# Airplane Static Stability in Maneuvering Flight with a Redefinition of the Aerodynamic Damping Derivatives

Narayan Ananthkrishnan

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Narayan Ananthkrishnan. Airplane Static Stability in Maneuvering Flight with a Redefinition of the Aerodynamic Damping Derivatives. 2024. hal-04709917

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# Airplane Static Stability in Maneuvering Flight with a Redefinition of the Aerodynamic Damping Derivatives

Narayan Ananthkrishnan<fnref n="1" />

Yanxiki Tech, 152 Clover Parkview, Koregaon Park, Pune 411001, India

It is commonly believed that an airplane in maneuvering flight enjoys a greater degree of static stability in the form of an additional aerodynamic damping effect due to its rotational velocity as against one in straight and level flight. Accordingly, the limiting condition for onset of instability in maneuvering flight is given by the Maneuver Point, which lies further aft of the Neutral Point for instability onset in straight and level flight. We show that this effect is, in fact, fictitious and arises from an erroneous modeling of the aerodynamic damping terms. When the aerodynamic damping derivatives are redefined in a physically meaningful manner in case of curved flight paths, this fictitious effect disappears, and there is no distinction between the Maneuver Point and the Neutral Point.

## Nomenclature

$CL$ = lift coefficient

$CL_α$ = derivative of lift coefficient with angle of attack

$CL_{δe}$ = derivative of lift coefficient with elevator deflection

$Cm_q$ = traditional pitch damping derivative

$Cm_{q1}$ = revised pitch damping derivative

$Cm_{q2}$ = pitch derivative due to flow curvature

$Cm_α$ = derivative of pitching moment coefficient with angle of attack

$Cm_{δe}$ = derivative of pitching moment coefficient with elevator deflection

$c$ = reference length, mean aerodynamic chord, m

$g$ = acceleration due to gravity, $m/s^2$

$h_{CG}$ = non-dimensional axial location of center of gravity (CG)

<footnote n="1">¹ Technical Consultant, Associate Fellow AIAA; akn.korea.19@gmail.com, akn@aero.iitb.ac.in</footnote>
$h_{NP}$ = non-dimensional axial location of neutral point (NP)

$L$ = lift force, N

$l_t$ = tail arm from airplane CG, m

$M$ = pitching moment, N.m

$n$ = normal acceleration, in g

$q, q_b$ = body-axis pitch rate, rad/s

$q_w$ = wind-axis pitch rate, rad/s

$\bar{q}$ = dynamic pressure, $N/m^2$

$S$ = reference area, wing planform area, $m^2$

$V$ = airplane velocity, m/s

$W$ = airplane weight, N

$X_B$ = body-fixed X axis

$Y_B$ = body-fixed Y axis

$\alpha$ = angle of attack, rad

$\gamma$ = flight path angle, rad

$\Delta C_L$ = incremental lift coefficient

$\Delta \alpha$ = induced angle of attack, rad

$\Delta \gamma$ = incremental flight path angle, rad/s

$\delta e$ = elevator deflection, rad

### Superscripts

(.) = derivative with time

* = trim state

## I.

## Introduction

THE concept of the Maneuver Point (MP) has been traced back to S. B. Gates (1942) and has since been faithfully reproduced in almost every textbook on airplane flight dynamics [e.g., see Ref. 1, pp. 60-62]. To understand the motivation for the Maneuver Point, we need to first state the definition of the Neutral Point (NP) and the associated concept of a Static Margin (SM). The Neutral Point is the aft limit of the airplane center of gravity (CG) beyond which the airplane no longer has static stability. As long as the CG is ahead of the NP, the change in lift
acting at the NP due to a perturbation in angle of attack works in such a manner as to create a restoring moment at the CG. The farther ahead of the NP that the CG is, the larger the moment arm and the greater the restoring moment, hence a larger degree of static stability. This can be quantified by the (non-dimensional) distance between the CG and the NP, which is called the Static Margin (SM). A positive SM implies a desirable negative restoring moment or a positive pitch stiffness — they are all equivalent concepts. Formally, this is represented by the stability derivative $Cm_α$ being negative — that is, an increase in angle of attack produces a nose-down pitching moment, and vice versa. This is the case when the airplane is in steady, level flight. An airplane in maneuvering flight, such as a vertical pull-up or a banked horizontal level turn, has a non-zero value of body-axis pitch rate $q_b$, which is supposed to provide an additional pitching moment due to the pitch damping effect, quantified by the pitch damping derivative $Cm_q$. Since $Cm_q$ is typically negative, this additional pitching moment, which is modeled as $Cm_q q_b$, is negative (nose down) for a pitch up, and positive (nose up) for a pitch down. When an airplane is perturbed during a pull-up maneuver, for example, this pitch damping effect supposedly contributes an additional pitching moment in addition to the restoring moment coming from the negative $Cm_α$. It is argued [1] that this additional pitching moment due to $Cm_q$ supplements the restoring moment from $Cm_α$, and hence for a maneuvering airplane the aft-most limit of the CG is not the Neutral Point (NP) but another point further aft, called the Maneuver Point (MP) (see Ref. [1], Fig. 3.2, p. 62). The difference between the NP and the MP is entirely due to the pitch damping derivative $Cm_q$; if $Cm_q$ were to become zero, then the NP and MP would be identical (see Ref. [1], Eq. 3.1,8, p. 62). The (non-dimensional) distance of the CG ahead of the MP is called the Maneuver Margin (MM), which is greater than the Static Margin (SM) applicable in case of steady, level flight. In other words, in a maneuvering flight, the pitch damping effect affords an additional degree of static stability (or restoring moment or pitch stiffness) over and above that provided by the Static Margin (SM).

From a physical perspective, it is difficult to explain how a damping effect could possibly provide (additional) stiffness or static stability; however, this discrepancy went unnoticed for decades. In fact, a similar anomaly arises in the derivation of the modal dynamics in pitch represented by the Short Period mode. The expression for the Short Period frequency features a term due to the pitch damping $Cm_q$, in addition to the expected effect due to $Cm_α$ (see Ref. [1], Eq. 6.3,15 (c), p. 175). When the modal dynamics is represented in second-order form (in analogy with a mass-damper-spring system), the pitch damping term appears as part of the spring (stiffness) effect (see Ref. [1], Eq. 6.3,14, p. 174). One may feel that this intrusion of a damping term into the stiffness (frequency) may be inadvertent, due to approximations made during the derivation of the modal parameters. However, increasingly rigorous
derivations of the literal approximations to the airplane rigid-body modes have shown that this damping effect on
modal stiffness cannot be eliminated [2, 3]. The puzzle was solved in 2004 when we [4] pointed out that the error
was due to the manner in which the pitch damping moment has been traditionally modeled; instead of being
modeled as $Cm_q q_b$, the effect ought to be modeled as $Cm_q(q_b - q_w)$, where $q_w$ is the wind-axis pitch rate. Then, in
case of a pull-up maneuver, where the body-axis pitch rate $q_b$ and velocity vector pitch rate $q_w$ are the same,
$(q_b - q_w)$ would be equal to zero, hence there would be no contribution due to pitch damping to the restoring
moment. However, whenever the aircraft pitches about an unchanged velocity vector, $q_w$ would be zero, and the
standard pitch damping moment $Cm_q q_b$ would be recovered. All the damping (rate) derivatives (in pitch, roll, and
yaw) were redefined and revised literal approximations for the modal parameters were obtained in Ref. [5]. The
revised results clearly showed that the pitch stiffness (short period frequency) no longer depended on the pitch
damping derivative, $Cm_q$ – the earlier dependence of short period frequency on $Cm_q$ was an aberration caused by
the error in the traditional modeling of the damping moment. A pedagogical presentation of the revised theory of
airplane small-perturbation dynamics has been made in Refs. [6, 7] and in a recent textbook by us [8].

The purpose of this article is to show that the concept of the Maneuver Point (MP) is likewise unjustified and
again a product of the previously flawed definition of the damping (rate) derivatives. When the corrected definition
of the damping (rate) derivatives as in Refs. [4-8] is used, the Maneuver Point is seen to be no different from the
Neutral Point. In other words, the Maneuver Point as a concept ceases to exist. In the rest of this article, the
traditional concept of the Maneuver Point is first written in terms of elevator deflection *per g*. Then the corrected
definition of the dynamic (rate) derivatives as presented in Refs. [4-8] is briefly described. Finally, the error in the
traditional concept of the Maneuver Point is pointed out and a revised expression for the elevator deflection *per g* is
presented.

## II.

### Traditional Maneuver Point Concept

The Neutral Point (NP) can be understood from an alternative viewpoint — in terms of the elevator deflection
required to generate a certain change in the airplane's trim state (either in terms of trim angle of attack $\alpha^*$ or trim lift
coefficient $CL^*$). In this view, a decreasing Static Margin (SM) makes the airplane trims more sensitive — that is, a
smaller change in elevator deflection is enough to create a given shift in its trim state. In the limit, as the CG
approaches the NP, the required elevator deflection tends to zero. Mathematically, the elevator deflection per trim
$CL$ can be written as [8, p. 134]:
$$ \frac{d\delta e^*}{dCL^*} = - \frac{Cm_\alpha}{Cm_{\delta e}CL_\alpha - Cm_\alpha CL_{\delta e}}. \qquad (1) $$

Rewriting $Cm_\alpha$ in terms of the Static Margin (SM):

$$ \frac{d\delta e^*}{dCL^*} = - \frac{CL_\alpha}{Cm_{\delta e}CL_\alpha - Cm_\alpha CL_{\delta e}}(h_{CG} - h_{NP}), \qquad (2) $$

where $(h_{CG} - h_{NP})$ is the Static Margin (SM). Clearly, as the SM approaches zero, the derivative on the left-hand side of Eq. (2) also tends to zero.

Along similar lines, one can define the elevator deflection per g (load factor n in a pull-up or push-down maneuver). As derived in Ref. [1] following the traditional approach (see Eq. 3.1,9, p. 62), the elevator deflection per g appears as:

$$ \frac{d\delta e^*}{dn^* - 1} = -[factor](h_{CG} - h_{MP}), \qquad (3) $$

where $(h_{CG} - h_{MP})$ is the Maneuver Margin (MM). In other words, in a maneuvering flight, the Maneuver Margin (MM) replaces the Static Margin (SM), with the MM typically being larger than the SM. Consequently, in the traditional view, the CG can move aft beyond the NP, until the MP is reached, before the elevator becomes infinitely sensitive to a change in airplane load factor.

### III. Corrected Definition of Dynamic (Rate) Derivatives

The traditional $Cm_q$ mechanism for an airplane (wing-body-tail configuration) arises when there is a body-axis pitch rate $q_b$ and the velocity vector does not reorient itself, i.e., $q_w = 0$. In that case, as sketched in Fig. 1, the horizontal tail experiences a velocity component $q_b l_t$ orthogonal to the velocity vector V, which results in an induced angle of attack $\Delta\alpha$ at the tail. The consequent change in tail lift creates a pitching moment about the airplane CG, which is the pitch damping effect.

In case of a pull-up maneuver where the airplane nose and the velocity vector rotate in sync (that is, the body-axis and wind-axis pitch rates are identical), such a pitch damping effect does not exist physically. This is even more starkly obvious in case of a level turn where, despite the presence of a body-axis pitch rate $q_b$, the horizontal tail clearly does not experience the induced velocity seen in Fig. 1 that is responsible for the pitch damping mechanism. That is, physically the pitch damping mechanism depicted in Fig. 1 is absent in these two examples of maneuvering flight. However, by modeling the pitch damping effect as $Cm_q q_b$, depending only on the body-axis pitch rate,
irrespective of the wind-axis rate (that is, the rate of reorientation of the velocity vector), a false pitch damping moment is introduced in the analysis in case of maneuvering flight such as pull-ups or turns. It is this spurious $Cm_q$ term in the analysis that is responsible for supposedly distinguishing the Maneuver Point from the Neutral Point.

<figure><img src="image_2.png" /><figcaption>Fig. 1 Traditional $Cm_q$ mechanism.</figcaption></figure>

The correct pitch damping model must discriminate between these two cases – it must admit the pitch damping mechanism in case of a pitch-up (nose rotates but velocity vector does not reorient) but deny it in case of a pull-up or a level turn maneuver. Following Refs. [4-8], this is possible if the pitch damping derivative is defined as follows:

$$C_{m1} = \frac{\partial Cm}{\partial(q_b - q_w)(c/2V)}, \qquad (4)$$

and the pitch damping term is modeled in the pitching moment equation as $Cm_{q1}(q_b - q_w)(c/2V)$, where $q_w$ is the wind-axis pitch rate. Then, in case of a pitch with $q_w = 0$, the pitch damping mechanism is correctly recovered, and in case of maneuvers where $(q_b - q_w) = 0$, the pitch damping mechanism does not introduce a spurious effect in the pitching moment equation.

In case of curved flight paths, an additional derivative is defined to account for the aerodynamic effect of flow curvature [8, pp. 172-174]:

$$C_{m2} = \frac{\partial Cm}{\partial q_w(c/2V)}, \qquad (5)$$

and a corresponding term $Cm_{q2}q_w(c/2V)$ is introduced in the analysis. However, this effect is usually negligible for most airplanes and may be of some slight significance in case of configurations where one lifting surface is at a height from the fuselage reference line, such as T-tail airplanes.
For a general maneuver with arbitrary values of body-axis and wind-axis pitch rates, $q_b$ and $q_w$ respectively, the correct way to model the pitch damping and flow curvature effects would be a combination of the two thus:

$$Cm_{q1}(q_b - q_w)(c/2V) + Cm_{q2}q_w(c/2V). \qquad (6)$$

## IV.

### Maneuver Point is the same as Neutral Point

Now we can derive an expression for the elevator deflection *per g* along similar lines as the derivation in Ref. [1] (pp. 60-62). Consider a pull-up maneuver where the airplane nose and velocity vector rotate in sync in the vertical plane. The centripetal acceleration causing the load factor is due to the difference between the lift and the weight, given as below:

$$(\Delta n - 1) \frac{W}{\bar{q}S} = CL_{\alpha}\Delta\alpha + CL_{q1}(q_b - q_w)\frac{c}{2V} + CL_{\delta e}\Delta\delta e. \qquad (7)$$

The pitching moment equation appears as:

$$\Delta \dot{q} \frac{I_{yy}}{\bar{q}Sc} = Cm_{\alpha}\Delta\alpha + Cm_{q1}(q_b - q_w)\frac{c}{2V} + Cm_{\delta e}\Delta\delta e. \qquad (8)$$

Under quasi-steady conditions for a pull-up maneuver where $(q_b - q_w) = 0$, the time derivative is set to zero and terms involving the pitch rate derivative fall out, giving:

$$(\Delta n - 1) \frac{W}{\bar{q}S} = CL_{\alpha}\Delta\alpha + CL_{\delta e}\Delta\delta e, \qquad (9)$$

$$0 = Cm_{\alpha}\Delta\alpha + Cm_{\delta e}\Delta\delta e. \qquad (10)$$

Eliminating $\Delta\alpha$ between Eqs. (9) and (10) results in the elevator deflection *per g*:

$$\frac{\Delta\delta e}{(\Delta n - 1)} = - \frac{Cm_{\alpha}(W/\bar{q}S)}{Cm_{\delta e}CL_{\alpha} - Cm_{\alpha}CL_{\delta e}}. \qquad (11)$$

The Maneuver Point, defined as the condition under which the derivative (or the fraction) on the left-hand side of Eq. (11) tends to zero, corresponds to $Cm_{\alpha} \to 0$, which is the same as the condition for the Neutral Point from Eq. (1). That is, the point where the elevator deflection *per g* vanishes (previously called the Maneuver Point) is none other than the Neutral Point.

In contrast, following the traditional version, one would obtain the expression for the elevator deflection *per g* (equivalent to Eq. 3.1,7 of Ref. [1], p. 62) as:

$$\frac{\Delta\delta e}{(\Delta n - 1)} = - \frac{Cm_{\alpha}(W/\bar{q}S) + Cm_{q1}(c/2V)CL_{\alpha}(g/V)}{Cm_{\delta e}CL_{\alpha} - Cm_{\alpha}CL_{\delta e}}, \qquad (12)$$
where the additional term in the numerator involving the pitch damping derivative $Cm_{q1}$ (typically negative) would
imply that the expression on the left-hand side tends to zero for a positive value of $Cm_{\alpha}$ — that is, when the CG is
aft of the Neutral Point (NP).

V.

**Conclusion**

It is shown that the so-called Maneuver Point which was meant to signify the aft limit of CG location to ensure positive pitch stiffness in maneuvering flight (e.g., pull-up or level turn) is no different from the Neutral Point. The erroneous distinction between the Maneuver Point and the Neutral Point in the traditional analysis is seen to be due to a faulty definition of the dynamic (rate) derivatives whereby a pitch damping term was claimed to provide an additional restoring moment. When the pitch damping derivative is redefined in a physically meaningful manner, the offending pitch damping term in the expression for the elevator deflection *per g* disappears, making it clear that the so-called Maneuver Point is nothing but the Neutral Point itself.

**References**

[1] Etkin, B., and Reid, L. D., *Dynamics of Flight: Stability and Control* (3rd ed.), Wiley, New York, 1996, Chaps. 3,6.

[2] Ananthkrishnan N., and Unnikrishnan, S., “Literal Approximations to Aircraft Dynamic Modes,” *Journal of Guidance, Control, and Dynamics*, Vol. 24, No. 6, 2001, pp. 1196-1203.

doi: 10.2514/2.4835

[3] Ananthkrishnan, N., and Ramadevi, P., “Consistent Approximations to Aircraft Longitudinal Modes,” *Journal of Guidance, Control, and Dynamics*, Vol. 25, No. 4, 2002, pp. 820-824.

doi: 10.2514/2.4952

[4] Ananthkrishnan, N., “Small-Perturbation Analysis of Airplane Flight Dynamics – A Reappraisal. I: Longitudinal Modes,” AIAA 2004-4950, AIAA Atmospheric Flight Mechanics Conference and Exhibit, August 2004.

doi: 10.2514/6.2004-4950

[5] Raghavan, B., and Ananthkrishnan, N., “Small-Perturbation Analysis of Airplane Dynamics with Dynamic Stability Derivatives Redefined,” *Journal of Aerospace Sciences and Technologies*, Vol. 61, No. 3, 2009, pp. 365-380.

doi: 10.61653/joast.v61i3.2009.554

[6] Ananthkrishnan, N., and Sinha, N. K., “A Simple, Correct Pedagogical Presentation of Airplane Longitudinal Dynamics,” *Journal of Aerospace Sciences and Technologies*, Vol. 65, No. 4, 2013, pp. 328-341.

doi: 10.61653/joast.v65i4.2013.873
[7] Ananthkrishnan, N., and Sinha, N. K., “A Simple, Correct Pedagogical Presentation of Airplane Lateral-Directional Dynamics,” *Journal of Aerospace Sciences and Technologies*, Vol. 65, No. 3, 2013, pp. 235-245.
doi: 10.61653/joast.v65i3.2013.844

[8] Sinha, N. K., and Ananthkrishnan, N., *Elementary Flight Dynamics with an Introduction to Bifurcation and Continuation Methods* (2nd ed.), CRC Press, Boca Raton, FL, 2022.