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{{Short description|Abrupt change in an aircraft's roll due to fuselage inertia}}
In aeronautics, '''inertia coupling''',<ref>Flightwise - Volume 2, Aircraft Stability and Control, Christopher Carpenter 1997, Airlife Publishing Ltd., {{ISBN|1 85310 870 7}}, p.336</ref> also referred to as '''inertial coupling'''<ref name="ALStabCtrl">{{wikicite|reference=Airplane Stability and Control - Second edition, Abzug and Larrabee, Cambridge University Press, {{ISBN|0-521-02128-6}}, p.109|ref={{harvid|Abzug|Larrabee}}}}</ref> and '''inertial roll coupling''',<ref>{{cite tech report |url=https://www.nasa.gov/centers/dryden/pdf/88484main_H-2106.pdf |title=Coupling Dynamics in Aircraft: A Historical Perspective|first1=Richard&nbsp;E.|last1=Day |institution=NASA Office of Management Scientific and Technological Information Program |location=Edwards, California |year=1997 |access-date=December 10, 2020|p=2|archive-url=https://web.archive.org/web/20050205051309/https://www.nasa.gov/centers/dryden/pdf/88484main_H-2106.pdf|archive-date=5 Feb 2005|number=Special publication 532}}</ref> is a potentially catastrophic phenomenon of [[high-speed flight]] in a [[Fineness ratio|long, thin]] aircraft, in which an intentional [[Aircraft principal axes|rotation of the aircraft about one axis]] prevents the aircraft's design from inhibiting other unintended rotations.<ref name="ALStabCtrl" />   The problem became apparent in the 1950s, when the first supersonic jet [[fighter aircraft]] and research aircraft were developed with narrow [[wingspan]]s, and caused the loss of aircraft and pilots before the design features to counter it (e.g. a big enough [[fin]]) were understood.<ref name="Hurt 315" />  

The term "inertia/inertial coupling" has been criticized as misleading, because the phenomenon is not solely an instability of inertial movement, like the [[Janibekov effect]].  Instead, the phenomenon arises because aerodynamic forces [[damping|react too slowly]] to track an aircraft's orientation.<ref name="Hurt 315" /><ref name=USAFTPTxt>{{cite book |date=April 1986 |title=Flying Qualities | volume=II: Part 2 |url=https://archive.org/details/DTIC_ADA170960/page/n3 |access-date=December 10, 2020 |publisher=[[Defense Technical Information Center]] |page=9.1 |via=the [[Internet Archive]] |id=ADA170960 |ref={{harvid|USAF|1986}}}}</ref>  At low speeds and thick air, [[Aerodynamics|aerodynamic]] forces match aircraft [[translation (physics)|translational velocity]] to orientation, avoiding the dangerous dynamical regime.  But at high speeds or thin air, the [[wing]] and [[empennage]] may not generate sufficient [[Force|forces]] and [[Moment of force|moments]] to stabilize the aircraft.<ref name="Hurt 315">{{cite book |last=Hurt |first=H. H. Jr. |title=Aerodynamics for Naval Aviators |date=January 1965 |publisher=U.S. Navy, Aviation Training Division |location=U.S. Government Printing Office, Washington D.C. |page=315 |id=NAVWEPS 00-80T-80 |orig-date=1960}}</ref>

==Description==
{{Technical|section|date=November 2024}}
Inertia coupling tends to occur in aircraft with a [[fineness ratio|long, slender]], high-density [[fuselage]].  A simple, yet accurate mental model describing the aircraft's [[mass distribution]] is a [[rhombus]] of [[point mass]]es: one large mass fore and aft, and a small one on each wing.  The [[inertia tensor]] that this distribution generates has a large [[Aircraft principal axes|yaw]] component and small [[Aircraft principal axes|pitch and roll]] components, with the pitch component slightly larger.{{Sfn|USAF|1986|pp=9.3-9.4}}

[[Euler's equations (rigid body dynamics)|Euler's equations]] govern the rotation of an aircraft.  When {{math|''&omega;''<sub>r</sub>}}, the [[angular velocity|angular rate of roll]], is [[control system|controlled]] by the aircraft, then the other rotations must satisfy <math display=block>\begin{align}
I_\text{y}\dot{\omega_\text{y}}&=(I_\text{p}-I_\text{r})\omega_\text{r}\omega_\text{p}+T_y \\
I_\text{p}\dot{\omega_\text{p}}&=-(I_\text{y}-I_\text{r})\omega_\text{r}\omega_\text{y}+T_p
\end{align}</math> where y, p, and r indicate yaw, pitch, and roll; {{mvar|I}} is the [[moment of inertia]] along an axis; {{mvar|T}} the external torque from aerodynamic forces along an axis; and [[Notation_for_differentiation#Newton's_notation|dots]] indicate [[time derivative]]s.{{sfn|Day|1997|p=53}}{{sfn|Phillips|1948|p=4}}   When aerodynamic forces are absent, this 2{{nbh}}variable [[system of equations|system]] is the equation of a [[simple harmonic oscillator]] with frequency {{math|(1-{{sfrac|''I''<sub>r</sub>|''I''<sub>p</sub>}})(1-{{sfrac|''I''<sub>r</sub>|''I''<sub>y</sub>}})''&omega;''{{su|b=r|p=2}}}}: a rolling [[Space Shuttle]] will naturally undergo small oscillations in pitch and yaw.  

Conversely, when the craft does not roll at all ({{math|''&omega;''<sub>r</sub>{{=}}0}}), the only terms on the right-hand side are the aerodynamic torques, which are ([[small angle approximation|at small angles]]) proportional to the craft's angular orientation {{mvar|&theta;}} to the [[freestream]] air.  That is: there are natural constants {{mvar|k}} such that an unrolling aircraft experiences{{sfn|Day|1997|p=53}}{{sfn|Phillips|1948|p=6}} <math display=block>\begin{align}
I_\text{y}\dot{\omega_\text{y}}&=T_y=-k_\text{y}I_\text{y}\theta_\text{y} \\
I_\text{p}\dot{\omega_\text{p}}&=T_p=-k_\text{p}I_\text{p}\theta_\text{p}
\end{align}</math>  

In the full case of a rolling aircraft, the connection between orientation and angular velocity is not entirely straightforward, because the aircraft is a [[rotating reference frame]].  The roll inherently exchanges yaw for pitch and vice-versa: <!--This may have an unimportant sign error: depending on how the axes are set up, \omega_r may need to be -\omega_r instead and vice-versa.--><math display=block>\begin{align}
\dot{\theta_\text{y}}&=\omega_\text{y}+\omega_\text{r}\theta_\text{p} \\
\dot{\theta_\text{p}}&=\omega_\text{p}-\omega_\text{r}\theta_\text{y}
\end{align}</math>  Assuming nonzero roll, [[Nondimensionalization|time can always be rescaled]] so that {{math|''&omega;''<sub>r</sub>{{=}}1}}.  The full equations of the body are then of two [[damping|damped]], coupled [[harmonic oscillators]]: <math display=block>\begin{align}
0&=\ddot{\theta_\text{y}}-(1+J_\text{y})\dot{\theta_\text{p}}+(k_\text{y}-J_\text{y})\theta_\text{y} \\
0&=\ddot{\theta_\text{p}}+(1-J_\text{p})\dot{\theta_\text{y}}+(k_\text{p}-J_\text{p})\theta_\text{p}
\end{align}</math> where <math display=block>\begin{align}
J_\text{y}&=\frac{I_\text{p}-I_\text{r}}{I_\text{y}} \\
J_\text{p}&=\frac{I_\text{y}-I_\text{r}}{I_\text{p}}
\end{align}</math>  But if {{math|''k''&approx;''J''}} in either axis, then the damping is eliminated and the system is [[unstable]].{{sfn|Day|1997|pp=1,53}}{{sfn|Phillips|1948|pp=7-9}}  

In dimensional terms (that is, unscaled time), instability requires {{math|''k''&approx;''J&omega;''<sub>r</sub>}}.  Since {{math|''I''<sub>r</sub>}} is small, <math display=block>J_\text{y}J_\text{p}\approx1</math>  In particular, one {{mvar|J}} is at least 1.  In thick air, {{mvar|k}} are too large to matter.  But in thin air and supersonic speeds, they decrease, and may become comparable to {{math|''&omega;''<sub>r</sub>}} during a rapid roll.<ref name=naca1948>{{cite tech report |url=http://naca.central.cranfield.ac.uk/reports/1948/naca-tn-1627.pdf |title=Effect of Steady Rolling on Longitudinal and Directional Stability |institution=National Advisory Committee for Aeronautics |number=1627 |first=William&nbsp;H |last=Phillips |location=Washington |page=2 |date=June 1948 |access-date=December 10, 2020 |archive-url=https://web.archive.org/web/20121129173013/http://naca.central.cranfield.ac.uk/reports/1948/naca-tn-1627.pdf |archive-date=29 Nov 2012}}</ref>  

Techniques to prevent inertial roll coupling include increased directional stability ({{mvar|k}}) and reduced roll rate ({{math|''&omega;''<sub>r</sub>}}).  Alternatively, the unstable aircraft dynamics may be [[mitigated]]: the unstable [[normal mode|modes]] require time to grow, and a sufficiently short-duration roll at limited angle of attack may allow recovery to a controlled state post-roll.{{Sfn|Hurt|1965|p=319}}

==Early history==
In 1948, William Phillips described inertial roll coupling in the context of [[missile]]s in an [[NACA]] report.<ref name=naca1948/> However, his predictions appeared primarily theoretical in the case of planes.{{sfn|Day|1997}}  The violent motions he predicted were first seen in the X-series research aircraft and Century-series fighter aircraft in the early 1950s. Before this time, aircraft tended to have greater width than length, and their mass was generally distributed closer to the [[center of mass]]. This was especially true for propeller aircraft, but equally true for early jet fighters as well. The effect became obvious only when aircraft began to sacrifice aerodynamic surface area to reduce drag, and use longer [[fineness ratio]]s to reduce supersonic drag. Such aircraft were generally much more fuselage-heavy, allowing gyroscopic effects to overwhelm the small control surfaces.

The roll coupling study of the [[X-3 Stiletto]], first flown in 1952, was extremely short but produced valuable data. Abrupt aileron rolls were conducted at Mach 0.92 and 1.05 and produced "disturbing" motions and excessive accelerations and loads.{{sfn|Day|1997|p=36}}

In 1953, inertial roll coupling nearly killed [[Chuck Yeager]] in the [[Bell X-1|X-1A]].<ref>{{cite web |author=Dr. James Young |title=The story of Chuck Yeager's wild ride in the Bell X-1A |url=http://www.chuckyeager.com/wild-ride-in-the-bell-x-1a |access-date=8 February 2015 |website=chuckyeager.com}}</ref>

Inertial roll coupling was one of three distinct coupling modes that followed one another as the rocket-powered [[Bell X-2]] hit Mach 3.2 during a flight on 27 September 1956, killing pilot [[Milburn G. Apt|Captain Mel Apt]].  Although [[flight simulator|simulators]] had predicted that Apt's maneuvers would produce an uncontrollable flight regime, at the time most pilots did not believe that the simulators accurately modeled the plane's flight characteristics.{{sfn|Day|1997|p=8}}

The first two production aircraft to experience inertial roll coupling were the [[F-100 Super Sabre]] and [[F-102 Delta Dagger]] (both first flown in 1953). The F-100 was modified with a larger vertical tail to increase its directional stability.{{sfn|Day|1997|p=39}} The F-102 was modified to increase wing and tail areas and was fitted with an augmented control system. To enable pilot control during dynamic motion maneuvers the tail area of the F-102A was increased 40%.

In the case of the [[F-101 Voodoo]] (first flown in 1954), a [[stability augmentation system]] was retrofitted to the A models to help combat this problem.

The [[Douglas Skyray]] was not able to incorporate any design changes to control inertial roll coupling and instead had restricted maneuver limits at which coupling effects did not cause problems.{{sfn|Abzug|Larrabee|p=119}}

The [[Lockheed F-104 Starfighter#Design|Lockheed F-104 Starfighter]] (first flown in 1956) had its [[stabilator]] (horizontal tail surface) mounted atop its vertical fin to reduce inertia coupling.

==See also==
* [[UPRT|Upset Prevention and Recovery Training]]

==References==
{{reflist}}

{{DEFAULTSORT:Inertia Coupling}}
[[Category:Aircraft aerodynamics]]
[[Category:Aviation risks]]
[[Category:Chuck Yeager]]