On the Motion of Airplanes Fixing a Century-Old Error in Flight Dynamics

N Ananthkrishnan & Nandan K Sinha

Recently, a hundred-year-old error at the heart of flight dynamics was discovered and fixed. Following this, the

basic theory of aircraft flight dynamics and its key results have undergone substantial change. The background

and the aftermath of this development have been concisely described in this article.

Keywords: Flight dynamics, error in Bryan’s model, corrected aerodynamic model, revised theory of aircraft flight

Believe it or not, the modern equations for simulating the motion of an airplane in flight were presented

over a century ago by Bryan1 in 1911 (see Box A) less than a decade after the Wright brothers’ pioneering

flight. These equations have been carried over virtually unchanged (but for minor notational differences)

to the present. In the interim, these equations have been used to analyze and predict aircraft dynamic

behavior, to simulate flight trajectories, and to design flight control laws – all quite successfully. Over the

years, dozens of textbooks by experienced practitioners in the field have appeared, all of them more or

less faithfully reproducing the equations by Bryan. If anything, Bryan’s equations can be claimed to have

stood the test of time.

Yet in an upcoming textbook2 (Fig.1), the present authors have pointed out an error in the model put

forward by Bryan – an error that has plagued flight dynamics for over a century but went unnoticed. In

the rest of this article, we shall try to convey in simple terms the source of this error and speculate on how

it may have arisen; we describe briefly how it has now been fixed; we explore the implications of this fix

for flight dynamics education and practice, and conclude with an outlook for scientific research in the

country. This article is written in as elementary and non-technical a manner as possible so as to appeal to

a wider audience.

The Crux of the Matter

Consider the sketch in Fig. 2 which shows two different kinds of aircraft motion – the aircraft in sub-

figure (a) is flying along a straight line (fixed velocity vector V) as it pitches nose-up (tail-down) as

marked by the clockwise arrow. In contrast, the sketch in sub-figure (b) shows an aircraft that is also

pitching nose-up (tail-down) but along a curved flight path such that the velocity vector V also pitches in

sync with the aircraft. Effectively, in case (b), the velocity vector V is always aligned with the XB axis

fixed to the aircraft whereas in case (a) the XB axis swings away from the velocity vector whose direction

is fixed. From the point of view of an inertial observer (one on the ground), the XB-Z

B axes (fixed to the

aircraft) in either case (a) or (b) appear to rotate in the same manner, even at the same rate if the pitching

1 G. H. Bryan, Stability in Aviation, Macmillan, 1911.

2 N. K. Sinha and N. Ananthkrishnan, Elementary Flight Dynamics with an Introduction to Bifurcation and

Continuation Methods, CRC Press, Taylor & Francis, 2013.

rate in the two cases is identical. The question is will the airplane experience the same aerodynamic forces

(and moments) in the two cases?

To answer this question, we must focus on the rear (tail) of the airplane where the horizontal stabilizer

(see Fig. 3) is usually located. What is the relative airflow that the tail (the horizontal stabilizer) sees in

each of these cases (a) and (b)? The aerodynamic forces (and moments) are a function of this relative

airflow.

In case (a), the tail (in fact, the entire airplane) by virtue of its forward speed V sees an equal and opposite

relative airflow of velocity V – this is marked in the inset to Fig. 2(a). Much the same happens in case (b)

too. Assuming the radius of curvature of the curved flight path to be much larger than the dimensions of

the airplane (which is quite reasonable), since the velocity vector V is always tangential to the curved

flight path sketched in Fig. 2(b), the relative airflow seen by the airplane is again equal to the velocity V

as marked in Fig. 2(b). However, in case (a), the tail experiences an additional component of relative

airflow arising due to the airplane’s pitching motion with a certain angular velocity (pitching rate) as

marked by the clockwise arrow in Fig. 2(a). The angular velocity gives the nose (forward of center of

gravity) sections an upward translational velocity, and likewise a downward translational velocity to the

tail (rear of center of gravity) sections. Thus, the horizontal stabilizer which has acquired a downward

translational velocity, labeled w, sees an upward component of the relative airflow, also equal to w, as

marked in Fig. 2(a). The net relative airflow in case (a) is the vector sum of V and w, as shown in Fig.

2(a), and this approaches the horizontal stabilizer at an incidence angle Δα relative to the direction of the

component V (which is the same as the flight direction). As is well known, relative airflow at an

incidence angle to an aerodynamic surface (such as the horizontal stabilizer) creates aerodynamic forces

(lift and drag), and since these act behind the airplane’s center of gravity, the forces will also produce a

moment (pitching moment) at the center of gravity. The point to note is that this second component of

relative airflow, w, is missing in case (b). This may be obvious by mere observation since at every point

of the curved flight path the velocity vector is always aligned along the XB axis – there is no “swinging

away” of the XB axis from the direction of V, as in case (a). Equivalently one may show that in case (b)

the velocity component w due to the angular velocity (pitching rate) is nullified by the simultaneous,

matching change in the orientation of the velocity V.

So, the answer to the question in the previous paragraph, “Will the airplane experience the same

aerodynamic forces (and moments) in the two cases?” is “No,” since the relative airflow seen by the

horizontal stabilizer in the two cases is not the same.

Bryan’s Error

Bryan assumed that the aerodynamic forces (and moments) were due to the pitching rate – the angular

velocity of the airplane (and the axes XB-Z

B attached to it) as seen by an observer on the ground. No

matter whether the airplane is flying along a straight line path (as in case (a)) or along a curved path (as in

case (b)), as long as the pitching rate with respect to the ground (inertial observer) is the same, the

aerodynamic forces (and moments) according to Bryan would be identical. Thus, as per Bryan, both the

airplanes in case (a) and case (b) would experience an effect due to the induced incidence angle Δα as

marked in Fig. 2(a), even though no such effect actually exists in case (b). Consequently, in every

analysis and simulation that follows Bryan’s model, as and when the airplane enters a curved flight path,

the aerodynamic forces (and moments) are wrongly calculated. This has been illustrated in Fig. 2 for a

curved flight path in the vertical plane where a pitching rate is present. A similar error is introduced when

the airplane flight path curves in the horizontal plane with a yawing rate.

Why did such an error arise and how could it have gone unnoticed for all these years? Bryan’s derivation

of the equations of airplane motion was based on the Newton’s law for forces (and moments) considering

three mutually orthogonal body-fixed axes (see Fig. 4) – three force components along the three axes and

three moment components about the three axes. The corresponding variables were the three components

(u,v,w) of the velocity V along the three axes and the three components (p,q,r) of the angular velocity ω

about the three axes. Bryan simply modeled the six force and moment components as a function of each

of the six velocity components – it was a mathematically convenient model but unfortunately not an

aerodynamically faithful one. But it must be remembered that those were the early days of aviation and

aerodynamic theory as applied to airplanes was not well developed yet. Bryan’s contribution remains a

landmark in aerospace science.

In later years, the dependence on the components (u,v,w) of velocity V was converted into a dependence

on more aerodynamically meaningful variables – the Mach number and two flow incidence angles called

the angle of attack and the sideslip angle. However, inexplicably, the dependence on the angular velocity

components (p,q,r) was left untouched. It is only now that the present authors have converted the angular

velocity components into meaningful aerodynamic variables as follows; for example, for the pitching rate:

Body-axis pitching rate = (Difference between body and relative wind pitching rates) + Relative wind

pitching rate

The relative wind pitching rate is the rate at which the flight path curves. The first of the two terms on the

right-hand side of this relation captures the effect in case (a) of Fig. 2 whereas the effect in case (b) is

represented by the second term on the right-hand side. The effect due to these two terms have now been

evaluated using the rules of aerodynamic theory and the (different) coefficients that multiply these terms

when they enter the equations of motion have also been obtained2. A similar correction is to be done for

the yawing rate as well.

We can only speculate on why this particular error went unnoticed. The years following Bryan’s

presentation witnessed some momentous events – the first world war, then two dramatic developments in

physics (quantum mechanics and the theory of relativity), followed by the second world war. Perhaps the

best theoretical minds were attracted to these exciting developments in physics. Possibly there was greater

emphasis on building and flying airplanes, less on delving into the underlying theory. In any case, it was

only after computers came into vogue in the decade following the second world war that Bryan’s model

could be tested in its entirety. By then, the model had been unquestioningly accepted and the clock was

never turned back. Certainly there were niggling issues with some of the results derived using Bryan’s

model but they were blamed on the approximations made during the derivation process or on errors in the

estimation or measurement of certain parameters; the model itself was considered sacrosanct.

Significance and Impact

How far does the discovery of this error and its correction impact the field? Is it a minor theoretical

quibble or is it a major game changer?

This change certainly goes to the heart of flight dynamics. To venture an analogy – imagine someone

spotting an error in Einstein’s theory of relativity and fixing it so that the expansion of the universe is

correctly explained without recourse to dark energy. What we have here is, perhaps, something along

those lines but on a much smaller scale. The changes are certainly quite significant. The very manner in

which the aircraft equations of motion are written is altered with some parameters redefined, some new

parameters introduced and the corresponding terms in the equations of motion modeled differently. Every

aircraft motion simulation code in use presently will need to be updated with the revised model. All the

redefined and newly introduced parameters will have to be estimated afresh. Some of these may need

wind tunnel and flight test experimentation and estimation procedures to be reevaluated.

The impact on pedagogy will be immediately noticeable. The entire theory of flight dynamics – the

approximate analytical expressions for frequency and damping, or time to halve/ double, for a dynamical

mode (see Box B) – has been reworked in our textbook2 following the revised model. The new results are

physically more meaningful and fix some key anomalies (see Box C) that arose earlier due to the use of

the Bryan model. The impact on aircraft design, flight simulation, flight testing, and flight control will be

immediate. Textbooks and reference works in all these subjects, besides those on flight dynamics itself,

will have to be rewritten taking the new model into account.

Scientific Reception

Needless to say, a development of this nature that turns a hundred years of conventional wisdom on its

head is not easy to digest. Among the initial feedback, some cautiously encouraging, some politely

indifferent, one negative comment stands out – “I cannot find anything wrong (in your work), but I am

sure there is a mistake.” Since that gentleman has not reverted for over 9 years now, presumably his

remark may be safely buried. Yet it is to the credit of the scientific community at large that no matter

what the background of the researchers, irrespective of whether their parent institutes support and

promote their work or not, every idea gets a fair hearing. That papers explaining and advocating the

revised model have been passing through peer review and getting accepted is evidence of this. When a

textbook2 based on this corrected model that rewrites almost every aspect of basic flight dynamics theory

is commissioned, the idea may be considered to have entered the mainstream. Then it is the authors’

responsibility to disseminate the development to the wider scientific community and to the public at large

(after all, we survive on taxpayer rupees). This article is a step in that direction.

Last Words

Nationally, India has taken giant strides in the development and use of aerospace technologies. However,

our contribution to aerospace scientific research has not been impressive (for example, see the data

compiled3). When it comes to path-breaking and original ideas, our share internationally plummets even

further to miniscule amounts. We therefore earnestly hope this article will reach out to young researchers

all over the country and embolden them to stick their neck out more often.

3 R. Ganguli, “A scientometric analysis of recent aerospace research,” Current Science, Vol. 95, No. 12, 25

th

December 2008, pp. 1670-1672.

Fig. 1: Elementary Flight Dynamics with an Introduction to Bifurcation and Continuation Methods, CRC

Press, Taylor and Francis, 2013.

Fig. 2: Two different kinds of aircraft motion: (a) Pitching motion about a fixed direction of flight

(velocity vector unchanged), and (b) Pull up – pitching motion coincident with curvature of velocity

vector.

Fig. 3: Parts of a conventional airplane (Source: NASA Glenn Research Center)

Fig 4: Airplane axes, forces and moments, and components of velocity V and angular velocity ω.

XE

YE

ZE

ZB

XB

YB

V

p q

r

u

v

w

OE

OB

R

Box A: GH Bryan (1864-1928) was a British professor who is credited with being the first to derive a

complete set of equations for the dynamics of an airplane in flight. It is presented in his book Stability in

Aviation, Macmillan (1911). The title page of this book is shown here. Besides the equations for the

dynamics of a rigid body (the airplane) in flight which by itself is not in dispute, Bryan included models

for the gravitational, propulsive and aerodynamic forces (and moments) acting on the airplane. A part of

his aerodynamic model has now been shown to be erroneous.

Box B: The airplane dynamic modes represent the fundamental types of motion that an airplane in

flight exhibits when disturbed from a steady flight condition. There are five fundamental modes – two for

motion in the vertical plane, called Short period and Phugoid, and three for motion out of the vertical

plane, called Roll, Dutch roll and Spiral. An evaluation of these modes is critical for understanding and

correcting the dynamical behavior of airplanes. Aircraft designers need this evaluation of the modes to set

a desired level of stability for the airplane during the design process. Pilots check this evaluation to judge

the ease and comfort with which an airplane can be flown. Control engineers use this evaluation to

configure control laws that enable an airplane to perform to its optimum while maintaining safety.

Box C: Short Period Anomaly is a suspicious result for the frequency of the Short period mode obtained

with the use of the Bryan model. Common sense suggests that the Short period mode frequency should be

identical to what is called the pitch stiffness parameter. But the existing result also includes a second term

dependent on the pitch damping which is difficult to reconcile with simple physical logic – how can a

damping parameter contribute to the stiffness (frequency)? With the revised model, this anomaly is

resolved and the Short period frequency is now identical to the pitch stiffness parameter. This is

considered to be one of the major successes of the revised model and appears to confirm its veracity. By

the way, the error that caused this anomaly is exactly what has been discussed in this article with

reference to Fig. 2.

Authors

N Ananthkrishnan (dr.akn.19@gmail.com) has in the past worked at NAL Bangalore and IIT Bombay;

he has been Visiting Assistant Professor of Aeronautics at Caltech, USA and Research Professor at

KAIST, Daejeon, South Korea. He is presently a consultant in the areas of atmospheric flight mechanics,

aerodynamics, air-breathing propulsion, and integrated modeling, simulation and design of aerospace

systems. He is Associate Fellow, AIAA and recipient of the Excellence in Teaching Award at IIT Bombay.

Nandan K Sinha (nandanks@iitm.ac.in) is Associate Professor at the Department of Aerospace

Engineering, IIT Madras. In the past, he has been a visiting postdoctoral scholar at the institute for

mechanics, Darmstadt, Germany, where he has taught and carried out research in the areas of mechanical

vibrations. His major areas of current teaching and research interests include flight mechanics, aircraft

design, and dynamics and control of nonlinear systems. He is a senior member of the American Institute

of Aeronautics and Astronautics.