Cyclo Curves- CYCLOGON- CYCLOID- HYPOTROCHOID- HYPOCYCLOID- EPITROCHOID- EPICYCLOID- EPICYCLIC GEARING

Cyclogon

A cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line.

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 slippage.

Tautochrone Curve

If r=a then:

Hypotrochoid

A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.

Hypocycloid

A hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.

R/r = k = a/b :

Epitrochoid

An Epitrochoid is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is a distance d from the center of the exterior circle.

Epicycloid

An epicycloid is a plane curve produced by tracing the path of a chosen point of a circle which rolls without slipping around a fixed circle. It is a particular kind of roulette.

R/r = k = a/b :

Epicyclic gearing

An epicyclic gear train consists of two gears mounted so that the center of one gear revolves around the center of the other. A carrier connects the centers 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 so that their pitch circles roll without slip. A point on the pitch circle of the planet gear traces an  Epicycloid curve. In this simplified case, the sun gear is fixed and the planetary gear(s) roll around the sun gear.

Gear ratio of standard epicyclic gearing Sun: The central gear Carrier: Holds one or more peripheral Planet gears, all of the same size, meshed with the sun

gear Annulus: An outer ring with inward-facing teeth that mesh with the planet gear or gears

 is the angular velocity of the Annulus, Sun gear, Planet gears and planet Carrier respectively, and is the Number of teeth of the Annulus, the Sun gear and each Planet gear respectively

if

Torque ratios of standard epicyclic gearing

In epicyclic gears, two speeds must be known, in order to determine the third speed. However, in a steady state condition, only one torque must be known, in order to determine the other two torques.

Thank you! seied Mahdi sadat hosseini