Appendix B
Barrowman Method of Calculating Normal ForceThe normal (lateral) force acting upon a particular rocket component such as rocket body, nosecone, conical transition (such as shoulder or boattail) and fins due to non-zero angle of attack is calculated as such:[Equation 1]
where:
As is seen by the last two terms of Equation 1, angle of attack is multiplied by the slope of the normal force coefficient. Slope is defined as the rise over run of a straight line tangent to the curve, at α=0, of the normal force coefficient versus angle of attack. This concept is illustrated in Figure 1. The slope of the straight line is change (Δ), or rise, in force coefficient over the corresponding change, or run, in attack angle. The curve of the function is seen to be nonlinear, with the slope gradually increasing as alpha increases. This demonstrates why this method of determining the normal force is valid for "small" angles of attack.
Figure 1: Concept of slope of force coefficient
The derivation of slope of the normal force coefficient at α = 0 ( C_{N α}) for the various rocket components is given in Resource 7. The derivations are worth reviewing to gain insight into the method and its assumptions.
Key assumptions applicable to the Barrowman method:
Normal Force Coefficient The expressions for normal force coefficient slope at α = 0 for various rocket components are summarized below. Units are per radian. Nosecone (C_{N α})_{N} = 2 Body (C_{N α})_{B} = 0 Conical Shoulder Conical Transition or Boattail
S_{1}, S_{2} = cross-sectional area, as indicated (in^{2} or mm^{2}). Finset
where N = 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. Resource 7 provides the derivations. The expressions for the C.P. location of various rocket components are summarized below. Units of 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. Conical Transition where d_{1} is the forward diameter of the transition, and d_{2} is the aft diameter of the transition. Finset
ExampleCalculate the normal forces acting on the example rocket shown below at an angle of attack of 10 degrees.Click for larger image. Use units of Newtons (N.) and millimetres (mm)
Angle of attack, α = 10 degrees = 0.1745 radians For any nosecone, (C_{N α})_{N} = 2 (per radian) and for any rocket body (C_{N α})_{B} = 0
For the conical shoulder
For the finset
For the boattail The normal force acting at each component is now calculated
At the nosecone C.P
At the conical shoulder C.P
At the boattail C.P
At the finset C.P The centre of pressure locations are next calculated
For the nosecone
For the conical shoulder
For the boattail
For the finset Results summary:
Important note: These are instantaeous forces that are reacted solely by mass inertia of the rocket acting under lateral and rotational accelerations. Resources: |