## Engineering Question #31

Mark Bodnar, a 33 year old male from Abilene, Texas, U.S.A. asks on July 15, 1999,

Can you help me find a simple definition of "inertial coupling" (if there is a simple definition). It's an aeronautics term. I did write to NASA a while ago, but the definition they sent was too complex.

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### John Jones answered on July 15, 1999

This is a tricky one; I've found several explanations, but they're like NASA's. I've only come across the expression `inertial coupling' in the context of vibrating machinery. In that context, coupling between two modes of vibration means that if you excite one mode of vibration, you also excite the other.

For example, you could have a lathe mounted on a spring at either end. Two modes of vibration available to the lathe are rotation around its centre of mass, and up-and-down motion of its centre of mass. If you place a hand somewhere along the axis of the lathe and bounce it up and down, you'll excite both modes of vibration. Now, in describing a mechanical system, such as a lathe, you generally introduce a coordinate system so you can describe the mechanics in terms of masses moving within that coordinate system. For example, with the lathe mentioned above, you could choose the height of its centre of gravity above the base plane, x1, and the angle between the lathe axis and the horizontal, as two coordinates, x2, and write a pair of equations of motion for the system's motion in these two directions. The equations will look like this: ( m11 m12) x1'' + (d11 d12)x1' + (k11 k12)x1 = 0 ( m21 m22) x2'' + (d21 d22)x2' + (k21 k22)x2 = 0 where m11, etc, are masses, d11, etc., are damping coefficients, and k11, etc., are spring constants. The presence of the _12 and _21 terms couples the two coordinates chosen. If the m12 and m21 terms are non-zero, while the d12, d21, k12 and k21 terms are zero, this is pure inertial coupling. It is always possible to choose a set of coordinates such that the equations of motion are uncoupled. You might try looking in a Vibrations text for more detail.

### Philip Joseph answered on August 3, 2005

I believe that inertial coupling (in the aeronautical sense) is what is also called "roll coupling", in which a maneuver in one axes will cause an uncommanded maneuver in one or two others.

From a page about the X-3 experimental plane at Answers.com: "On October 27, 1954, Walker made an abrupt left roll at Mach 0.92 and an altitude of 30,000 feet. The X-3 rolled as expected, but also pitched up 20 degrees and yawed 16 degrees. The aircraft gyrated for five seconds before Walker was able to get it back under control. He then set up for the next test point. Walker put the X-3 into a dive, accelerating to Mach 1.154 at 32,356 feet, where he made an abrupt left roll. The aircraft pitched down and reached a g-loading of -6.7, then pitched upward to +7 Gs. At the same time, the X-3 sideslipped, resulting in a loading of 2 Gs. Walker managed to bring the X-3 under control and successfully landed... Walker and the X-3 had experienced "roll coupling," in which a maneuver in one axes will cause an uncommanded maneuver in one or two others... The principal contribution of the X-3 was its data on inertia coupling - a potentially violent divergence from the intended flight path when executing an abrupt maneuver which had afflicted the X-1 and X-2 aircraft."

### Francis Dudley, mechanical engineer, Arlington, Texas answered on August 16, 2007

Here is an explanation of roll coupling that can be intuitively understood. Imagine any aircraft with a weight attached to the tip of its vertical fin, i.e., high and to the rear.  Now imagine another weight attached to its lowered nosewheel, i.e., low and forward.  If the weights are suitably chosen, the aircraft could still be in balance and could still fly.  If you tried to roll such an aircraft quickly, however, the two weights would tend to fly out, relative to the roll axis, and this would result in the nose of the aircraft pitching down. ("Down", that is, relative to the aircraft.  It would actually move out relative to the roll axis).  As this happened, gyroscopic precession would kick in and the nose would move to one side also, depending on the direction of roll.

Aircraft don't have weights attached to them in this manner, of course, but you can imagine that the various masses in the fuselage are not neatly arranged along the roll axis. Something similar happens to rotating components in machinery if they are not dynamically ballanced.

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