# Longitudinal Stability Criteria for a Propeller-Driven Aircraft

Authors: Gil Iosilevskii
Journal: International Journal of Aerospace Engineering
 (2010)

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## 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.


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## Body

## 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 perturbationsx=[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
(1)B⋅ẋ=A⋅x,
with(2)A=[Cm,uCm,αCm,q0CL,u+2CLCD+CL,αCL,q-2μγCLCT,u-CD,u+2γCLCL-CD,α0-CL0010],(3)B=[0-Cm,α̇iy00-(2μ+CL,α̇)002μ0000001];
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) “iy” and “u” replace “Îy” and “ΔV̂” for the reduced moment of inertia and the reduced velocity.Settingx=x0est 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,(4)as4+bs3+cs2+ds+e=0.
The coefficients in (4) can be identified with products of the elements of the matrices A and B, but except for (5)a=2iyμ(2μ+CL,α̇),
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 derivativesCm,•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 (6)Cm,•=CM,•-CL,•(xcg-xn)
of their neutral-point-referred counterparts, CM,•, and lift coefficient derivatives, CL,•. Here, xcg and xn are the center-of-gravity and neutral-point positions, and “•” stands for “α,” “α̇,” “q,” and “u.” Among the neutral-point-referred moment derivatives,  (7)CM,α=0,(8)CM,q=2μ(xm-xn),(9)CM,u=-2CL(xs-xn),
by definitions of the neutral (xn), maneuver (xm), and neutral-speed-stability (xs) points; the last pair follows equations (6.10, 8) and (6.4, 26) of [1] by (6).These substitutions do simplify the expressions forb,…,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
(10)CD,CD,u,CD,α,CT,u,CL,u,CM,u,γareoftheorder10-1,CLisoftheorder100,CL,α,CL,α̇,CL,q,CM,α̇andCM,qareoftheorder101,μisoftheorder102,iyisoftheorder102or103.
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  10round(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 (11)b=2μ(iy(CL,α+O(10-1))-(2μ-CL,q)(CM,q+CM,α̇)+2μ(CL,q+CL,α̇)(xcg-xm)),(12)c=4μ2CL,α(xcg-xm′)(1+O(10-3)),(13)d=-CL((CM,q+CM,α̇)(2CL+CL,u)+γCL,αCM,q+O(10-1))+2CL(xcg-xs)((CL,q+CL,α̇)CL+2μCD,α+O(1))-2μ(xcg-xm)×(CL,α(CT,u-CD,u+3γCL)+2CD,αCL+O(10-2))+O(10-2),(14)e=2CL2CL,α(xcg-xs′)(1+O(10-2)),
where(15)xm′=xm+(xm-xn)O(10-2iy/μ),(16)xs′=xs+(xs-xn)O(10-2).
Since xm-xn and xs-xn are of the order 10−1 by (8)–(10), therefore xs′=xs+O(10-3) and xm′=xm+O(10-3iy/μ). Full expression for xs′ can be found in (A.7)–(A.9).It can be verified by direct substitution that the ratiob/a is of the order 10−1, whereas the ratios c/a, d/a, and e/a are of the orders  10(xcg-xm′)iy-1, 10−3iy-1, and 10-3iy-1(xcg-xs′), respectively. Hence, as long as (17)xcg-xmisoftheorder10-1,
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,(18a)g=CL,αCL2(i2X̅+(i1+i3-i̅y)+(i0-i3+i̅y)X̅-1)
and(18b)g=CL,αCL2X̅-1X̅×(i2(X̅-1)2+(2i2+i1+i3)(X̅-1)+(i0+i1+i2)X̅-1-i̅y),will serve the needs of the following discussion. Here, (19)X̅=xcg-xmxs-xm,(20)i0=4μkqxn-xmCL,α=-2kqCM,qCL,α,(21)i1=2μCL,αCL((kqCL,u+γCL,α)(xn-xm)-2CD,α(xs-xm)(kqCL,u+γCL,α)),(22)i2=2μCL2(CD,u-CT,u-3γCL)(xs-xm),(23)i3=i0CL,q2μ=-kqCM,qCL,qμCL,α,(24)i̅y=iyμ,
and in which(25)kq=1+CM,α̇CM,q.By the Routh criterion, the number of roots in the right-half-plane equals the number of the sign alternations inS. Since, invariably, CM,q<0, CM,α̇<0, and CL,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 xcg>xm′ (see (12)), or, approximately,(26a)xcg>xm.
In view of (17) this condition is almost redundant. We shall assume that it is satisfied unconditionally. The second of these conditions, (26b)g>0,
will be discussed in the following sections. The last of these conditions yields xcg>xs′ (see (14)), or, approximately,(26c)xcg>xs.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 X̅>1 or X̅<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 i̅y,(27)i̅y<J̅(X̅),
where(28)J̅(X̅)=i2(X̅-1)+(2i2+i1+i3)+i0+i2+i1X̅-1.
Conversely, given i̅y, the conjunction of (26b) and (18a) sets limits on X̅: (29a)ifi2>0andi̅y≥J̅0,thenX̅-(i̅y)<X̅<0or0<X̅<X̅-(i̅y)or0<X̅+(i̅y)<X̅;(29b)ifi2>0andi̅y<J̅0,then0<X̅;(29c)ifi2<0,thenX̅<X̅+(i̅y)<0or0<X̅<X̅-(i̅y).Here,(30)X̅±(i̅y)=-(i1+i3-i̅y)±(i1+i3-i̅y)2-4i2(i0-i3+i̅y)2i2
are the values of X̅ for which g=0, whereas(31)J̅0=i1+i3+2i2+2i2(i0+i1+i2)
is the value of i̅y for which the square root in (30) vanishes with i2>0. Note that J̅0=minX̅>1J̅(X̅), 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 relation(32)CD=CD0+KCL2;
neglect CD,u, CL,u, and CM,u; and set(33)CT,u=-kT(CD+γCL),
where kT 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 CD0, K, μ, and the relevant stability derivatives are listed in Table 1.Table 1
Dimensionless stability derivatives of the model aircraft.
QuantityValueQuantityValueCLvariableCM,u0CL,α6.3CD00.03CL,α̇1K0.03CL,q4.5CD,α0.38CLCL,u0CD,u0CM,αvariableCT,uvariableCM,α̇−4.0μ150CM,q−20.0iyvariableUsing this data, equation (4) was solved numerically with many combinations of lift coefficient, stability margin (CM,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 J̅(X̅)=i̅y (see (27)) using (28), (8), and (19)–(25); they appear as thin lines on the same figure. The fit between the two seems fair.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 ofJ̅(X̅)=i̅y.
(a)
level flight,kT=2(b)
8 deg climb,kT=2(c)
level flight,kT=3(d)
power off glideWith nou-derivatives, xs=xs′=xn, by (9) and (16), and since CM,q<0, therefore xs-xm>0, by (8). Thus, X̅>1, by (19) and (26a).  Moreover,(34)i1=-CM,qCL(γ-2CD,αCL,α),(35)i2=-CM,qCL(kTCDCL+(kT-3)γ),
by (21) and (22).WithkT=2 (an ideal jet), i2 turns negative once γ exceeds 2CD/CL. In this case (29c) applies, inferring that in a (sufficiently) steep climb, the aircraft turns unstable with X̅>X̅-(i̅y). 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.WithkT=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 J̅(X̅) 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=-CD/CL. Indeed, the curves of constant iy in Figure 1(d) are lower than those in Figure 1(c).In a glide,(36)J̅(X̅)=-KCM,q((3+3CD0KCL2)(X̅-1)+(1+5CD0KCL2+2kqKCL,αCL,q2μ)+(2kqKCL,α-2+2CD0KCL2)1X̅-1),
by (28), (32)–(35), (20), and (23). Since X̅>1 (see above), J̅(X̅) decreases with increasing lift coefficient. Indeed, the lines of constant iy in Figure 1(d) are higher for lower values of iy. 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

LetJ̅1=min(X̅>1,CL>0)J̅(X̅) be the minimum of J̅(X̅) in a glide with respect to both X̅ and CL.  Under the same assumptions as those of the preceding section (the parabolic drag polar and the neglect of the u-derivatives) it is given by(37)J̅1=-2CM,qP,
where(38)P=K(12+kqKCL,αCL,q2μ+6kqKCL,α-6).
This minimum corresponds to CL→∞ (which is, of course, beyond reach) and(39)X̅=1+2kq/(KCL,α)-2+2CD0/(KCL2)3+3CD0/(KCL2).
Clearly, an aircraft which is designed with sufficiently small moment of inertia to satisfy i̅y<J̅1 and with sufficiently forward center-of-gravity to satisfy (26c) (and (26a)) will be stable at all flight conditions.It is shown in AppendixB that P depends mainly on the aspect ratio of the wing. Since the aspect ratio is dictated by performance (rather than stability) requirements, CM,q remains the main design parameter affecting the maximal moment of inertia for which the aircraft is stable.For a conventional subsonic aircraft,CM,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,(40)CM,q=-2at(xn-xt)2(StSw),
provides a lower bound on |CM,q|. Here, St/Sw is the tail to wing areas ratio, xn-xt is the distance between the aircraft neutral point and the aerodynamic center of the tail, and at is the lift slope coefficient of the tail  [1, page 269].i̅y can be related with longitudinal (reduced) radius of gyration, ry. In fact, i̅y=4ry2 (see the Nomenclature). Hence, one may set J̅1=4R12 and use R1 as a conservative upper bound of ry;(41)R1=(xn-xt)atPStSw,
by (37), (38), and (40). With at=5, St/Sw=6, and the data of our model aircraft, R1 equals, approximately, 0.4(xn-xt)—almost half the distance between the tail and the center-of-gravity (it is implied here that (xcg-xn)≪(xn-xt)). 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 J̅1 from (37) may provide an early warning when longitudinal inertia becomes an issue.



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