# Barrowmen Equations Based on Varying Angles of Attack

#### markfer20

##### New Member
Hello, I have a question regarding Barrowmen's questions to solve for the center of pressure of each rocket's component. Is there a version of his set of equations that changes the stability derivatives based on the angle of attack of the rocket? I'm attempting to simulate the case of a rocket having a permanent angle deflection along its length (~2 degrees), and thus seeing if the center of pressure/stability changes are considerable. Regardless, thank you for your time and effort!

#### mjennings

##### Well-Known Member
The low angle of attack is an assumption Barrowmen made to get the equations so it is not as simple as resolving the equations. 2 degrees is a fairly small angle unless your rocket is very long, it may not have much effect depending on what it is. I'm not exactly sure what you are simulating. Is it the whole rocket at an AoA of 2 degrees or a component installed at a 2 degree angle?

#### markfer20

##### New Member
The low angle of attack is an assumption Barrowmen made to get the equations so it is not as simple as resolving the equations. 2 degrees is a fairly small angle unless your rocket is very long, it may not have much effect depending on what it is. I'm not exactly sure what you are simulating. Is it the whole rocket at an AoA of 2 degrees or a component installed at a 2 degree angle?
The rocket's length is approximately 15 feet. I'm trying to simulate the rocket as being bent, such that successive installed sections at minor angles have the nose cone at 2 degrees from the vertical. Does that make sense?

• OverTheTop

#### Alan15578

##### Well-Known Member
The rocket's length is approximately 15 feet. I'm trying to simulate the rocket as being bent, such that successive installed sections at minor angles have the nose cone at 2 degrees from the vertical. Does that make sense?
The attached file may give you a clue, but your problem is different. You will need to find the trim angle of a stable rocket with misaligned components.

#### Attachments

• Superroc Design.pdf
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#### UhClem

##### Well-Known Member
Barrowman linearized the equations (We engineers will linearize a system around an operating point at the drop of a hat because non-linear systems are a PITA.) around zero angle of attack. Assuming a nice linear relationship between angle of attack and lift. For a small angle like 2 degrees, you need do nothing on that front.

You do have to figure out how to handle different bits of the rocket having different angles of attack on your own. Hint: whenever you see the symbol "alpha" that is angle of attack.

#### Chuck Rogers

##### Well-Known Member
As Allan15578 noted, you'll have to calculate the trim angle of attack. Take the CNalpha's for each of the rocket components, take the CNalpha for the bent nose, and varying angle of attack (with the nose cone being at 2 deg higher angle of attack than the other rocket components). With the CNalpha's and angle of attack (alpha) for each component you can determine the Normal Force for each component, and then determine the total angle of attack for the rocket where the Normal Forces balance. This is the trim angle of attack.

I kind of know where this is going; aerodynamic loads. Using the CNalpha's for all of the other components, and the CNalpha for the nose, and the fact that the other components and the nose will see different angles of attack (the nose is bent), you can turn the CNalpha's into Normal Forces on each component, and then you can calculate the bending force on the body tube.

Also note that after doing these CNalpha and Normal Force calculations, and coming up with the trim angle of attack, using the Normal Forces check the Center of Pressure (CP) location, this could be doing some interesting things to the CP location.

Also, you can run the rocket on the RASAero II software at the trim angle of attack using the Run Test feature (see the RASAero II Users Manual Pages 86-92), and assume the entire rocket is at the trim angle of attack (ignoring that the bent nose is higher), then you will have an estimate for the increase in drag.

Note that you are doing all this with the Barrowman Method. For running RASAero II with the trim angle of attack you can run with the Barrowman Method, and then run it with the Rogers Modified Barrowman Method which includes viscous crossflow on the body for a more accurate CD at the angle of attack.

Charles E. (Chuck) Rogers
Rogers Aeroscience

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#### G_T

##### Well-Known Member
If parts aren't lined up coaxially, and if the rocket does not have some rotation rate, it's going to fly in an arc. Your only options at that point are active stabilization or spin. Spin evens out the direction of the arc over time.

Changes in stability aren't the root of the problem and finding that it is still stable doesn't prevent it from arcing over. Stable does not mean stays pointed in the same direction. Not under the assumption of a non-colinear airframe.

Gerald

#### Alan15578

##### Well-Known Member
As Allan15578 noted, you'll have to calculate the trim angle of attack. Take the CNalpha's for each of the rocket components, take the CNalpha for the bent nose, and varying angle of attack (with the nose cone being at 2 deg higher angle of attack than the other rocket components). With the CNalpha's and angle of attack (alpha) for each component you can determine the Normal Force for each component, and then determine the total angle of attack for the rocket where the Normal Forces balance. This is the trim angle of attack.

I kind of know where this is going; aerodynamic loads. Using the CNalpha's for all of the other components, and the CNalpha for the nose, and the fact that the other components and the nose will see different angles of attack (the nose is bent), you can turn the CNalpha's into Normal Forces on each component, and then you can calculate the bending force on the body tube.

Also note that after doing these CNalpha and Normal Force calculations, and coming up with the trim angle of attack, using the Normal Forces check the Center of Pressure (CP) location, this could be doing some interesting things to the CP location.

Also, you can run the rocket on the RASAero II software at the trim angle of attack using the Run Test feature (see the RASAero II Users Manual Pages 86-92), and assume the entire rocket is at the trim angle of attack (ignoring that the bent nose is higher), then you will have an estimate for the increase in drag.

Note that you are doing all this with the Barrowman Method. For running RASAero II with the trim angle of attack you can run with the Barrowman Method, and then run it with the Rogers Modified Barrowman Method which includes viscous crossflow on the body for a more accurate CD at the angle of attack.

Charles E. (Chuck) Rogers
Rogers Aeroscience
Nice.

I would not expect a 2 degree missalignment to be a serious problem. but it is good to see someone doing some due diligence. The trim angle should be inversely proportional to the static margin, so increasing fin area to increase the static margin will decrease the trim angle. The usual method for dealing with asymmetry is to induce a suitable amount of roll, but this can also lead to some cork screw flights. If breaking things out for loads analysis, remember that the Barrowman CNalpha is per radian, not per degree.

#### markfer20

##### New Member
As Allan15578 noted, you'll have to calculate the trim angle of attack. Take the CNalpha's for each of the rocket components, take the CNalpha for the bent nose, and varying angle of attack (with the nose cone being at 2 deg higher angle of attack than the other rocket components). With the CNalpha's and angle of attack (alpha) for each component you can determine the Normal Force for each component, and then determine the total angle of attack for the rocket where the Normal Forces balance. This is the trim angle of attack.

I kind of know where this is going; aerodynamic loads. Using the CNalpha's for all of the other components, and the CNalpha for the nose, and the fact that the other components and the nose will see different angles of attack (the nose is bent), you can turn the CNalpha's into Normal Forces on each component, and then you can calculate the bending force on the body tube.

Also note that after doing these CNalpha and Normal Force calculations, and coming up with the trim angle of attack, using the Normal Forces check the Center of Pressure (CP) location, this could be doing some interesting things to the CP location.

Also, you can run the rocket on the RASAero II software at the trim angle of attack using the Run Test feature (see the RASAero II Users Manual Pages 86-92), and assume the entire rocket is at the trim angle of attack (ignoring that the bent nose is higher), then you will have an estimate for the increase in drag.

Note that you are doing all this with the Barrowman Method. For running RASAero II with the trim angle of attack you can run with the Barrowman Method, and then run it with the Rogers Modified Barrowman Method which includes viscous crossflow on the body for a more accurate CD at the angle of attack.

Charles E. (Chuck) Rogers
Rogers Aeroscience
What is the trim angle conceptually, with regards to a rocket? I did some internet searching and while I understand it slightly it's still a bit confusing. Also, should the normal forces be acting on the CP of the entire rocket? I'm also confused how the normal forces are contributing to the change in the rocket's CP location.

#### Chuck Rogers

##### Well-Known Member
What is the trim angle conceptually, with regards to a rocket? I did some internet searching and while I understand it slightly it's still a bit confusing. Also, should the normal forces be acting on the CP of the entire rocket? I'm also confused how the normal forces are contributing to the change in the rocket's CP location.

You'll have to calculate the Normal Force for each component. You'll have the CNalpha for each component, including the nose cone, but the nose cone will be at a different angle of attack than the rest of the rocket (it will be 2 deg higher). Remember, as Alan15578 noted, the angle of attack used in the Barrowman Equations is in radians, not degrees.

Again, you can't just add up or balance the CNalpha's as usual. You'll have to add up and balance the Normal Forces for the rocket components.

The angle of attack where the Normal Forces front and rear balance, that is the trim angle of attack. With no wind or other disturbances your rocket would fly at that angle of attack, not zero degrees angle of attack like (theoretically) a regular rocket. With wind or other disturbances your rocket will oscillate around that angle of attack, not oscillate around zero degrees angle of attack like a regular rocket.

Charles E. (Chuck) Rogers
Rogers Aeroscience

#### G_T

##### Well-Known Member
It would not be that simple. If the rocket is shaped like a banana with fins, said fins lined up with their end of the banana, it will fly a curve if there is no spin. The pitching moment won't be zero. Thrust line is off-axis relative to center of mass. Nosecone end generates side force due to pointing direction. Center of mass is also off-axis the thrust line towards the same direction the nosecone will be pushing. Fins generate a side force due to misalignment with CM. All four effects contribute in the same direction to a net torque about the center of mass, resulting in a non-zero pitch rate.

Gerald

#### Chuck Rogers

##### Well-Known Member
It would not be that simple. If the rocket is shaped like a banana with fins, said fins lined up with their end of the banana, it will fly a curve if there is no spin. The pitching moment won't be zero. Thrust line is off-axis relative to center of mass. Nosecone end generates side force due to pointing direction. Center of mass is also off-axis the thrust line towards the same direction the nosecone will be pushing. Fins generate a side force due to misalignment with CM. All four effects contribute in the same direction to a net torque about the center of mass, resulting in a non-zero pitch rate.

Gerald

I should have noted that my statement was for power-off (to keep things simple). For power-on as you note you'd have to include the thrust misalignment (it's a force, it would be included along with the Normal Forces), and its effect would vary with the thrust curve.

You'd have to work the problem in detail to see what would happen. Doing the hand calculations you'd figure out what would happen. My opinion is at some point the fins would be at a high enough angle of attack (not exceeding the fin stall angle of attack) to overpower the misaligned nose, and the thrust misalignment; the rocket would have a trimmed angle of attack. Airplanes have trimmed angles of attack, unless they are unstable. We're assuming that the rocket is still statically stable.

The rocket would indeed fly a curved trajectory, at the trimmed angle of attack the continuous Total Normal Force (i.e., the Lift) would continuously turn the rocket trajectory to one side.

Charles E. (Chuck) Rogers
Rogers Aeroscience

#### G_T

##### Well-Known Member
Rocket stability is basically CG in front of CP along the direction of travel, with CM zero. Try to fly it horizontally and you'd find it generally doesn't fit the criteria for stability for horizontal flight. It's not at all the same thing. Rocket stability is more like throwing a dart. It arcs over into the ground, point first. With a plane, if nose-down relative to the horizontal flight position, the plane speeds up. Due to the CM having a negative value (inverse relationship to angle of attack, to a linear approximation for small alpha) the increased nose-lifting torque as speed increases brings the nose back up. This increases the angle of attack, increases the lift, and it slows back down. You may get a phugoid; you may get restoration of horizontal(ish) flight, or you may get stall and divergence. Any of those could happen with a plane that is stable in horizontal flight, possibly depending on the magnitude of the disturbance for the third case. It depends on the details.

If the typical rocket had the stability of a plane, it wouldn't come in ballistic if recovery fails. It would glide away somewhere.

Gerald

PS - Somewhat poorly phrased. It's been a while.

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#### Chuck Rogers

##### Well-Known Member
Let's go through the calculations. Actually, this will answer markfer20's original question on how to do the calculations.

Using Centuri TIR-33;

Using the Javelin rocket example on Pages 21-23 of TIR-33, the Javelin Nose and Fin fb (Fin in the Presence of the Body) CNalpha and CP are:    The Total Rocket CP is 11.3 inches. We will assume that the Nose is bent up 2.0 degrees, so, as an example, when the rocket is at an angle of attack of 2.0 deg, the local angle of attack for the Nose will be 4.0 deg. The local angle of attack for the Fins will be 2.0 deg.

The Moment Reference will be about the Center of Gravity (CG). Assuming 1.0 Caliber Stabilit Margin, TIR-33 has the Javelin CG at: The Moments are balanced about the CG, using the Nose Contribution and the Fin Contribution. The moment arms are the distances from the component CP's to the Rocket CG:

Note, that as will be seen, we can't balance the CNalpha's, we have to balance the Normal Forces.

The Nose and Fin Normal Forces are: Note that 1/2 rho V^2 (dynamic pressure) and Aref (the Reference Area) will be on both sides of the equation, and will drop off.

With a positive angle of attack, the moment from the Nose tries to rotate the rocket clockwise, the moment from the Fins tries to rotate the rocket counterclockwise. If the two moments are equal, there is no rotation of the rocket. This will occur at the trim angle of attack.

Balancing the moments:

(CNalphaNose x (aoa + 2 deg)) x (Xcg - NoseCP) = (CNalphaFin)fb x aoa x (FinCP-Xcg)

Putting in the numbers, converting 2 deg to radians (the Barrowman CNalpha's are per radian), 2 deg = 0.03491 radians,

(2.0 x (aoa + 0.03491)) x (10.55 - 1.68) = (33.9 x aoa) x (11.86 - 10.55)

(17.74 x aoa) + 0.6193 = (44.409 x aoa)

0.6193 = 26.669 aoa

aoa = 0.02322 radians = 1.33 deg

So the trim angle of attack will be at an angle of attack of 1.33 deg.

Note that TIR-33 has an Appendix (Appendix 12 on Page 35) providing a proof on why we normally can use CNalpha to replace Normal Force when doing these calculations. Of course this time, since the bent Nose had a different local angle of attack than the Fins, we had to use the actual Normal Force, not the Cnalpha's.

Again, in summary, the trim angle of attack is 1.33 deg. Absent any other disturbing forces, the rocket will fly at an angle of attack of 1.33 deg. Because of the continuous Normal Force, lift is being generated. The rocket will fly a slowing arcing trajectory because of the continuous (small) lift.

Now addressing Thrust misalignment. The Thrust is now misaligned by only 1.33 deg (the trim angle of attack). The normal component of the misaligned thrust will only be 2.3% of the total thrust. One could argue also how much the nozzle might be misaligned anyway, a 1.33 deg misalignment might not be so large compared to the expected variation of the rocket nozzle alignment in our rocket motors.

In summary, the rocket will fly, it will just fly a gently arcing trajectory. I could actually predict the trajectory using Normal Force from the trim angle of attack, but again a 1.33 deg trim angle of attack is very small.

In TIR-33 on Page 36 a projected "Banana Rocket" is included. Using the 1/2 deg (0.5 deg) deflection of the tail, and balancing the moments again, we get (0.5 deg converted to 0.0087266 radians)

(CNalphaNose x (aoa + 2 deg)) x (Xcg - NoseCP) = ((CNalphaFin)fb x (aoa - 0.5 deg)) x (FinCP-Xcg)

(CNalphaNose x (aoa + 0.03491)) x (Xcg - NoseCP) = ((CNalphaFin)fb x (aoa - 0.0087266)) x (FinCP-Xcg)

Note that the bent Nose gets 2.0 deg added to the angle of attack, the bent Tail (Fins) gets 0.5 deg subtracted from the angle of attack.

Continuing, again using the Javelin numbers:

(2.0 x (aoa + 0.03491)) x (10.55 - 1.68) = (33.9 x (aoa - 0.0087266)) x (11.86 - 10.55)

(17.74 x aoa) + 0.6193 = (44.409 x aoa) - 0.3875396

1.00684 = 26.669 aoa

aoa = 0.037753 radians = 2.163 deg

So for the "Banana Rocket" the trim angle of attack will be at an angle of attack of 2.163 deg. Still a pretty small trim angle of attack.

Charles E. (Chuck) Rogers

#### Chuck Rogers

##### Well-Known Member
Rocket stability is basically CG in front of CP along the direction of travel, with CM zero. Try to fly it horizontally and you'd find it generally doesn't fit the criteria for stability for horizontal flight. It's not at all the same thing. Rocket stability is more like throwing a dart. It arcs over into the ground, point first. With a plane, if nose-down relative to the horizontal flight position, the plane speeds up. Due to the CM having a negative value (inverse relationship to angle of attack, to a linear approximation for small alpha) the increased nose-lifting torque as speed increases brings the nose back up. This increases the angle of attack, increases the lift, and it slows back down. You may get a phugoid; you may get restoration of horizontal(ish) flight, or you may get stall and divergence. Any of those could happen with a plane that is stable in horizontal flight, possibly depending on the magnitude of the disturbance for the third case. It depends on the details.

If the typical rocket had the stability of a plane, it wouldn't come in ballistic if recovery fails. It would glide away somewhere.

Gerald

PS - Somewhat poorly phrased. It's been a while.

Perhaps my "all airplanes have a trim angle of attack" comment started a mix-up here between airplane dynamics and rocket dynamics. My point was you can have some very complex aircraft shapes, they will have a trim angle of attack. We're obviously talking about rocket dynamics here.

A fin-stabilized rocket flies by oscillating around the angle of attack where the moments are zero. The angle of attack has an excursion below or above that point, if the rocket is stable, the moments return the rocket towards the condition where the moments are zero. And then it overshoots on the other side, etc., etc.

It just turns out that since our rockets are symmetrical, they oscillate around an angle of attack of zero, where there are no aerodynamic moments.

It turns out in this case for a rocket with a bent nose, the angle of attack where the moments are zero is non-zero, it is an angle of attack of 1.33 deg.

Going through the rocket dynamics equations, there is no reason why the rocket wouldn't have the same dynamic response. Instead of oscillating around an angle of attack of zero, it would oscillate around an angle of attack of 1.33 deg.

If the symmetric rocket is launched with no wind, no misalignments, etc., then it will fly straight up. The rocket with the bent nose, again assuming no wind and no misalignments, will have a small continuous Normal Force (Lift), which will slowly turn the trajectory into a slow arc.

Charles E. (Chuck) Rogers

#### Alan15578

##### Well-Known Member
Now addressing Thrust misalignment. The Thrust is now misaligned by only 1.33 deg (the trim angle of attack). The normal component of the misaligned thrust will only be 2.3% of the total thrust. One could argue also how much the nozzle might be misaligned anyway, a 1.33 deg misalignment might not be so large compared to the expected variation of the rocket nozzle alignment in our rocket motors.
I have not checked your math, but it seems like a pretty good explanation. However, I think you could have provided a better explanation for thrust misalignment. Strictly speaking, the trim angle of attack is not a thrust misalignment. You need to consider the thrust moment about the CG from a thrust misalignment. This moment will be balanced by the aerodynamic moment when the stable rocket rotates to the trim position. This trim angle is more complicated. It can be quite large at low speed. and rather low at high speed, and of course all the other asymmetrys and misalignments should be considered as well.

I have seen nozzles visibly misaligned before firing, and even nozzles asymmetrically eroded during and after firing. The effects can be mitigated by using a larger static margin, and well chosen fin cant (roll rate). I really want to know the actual expected variation, and worst case variation, of effective nozzle alignment in our certified motors.

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#### Chuck Rogers

##### Well-Known Member
I have not checked your math, but it seems like a pretty good explanation. However, I think you could have provided a better explanation for thrust misalignment. Strictly speaking, the trim angle of attack is not a thrust misalignment. You need to consider the thrust moment about the CG from a thrust misalignment. This moment will be balanced by the aerodynamic moment when the stable rocket rotates to the trim position. This trim angle is more complicated. It can be quite large at low speed. and rather low at high speed, and of course all the other asymmetrys and misalignments should be considered as well.

I agree. My basic point was that "it's small" for this 2.0 deg bent nose example. And as you note, in this example technically the thrust is not misaligned. If only the nose is bent, then the thrust goes through the Center of Gravity (CG).

As you note, the actual typical example of a straight (non-bent) rocket, with the thrust misaligned from a misaligned nozzle, is an interesting case. In the past the Tripoli Class 3 Committee has used a 1-sigma input for the uncertainty in the Thrust Axis of 0.2 deg for the various Monte-Carlo dispersion programs.

Charles E. (Chuck) Rogers