Richard Nakka’s Experimental Rocketry Web Site
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Introduction
to Rocket Design
Appendix C
Calculating Centre of Pressure
Introduction
There are plenty of good software apps that do a fine job of
calculating the Centre of Pressure (C.P.) of your rocket, such as AeroLab, RASAero, OpenRocket, Barrowman.xls
and RockSim. Why not simply use one of these
apps and be done with it? In fact, I regularly use AeroLab
and RASAero. Engineers (and other technical professionals)
have a saying “garbage
in, garbage out”. Notwithstanding inadvertent typos when entering data (easy to
do, that’s why we check and double check), we can innocently enter wrong or otherwise inappropriate data if we do not fully
understand what is expected and what assumptions are made by a software
algorithm.
In my profession as an aerospace Stress
Engineer, one of the most useful
tools is finite element analysis (FEA). Although
a Stress Engineer relies most heavily on hand
calculations, certain structures are best analyzed using FEA, such as those
with complex geometry or highly redundant load paths. FEA will always give you
an answer (unless you screw up something when creating your model), usually in
the form of a nicely-coloured stress plot which looks most
impressive. But can we trust the results?
There are two important means to give us confidence in such
results. One, do a sanity
check. This is where hand
calculations are indispensible. But to
do hand calculation checks, we must understand the underlying principles or algorithms, upon which our
software results are based. Two, understanding the basic principles helps us
with the design of a rocket. In particular, why does the
C.P. lies at a certain location? What features of a rocket design influences
its location? How can we design a rocket, in an effective and efficient manner,
to achieve a suitable C.P. that will help provide us with the stability margin
that we seek? Additionally, what are the assumptions and
limitations underlying a particular
method? In this appendix, I am presenting means of calculating the C.P. of a
rocket, whether it be a model rocket, HiPower or EX rocket. At the very least,
the information that follows will help the rocket designer to be better
equipped to use the software apps in a more informed and capable manner with
reduced likelihood of getting garbage
results.
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Barrowman Method
In August of 1966, James S. Barrowman and Judith A. Barrowman
presented a research and development project at NARAM-8 entitled “The Theoretical Prediction of the Center of Pressure” (Res.C1).
The objective of this seminal report was to derive the subsonic theoretical
centre of pressure equations for a general rocket and present equations in a
simplified manner to be of practical use to the rocket modeler. To verify their
results, Barrowman conducted fifteen flight tests of a representative
model rocket. The centre of gravity (C.G.) was changed from flight to flight
until unstable flight was achieved, taken as identifying the location of the
C.P.
It is worth mentioning that this was followed up in 1967 when James
Barrowman (then of NASA's Sounding Rocket Branch) submitted a document entitled
“The Practical Calculation of the Aerodynamic
Characteristics of Slender Finned Vehicles” (Res.C2) as his Master's
thesis to the School of Engineering and Architecture of the Catholic University
of America. The document expanded on his earlier work, and presented a practical
method for determining the roll forcing moment coefficient derivative, roll
damping moment coefficient derivative, pitch damping moment coefficient
derivative and the drag coefficient of a rocket for both subsonic and
supersonic velocities, and at small angles of attack, to a high order of
accuracy.
The method of
calculating C.P. location, presented in Barrowman’s NARAM-8 report, is given
here. First, the assumptions guiding the method are given.
1.
Flow
over rocket is potential
flow.
In reality, there will be some friction, as air has viscosity. Flow will not be
perfectly smooth, as all rockets have some protruberances such as fasteners,
joints, etc. The net affect is assumed to be minor with respect to accuracy of
the method.
2.
Flow
over rocket is steady-state and subsonic.
As indicated in Resource 6, the Barrowman method can be used for velocities
somewhat greater than mach one. However, the results generally become
unconservative. To compensate, larger safety margins should be used when
applying the results to supersonic rocket design. Resource 6 provides methods
to deal with higher mach numbers. However, for most EX rockets, the critical moment
for stability is immediately after clearing the launch rail, when velocity is
at a minimum, for two reasons: 1) The C.G. is furthest aft
and moves forward as propellant burns away, thereby increasing the stability
margin; 2) cross-wind has greatest influence on the effective angle-of-attack.
3.
Angle-of-attack
is small.
The Barrowman method utilizes a force coefficient slope at zero degree angle-of-attack
(see Fig.2). As angle-of-attack increases, the slope changes at an increasing
rate. The degree to which the slope changes will be different for each rocket
component and is not feasible to quantify. Barrowman suggests 10 degrees as a
maximum angle-of-attack for the method to remain accurate. Resource 6 discusses
how to deal with larger angles of attack.
4.
Nosecone
tip is sharp
If a nosecone has a rounded, rather than sharp tip, base the calculations on
the theoretical nosecone geometry assuming the rounded tip is not present.
5.
Rocket
finset consists of either 3 or 4 fins
Originally the method was derived for rockets with 3 or 4 fins. However,
Barrowman expanded the method to include 6 fin rockets.
6.
Fins
are thin flat plates
Aerofoiled fins possess a centre of pressure that is at a location different
than that of flat plates. This should be borne in mind when assessing a rocket
with aerofoiled fins.
7. The cylindral body produces zero lift
The rocket body will generate lift
as the angle-of-attack increases, thereby changing the location of the C.P from
that predicted. This effect is more pronounced with long slender rockets.
8. Rocket is a rigid
body
For a hobby-sized rocket, this is
generally an accurate assumption, as body tubes are stiff and offer little
flexing.
It is important to note that these assumptions were made by
Barrowman in order to simplify solution of the pertinent equations and to remove
complexities in the resulting method. It does not
mean that violating any of the assumptions invalidates the method. Rather,
deviations from the assumptions, as long as they are reasonably minor, may
reduce the accuracy and/or confidence in the result to some degree. For
example, a sharp nosecone is rarely used on hobby rockets; there is usually
some bluntness, for practical and safety reasons. Barrowman assumed a sharp
nosecone such that the tip area is zero.
This served to simplify
the resulting expressions for the normal force coefficient slope and x-bar applicable to
the nosecone. Using a rounded nosecone on your rocket does not mean the method
of calculating C.P. will be invalid. However, when using the Barrowman method,
your nosecone should be idealized as being sharp (i.e. remove
the bluntness). This applies to any shape of nosecone,
not just straight cones.
The Barrowman method
works as follows. The C.P. of a rocket is calculated as the aerodynamic balance
point, analagous to the gravitational balance point relating to a rocket’s
centre of gravity (C.G.). Just as the C.G. can be calculated as a weighted average
of the positions of all the components that make up the rocket’s mass, the C.P.
can be calculated as a weighted average of the positions of all the components
that contribute a normal force (lift) acting upon a component’s surface.
Basically, you sum up the contribution of each component’s normal force factored
by its position, then divide by the sum of the normal forces of the components:
The normal force (N), also commonly referred to as lift,
is the force that results from air flow acting on a component at some angle-of-attack.
The force acts normal to a plane along the rocket’s longitudinal axis (or
put another way, perpendicular to the longitudinal axis). The position of any particular part (x)
is with respect to some chosen reference point, or datum. For a rocket, the
datum for calculating C.P. is traditionally the tip of the nosecone with
positive x aftward along the rocket’s
longitudinal (roll) axis. This is known as nose-to-tail
axis system. However, the datum can be any convenient location, such as the aft
end of the body with positive x
forward. This is tail-to-nose
axis system. These two possible coordinate systems are shown in Figure 1.
Figure 1: Two possible coordinate systems for
determing C.P.
Normal force is given by:
with
units of force being pounds-force (lbf) or Newtons. Subscript c refers to the particular component of
interest (such as nosecone or fin). CNc is the normal
force coefficient as shown in Figure 2, which is seen to be equal to
zero at zero angle-of-attack.
Figure 2: Concept of slope of normal force
coefficient
q = air flow dynamic
pressure = ½ ρ V2,
lbf/in2 or N/m2
A = reference area upon which CN α is based,
usually the nosecone base area; A
= ¼ π D2 where D = nosecone base diameter, in. or m.
V = airflow velocity, ft/sec or
m/s
At small angles of attack we may
approximate CN to be linear with angle-of-attack, such that
CN » a
(CN
α)
As such, normal force is alternatively given by:
where:
(CN α) = slope
of the normal force coefficient at α = 0 (also referred to as stability derivative), and can be written as (∂CN/∂α
| α=0), per radian. See Figure 2.
α = effective angle-of-attack,
radians
The normal force (N) in Equation 1 can be replaced by CNa, as the other terms in Eqn.2a cancel out:
Or in summation notation:
Now that we have the equation that allows us to calculate the
location of the C.P., we need to determine the terms in the equation for each
of the rocket components. The values (or expressions) for CN α and x
for the nosecone, body, conical shoulder, boattail and finset are summarized
below. CN α
is at α = 0 and units are per radian. Resource C1 provides the
derivation for each of these terms.
Normal Force
Coefficient Slope
Nosecone (CN α)N = 2
Body (CN α)B = 0
Conical Shoulder
Conical Transition or Boattail
S1, S2 =
cross-sectional area, as indicated (in2 or mm2).
d = reference length = diameter at base
of nosecone (in. or mm)
The expressions for conical shoulder and conical boattail are identical. It is
important to note that, for a boattail, S2
is less than S1. As such, the value of (CN α)B
is negative, meaning the normal force acts in an opposite
sense to a conical shoulder.
The expression for conical shoulder/boattail can be simplified by incorporating
the expressions for areas S1 =
¼ π d12 and S2 = ¼ π d22,
giving:
Finset
where Nf = number of fins in finset (3, 4 or 6). Theta (θ) and l are the mid-chord sweep angle and mid-chord length, respectively. Interference coefficient f=1 for 3 or 4 fins, f=0.5 for 6 fins.
Location of Centre
of Pressure
The normal force resultant,
as calculated above, acts at the centre of
pressure (C.P.) of each of the components. The expressions for the
C.P. location of various rocket components are summarized below. Units are
inches or mm.
Nosecone
Note that if the nosecone tip is radiused, rather than sharp, the reference line from which the nosecone length and C.P. location is obtained by extrapolating the profile of the nosecone curve as shown (this is to be consistent with assumption #4).
Conical Transition
where d1 is the forward diameter of the transition, and d2 is the aft diameter of the transition.
Finset
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Res.C1 provides useful charts that graph CN α and location of C.P. for conical transitions (shoulder and boattail) and fins as a function of component geometry. The charts are of particular value when setting out to design each of these rocket components.
Appendix B provides an example of calculating these terms for a typical EX rocket and the resulting normal forces applied to the rocket. The result of this example is shown in Figure 3.
Figure
3: Example rocket indicating normal forces
resulting from angle-of-attack
The normal force and centre of pressure locations
for each of the components is listed below, with the distance being relative
the the tip of the nosecone:
Nosecone NN = 76 N. xN = 93.2 mm
Conical Shoulder NCS = -27.5 N. xCS = 636 mm
Finset NF = 347 N. xF = 1202 mm
Boattail NCB = -27.1 N. xCB = 1273 mm
The centre of pressure is calculated using Equation
1:
=1010 mm
Alternatively, Equation 1a may be used to calculate
the rocket C.P.
Nosecone (CN
α)N = 2 per rad. xN = 93.2 mm
Conical Shoulder (CN
α)CS = -0.72 per rad. xCS = 636 mm
Finset (CN
α)F = 9.1 per rad. xF = 1202 mm
Boattail (CN
α)CB = -0.71 per rad. xCB = 1273 mm
=1010 mm
which, as expected, gives the same answer. This
demonstates that it is not necessary to calculate the normal force, solely the
coefficent CN
α. is required to determine the C.P.
location of the rocket. In fact, the normal force is zero
at zero angle-of-attack, so Equation 1 is of no use to us, however, Equation 1a
is useful as the slope of the CN
as a function of angle-of-attack is non-zero at
zero angle-of-attack (ref. Figure 2).
It is both interesting and useful to assess the pitching torque generated by the various
components of the example rocket. The pitching torque, which acts about the
C.G. of the rocket, is given by:
Tp = NC ´ xc
The location of the C.G.
for the example rocket is not given. However, if we assume a stability margin
of S.M. = 1.5 calibres, the C.G. location will be:
xCG = 1010 – (1.5) 60 = 920
mm. This gives the following pitching torque for the rocket components:
Nosecone TpN = 76 ´ (920 - 93.2) = 62,837 N-mm
Conical Shoulder TpCS = -27.5 ´ (920 – 636) = -7810 N-mm
Finset TpF = 347 ´ (920 – 1202) = -97,854 N-mm
Boattail TpCB = -27.1 ´ (920 – 1273) = 9566
N-mm
Based on the sign convention, a positive value of pitching torque
is a destabilizing torque and a negative
value is a stabilizing torque. This is
illustrated in Figure 4. The fins generate the largest torque and, as expected,
it is a stabilizing torque. What is of particular interest here is that the
nosecone generates a significant destabilizing torque. This is primarily a
consequence of its appreciable distance from the C.G. Also, the shape and
aspect ratio of the nosecone comes into play. A short conical nosecone will
produce less of a destabilizing torque than a long ellipsoidal shape. The
boattail also generates a destabilizing torque.
Figure
4: Stabilizing versus destabilizing torque
Modified Barrowman Methods
Resource C4 provides an extension, or modification, to the Barrowman
method that takes into account C.P. variation with angle-of-attack. The
Barrowman method ignores lift due to the rocket
body. Barrowman justified this assumption based on test data that
indicated that the lift force on a cylindrical body is tiny for small
angle-of-attack, as shown in Figure 5 excerpted from the Barrowman report Res.
C1.
Figure
5: Cylinder lift vs angle-of-attack
Note: Although
labeled as force (N.), the vertical axis scale represents the lift and drag
coefficients, as seen in the referenced figure.
When a cylinder is at a small angle to the oncoming flow, it generates negligible
lift due to symmetrical pressure distribution around its surface. As the angle-of-attack
increases, the pressure distribution becomes asymmetrical, leading to lift
generation. Experiments have suggested
that rockets with particularly long slender bodies, body lift may not be
negligible, especially at larger angle-of-attack. Robert Galejs, in his article
“Wind Instability -- What Barrowman Left Out” (Res.C4),
describes an extension to the Barrowman method that models C.P. variation with
angle-of-attack by accounting for body lift. The body lift force may be
expressed as:
N = q K Ap a 2
where K
is a constant with a value between 1.0 and 1.5 and Ap is the body planform area, which includes the nose, body, transitions
and boattail (not fins). The lift force acts at the centre of the body planform
area. Putting this force into the Barrowman equation format (Eqn.2a), one
factor of a is factored out leaving body force as a linear
variation with angle-of-attack. The coefficient of body lift is:
where D
= nosecone base diameter, and noting that this CNa2 is a function of alpha, unlike the other terms in the Barrowman
equation which are considered to be independent of alpha. Also note a is a close approximation of sina.
Example C1 describes how this method is used to calculate
C.P. for the Xi-41 rocket at various angles-of-attack.
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RASAero has a feature deemed Rogers Modified Barrowman method for
calculating centre of pressure. Details of this particular method are:
1) CNa of the combined nosecone and body at low angles of attack is included,
versus the Barrowman method whereby solely the nosecone is included. The effect
of adding combined body and nosecone at low angles of attack is an increase in CNa, and the C.P. of the nosecone/body combination, at low angles of
attack, is moved aft towards the bottom of the nosecone (top of the body tube).
This results in a net improvement in predicted stability, as it moves the
rocket C.P. aft compared to Barrowman.
2) The body in the presence of fins interference factor (Kbf), which was left out of the
Barrowman method, is included.
3) The viscous crossflow increase in CN
with angle-of-attack is included. (described by Galejs) using an advanced
method for the viscous crossflow CN
calculations (Jorgensen Viscous
Crossflow method, which is also used in Missile DATCOM ). The Rogers Modified
Barrowman method is used for subsonic flow only.
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Stability at Velocity above Mach 1
The C.P. of a rocket moves forward with increasing mach number. This is
a destabilizing effect. This is a consequence of both changes in the normal
force coefficient and C.P. position for the rocket components. For supersonic (HiPer) rockets powered by a solid
propellant motor, the stability margin will in most cases not be
reduced to an unsafe level. This is because the most critical instant, or design point, for static stability is at
liftoff, after the rocket clears the launch rail. The rocket is moving at a
relatively slow velocity and the C.G. of the rocket is at its aftmost position.
Any cross-wind at liftoff will result in a non-zero angle-of-attack which will
tend to destabilize the rocket. As the propellant quickly burns away, the C.G.
will shift forward significantly, thereby improving the stability margin and
compensating for the shift in C.P. as the mach number increases. This may not
be the case for hypersonic (ViPer) rockets, as the C.P. shift at hypersonic velocity can be quite
severe. Hybrid or liquid propellant rockets may need to consider the forward
shift in C.P. as the movement of the C.G. point, as the propellant burns, will
not necessarily be stability enhancing, as is the case with solid propellant
motors. Resources C6 and C9 provide information on the topic of static
stability at higher mach numbers.
For supersonic flow, RASAero utilizes a combination of USAF
DATCOM and Missile DATCOM methods, and for hypersonic
flow, modified Newtonian theory and Missile DATCOM methods are used.
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1. Using
the Barrowman method, calculate the C.P. location of the Xi-41 rocket and
compare to RASAero prediction
2. Compare
the Barrowman C.P. to the Rogers Modified Barrowman C.P. as predicted by
RASAero.
3. Calculate
the C.P. based on the Galejs modified Barrowman method which includes the body
lift term at non-zero angle-of-attack.
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Resources
Res.C1 The Theoretical
Prediction of the Center of Pressure, James S. Barrowman & Judith A.Barrowman,
Research and Development Project, NARAM-8, August 18, 1966
Res.C2 The Practical
Calculation of the Aerodynamic Characteristics of Slender Finned Vehicles Original Barrowman
report
Res.C3 The Modified Barrowman Method
with Applications to the OPDAMS and PDAMS vehicles, Charles E. Hall, Jr.,
April 1989, Technical Report RD-RE-88-7, U.S.Army Missile Command (Redstone
Arsenal, Alabama)
Res.C4 Wind Instability -- What
Barrowman Left Out, Robert J Galejs
Res.C5 Extending The
Barrowman Method For Large Angles Of Attack, Edward V. LaBudde, Research
and Development Project submitted at NARCON March 1999
Res.C6 Rocketry Aerodynamics , Rick Newlands (AspireSpace Technical Paper)
Res.C7 Design of Aerodynamically
Stabilized Free Rockets, MIL-HDBK-762, July 1990
Res.C8 Static
Stability Investigation of a Single-Stage Sounding-Rocket at Mach Numbers from
0.60 to 1.20, NASA TN D-4013, J.C.Ferris, June 1967
Res.C9 Static
Stability Investigation of a Sounding-Rocket Vehicle at Mach Numbers from 1.50
to 4.63, NASA TN D-4014, C.D.Babb & D.E.Fuller, June 1967
Res.C10 Estimating the
dynamic and aerodynamic paramters of passively controlled high power rockets
for flight simulaton, S.Box, C.M.Bishop,
H.Hunt, Feb.2009
Res.C11 OpenRocket technical documentation (2013-05-10) (PDF
1.4MB)
Last updated March 13,
2025
Originally posted March
13, 2025