Longitudinal Stability Criteria for a Propeller-Driven Aircraft

Abstract

The Routh criterion is used to assess longitudinal dynamic stability of a propeller-driven aircraft. Under a few plausible assumptions on possible ranges of the pertinent stability derivatives, it reduces to a pair of simple conditions imposing a traditional aft limit (the forward of the maneuver and the neutral-speed-stability points) on the center-of-gravity position and an upper limit on the longitudinal moment of inertia. It is demonstrated that most aircraft have sufficiently small inertia to remain stable as long as their center-of-gravity is properly placed. At the same time, sailplane-like aircraft (as, e.g., long endurance UAVs), with an engine installed at the rear extremity of the aircraft, may have sufficiently high inertia to become unstable regardless of their center-of-gravity placement.

1. Introduction

Linear analysis of the longitudinal dynamics of a rigid aircraft can be found practically in any textbook on flight mechanics. In a nutshell, it can be recapitulated as follows. Assuming constant-density atmosphere, flat nonrotating Earth, zero thrust angle, and no control inputs, the equations governing small perturbations 𝐱=[𝑢,𝛼,𝑞,𝜃]^T of airspeed, angle-of-attack, pitch rate and pitch angle from their respective values in a trimmed straight-and-almost-level flight can be written as 𝐁⋅𝐱̇=𝐀⋅𝐱, (1) with𝐶𝐀=^m,𝑢𝐶_m,𝛼𝐶_m,𝑞0𝐶_L,𝑢+2C_L𝐶_D+𝐶_L,𝛼𝐶_L,𝑞−2𝜇𝛾𝐶_L𝐶_T,𝑢−𝐶_D,𝑢+2𝛾𝐶_L𝐶_L−C_D,𝛼0−𝐶_L0010, (2)𝐁=0−𝐶_m,̇𝛼𝑖_𝑦00−2𝜇+𝐶_L,̇𝛼002𝜇0000001; (3) see equation (5.13,16) in [1]. The notation here is the same as in [1] with three exceptions: (i) an overdot replaces “𝐷” to designate a derivative with respect to reduced time; (ii) the subscript “e” is omitted from unperturbed (equilibrium) values of the flight-path angle and the lift and drag coefficients; (iii) “𝑖_𝑦” and “𝑢” replace “𝐼_𝑦̂” and “Δ𝑉” for the reduced moment of inertia and the reduced velocity.

Setting  𝐱=𝐱₀e^𝑠𝑡 in (1), where 𝑡 is the (reduced) time and 𝐱₀ and 𝑠 are obvious time-independent quantities, leads to the generalized eigenvalue problem (𝑠𝐁−𝐀)⋅𝐱₀=0, and, in turn, to the fourth order characteristic equation,𝑎𝑠⁴+𝑏𝑠³+𝑐𝑠²+𝑑𝑠+𝑒=0. (4) The coefficients in (4) can be identified with products of the elements of the matrices 𝐀 and 𝐁, but except for 𝑎=2𝑖_𝑦𝜇2𝜇+𝐶_L,̇𝛼, (5) all other coefficients are rather unwieldy (Appendix A). Of course, the aircraft is stable if all the roots of (4) are in the left half-plane.

Equation (4) has no analytical solution for 𝑠, and therefore it has to be solved either approximately [2] or numerically. Each approach has its virtues, but in many cases the roots of (4) are not really needed—what is needed is the knowledge whether the aircraft is stable or not. In principle, stability of a system (the aircraft, in this case) can be assessed without actually solving its characteristic equation—by the Routh criterion, the number of roots of (4) which are in the right-half-plane equals the number of sign alternations in the five-member sequence 𝑆={𝑎,𝑏,𝑓,𝑔,𝑒}, where 𝑓=𝑐−𝑎𝑑/𝑏 and 𝑔=𝑑−𝑏𝑒/𝑓[3, pages 253–255]. In practice, however, this approach is hardly ever used with (4)—the unwieldiness of the expressions for the members of 𝑆 renders it rather unattractive as compared with the instant numerical solution of the generalized eigenvalue problem leading to (4).

Notwithstanding the majority approach, we suggest that for a propeller-driven aircraft, an application of the Routh criterion to (4) yields a pair of very simple conditions with immediate design implications. One of these conditions sets a traditional aft limit (the forward of the maneuver and the neutral-speed-stability points) on the center-of-gravity position; the other sets an upper limit on the longitudinal moment of inertia. The details follow.

2. Approximate Stability Criterion

As a first step, we replace the moment derivatives 𝐶_m,•in (2) and (3) (or, to the same end, in (A.1)–(A.5) of Appendix A), all referred to the center of gravity, with combinations 𝐶_m,•=𝐶_M,•−𝐶_L,•𝑥_cg−𝑥_n (6) of their neutral-point-referred counterparts, 𝐶_M,•, and lift coefficient derivatives, 𝐶_L,•. Here, 𝑥_cg and 𝑥_n are the center-of-gravity and neutral-point positions, and “•” stands for “𝛼,” “𝛼̇,” “𝑞,” and “𝑢.” Among the neutral-point-referred moment derivatives, 𝐶_M,𝛼𝐶=0,_M,𝑞=2𝜇𝑥_m−𝑥_n𝐶_M,𝑢,=−2𝐶_L𝑥_s−𝑥_n, (7)(8)(9) by definitions of the neutral (𝑥_n), maneuver (𝑥_m), and neutral-speed-stability (𝑥_s) points; the last pair follows equations (6.10, 8) and (6.4, 26) of [1] by (6).

These substitutions do simplify the expressions for 𝑏,…,𝑒, but essentially do not reduce the number of terms comprising them. Hence, as a next step, we sort these terms by their magnitudes and retain a few dominant ones only. For a typical subsonic aircraft in an almost level flight, invariably 𝐶_D,𝐶_D,𝑢,𝐶_D,𝛼,𝐶_T,𝑢,𝐶_L,𝑢,𝐶_M,𝑢,𝛾areoftheorder10⁻¹,𝐶_Lisoftheorder10⁰,𝐶_L,𝛼,𝐶_L,̇𝛼,𝐶_L,𝑞,𝐶_M,̇𝛼and𝐶_M,𝑞areoftheorder10¹,𝜇isoftheorder10²,𝑖_𝑦isoftheorder10²or10³. (10) The word “order” can be understood here (and below) in its intuitive sense; nevertheless, we shall remove possible ambiguity and define the order 𝑂(𝑎) of a quantity 𝑎 as 10^{round(log10|𝑎|)}. Thus, 4 and 30 are both of the order 10¹.

Sorting out the terms in (A.2)–(A.5) by (10), and discarding all those that are of the order 10⁻² (or smaller) as compared with the leading ones, leaves 𝑏=2𝜇𝑖_𝑦𝐶_L,𝛼+𝑂10⁻¹−2𝜇−𝐶_L,𝑞𝐶_M,𝑞+𝐶_M,̇𝛼+2𝜇𝐶_L,𝑞+𝐶_L,̇𝛼𝑥_cg−𝑥_m,𝑐=4𝜇²𝐶_L,𝛼𝑥_cg−𝑥^′_m1+𝑂10⁻³,𝑑=−𝐶_L𝐶_M,𝑞+𝐶_M,̇𝛼2𝐶_L+𝐶_L,𝑢+𝛾𝐶_L,𝛼𝐶_M,𝑞+𝑂10⁻¹+2𝐶_L𝑥_cg−𝑥_s𝐶_L,𝑞+𝐶_L,̇𝛼𝐶_L+2𝜇𝐶_D,𝛼(1)𝑥+𝑂−2𝜇_cg−𝑥_m×𝐶_L,𝛼𝐶_T,𝑢−𝐶_D,𝑢+3𝛾𝐶_L+2𝐶_D,𝛼𝐶_L+𝑂10⁻²+𝑂10⁻²,𝑒=2𝐶²_L𝐶_L,𝛼𝑥_cg−𝑥^′_s1+𝑂10⁻², (11)(12)(13)(14) where𝑥^′_m=𝑥_m+𝑥_m−𝑥_n𝑂10⁻²𝑖_𝑦𝑥/𝜇,_′s=𝑥_s+𝑥_s−𝑥_n𝑂10⁻². (15)(16) Since 𝑥_m−𝑥_n and 𝑥_s−𝑥_n are of the order 10⁻¹ by (8)–(10), therefore 𝑥^′_s=𝑥_s+𝑂(10⁻³) and 𝑥^′_m=𝑥_m+𝑂(10⁻³𝑖_𝑦/𝜇). Full expression for 𝑥^′_𝑠 can be found in (A.7)–(A.9).

It can be verified by direct substitution that the ratio 𝑏/𝑎 is of the order 10⁻¹, whereas the ratios 𝑐/𝑎, 𝑑/𝑎, and 𝑒/𝑎 are of the orders 10(𝑥_cg−𝑥^′_m)𝑖_𝑦⁻¹, 10⁻³𝑖_𝑦⁻¹, and 10⁻³𝑖_𝑦⁻¹(𝑥_cg−𝑥^′_s), respectively. Hence, as long as 𝑥_cg−𝑥_misoftheorder10⁻¹, (17) the third and the fourth terms in the Routh sequence, 𝑓=𝑐−𝑎𝑑/𝑏 and 𝑔=𝑑−𝑏𝑒/𝑓, can be approximated with 𝑐 and 𝑑−𝑏𝑒/𝑐; the error of these approximations should be of the order of 10⁻² relative to the respective leading terms. Explicit expression for 𝑐 is found in (12); explicit (approximate) expression for 𝑔 follows from (8) and (11)–(16). It can be regrouped in various forms; the following two,𝑔=𝐶_L,𝛼𝐶²_L𝑖₂𝑋+𝑖₁+𝑖₃−𝑖_𝑦+𝑖₀−𝑖₃+𝑖_𝑦𝑋⁻¹ (18a) and𝑔=𝐶_L,𝛼𝐶²_L𝑋−1𝑋×𝑖₂𝑋−1²+2𝑖₂+𝑖₁+𝑖₃𝑋−1+𝑖₀+𝑖₁+𝑖₂−𝑋−1𝑖_𝑦, (18b)will serve the needs of the following discussion. Here, 𝑥𝑋=^cg−𝑥_m𝑥_s−𝑥_m𝑖₀,=4𝜇𝑘_𝑞𝑥_n−𝑥_m𝐶_L,𝛼=−2𝑘_𝑞𝐶_M,𝑞C_𝐿,𝛼𝑖₁,=2𝜇𝐶_L,𝛼𝐶_L𝑘_𝑞𝐶_L,𝑢+𝛾𝐶_𝐿,𝛼𝑥_n−𝑥_m−2𝐶_D,𝛼𝑥_s−𝑥_m𝑖₂,=2𝜇𝐶²_L𝐶_D,𝑢−𝐶_T,𝑢−3𝛾𝐶_L𝑥_s−𝑥_m𝑖₃,=𝑖₀𝐶_L,𝑞𝑘2𝜇=−^𝑞𝐶_M,𝑞𝐶_L,𝑞𝜇𝐶_L,𝛼,𝑖_𝑦=𝑖_𝑦𝜇, (19)(20)(21)(22)(23)(24) and in which𝑘_𝑞𝐶=1+^M,̇𝛼𝐶_M,𝑞. (25)

By the Routh criterion, the number of roots in the right-half-plane equals the number of the sign alternations in 𝑆. Since, invariably, 𝐶_M,𝑞<0, 𝐶_M,̇𝛼<0, and 𝐶_L,𝛼>0, the first two members of 𝑆, 𝑎, and 𝑏, are positive by (5), (11), (10) and (17). Hence, the aircraft is stable only if the remaining members, 𝑓, 𝑔, and 𝑒, are positive as well. The first of these conditions yields 𝑥_cg>𝑥^′_m (see (12)), or, approximately,𝑥_cg>𝑥_m. (26a) In view of (17) this condition is almost redundant. We shall assume that it is satisfied unconditionally. The second of these conditions, 𝑔>0, (26b) will be discussed in the following sections. The last of these conditions yields 𝑥_cg>𝑥^′_s (see (14)), or, approximately,𝑥_cg>𝑥_s. (26c)It suggests that the center of gravity should be placed forward of the neutral speed stability point. There is nothing new in this result [1, page 335], but it is satisfying to obtain it here. In the analysis of (26b) we shall assume that this condition is satisfied as well. In conjunction with (26a) and (19) it implies that either 𝑋>1 or 𝑋<0.

Note that failing (26b), but not (26a) and (26c), implies two sign alternations in the Routh sequence and hence two divergent (phugoid) roots; failing (26c) only, or both (26c) and (26b), implies one divergent root.

3. Condition (26b)

The consequences of (26b) can be inferred from either (18a) or (18b), depending on the variable that is considered given. For example, the conjunction of (26b) and (18b) sets an upper limit on 𝑖_𝑦,𝑖_𝑦<𝐽𝑋, (27) where𝐽𝑋=𝑖₂𝑋−1+2𝑖₂+𝑖₁+𝑖₃+𝑖₀+𝑖₂+𝑖₁.𝑋−1 (28) Conversely, given 𝑖_𝑦, the conjunction of (26b) and (18a) sets limits on 𝑋: if𝑖₂>0and𝑖_𝑦≥𝐽₀,then𝑋₋𝑖_𝑦<𝑋<0or0<𝑋<𝑋₋𝑖_𝑦or0<𝑋₊𝑖_𝑦<𝑋;if𝑖₂>0and𝑖_𝑦<𝐽₀,then0<𝑋;if𝑖₂<0,then𝑋<𝑋₊𝑖_𝑦<0or0<𝑋<𝑋₋𝑖_𝑦. (29a)(29b)(29c)Here,𝑋_±𝑖_𝑦=−𝑖₁+𝑖₃−𝑖_𝑦±𝑖₁+𝑖₃−𝑖_𝑦²−4𝑖₂𝑖₀−𝑖₃+𝑖_𝑦2𝑖₂ (30) are the values of 𝑋 for which 𝑔=0, whereas𝐽₀=𝑖₁+𝑖₃+2𝑖₂+2𝑖₂𝑖₀+𝑖₁+𝑖₂ (31) is the value of 𝑖_𝑦 for which the square root in (30) vanishes with 𝑖₂>0. Note that 𝐽₀=min_𝑋>1𝐽(𝑋), by (28).

4. An Example

The use of (27) and (29) can be elucidated on a practical example. Consider a fictitious long-endurance UAV. For this aircraft we adopt a standard drag-lift relation𝐶_D=𝐶_D0+𝐾𝐶²_L; (32) neglect 𝐶_D,𝑢, 𝐶_L,𝑢, and 𝐶_M,𝑢; and set𝐶_T,𝑢=−𝑘_T𝐶_D+𝛾𝐶_L, (33) where 𝑘_T is the scaling exponent of the propulsion system. It equals 2 for an ideal (constant thrust) jet and 3 for an ideal (constant power) propeller in [1, page 263]. Values of 𝐶_D0, 𝐾, 𝜇, and the relevant stability derivatives are listed in Table 1.

Table 1 

Dimensionless stability derivatives of the model aircraft.

Using this data, equation (4) was solved numerically with many combinations of lift coefficient, stability margin (𝐶_M,𝛼), and reduced moment of inertia. Differentiating the stable and unstable combinations yielded the stability boundaries drawn by the thick lines in Figure 1. Approximate boundaries have been constructed as contours 𝐽(𝑋)=𝑖_𝑦 (see (27)) using (28), (8), and (19)–(25); they appear as thin lines on the same figure. The fit between the two seems fair.

(a) level flight, 𝑘T=2

(b) 8 deg climb, 𝑘T=2

(c) level flight, 𝑘T=3

(d) power off glide

Figure 1 

Stability boundaries for the model aircraft. Combinations of lift coefficient and stability margin are unstable if situated to the right (b) or above (a, c, d) the respective line. Thick lines mark the exact boundaries obtained from multiple solutions of (4); thin lines are solutions of𝐽(𝑋)=𝑖𝑦.

With no 𝑢-derivatives, 𝑥_s=𝑥^′_s=𝑥_n, by (9) and (16), and since 𝐶_M,𝑞<0, therefore 𝑥_s−𝑥_m>0, by (8). Thus, 𝑋>1, by (19) and (26a). Moreover,𝑖₁𝐶=−^M,𝑞𝐶_L𝐶𝛾−2^D,𝛼𝐶_L,𝛼,𝑖₂𝐶=−^M,𝑞𝐶_L𝑘_T𝐶_D𝐶_L+𝑘_T−3𝛾, (34)(35) by (21) and (22).

With 𝑘_T=2 (an ideal jet), 𝑖₂ turns negative once 𝛾 exceeds 2𝐶_D/𝐶_L. In this case (29c) applies, inferring that in a (sufficiently) steep climb, the aircraft turns unstable with 𝑋>𝑋₋(𝑖_𝑦). This behavior is explicit in Figure 1(b). An exhaustive discussion of the stability of a climbing jet can be found on pages 329–346 in [1] and hence will not be repeated here.

With 𝑘_T=3 (an ideal propeller), 𝑖₂ is positive unconditionally. In this case, (27) applies, inferring that an ideal-propeller-driven aircraft turns unstable for sufficiently large moment of inertia. This behavior is explicit in Figures 1(c) and 1(d). Since 𝑖₀, 𝑖₂, and 𝑖₃ are independent of 𝛾 by (20), (23), and (35) and 𝑖₁ increases with 𝛾 by (34), it follows from (28) that 𝐽(𝑋) increases with 𝛾 as well. Consequently, the lowest (and hence the most constricting) limit on the longitudinal moment of inertia of a propeller-driven aircraft is obtained in a power-off glide, where 𝛾=−𝐶_D/𝐶_L. Indeed, the curves of constant 𝑖_𝑦 in Figure 1(d) are lower than those in Figure 1(c).

In a glide,𝐽𝑋=−𝐾𝐶_M,𝑞3+3𝐶_D0𝐾𝐶²_L+𝑋−11+5𝐶_D0𝐾𝐶²_L+2𝑘_𝑞𝐾𝐶_L,𝛼𝐶_L,𝑞+2𝜇2𝑘_𝑞𝐾𝐶_L,𝛼−2+2𝐶_D0𝐾𝐶²_L1,𝑋−1 (36) by (28), (32)–(35), (20), and (23). Since 𝑋>1 (see above), 𝐽(𝑋) decreases with increasing lift coefficient. Indeed, the lines of constant 𝑖_𝑦 in Figure 1(d) are higher for lower values of 𝑖_𝑦. Accordingly, the most constricting limit on the moment of inertia in a glide is obtained when the aircraft flies at the highest possible lift coefficient.

5. Maximal Moment of Inertia

Let 𝐽₁=min_{(𝑋>1,𝐶L>0)}𝐽(𝑋) be the minimum of 𝐽(𝑋) in a glide with respect to both 𝑋 and 𝐶_L. Under the same assumptions as those of the preceding section (the parabolic drag polar and the neglect of the 𝑢-derivatives) it is given by𝐽₁=−2𝐶_M,𝑞𝑃, (37) where1𝑃=𝐾2+𝑘_𝑞𝐾𝐶_L,𝛼𝐶_L,𝑞+2𝜇6𝑘_𝑞𝐾𝐶_L,𝛼−6. (38) This minimum corresponds to 𝐶_L→∞ (which is, of course, beyond reach) and𝑋=1+2𝑘_𝑞/𝐾𝐶_L,𝛼−2+2𝐶_D0/𝐾𝐶²_L3+3𝐶_D0/𝐾𝐶²_L. (39) Clearly, an aircraft which is designed with sufficiently small moment of inertia to satisfy 𝑖_𝑦<𝐽₁ and with sufficiently forward center-of-gravity to satisfy (26c) (and (26a)) will be stable at all flight conditions.

It is shown in Appendix B that 𝑃 depends mainly on the aspect ratio of the wing. Since the aspect ratio is dictated by performance (rather than stability) requirements, 𝐶_M,𝑞 remains the main design parameter affecting the maximal moment of inertia for which the aircraft is stable.

For a conventional subsonic aircraft, 𝐶_M,𝑞 is associated mainly with the contribution of its horizontal tail; the contribution of the other parts is small and has the same (negative) sign as the tail. Hence, the contribution of the tail,𝐶_M,𝑞=−2𝑎_t𝑥_n−𝑥_t²𝑆_t𝑆_w, (40) provides a lower bound on |𝐶_M,𝑞|. Here, 𝑆_t/𝑆_w is the tail to wing areas ratio, 𝑥_n−𝑥_t is the distance between the aircraft neutral point and the aerodynamic center of the tail, and 𝑎_t is the lift slope coefficient of the tail [1, page 269].

𝑖_𝑦 can be related with longitudinal (reduced) radius of gyration, 𝑟_𝑦. In fact, 𝑖_𝑦=4𝑟²_𝑦 (see the Nomenclature). Hence, one may set 𝐽₁=4𝑅²₁ and use 𝑅₁ as a conservative upper bound of 𝑟_𝑦;𝑅₁=𝑥_n−𝑥_t𝑎_t𝑃𝑆_t𝑆_w, (41) by (37), (38), and (40). With 𝑎_t=5, 𝑆_t/𝑆_w=6, and the data of our model aircraft, 𝑅₁ equals, approximately, 0.4(𝑥_n−𝑥_t)—almost half the distance between the tail and the center-of-gravity (it is implied here that (𝑥_cg−𝑥_n)≪(𝑥_n−𝑥_t)). This result marks out a single configuration which may be at risk of becoming unstable because of the excessive inertia. This is the configuration with an aft engine, counterbalanced by the equipment in the nose. All conventional configurations of propeller-driven aircraft should be stable as long as their center-of-gravity is properly placed.

6. Concluding Remarks

The key element underlying the preceding derivations is that, under assumptions (10) and (17), the coefficients 𝑎,…,𝑒 of the characteristic polynomial can be approximated by a few dominant terms only. These dominant terms were used in (11)–(14) and (18) to approximate the respective members in the Routh sequence, leading to stability conditions (26b) and (26c). The former sets an upper limit on the longitudinal moment of inertia; the latter sets an aft limit on the center-of-gravity position.

Perhaps the most conspicuous result of this study is that most configurations of propeller-driven aircraft are practically unlimited by inertia and hence are stable as long as their center-of-gravity is placed forward of the neutral-speed-stability (and maneuver) points—condition (26c). At the same time, aircraft with high-aspect ratio wing and an aft engine are at risk of becoming unstable because of an excessive inertia—condition (26b). The critical moment of inertia (28) depends on the center-of-gravity position and the flight conditions, but 𝐽₁ from (37) may provide an early warning when longitudinal inertia becomes an issue.

Appendices

A. Coefficients of the Characteristic Polynomial

The coefficients in (4) are 𝑎=2𝑖_𝑦𝜇2𝜇+𝐶_L,̇𝛼𝐶,𝑏=2𝜇−2𝜇_m,𝑞+𝐶_m,̇𝛼+𝑖_𝑦𝐶_L,𝛼+𝐶_D+𝐶_m,̇𝛼𝐶_L,𝑞−𝐶_L,̇𝛼𝐶_m,𝑞−𝑖_𝑦2𝜇+𝐶_L,̇𝛼𝐶_T,𝑢−𝐶_D,𝑢+2𝛾𝐶_L,𝑐=2𝜇−2𝜇−𝐶_𝐿,𝑞𝐶_m,𝛼−𝐶_m,𝑞𝐶_L,𝛼+𝐶_D+𝛾𝐶_L+𝐶_T,𝑢−𝐶_D,𝑢+3𝛾𝐶_L𝐶_m,𝑞+𝐶_m,̇𝛼+𝑖_𝑦2𝐶_L+𝐶_L,𝑢𝐶_L−𝐶_D,𝛼−𝐶_L,𝛼+𝐶_D𝐶_T,𝑢−𝐶_D,𝑢+2𝛾𝐶_L−𝐶_T,𝑢−𝐶_D,𝑢+2𝛾𝐶_L𝐶_m,̇𝛼𝐶_L,𝑞−𝐶_m,𝑞𝐶_L,̇𝛼,𝑑=−2𝐶_L+𝐶_L,𝑢𝐶_L𝐶_m,𝑞+𝐶_m,̇𝛼−𝐶_m,𝑞𝐶_D,𝛼+𝐶_T,𝑢−𝐶_D,𝑢+2𝛾𝐶_L×−𝛾𝐶_L𝐶_m,̇𝛼+𝐶_D+𝐶_L,𝛼𝐶_m,𝑞+2𝜇−𝐶_L,𝑞𝐶_m,𝛼+𝐶_m,𝑢𝐶_L𝐶_L,𝑞+𝐶_L,̇𝛼+2𝜇−𝐶_L,𝑞𝐶_D,𝛼+2𝜇𝐶_L𝛾𝐶_m,𝛼,𝑒=−𝐶_m,𝛼𝐶_L2𝐶_L+𝐶_L,𝑢+𝐶_T,𝑢−𝐶_D,𝑢+2𝐶_L𝛾𝛾+𝐶_m,𝑢𝐶_L𝐶_L,𝛼+𝐶_D+𝐶_L−𝐶_D,𝛼𝛾. (A.1)(A.2)(A.3)(A.4)(A.5)

Using (6), (7), and (9), the last expression can be recast as𝑒=2𝐶²_L𝐶_L,𝛼𝑥_cg−𝑥^′_s×𝐶1−^D𝐶_L,𝑢2𝐶_L𝐶_L,𝛼𝐶+𝛾𝛾−^L,𝑢𝐶_L−𝐶_D,𝛼2𝐶_L𝐶_L,𝛼+𝐶_T,𝑢−𝐶_D,𝑢2𝐶_L, (A.6) where𝑥^′_s=𝑥_𝑛−𝐶_M,𝑢2𝐶_L𝐶₁𝐶₂, (A.7) and in which𝐶₁𝐶=1−^D𝐶_L+𝐶_L,𝑢𝐶_L𝐶_L,𝛼𝐶+2𝛾𝛾−^L−𝐶_D,𝛼𝐶_L+𝐶_L,𝑢2𝐶_L𝐶_L,𝛼+𝐶_T,𝑢−𝐶_D,𝑢2𝐶_L,𝐶₂𝐶=1−^D𝐶_L,𝑢2𝐶_L𝐶_L,𝛼𝐶+𝛾𝛾−^L−𝐶_D,𝛼𝐶_L,𝑢2𝐶_L𝐶_L,𝛼+𝐶_T,𝑢−𝐶_D,𝑢2𝐶_L (A.8)(A.9)

B. Parameter 𝑃

In this appendix we seek the leading-order approximation for 𝑃, which was defined in (38). It depends on four parameters:𝑘_𝑞, 𝐶_L,𝛼, 𝐾, and the ratio 𝐶_L,𝑞/𝜇. The former is given by (25). Neglecting the respective contributions of the horizontal tail, and assuming that the wing loading is elliptic, the simplest approximations for the next two are𝐶_L,𝛼=2𝜋𝐴_w𝐴_w,1+2𝐾=𝜋𝐴_w, (B.1)(B.2) where 𝐴_wis the aspect ratio of the wing [4, page 130]. With these, 𝐶𝑃=^L,𝑞𝐶4𝜋𝜇1+^M,̇𝛼𝐶_M,𝑞+1𝜋3𝐴_w+3𝐴_w𝐶_M,̇𝛼𝐶_M,𝑞+6𝐴²_w𝐶_M,̇𝛼𝐶_M,𝑞+12𝜋𝐴_w𝐶1+^L,𝑞𝜇+𝐶_L,𝑞𝜇𝐶_M,̇𝛼𝐶_M,𝑞. (B.3)

The ratio 𝐶_M,̇𝛼/𝐶_M,𝑞 is, approximately, the downwash derivative at the tail [1, pp. 269, 284], about 0.3. The ratio 𝐶_L,𝑞/𝜇 for a conventional subsonic aircraft (ultralights excluded) is of the order of 10⁻¹—see (10) and (10); the aspect ratio 𝐴_w is of the order of 10¹. Hence, the leading-order (10⁻¹) term in (B.3) is 𝜋⁻¹3𝐴_𝑤⁻¹.

Nomenclature

𝑎t:	Lift slope coefficient of the horizontal tail
𝑎,…,𝑒:	Coefficients in the characteristic equation
𝐴w:	Aspect ratio of the wing
𝑐w:	Mean aerodynamic chord of the wing
𝐶D,𝐶L,𝐶T:	Drag, lift, and thrust coefficients at trim; the unit is (1/2)𝜌𝑉2𝑆w
𝐶D0:	Parasite (zero lift) drag coefficient
𝐶m,𝐶M:	Pitching moment coefficients; the former is referred to the center of gravity, the latter is referred to the neutral point; the unit is (1/2)𝜌𝑉2𝑆w𝑐w
𝑓,𝑔:	The third and the fourth terms in the Routh sequence
𝑖0,…,𝑖3:	Combinations of stability derivatives
𝑖𝑦,𝑖𝑦:	Reduced moment of inertia; the units are (1/8)𝜌𝑆w𝑐3w and (1/8)𝜌𝑆w𝑐3w𝜇
𝐽(𝑋):	Critical value of 𝑖𝑦 for a given 𝑋
𝐽0, 𝐽1:	min𝑋>1𝐽(𝑋) and min(𝑋>1,𝐶𝐿>0)𝐽(𝑋), respectively
𝑘𝑞:	Combination of the pitch damping derivatives defined in  (25)
𝑘T:	Scaling exponent of the thrust coefficient variation with airspeed at constant throttle
𝐾:	Induced drag coefficient
𝑃:	Parameter defined in  (38)
𝑞:	Reduced pitch rate, 𝜃̇
𝑟𝑦:	Reduced radius of gyration, (1/2)𝑖𝑦; the unit is 𝑐w
𝑅1:	An upper stability limit for 𝑟𝑦
𝑆w, 𝑆t:	Wing and tail areas
𝑡:	Reduced time; the unit is 𝑐w/(2𝑉)
𝑢:	Reduced airspeed perturbation from trim; the unit is 𝑉
𝑉:	Airspeed at trim
𝑥cg:	Reduced longitudinal center-of-gravity position; the unit is 𝑐w
𝑥m, 𝑥n, 𝑥s:	Maneuver, neutral, and neutral-speed-stability points; the unit is 𝑐w
𝑋:	Reduced maneuver margin; the unit is 𝑥s−𝑥m
𝑋±(𝑖𝑦):	Critical values of 𝑋 for a given 𝑖𝑦
𝛼:	Angle of attack, measured from trim
𝛾:	Flight path angle at trim
𝜃:	Pitch angle, measured from trim
𝜇:	Reduced mass; the unit is (1/2)𝜌𝑆w𝑐w
𝜌:	Air density.

Overheads

…•:	(overdot) a derivative with respect to the reduced time.
…:	(overbar) a nonstandard reduced quantity.

Subscripts

…,•:	Partial derivative with respect to •, computed at trim.