

1. The wing must be inflated and pressurized by ram air prior to each takeoff.

2. The aircraft uses a pendulum configuration, where the cart hangs about 20 feet below the wing, connected via flexible suspension lines.

3. The wing is at a relatively fixed angle with the suspension lines and flies at a relatively constant speed. Other aircraft categories allow pilots to change the speed of the aircraft, but the powered parachute airspeed remains within a very small range. A powered parachute can be a single place ultralight flying vehicle, a single place light-sport aircraft, or a multi-place light-sport aircraft. The common acronyms for this

vehicle/aircraft are PPC (powered parachute), PPCL (powered parachute land) or PPCS (powered parachute sea).

A light-sport aircraft PPC used for sport and private flying must be registered with an

FAA N-number, have an airworthiness certificate, a pilot's operating handbook (POH), and/or limitations with a weight and balance document aboard. The aircraft must be maintained properly by the aircraft owner or other qualified personnel and have the aircraft logbooks available for inspection. Dual controls are required in the aircraft for training.

FIGURE 34 illustrates a perspective view of the fuselage in FIGURE 30 with rotating wings attached to the nose of the fuselage. FIGURE 35 illustrates a perspective view of the aircraft in FIGURE 34 with wings rotated into a rotary wing helicopter type lift or forward thrust configuration, which provides a tip-path thrust system with forces and aerodynamics around the same circular path. FIGURE 36 illustrates a perspective view of the aircraft in FIGURES 34 and 35 with additional rotary wings to add to the capacity of the aircraft, including a fuselage that rotates with each independent rotary wing assemble, providing additional X wing four rotary wings on back to stand the aircraft on the backs of the wings during lading or takeoff. FIGURE 37 illustrates a perspective view of the aircraft in FIGURE 36 with the front two sets of rotary wings position to rotate in opposite directions. FIGURE 38 illustrates a side view of a wing aerodynamic profile with a hinge at the front of the wing to pivot the two halves of the wings around the hinge. FIGURE 39 illustrates a perspective view of the aircraft wing in FIGURE 37 pivoted around hinge 180-degree angle from a 90-degree angle from the original wing positions, which provides a maximum gliding function for aircraft in FIGURES 34 and 37. FIGURE 40 illustrates a perspective view of the aircraft in FIGURE 38 providing wings that slide open rather than hinge.

FIGURE 34 illustrates perspective view of an aircraft fuselage and wing assembly

200, providing wings 218 and 219 on a fuselage 201 with tip-path propulsion rotary wing propellers on the ends of the wings 218 and 219. Wings 218 and 219 rotate in FIGURE 35 around rotating cylinder 201 to provide a helicopter type rotary wing assembly 200 to convert from horizontal flight by rotating the wings near 180-degrees apart from FIGURE 34 to rotary wings in FIGURE 35.

FIGURE 36 and 37 illustrates an aircraft fuselage and wing assembly 200, provided three rotating sections 201a, 201b, and 201c of the fuselage 200. Wings 218 and 219, including rotating wing propellers to provide rotating wing propeller in FGURE 35. In FIGURE 37, rotating wings 218 and 219, including rotating wing propellers provided rotating wing propeller around rotating fuselage hub 201c. Wings 216 and 211, including rotating wing propellers provided rotating wing propeller around rotating fuselage hub 201b. Wings 211. 212, 213 and 214, including rotating wing propellers provided rotating wing propeller around rotating fuselage hub 201a, which also can be stable cross to stand the aircraft up, pointing the fuselage vertical into a liftoff or landing position. FIGURE 36 is a stable flight positions to fly horizontal to the ground. These wings in FIGURES 34, 35, 36, and 37 can unfold into a parafoil or parachute, which may have optional controls to phase the aircraft into a glider without power or minimized power to direct the aircraft to a destination. Fuel or electric energy will be saved for applications, like product delivery or drone application. Future air taxi travel services would be much safer, if the aircraft phased from high speed into a glider when energy input was withdrawn, just like parachuting down to destination on every flight. Fuselage in FIGURES 28 and 30 has a top 160c with nose 151c and tail 159c that separates from at a fixed distance from the fuselage 160b, nose 151a and 151b, including tail 159a and 159b into a parachute or paragliding type structure. FIGURE 29 fuselage 160b has a hinge 170, also illustrated in FIGURE 30 that the fuselage rotates around with opening facing the ground expanding the aerodynamics toward a parachute or glider function. FIGURE 30 fuselage is illustrated in FIGURE 32 with nose half 151b and tail half 159b rotated around fuselage center- point 161 providing a glider or parachute function relative to landing without power, or minimum power input on the decent to a destination.

Roller chain and sprocket Chain drive is a way of transmitting mechanical power from one place to another. It is often used to convey power to the wheels of a vehicle, particularly bicycles and motorcycles. It is also used in a wide variety of machines besides vehicles.

Roller chain and sprocket Chain drive as a way of transmitting mechanical power from one place to another, which a triangular addition provides an reciprocating motion

translational function to compress and expand piston cylinder assembly, stretch and compress piezoelectric material to drive the chain assembly or energy that moves the chain assembly can generate electricity.

Most often, the power is conveyed by a roller chain, known as the drive chain or transmission chain, passing over a sprocket gear, with the teeth of the gear meshing with the holes in the links of the chain. The gear is turned, and this pulls the chain putting mechanical force into the system. Another type of drive chain is the Morse chain, invented by the Morse Chain Company of Ithaca, New York, United States. This has inverted teeth.

Sometimes the power is output by simply rotating the chain, which can be used to lift or drag objects. In other situations, a second gear is placed and the power is recovered by attaching shafts or hubs to this gear. Though drive chains are often simple oval loops, they can also go around corners by placing more than two gears along the chain; gears that do not put power into the system or transmit it out are generally known as idler-wheels. By varying the diameter of the input and output gears with respect to each other, the gear ratio can be altered. For example, when the bicycle pedals' gear rotates once, it causes the gear that drives the wheels to rotate more than one revolution.

Chains versus belts

Roller chain and sprockets is a very efficient method of power transmission compared to (friction-drive) belts, with far less frictional loss.

Although chains can be made stronger than belts, their greater mass increases drive train inertia.

Drive chains are most often made of metal, while belts are often rubber, plastic, urethane, or other substances. Although this invention teaches chain link belts with composite system integration into belt material or with fasteners embedder in the belt can serve the same function as this chain roller triangular system. There is no limit to the radial distance compression and expansion can occur, so it is only a matter of material strength and stress that determines the distance from the center of a sprocket. Sprocket links and sprockets can be made into any size or width. In FIGURE 56 sprocket 301 could have structures on the surface along the lines illustrated (including full 360-degrees) on the sprocket to project mechanical contact with sprocket systems: electrical probe connections, a mechanical leverage against a chain-link system, like engines of piezoelectric wafers, motors, generators, and the sprocket could be a turbo fan, or turbo blade, fan or water propeller. A second function can be placed on the sprocket, like helicopter blades within the sprocket. Systems would be extending perpendicular to the sprocket's surfaces, and possibly adding the mechanical motion of devices on the chain links. This invention teaches mounting functions in the direction of the rotation axis on the face of the sprocket, extending into the chain-links, and there is no limit to the radial distance from the CenterPoint that a system can be added, only limited by stress and durability.

Drive belts can slip unless they have teeth, which means that the output side may not rotate at a precise speed, and some work gets lost to the friction of the belt as it bends around the pulleys. Wear on rubber or plastic belts and their teeth is often easier to observe, and chains wear out faster than belts if not properly lubricated.

One problem with roller chains is the variation in speed, or surging, caused by the acceleration and deceleration of the chain as it goes around the sprocket link by link. It starts as soon as the pitch line of the chain contacts the first tooth of the sprocket. This contact occurs at a point below the pitch circle of the sprocket. As the sprocket rotates, the chain is raised up to the pitch circle and is then dropped down again as sprocket rotation continues. Because of the fixed pitch length, the pitch line of the link cuts across the chord between two pitch points on the sprocket, remaining in this position relative to the sprocket until the link exits the sprocket. This rising and falling of the pitch line is what causes chordal effect or speed variation.

In other words, conventional roller chain drives suffer the potential for vibration, as the effective radius of action in a chain and sprocket combination constantly changes during revolution ("Chordal action"). If the chain moves at constant speed, then the shafts must accelerate and decelerate constantly. If one sprocket rotates at a constant speed, then the chain (and probably all other sprockets that it drives) must accelerate and decelerate constantly. This is usually not an issue with many drive systems; however, most motorcycles are fitted with a rubber bushed rear wheel hub to virtually eliminate this vibration issue. Toothed belt drives are designed to avoid this issue by operating at a constant pitch radius.

Chains are often narrower than belts, and this can make it easier to shift them to larger or smaller gears in order to vary the gear ratio. Multi-speed bicycles with derailleurs make use of this. Also, the more positive meshing of a chain can make it easier to build gears that can increase or shrink in diameter, again altering the gear ratio. However, some newer synchronous belts claim to have "equivalent capacity to roller chain drives in the same width".

Both can be used to move objects by attaching pockets, buckets, or frames to them; chains are often used to move things vertically by holding them in frames, as in industrial toasters, while belts are good at moving things horizontally in the form of conveyor belts. It is not unusual for the systems to be used in combination; for example, the rollers that drive conveyor belts are themselves often driven by drive chains.

Drive shafts are another common method used to move mechanical power around that is sometimes evaluated in comparison to chain drive; in particular belt drive vs chain drive vs shaft drive is a key design decision for most motorcycles. Drive shafts tend to be tougher and more reliable than chain drive, but the bevel gears have far more friction than a chain. For this reason, virtually, all high-performance motorcycles use chain drive, with shaft-driven arrangements generally used for non-sporting machines. Toothed-belt drives are used for some (non-sporting) models.

Use in vehicles

In engines: Internal combustion engines often use a timing chain to drive the camshaft(s). This is an area in which chain drives frequently compete directly with timing belt drive systems, particularly when the engine has one or more overhead camshafts, and provides an excellent example of some of the differences and similarities between the two approaches. For this application, chains last longer, but are often harder to replace, as they must be enclosed in a space into which lubricating oil can be introduced. Being heavier, the chain robs more power, [dubious - discuss] but is also less likely to fail. The camshaft of a four-stroke engine rotates at half crankshaft speed, so the camshaft sprocket has twice as many teeth as the crankshaft sprocket. Less common alternatives to timing chain drives include spur gears or bevel gears combined with a shaft.

FIGURE 59 illustrates an aircraft wing aerodynamic, provided with a chain roller 410 and sprocket (not shown) to move the wing surface 411 around the aircraft wing, which may be used to position the aircraft on a terrestrial surface too. The surface material has a cable (metal, polymer, composite, or nano graphene cables) threaded through the surface of 410 sandwiched between the chain roller links 410.

FIGURE 60 illustrates a ball joint 420 that has the freedom to rotate around its axis 360-degrees in ball joint socket 421, which differentiates it from a prior art chain link, and a 25- degree angle is possible in this view providing the ability to curve around a chain roller sprocket designed to receive the ball shape. Surface 425 between two ball joint links 420 limit the angle in this view however other joint can replace this example to twist sprockets at any angle to optimize an application. This link can also be made triangular, but without limits of rotation in socket 421, again 360-degrees.

FIGURE 61 illustrates an end view of FIGURE 62 which is a propulsion ball 431 in the airduct 432 on the surface of an aircraft or migrating to the surface of an engineering mobility system.

FIGURE 62 illustrates a perspective view of FIGURE 61 for viewing the propulsion ball 431 entering the left end of the firing tube 434 and a second propulsion ball 431 is exiting the firing propulsion tube 434 into the open-air propulsion chamber 432 provide a curved edge on the chamber to add to the aerodynamic flow and restricting the propulsion ball from escaping from the open air chamber 432 to move air aerodynamically on the surface of an aircraft. Propulsion balls can be made of a soft material with aerodynamic bladed shapes, including electric circuits that can move the propulsion ball back to the firing propulsion tube 434 to fire down the axis 435 into open air duct 432 again. Propulsion balls can be electric capacitors, act like motors, contain chemicals to clean and air flow, and travel in FIGURE 21 through 27 helix pathways 147 and 148. FIGURE 12 illustrates how two balls starting out on the top of the cycloid curves illustrated, reach the bottom center of the curve at the same time, even though the balls started at different distances from the midpoint of the cycloid curve, which this invention applies in FIGURE 61 and 62 airducts when it provides optimization on wings fuselages or internally to return the propulsion ball to the firing tube. Firing tube can be electrically wired to propel the ball or explosive fuels, steam, and any force projecting source, like an just compressed air pulsed.

The present invention has been described in relation to a preferred embodiment and several alternative preferred embodiments. One of ordinary skill, after reading the foregoing specification, may be able to affect various other changes, alterations, and substitutions or equivalents thereof without departing from the concepts disclosed. It is therefore intended that the scope of the Letters Patent granted hereon be limited only by the definitions contained in the appended claims and equivalents thereof.