Epicyclic gearing

An epicyclic gear train consists of two gears mounted so that the centre of one gear revolves around the centre of the other. A carrier connects the centres of the two gears and rotates to carry one gear, called the planet gear, around the other, called the sun gear; the planet and sun gears mesh. A point on the pitch circle of the planet gear traces an epicycloid curve. In this simplified case, the sun gear is the planetary gear roll around the sun gear. An epicyclic gear train can be assembled so the planet gear rolls on the inside of the pitch circle of a fixed, outer gear ring, or ring gear, sometimes called an annular gear. In this case, the curve traced by a point on the pitch circle of the planet is a hypocycloid; the combination of epicycle gear trains with a planet engaging both a sun gear and a ring gear is called a planetary gear train. In this case, the ring gear is fixed and the sun gear is driven. Epicyclic gears get their name from their earliest application, the modelling of the movements of the planets in the heavens.

Believing the planets, as everything in the heavens, to be perfect, they could only travel in perfect circles, but their motions as viewed from Earth could not be reconciled with circular motion. At around 500 BC, the Greeks invented the idea of epicycles, of circles travelling on the circular orbits. With this theory Claudius Ptolemy in the Almagest in 148 AD was able to predict planetary orbital paths; the Antikythera Mechanism, circa 80 BC, had gearing, able to approximate the moon's elliptical path through the heavens, to correct for the nine-year precession of that path. Epicyclic gearing or planetary gearing is a gear system consisting of one or more outer gears, or planet gears, revolving about a central, or sun gear; the planet gears are mounted on a movable arm or carrier, which itself may rotate relative to the sun gear. Epicyclic gearing systems incorporate the use of an outer ring gear or annulus, which meshes with the planet gears. Planetary gears are classified as simple or compound planetary gears.

Simple planetary gears have one sun, one ring, one carrier, one planet set. Compound planetary gears involve one or more of the following three types of structures: meshed-planet, stepped-planet, multi-stage structures. Compared to simple planetary gears, compound planetary gears have the advantages of larger reduction ratio, higher torque-to-weight ratio, more flexible configurations; the axes of all gears are parallel, but for special cases like pencil sharpeners and differentials, they can be placed at an angle, introducing elements of bevel gear. Further, the sun, planet carrier and ring axes are coaxial. Epicyclic gearing is available which consists of a sun, a carrier, two planets which mesh with each other. One planet meshes with the sun gear. For this case, when the carrier is fixed, the ring gear rotates in the same direction as the sun gear, thus providing a reversal in direction compared to standard epicyclic gearing. In the 2nd-century AD treatise Almagest, Ptolemy used rotating deferent and epicycles that form epicyclic gear trains to predict the motions of the planets.

Accurate predictions of the movement of the Sun and the five planets, Venus, Mars and Saturn, across the sky assumed that each followed a trajectory traced by a point on the planet gear of an epicyclic gear train. This curve is called an epitrochoid. Epicyclic gearing was used in the Antikythera Mechanism, circa 80 BCE, to adjust the displayed position of the moon for the ellipticity of its orbit, for the apsidal precession of its orbit. Two facing gears were rotated around different centers, one drove the other not with meshed teeth but with a pin inserted into a slot on the second; as the slot drove the second gear, the radius of driving would change, thus invoking a speeding up and slowing down of the driven gear in each revolution. Richard of Wallingford, an English abbot of St Albans monastery is credited for reinventing epicyclic gearing for an astronomical clock in the 14th century. In 1588, Italian military engineer Agostino Ramelli invented the bookwheel, a vertically-revolving bookstand containing epicyclic gearing with two levels of planetary gears to maintain proper orientation of the books.

The gear ratio of an epicyclic gearing system is somewhat non-intuitive because there are several ways in which an input rotation can be converted into an output rotation. The three basic components of the epicyclic gear are: Sun: The central gear Carrier: Holds one or more peripheral Planet gears, all of the same size, meshed with the sun gear Ring or Annulus: An outer ring with inward-facing teeth that mesh with the planet gear or gearsThe overall gear ratio of a simple planetary gearset can be calculated using the following two equations, representing the sun-planet and planet-ring interactions respectively: N s ω s + N p ω p − ω c = 0

Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object. For example, the equations x = cos t y = sin t form a parametric representation of the unit circle, where t is the parameter: A point is on the unit circle if and only if there is a value of t such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: =. Parametric representations are nonunique, so the same quantities may be expressed by a number of different parameterizations. In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, the number of equations being equal to the dimension of the space in which the manifold or variety is considered.

Parametric equations are used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is labeled t. Parameterizations are non-unique. In kinematics, objects' paths through space are described as parametric curves, with each spatial coordinate depending explicitly on an independent parameter. Used in this way, the set of parametric equations for the object's coordinates collectively constitute a vector-valued function for position; such parametric curves can be integrated and differentiated termwise. Thus, if a particle's position is described parametrically as r = its velocity can be found as v = r ′ = and its acceleration as a = r ″ =. Another important use of parametric equations is in the field of computer-aided design. For example, consider the following three representations, all of which are used to describe planar curves; the first two types are known as non-parametric, representations of curves.

In particular, the non-parametric representation depends on the choice of the coordinate system and does not lend itself well to geometric transformations, such as rotations and scaling. These problems can be addressed by rewriting the non-parametric equations in parametric form. Numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclid's parametrization of right triangles such that the lengths of their sides a, b and their hypotenuse c are coprime integers; as a and b are not both one may exchange them to have a and the parameterization is a = 2 m n, b = m 2 − n 2, c = m 2 + n 2, where the parameters m and n are positive coprime integers that are not both odd. By multiplying a, b and c by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths. Converting a set of parametric equations to a single implicit equation involves eliminating the variable t from the simultaneous equations x = f, y = g

Cardioid

A cardioid is a plane curve traced by a point on the perimeter of a circle, rolling around a fixed circle of the same radius. It can be defined as an epicycloid having a single cusp, it is a type of sinusoidal spiral, an inverse curve of the parabola with the focus as the center of inversion. The name had been the subject of study decades beforehand. Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid, In three dimensions, the cardioid is shaped like an apple centred around the microphone, the "stalk" of the apple. Let a be the common radius of the two generating circles with midpoints, φ the rolling angle and the origin the starting point. One gets the parametric representation: x = 2 a ⋅ cos φ, y = 2 a ⋅ sin φ, 0 ≤ φ < 2 π and the representation in polar coordinates: r = 2 a. Introducing the substitutions cos φ = x / r and r = x 2 + y 2 one gets after removing the square root the implicit representation in cartesian coordinates: 2 + 4 a x − 4 a 2 y 2 = 0.proof for the parametric representation The proof can be done using complex numbers and their common description as complex plane.

The rolling movement of the black circle on the blue one can be split into two rotations. In the complex plane a rotation around point 0 by an angle φ can be performed by the multiplication of a point z by e i φ. Hence the rotation Φ + around point a is: z ↦ a + e i φ, rotation Φ − around point − a is: z ↦ − a + e i φ. A point p of the cardioid is generated by rotating the origin around point a and subsequent rotating around − a by the same angle φ: p = Φ − = Φ − = − a + e i φ = a. Herefrom one gets the parametric representation above: x = a = 2 a ⋅ cos

Geometry

Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.

He studied the sp

Circle

A circle is a simple closed shape. It is the set of all points in a plane; the distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior. A circle may be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. A circle is a plane figure bounded by one line, such that all right lines drawn from a certain point within it to the bounding line, are equal; the bounding line is called the point, its centre. Annulus: a ring-shaped object, the region bounded by two concentric circles.

Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle. Centre: the point equidistant from all points on the circle. Chord: a line segment whose endpoints lie on the circle, thus dividing a circle in two sements. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; this is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, its length is twice the length of a radius. Disc: the region of the plane bounded by a circle. Lens: the region common to two overlapping discs. Passant: a coplanar straight line that has no point in common with the circle. Radius: a line segment joining the centre of a circle with any single point on the circle itself. Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.

Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to. Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Tangent: a coplanar straight line that has one single point in common with a circle. All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries; the word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, meaning "hoop" or "ring".

The origins of the words circus and circuit are related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand; the circle is the basis for the wheel, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Early science geometry and astrology and astronomy, was connected to the divine for most medieval scholars, many believed that there was something intrinsically "divine" or "perfect" that could be found in circles; some highlights in the history of the circle are: 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π. 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed explanation of the circle.

Plato explains the perfect circle, how it is different from any drawing, definition or explanation. 1880 CE – Lindemann proves that π is transcendental settling the millennia-old problem of squaring the circle. The ratio of a circle's circumference to its diameter is π, an irrational constant equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: C = 2 π r = π d; as proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2. Equivalently, denoting diameter by d, A r e

Deferent and epicycle

In the Hipparchian and Ptolemaic systems of astronomy, the epicycle was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon and planets. In particular it explained the apparent retrograde motion of the five planets known at the time. Secondarily, it explained changes in the apparent distances of the planets from the Earth, it was first proposed by Apollonius of Perga at the end of the 3rd century BC. It was developed by Apollonius of Perga and Hipparchus of Rhodes, who used it extensively, during the 2nd century BC formalized and extensively used by Ptolemy of Thebaid in his 2nd century AD astronomical treatise the Almagest. Epicyclical motion is used in the Antikythera mechanism, an ancient Greek astronomical device for compensating for the elliptical orbit of the Moon, moving faster at perigee and slower at apogee than circular orbits would, using four gears, two of them engaged in an eccentric way that quite approximates Kepler's second law.

Epicycles worked well and were accurate, because, as Fourier analysis showed, any smooth curve can be approximated to arbitrary accuracy with a sufficient number of epicycles. However, they fell out of favour with the discovery that planetary motions were elliptical from a heliocentric frame of reference, which led to the discovery that gravity obeying a simple inverse square law could better explain all planetary motions. In both Hipparchian and Ptolemaic systems, the planets are assumed to move in a small circle called an epicycle, which in turn moves along a larger circle called a deferent. Both circles rotate clockwise and are parallel to the plane of the Sun's orbit. Despite the fact that the system is considered geocentric, each planet's motion was not centered on the Earth but at a point away from the Earth called the eccentric; the orbits of planets in this system are similar to epitrochoids. In the Hipparchian system the epicycle revolved along the deferent with uniform motion. However, Ptolemy found that he could not reconcile that with the Babylonian observational data available to him.

The angular rate at which the epicycle traveled was not constant unless he measured it from another point which he called the equant. It was the angular rate at which the deferent moved around the point midway between the equant and the Earth, constant, it was the use of equants to decouple uniform motion from the center of the circular deferents that distinguished the Ptolemaic system. Ptolemy did not predict the relative sizes of the planetary deferents in the Almagest. All of his calculations were done with respect to a normalized deferent, considering a single case at a time; this is not to say that he believed the planets were all equidistant, but he had no basis on which to measure distances, except for the Moon. He ordered the planets outward from the Earth based on their orbit periods, he calculated their distances in the Planetary Hypotheses and summarized them in the first column of this table: Had his values for deferent radii relative to the Earth–Sun distance been more accurate, the epicycle sizes would have all approached the Earth–Sun distance.

Although all the planets are considered separately, in one peculiar way they were all linked: the lines drawn from the body through the epicentric center of all the planets were all parallel, along with the line drawn from the Sun to the Earth along which Mercury and Venus were situated. That means. Babylonian observations showed that for superior planets the planet would move through in the night sky slower than the stars; each night the planet appeared to lag a little in what is called prograde motion. Near opposition, the planet would appear to reverse and move through the night sky faster than the stars for a time in retrograde motion before reversing again and resuming prograde. Epicyclic theory, in part, sought to explain this behavior; the inferior planets were always observed to be near the Sun, appearing only shortly before sunrise or shortly after sunset. Their apparent retrograde motion occurs during the transition between evening star into morning star, as they pass between the Earth and the Sun.

When ancient astronomers viewed the sky, they saw the Sun and stars moving overhead in a regular fashion. They saw the "wanderers" or "planetai"; the regularity in the motions of the wandering bodies suggested that their positions might be predictable. The most obvious approach to the problem of predicting the motions of the heavenly bodies was to map their positions against the star field and to fit mathematical functions to the changing positions; the ancients worked from a geocentric perspective for the simple reason that the Earth was where they stood and observed the sky, it is the sky which appears to move while the ground seems still and steady underfoot. Some Greek astronomers speculated that the planets orbited the Sun, but the optics necessary to provide data that would convincingly support the heliocentric model did not exist in Ptolemy's time and would not come around for over fifteen hundred years after his time. Furthermore, Aristotelian physics was not designed with these sorts of calculations in mind, Aristotle's phi

Cycloid

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve; the cycloid, with the cusps pointing upward, is the curve of fastest descent under constant gravity, is the form of a curve for which the period of an object in descent on the curve does not depend on the object's starting position. The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians. Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery cited similar work by the Syrian philosopher Iamblichus as evidence that the curve was known in antiquity. English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa, but subsequent scholarship indicates Wallis was either mistaken or the evidence used by Wallis is now lost.

Galileo Galilei's name was put forward at the end of the 19th century and at least one author reports credit being given to Marin Mersenne. Beginning with the work of Moritz Cantor and Siegmund Günther, scholars now assign priority to French mathematician Charles de Bovelles based on his description of the cycloid in his Introductio in geometriam, published in 1503. In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel. Galileo originated the term was the first to make a serious study of the curve. According to his student Evangelista Torricelli, in 1599 Galileo attempted the quadrature of the cycloid with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them, he discovered the ratio was 3:1 but incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible. Around 1628, Gilles Persone de Roberval learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieri's Theorem.

However, this work was not published until 1693. Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Mersenne passed these results along to Galileo, who gave them to his students Torricelli and Viviana, who were able to produce a quadrature; this result and others were published by Torricelli in 1644, the first printed work on the cycloid. This led to Roberval charging Torricelli with plagiarism, with the controversy cut short by Torricelli's early death in 1647. In 1658, Blaise Pascal had given up mathematics for theology but, while suffering from a toothache, began considering several problems concerning the cycloid, his toothache disappeared, he took this as a heavenly sign to proceed with his research. Eight days he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons.

Pascal and Senator Carcavy were the judges, neither of the two submissions were judged to be adequate. While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid. Wallis published Wren's proof in Wallis's Tractus Duo, giving Wren priority for the first published proof. Fifteen years Christiaan Huygens had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse a segment of an inverted cycloidal arch in the same amount of time, regardless of its starting point. In 1686, Gottfried Wilhelm Leibniz used analytic geometry to describe the curve with a single equation. In 1696, Johann Bernoulli posed the brachistochrone problem, the solution of, a cycloid; the cycloid through the origin, with a horizontal base given by the line y = 0, this line being known as the x-axis, generated by a circle of radius r rolling over the "positive" side of the base, consists of the points, with x = r y = r where t is a real parameter, corresponding to the angle through which the rolling circle has rotated.

For given t, the circle's centre lies at x = rt, y = r. Solving for t and replacing, the Cartesian equation is found to be: x = r cos − 1 − y. An equation for the cycloid of the form y = f with a closed-form expression for the right-hand side is not possible; when y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps, where it hits the x-axis, with the derivative tending toward ∞ or