DESIGN OF CYCLOIDS, HYPOCYCLOIDS AND EPICYCLOIDS CURVES
WITH DYNAMIC GEOMETRY SOFTWARE. ENGINEERING APPLICATIONS
M. Flórez, M.V. Carbonell, E. Martínez
Dpto. Física y Mecánica. Escuela Técnica Superior de Ingenieros Agrónomos. Universidad Politécnica de Madrid (España)
Objetives
To draw the mechanical curves most used in engineering
(cycloid, hypocycloid, epicycloids) by using the Geogebra
software.
Study some engineering applications of these mechanical
curves, the planetary gear trains, and the kinematic
requirements.
The Geogebra Software
Cycloid curves
The cycloids curves, are planar curves obtained by the motion of a point
of a circle or a line that rolls without slipping on another circle or a line.
The line or moving circle is called wheel of rolling circle (it is also called
roulette), and the line or circle on moving the wheel is the base of fixed
circle.
When the rolling circle is rotating on another circumference, the path
described is an epicycle;.
If the rolling circle is outside the circumference, the path obtained is an
epicycloid and if the rolling circle is inside the circumference the path
obtained is a hypocycloid (the Greek epi and hypo mean exterior and
interior respectively).
The cycloid curve is defined as the path described by a circumference point
which is rolling without slipping on a straight line. The parametric equation of
cycloid are
x  R(  sen )
y  R(1  cos  )
Hypocycloids are particular cases of cycloid is the hypocycloid ("hypo"beneath); in this case the rolling circle rotates on the inside of another circle.
When a circle of radius r rolls along the inside of a circle of radius R, the form of
the trajectories of the boundary points of the rolling circle (the “wheel”) depends
on n = R/r and they have n cusps and n arches. The parametric equation of
hypocycloid is
 Rr 
x  ( R  r )cos  r cos 

 r

 Rr 
y  ( R  r )sen  rsen 

 r

n=5
Hypocycloid with a ratio R/r=5 and, consequently ,there are five cusps.
After opening the applet, the value of n can be changed and the number
of cusps should be then modified.
When n=3, the particular curve is called the deltoid (analogous to the Greek
letter) and when n =4 is called asteroid curve (analogous to a star).
n=3
n=4
Epicycloids are curves traced when the rolling circle rotates outside the
circumference of another circle without slipping ("epi"- upon).
A circle of radius r rolling along the outside of a circle of radius R and the
form depends on the ratio n=R/r.
x  ( R  r)cos  r cos ( R  r) / r 
y  ( R  r)sen  rsen( R  r) / r 
Epicycloid with a ratio R/r=5 and, consequently ,there are five cusps
After opening the applet, the P point should be moved and the path
described by the point A belonging to the periphery of the rotating circle will
be drawn.
Engineering applications: Planetary gear train
A planetary gear train is formed by a fixed ring gear, a , a
planetary gear, b , and a planetary carrier, c .
When the carrier acts as the input link and rotates one
revolution, a point on the pitch circle of the planetary gear
will describe a hypocycloid/epicycloid path.
Hypocycloid planetary gear train is formed by a fixed ring gear, a, a planetary
gear,b, and a planetary carrier,c. When the carrier acts as the input link and
rotates one revolution, a point on the pitch circle of the planetary gear will
describe a hypocycloid path.
The relationship between the angular velocities of the ring gear (in this case the
ring gear is fixed and the a=0), the planetary gear (b) and the planetary carrier
c in an n-cusped hypocycloid planet gear train is
  T r
  n
  T r
b
c
a
a
a
c
b
b
  (n  1)  n´
b
c
c
In a planetary hypocycloid system, the singular link has the n′ = (n −1) multiples
angular velocity as the binary link but rotates at an opposite direction.
The main feature of a hypocycloid mechanism is that a fixed point A on the
singular link may describe a hypocycloid when the binary link is rotated.
Three-cusped hypocycloid gear train (n=3) . By changing the number of
cusps of the path traced by the point A, the radii will change
n=3
An epicycloid planetary gear train is formed by a fixed ring gear, a , a
planetary gear, b , and a planetary carrier, c . In this case the length of the
carrier will be greater than the radius of the fixed ring gear.
The relationship between the angular velocities of the ring gear (in this
case the ring gear is fixed and the a=0), the planetary gear (b) and the
planetary carrier c in an n-cusped hypocycloid planet gear train is
  T
r
    n
  T
r
b
c
a
a
a
c
b
b
In an epicycloid system, the singular link has the (n +1) multiples angular
velocity as the binary link but rotates at an uniform direction
The main feature of a epicycloids mechanism is that a fixed point A on the
singular link may describe an a epicycloids when the binary link is rotated.
n=4
All the simulations have been generated with the Geogebra software. Authors
are members of the Educational Innovation Group of the Technical University
of Madrid “New Trends in Physics Teaching”.
The authors are grateful to the Universidad Politécnica de Madrid for the
financial support of projects about Innovation and Teaching
Thanks for your attention