7.3 Rotary Engine Geometry
The major elements of the rotary engine–the housing and the rotor– are shown in
cross-section in Figure 7.2. The housing inner surface has a mathematical form known
as a trochoid or epitrochoid. A single-rotor engine housing may be thought of as two
parallel planes separated by a cylinder of epitrochoidal cross-section. Following the
notation of Figure 7.5, the parametric form of the epitrochoid is given by
x = e cos 3
+ Rcos
[ft | m]
(7.1a)
y = e sin 3
+ R sin
[ft | m]
(7.1b)
where e is the eccentricity and R is the rotor center-to-tip distance. For given values of
e and R, Equations (7.1) give the x and y coordinates defining the housing shape when
is varied from 0 to 360 degrees.
The rotor shape may be thought of as an equilateral triangle, as shown in Figures
7.2 and 7.4 (flank rounding and other refinements are discussed later in the chapter).
Because the rotor moves inside the housing in such a way that its three apexes are in
constant contact with the housing periphery, the positions of the tips are also given by
equations of the form of Equations (7.1):
x = e cos 3
+ R cos(
+ 2n
*
/3)
[ft | m]
(7.2a)
y = e sin 3
+ R sin(
+ 2n
*
)
[ft | m]
(7.2b)
where n = 0, 1, or 2, the three values identifying the positions of the three rotor tips,
each separated by 120°. Because R represents the rotor center-to-tip distance, the
otion of the center of the rotor can be obtained from Equations (7.2) by setting R = 0.
The equations and Figure 7.5 indicate that the path of the rotor center is a circle of
radius e.
Note that Equations (7.1) and (7.2) can be nondimensionalized by dividing through
by R. This yields a single geometric parameter governing the equations, e/R, known as
the eccentricity ratio. It will be seen that this parameter is critical to successful
performance of the rotary engine.
The power from the engine is delivered to an external load by a cylindrical shaft.
The shaft axis coincides with the axis of the housing, as seen in Figure 7.2. A second
circular cylinder, the eccentric, is rigidly attached to the shaft and is offset from the
shaft axis by a distance, e, the eccentricity. The rotor slides on the eccentric. Note that
the axes of the rotor and the eccentric coincide. Gas forces exerted on the rotor are
transmitted to the eccentric to provide the driving torque to the engine shaft and to the
external load.
The motion of the rotor may now be understood in terms of the notation of Figure
7.5. The line labeled e rotates with the shaft and eccentric through an angle 3
, while
the line labeled R is fixed to the rotor and turns with it through an angle
about the
moving eccentric center. Thus the entire engine motion is related to the motion of these
two lines. Clearly, the rotor (and thus line R) rotates at one-third of the speed of the
shaft, and there are three shaft rotations for each rotor revolution.