As any object moves through a fluid, the
velocity
of the fluid varies around the
surface of the object. The variation of velocity produces a variation of
pressure on the surface of the object
as shown by the the thin red lines on the figure.
Integrating the pressure times the surface area around the body determines the
aerodynamic force
on the object.
We can consider this single force to act through the average location of
the pressure on the surface of the object. We call the
average location of the pressure variation the
center of pressure
in the same way that we call the
average location of the
weight
of an object the
center of gravity.
The aerodynamic force can then be resolved into two components,
lift
and
drag,
which act through the center of pressure in flight.
Determining the center of pressure is very important
for any flying object.
To determine the
stability
of a rocket
it is necessary to know the location of the center of
pressure relative to the center of gravity.
How do engineers determine the location of the center of
pressure for a rocket which they are designing?
In general,
determining the center of pressure (cp) is a very complicated
procedure because the pressure changes around the object.
Determining the center of pressure requires the use of calculus
and a knowledge of the pressure distribution around the body.
We can characterize the pressure variation around the surface as a
function
p(x) which indicates that the pressure depends on the distance x from
a reference line usually taken as the leading edge of the object.
If we can determine the form of the function, there are methods to
perform a calculus integration of the equation.
We will use the symbols "S[ ]dx" to denote the integration of a continuous
function. Then the center of pressure can be determined from:
cp = (S[x * p(x)]dx) / (S[p(x)]dx)
If we don't know the actual functional form,
we can numerically integrate the equation using a spreadsheet
by dividing the distance into a number of small distance segments and
determining the average value of the pressure over
that small segment. Taking the sum of the average value times the distance times
the distance segment
divided by the sum of the average value times the distance segment will produce
the center of pressure.
There are several important problems to consider when determining the center of
pressure for an object. As we change angle of attack, the pressure at every
point on the object changes.
And, therefore, the location of the center of pressure changes as well.
The movement of the center of pressure caused a major problem for early
wing designers because the amount (and sometimes the direction)
of the movement was different for different designs.
In general, the pressure variation around the object also imparts a
torque,
or "twisting force", to the rocket.
If an object is not restrained in some way it will
rotate
as it moves through the air.
(As a further complication, the center of pressure also moves because of
viscosity and compressibility
effects on the flow field. But let's save that discussion for another page.)
To resolve some of these design problems, engineers prefer to
characterize the forces on an object by the aerodynamic force, described
above, coupled with an aerodynamic moment to account for the torque.
It was found both experimentally and analytically that, if the aerodynamic
force is applied at a location 1/4 of the length
back from the leading edge on most low speed objects,
the magnitude of the aerodynamic moment remains nearly constant
with angle of attack.
Engineers call the location where the aerodynamic moment remains constant the
aerodynamic center.
Using the aerodynamic center as the location where the aerodynamic
force is applied eliminates the problem of the movement of the center of pressure
with angle of attack in aerodynamic analysis.
(For supersonic airfoils, the aerodynamic center is nearer the 1/2 chord
location.)
When computing the stability of a rocket, we usually
apply the aerodynamic forces at the aerodynamic center of airfoils and compute
the center of pressure of the vehicle as an areaweighted average of the
centers of the components.
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