Question for the aerospace engineers among us.

[pathology_doc]

Image source: Missile Design and Systems Engineering, by Gene Fleeman, AIAA Press 2012. Fair use claimed for posting of limited amount of source text for discussion and/or analysis.

Question/discussion below the image.



So, the equation in the box at the top is for the missile static margin. As I understand it, the aerodynamic centre should be behind the centre of pressure for inherent stability - i.e. measured from the tip of the nose, Xac should be greater and Xac - Xcg should be positive. By those lights, a tail which is larger in area and/or has a higher Cn slope should give greater stability.

However, the (Xcg - Xac) term for the tail is intrinsically negative, since the position of the tail Xac is always going to be greater than that of the CG, and thus the term will always exert a destabilizing effect. Which it shouldn't.

I can understand why the coefficient on the body term is negative, since the body lift exerted at the nose is always destabilizing (magnifies any inadvertent pitch). But the tail term? This mystifies me. I feel like there should be an absolute value bar around it, or it should be Xac - Xcg, which is still the moment arm, but reflects in its sign the effect on stability.

Compounding my confusion are the graph and the text box labelled "Example rocket baseline". These show (Xcg - Xac) divided not by the diameter of the rocket but by its LENGTH, a completely different state of affairs.

Every other example of centre of pressure calculation that I've found online so far is calculated from the rocket's nose as datum, so comparison is impossible.

I've been in communication with the author, but while he's helpful, he's also a busy man and can take a while to get back to me. Thought I would bring it here and see if anyone could help.

What am I missing/overlooking?

 

[Desertfox]

The datum should be the nose. My guess is you are missing a negative sign somewhere..

 

[martinbayer]

To take any guesswork out of this discussion, I recommend to contact Mr. Fleeman directly at https://www.linkedin.com/in/eugene-fleeman-15843316/, which also provides his email address as [email protected].

 

[pathology_doc]

To take any guesswork out of this discussion, I recommend to contact Mr. Fleeman directly at https://www.linkedin.com/in/eugene-fleeman-15843316/, which also provides his email address as [email protected].

I'm currently waiting for him to reply to another e-mail before I bother him yet again!

I just figured I should get a second opinion and make absolutely sure that it's not me who's wrong! Pointless to waste his time if in fact it's me who has missed something blindingly obvious.

 

[amadolabonte]

Image source: Missile Design and Systems Engineering, by Gene Fleeman, AIAA Press 2012. Fair use claimed for posting of limited amount of source text for discussion and/or analysis.

Question/discussion below the image.

View attachment 803258

So, the equation in the box at the top is for the missile static margin. As I understand it, the aerodynamic centre should be behind the centre of pressure for inherent stability - i.e. measured from the tip of the nose, Xac should be greater and Xac - Xcg should be positive. By those lights, a tail which is larger in area and/or has a higher Cn slope should give greater stability.

However, the (Xcg - Xac) term for the tail is intrinsically negative, since the position of the tail Xac is always going to be greater than that of the CG, and thus the term will always exert a destabilizing effect. Which it shouldn't.

I can understand why the coefficient on the body term is negative, since the body lift exerted at the nose is always destabilizing (magnifies any inadvertent pitch). But the tail term? This mystifies me. I feel like there should be an absolute value bar around it, or it should be Xac - Xcg, which is still the moment arm, but reflects in its sign the effect on stability.

Compounding my confusion are the graph and the text box labelled "Example rocket baseline". These show (Xcg - Xac) divided not by the diameter of the rocket but by its LENGTH, a completely different state of affairs.

Every other example of centre of pressure calculation that I've found online so far is calculated from the rocket's nose as datum, and while I was searching I even came across unrelated stuff like fast way to make cash in houston, so comparison is impossible.

I've been in communication with the author, but while he's helpful, he's also a busy man and can take a while to get back to me. Thought I would bring it here and see if anyone could help.

What am I missing/overlooking?

If the missile pitches up slightly, the tail being behind the CG sees that angle of attack and generates lift upward. An upward force acting behind the center of gravity creates a nose down pitching moment. A nose down moment opposes the original pitch up disturbance. That is static stability. In the math, a restoring nose down moment is defined as negative. Since the tail force is positive lift acting at a negative moment arm CG minus tail AC, their product is negative. Negative moment slope equals stabilizing. So the term looks negative not because the tail is destabilizing, but because the sign convention defines restoring moments as negative. The physics is correct and the sign simply reflects the coordinate system being used.