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What is the difference between "x" and "+" configuration tail fins, and what are the advantages of each in free flight? (for example JDAM missiles)

enter image description here

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    $\begingroup$ The difference is a rotation by 45°. $\endgroup$ – Peter Kämpf Apr 2 '17 at 11:45
  • $\begingroup$ @PeterKämpf you mean the max rotation of "x" configuration will be 45° ,or max rotation in "+" configuration will be 45° $\endgroup$ – Michael Apr 3 '17 at 6:19
  • $\begingroup$ @PeterKämpf: Very funny, but you're obviously missing something. The drawings imply that the tail fins operate differently (except for roll) in both configurations, so as to move the missile only in the vertical and horizontal axis in any of the configurations. So the question is valid if these different configurations do indeed exist. OP: Do you have the source where the drawing is taken from? $\endgroup$ – Scrontch Apr 3 '17 at 15:20
  • $\begingroup$ @Scrontch: Missiles are stabilised by rolling. Yes, the effectiveness of the control surface varies slightly with roll angle, but in reality commands are almost always a mixture of pitch and yaw, and constantly changing due to the rolling motion. This is no either - or situation. $\endgroup$ – Peter Kämpf Apr 3 '17 at 16:04
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    $\begingroup$ @PeterKämpf: Missiles are stabilised by rolling. Do you have any sources for that? From what i was able to pick up, i conclude this is NOT true. The vast majority of missiles (MANPADS are an exception) are roll-stabilized, i.e. they actively counter-act roll to maintain a fixed longitudinal plane. (Which of course can either be the + or the X plane, with respect to the control fins - which brings us back to the question) cf. maritime.org/doc/missile/part1.htm, section 5A2 books.google.fr/books?id=ubcczZUDCsMC, section 7-2 $\endgroup$ – Scrontch Apr 4 '17 at 8:39
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Mounting missiles below a wing or stocking them in weapon bays is much easier with X-Configured missiles. Look at these photos:

K-5M Missile Underwing missile and attachmentSource

AIM-120 Missile Weapon bay with missilesSource

"+" mounted missiles would need more complex hard points and larger weapon bays.

Once fired the configuration doesn't matter anymore as all four control surfaces are continuously used to guide and stabilize the missile along its 3d trajectory.

If the missile only moves in the horizontal or vertical plane, the X configuration is better. For a fixed force F the angle of attack of the control surfaces ($\alpha_{cs}$) is smaller and can therefore be set faster, making the missile more agile. Assuming* a linear relationship between the the control surface force and its angle of attack:

$$\alpha_{csx} = \frac{F}{q \cdot S \cdot C_{L\alpha}}\cdot \frac{\sqrt{2}}{4}$$ $$\alpha_{cs+} = \frac{F}{q \cdot S \cdot C_{L\alpha}}\cdot \frac{1}{2}$$

For a quadratic relationship* between the drag ($D_{cs}$) and the angle of attack of the control surfaces there is no difference in the total drag.

$$D_{cs+} = q \cdot S \cdot (4\cdot C_{D0}+2 \cdot C_{D\alpha} \alpha_{cs+}^2) = 4\cdot q \cdot S \cdot C_{D0} + \frac{1}{2} C_{D\alpha} \cdot (\frac{F}{q \cdot S \cdot C_{L\alpha}})^2$$

$$D_{csx} = q \cdot S \cdot (4\cdot C_{D0}+4 \cdot C_{D\alpha} \alpha_{csx}^2) = 4\cdot q \cdot S \cdot C_{D0} + \frac{1}{2} C_{D\alpha} \cdot (\frac{F}{q \cdot S \cdot C_{L\alpha}})^2$$

*If someone has better assumptions for lift and drag please comment or correct the post.

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    $\begingroup$ Note that some missiles, such as the AIM-7 Sparrow use the upper fin for mounting. They have moved away from that, though. Maybe due to occurrences like this $\endgroup$ – TomMcW Apr 2 '17 at 20:55
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    $\begingroup$ @TomMcW I'm loving that link $\endgroup$ – Pugz Apr 3 '17 at 1:24
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    $\begingroup$ @jwenting i appreciate your help, but which one do you prefer for maneuvering "x" or "+" and why ? $\endgroup$ – Michael Apr 3 '17 at 7:03
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    $\begingroup$ @jwenting Yes, that's why I wrote that once fired the configuration doesn't matters anymore: it will fly following some 3d-trajectory combining pitch, yaw and roll movements in unknown amounts depending on the target (airplane) and disturbances. $\endgroup$ – Gypaets Apr 3 '17 at 7:09
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    $\begingroup$ @TomMcW great story on the link! $\endgroup$ – PJNoes Apr 4 '17 at 17:28
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The difference between cross and plus configuration has nothing to do with equations of motion. The difference is just in control system implementation and transferring the control surface deflections to equivalent 3 plane surfaces. for example cross config can be controlled better, while plus config in simpler. or: for optimum control efort for a specified da dr de: in plus: d1= da-dr in cross:d1=da+de-dr

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