[American Institute of Aeronautics and Astronautics AIAA Atmospheric Flight Mechanics Conference and Exhibit - Keystone, Colorado ()] AIAA Atmospheric Flight Mechanics Conference and

Analytical Criterion for Onset of Departure in

Inertia-Coupled Roll Maneuvers of Airplanes

Amrita Mahale∗ and N. Ananthkrishnan†

Indian Institute of Technology Bombay, Mumbai 400076, India

Inertia coupling is the phenomenon by which gyroscopic moments lead to cross-coupling

of dynamic modes in a rolling aircraft. Inertia coupling in roll maneuvers results in nonlin-

ear behavior, such as sudden jump in the roll rate, causing heavy loads on control surfaces

and potential loss of pilot control, apart from yaw and pitch divergence. Autorotation is

also a common consequence. It is more pronounced for aircraft with low inertia in roll,

making modern fighter aircraft with slender long fuselages and short wings prime candi-

dates. Existing analytical studies of inertia-coupled roll maneuvers make the simplifying

assumption of steady rolling. Pitch and yaw divergence can be predicted, but there is no

known analytical technique for the prediction of jump phenomena. This paper refines the

existing theories on inertia coupling by incorporating the effects of perturbations in roll

rate. The first-known analytical criterion for the prediction of jump instabilities has also

been obtained.

Nomenclature

A dynamics matrix of fourth-order model

b input matrix of fourth-order model

C, D, E inertia coupling coefficient in pitch, yaw, and roll, respectively

g acceleration due to gravity

Ix, Iy, Iz roll, pitch, and yaw inertia, respectively

L, M, N total moments about body roll, pitch, and yaw axes, respectively

lβ , lp.. dimensional stability derivatives of L with respect to β, p...

mα, mq.. dimensional stability derivatives of M with respect to α, q...

nβ , nr.. dimensional stability derivatives of N with respect to β, r...

m′

α, m′

q inertia-averaged stability derivatives of M with respect to α and q

n′

β , n′

r inertia-averaged stability derivatives of N with respect to β and r

P1, P2 critical roll rates

P, Q, R total value of roll, pitch, and yaw rates, respectively

P0, Q0, R0 trim values of roll, pitch, and yaw rates, respectively

p, q, r perturbation in roll, pitch, and yaw rates, respectively

ps static residual value of roll rate perturbation

V aircraft speed

α, β angles of attack and sideslip, respectively

δa, δe, δr aileron, elevator, and rudder deflections, respectively

φ roll attitude angle

θ pitch attitude angle

∗Undergraduate Student, Department of Aerospace Engineering; [email protected]. Student Member AIAA.†Associate Professor, Department of Aerospace Engineering; [email protected]. Senior Member AIAA

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American Institute of Aeronautics and Astronautics

AIAA Atmospheric Flight Mechanics Conference and Exhibit21 - 24 August 2006, Keystone, Colorado

AIAA 2006-6267

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Superscripts

( ˙ ) derivative with respect to time, t

(¨) double derivative with respect to time, t

I. Introduction

Inertia coupling is the phenomenon by which gyroscopic moments lead to cross-coupling of dynamic modesin a rolling aircraft. Inertia coupling in roll maneuvers results in nonlinear behavior, such as sudden jump

in the roll rate, causing heavy loads on control surfaces and potential loss of pilot control, apart from yaw andpitch divergence. Autorotation is also a common consequence. It is more pronounced for aircraft with lowinertia in roll, making modern fighter aircraft with slender long fuselages and short wings prime candidates.There are two kinds of critical roll rates that arise due to roll coupling: one associated with yaw (or pitch)divergence, and the other related to jump phenomenon.

One of the possible consequences of the interaction of gyroscopic forces with the aerodynamics of therolling aircraft is the occurrence of autorotation. Autorotation is a condition in which the aircraft locks intosteady rolling even after the ailerons are centered. It manifests itself as a sudden ‘jump’ in the response asthe motion turns unstable and is attracted to a new stable state.

Analytical studies of the aircraft inertia coupling problem, also known as the roll coupling problem,started with the work of Phillips1 in 1948. He restricted himself to the treatment of steady rolling. Rhoadsand Schuler2 showed that Phillips’ critical roll rates could be obtained from the steady-state solutions of anapproximate set of equations of motion, now known as the pseudosteady state (PSS) equations of motion.This method also predicted steady-state values of other variables and permitted the use of control input, andnot the roll rate, as the independent parameter. However, this method could not predict peak disturbancesreliably. This method also (incorrectly) related Phillips’ critical roll rate to the phenomenon of autorotation.Hacker and Oprisiu3 have reviewed the early work on inertia-coupled roll maneuvers.

The work by Schy and Hannah,4 however, brought about a dramatic change in the approach to theanalysis of the problem. They computed PSS roll rate solutions for an aircraft with a given linear aerodynamicmodel as a function of aileron deflection. They demonstrated the existence of multiple PSS roll rate solutionsfor a given aileron input. A jump in the roll rate was seen to occur at a critical aileron deflection. This wasassociated with the folding of one of the PSS solutions. Young et al.5 extended this work with a nonlinearaerodynamic model. They observed that jump phenomenon was mainly caused by the inertia couplingterms in the aircraft equations of motion, with the nonlinear aerodynamic terms playing an insignificantrole. Ananthkrishnan and others6, 7, 8, 9 used bifurcation theory and continuation methods to study the jumpphenomenon in detail. They observed that prevention of jump could be achieved by limiting the growth insideslip developed during roll by manipulating the rudder suitably.

Though a lot of effort has gone into obtaining computational solutions of inertia-coupled roll maneuvers,there is a need for analytical criteria for the prediction of instabilities. Early analytical work assumed steadyrolling. An analysis of the complete fifth-order model including roll rate perturbations is not available. Thus,there is a need for a technique that can incorporate roll perturbations into the analysis of roll maneuvers.

Tradionally, aircraft dynamic modes have been decoupled into two independent sets: longitudinal modes,called the short period and phugoid; and lateral modes, called the roll, dutch roll and spiral. To describethese modes, literal expressions for individual modes have been developed. Simplified literal approximationsare found in several textbooks.10, 11 Ananthkrishnan and Unnikrishnan,12 however, found errors in someof these approximations. Using a two-time scale decomposition, they showed that perturbations in thevariables associated with the fast modes cannot be equated to zero, as there could be a static residual valueof perturbation in these variables. A similar procedure employing fast-slow decomposition can be used toanalyze the roll-coupling problem.

This paper exposes the shortcomings of existing theories in understanding the two types of critical rollrates. Literal approximations to coupled roll-pitch-yaw dynamic modes have been derived to obtain ananalytical criterion for prediction of jump.

II. Pseudosteady-State (PSS) Approach to the Roll-Coupling Problem

The pseudosteady-state (PSS) approach is used to calculate the steady states of the approximate equa-tions of motion. This approach neglects the effects of varying weight components in body axes. This approach

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Table 1. Predicted values of critical roll rate from Eq. (5).

Critical roll rate P1 P2

Aircraft A (none) (none)

Aircraft B 2.98 rad/sec 4.78 rad/sec

was first used by Schy and Hannah4 to predict jump phenomena in roll-coupled maneuvers of airplanes.The approximate nonlinear equations of motion are presented in Appendix A.1. These have been used to

calculate all time histories in this work. They assume that the speed is constant, that the incidence angles(α and β) are small, and that the aerodynamics is linear. Principal axes are used. The equations for thePSS solutions are obtained by dropping the g/V terms in the force equations (which contain φ and θ) andthe φ and θ equations. The gravity term is related to the spiral and the phugoid modes. These slow modesare not of much importance in the roll-coupling analysis. The resulting fifth-order model (Appendix A.2) issolved for pseudosteady states by setting all time derivatives to zero.

A. Solution for Pseudosteady States

Leaving aside the rolling equation, the other four equations are linear in α, β, q and r.4 In vector matrixform, this fourth-order system can be written as follows:

x = Ax + bu + b1P0 (1)

where x = [α, β, q, r]T is the vector of state variables and u = [δe, δr]T is the vector of control inputs. The

coefficient matrices of Eq. (1) are given by:

A =

zα −P0 1 0P0 yβ 0 −1mα 0 mq CP0

0 nβ −DP0 nr

, b =

0 00 0

mδe0

0 nδr

, b1 =

000np

(2)

For PSS solutions, x is set to zero in Eq. (1) giving:

x0 = −A−1[bu + b1P0] (3)

= −adj[A]

det[A][bu + b1P0] (4)

As det[A] → 0, trim values of the four states α0, β0, Q0 and R0 blow up. det[A] is given by a biquadraticin P0. The roots of this biquadratic equation are known as the critical roll rates for pitch-yaw divergence.1

Thus, divergence is exhibited when the roll rate P0 satisfies the equation below:

P 40 + [(m′

α + n′

β) + yβzα − m′

qn′

r]P20 + [m′

αn′

β + m′

αn′

ryβ − m′

qn′

βzα − m′

qn′

ryβzα] = 0 (5)

where the primed quantities are given by m′

α = mα/C, m′

q = mq/C, n′

β = nβ/(−D) and n′

r = nr/(−D).Two different aircraft have been used for the present analysis, whose aerodynamic data are provided in

Appendix A.3. Equation (5) has been used to predict the two values of critical roll rates for these aircraft,which are given by P1 and P2. These values are provided in Table A. Equation (5) predicts no divergencefor Aircraft A and a pair of critical roll rates for Aircraft B at which divergence occurs.

B. Computations using a Continuation Algorithm

To corroborate the predicted values of the critical roll rates for the above aircraft, PSS solutions for anunconstrained rolling motion have been computed for δe = 2 deg and δr = 0 and varying aileron deflectionδa, by using the AUTO200013 continuation algorithm. The PSS solutions so obtained for roll maneuvers ofAircraft A and B are plotted in Figs. 1 and 2, respectively. The solution branch passing through zero rollrate for zero aileron deflection is called the primary branch. The continuation plots for the other trim statesof Aircraft A and Aircraft B are plotted in Figs. 3 and 4, respectively.

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Figure 1. PSS solutions as a function of aileron deflection for δe = 2 deg and δr = 0 for Aircraft A (full lines: stable equilibria;dashed lines: unstable equilibria)

For Aircraft A (Fig. 1), a saddle-node bifurcation (or turning point) of the primary solution branch isseen at δa ≈ −4 deg. The solution branch is then unstable and regains stability after a second saddle-nodebifurcation at δa ≈ 8 deg.

In the continuation plot for Aircraft B in Fig. 2, the primary solution branch loses stability at a saddle-node bifurcation at δa ≈ −12 deg. This unstable branch, however, does not regain stability. Figure 4indicates that all trim values other than the roll rate (α0, β0, Q0 and R0) slowly diverge for increasingpositive values of δa, corresponding to P0 = 3 rad/sec approximately. The new solution branch in Fig. 2undergoes three more saddle-node bifurcations (at points labeled b, c and d) before divergence is exhibitedagain. The new value of roll rate at which trim values blow up is P0 ≈ 4.8 rad/sec. The two values of P0 atwhich divergence is observed match closely with those predicted in Table A, using Eq. (5). However, Eq. (5)cannot predict the occurrence of the saddle-node bifurcations for either Aircraft A or B. For Aircraft B, thevalues of roll rate at which divergence occurs, as predicted by Table A, are never reached, as a jump occursbefore that. It is important to predict the occurrence of the saddle-node bifurcation as it is the first valueof roll rate at which stability is lost, i.e., a jump occurs here. A jump leads to undesirably high values of αand β, from which recovery may be impossible.

III. Simulations

The PSS solutions are steady state solutions of the approximate equations of motion for different controlinputs. To verify the accuracy of the PSS solutions, simulations of the time histories of the complete seventh-order model must be carried out.

The response was evaluated at different values of aileron input for Aircraft A. The response of the rollrate to different values of aileron input is illustrated in Fig. 5. It must be noted that the roll rates have not

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Figure 2. PSS solutions as a function of aileron deflection for δe = 2 deg and δr = 0 for Aircraft B (full lines: stable equilibria;dashed lines: unstable equilibria; filled squares: Hopf bifurcation points)

settled to their steady-state values. However, the steady state values match closely with those predicted byAUTO2000. An interesting observation is that there is a sudden increase in value of roll rate at δa = −4deg, which is just to the left of the saddle-node bifurcation of the PSS model, as shown in Fig. 1.

Figure 6 shows the response of the roll rate of Aircraft B to different values of aileron input. Again, thesteady state values agree closely with the predicted PSS solutions. There is a sudden increase in value ofsteady state roll rate at δa = −12 deg, which is just to the left of the saddle-node bifurcation in the PSSsolutions as seen in Fig. 2.

Thus, the issue of the two types of ‘critical’ roll rates can be resolved as follows:

1. The saddle-node bifurcations are points of jump in roll response (which can lead to autorotation).These correspond to a specific value of aileron input.

2. The value of roll rate at which all trim values blow up is in some sense the critical roll rate that isreferred to in the literature,1 where pitch and yaw divergence are exhibited. This does not correspondto any particular value of aileron input.

Early research3 used an equivalent of the fourth-order PSS model to comment on the stability of theaircraft. If one looks at the characteristic equation of the fourth-order matrix A in Eq. (2), it can be seenthat its determinant is the constant term in this equation. As the eigenvalues of a matrix cross the imaginaryaxis at the origin to enter the right half-plane, the constant term in its characteristic equation goes to zero.The value of the determinant of matrix A being zero was related to the crossing of its eigenvalues to the righthalf plane, and hence, to the inception of instability. This reasoning, however, is fallacious. The stability ofthe system depends on the eigenvalues of the complete fifth-order system. The fourth-order model cannot

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(a) Angle of Attack, α (b) Sideslip, β

(c) Pitch rate, q (d) Yaw Rate, r

Figure 3. PSS trim states for Aircraft A

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(a) Angle of Attack, α (b) Sideslip, β

(c) Pitch rate, q (d) Yaw Rate, r

Figure 4. PSS trim states for Aircraft B

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Figure 5. Roll response of Aircraft A for various aileron inputs with δe = 2 deg

be used to predict stability. It can just be used to compute trim values of state variables α0, β0, Q0 and R0,given a trim value of roll rate P0.

To comment on the stability of the system, one needs to look at the entire fifth-order model. In otherwords, perturbation in roll must be considered to predict the stability of the system.

IV. Analytical Instability Criterion

Literal approximations have traditionally been used to describe decoupled aircraft dynamic modes.10

They are usually derived by considering simplifying assumptions based on the physics of the modes. However,for a long time, many of these assumptions appeared to be ad hoc, and often yielded inaccurate results.Ananthkrishnan and Unnikrishnan12 devised a formal procedure for deriving literal approximations to theaircraft dynamic modes. They also pointed out a major flaw in previous derivations that was responsiblefor the poor approximations to the slow modes. A similar methodology is used here to derive an analyticalinstability criterion for onset of jump phenomenon in inertia-coupled roll maneuvers.

A. Linearized Equations in Second-Order Form

The trim values of the states and controls (of the fifth-order system) are xT = [α0, β0, P0, Q0, R0] anduT = [δa, δe, δr = 0]

The linearized equations for the aircraft dynamics are obtained by considering small perturbations in the

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Figure 6. Roll response of Aircraft B for various aileron inputs with δe = 2 deg

states with the control parameters fixed at their trim values. The state variables are written as a sum ofthe trim value and a small perturbation, for example, Q = Q0 + q. Putting these state variables in the PSSmodel in Appendix A.2, small-perturbation equations in the five state variables can be obtained. These fivelinearized dynamic equations will have perturbed state variables as δxT = [δα, δβ, δp, δq, δr]. For the sake ofconvenience, the δ has been dropped from these symbols and α, β, p, q and r shall denote the perturbationsin trim values of state variables.

Using this notation, the small-perturbation linearized equations obtained can now be written as follows:

α = q − P0β − pβ0 + zαα (6)

β = P0α + pα0 − r + yββ (7)

p = lpp + lββ + lrr − EQ0r − EqR0 (8)

q = mqq + mαα + CP0r + CpR0 (9)

r = nrr + nββ + npp − DP0q − DpQ0 (10)

This fifth-order system can be rewritten as a pair of second-order equations and one first-order equation.In order to do this, q and r must first be expressed in terms of α, β, p and their derivatives, as follows, byrearranging terms in Eqs. (6) and (7):

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q = α + P0β + pβ0 − zαα (11)

r = P0α + pα0 − β + yββ (12)

Using Eqs. (11) and (12), Eqs. (6) and (9) are combined to obtain the coupled longitudinal equationsin second-order form. Similarly, Eqs. (7) and (10) are combined to obtain the coupled lateral equations insecond order form.

α − (zα + mq)α − (mα − mqzα + CP 20 )α + P0(C + 1)β − P0(mq + Cyβ)β

+ pβ0 − (mqβ0 − CR0 − CP0α0)p = 0 (13)

β − (nr + yβ)β + (nβ + nryβ − DP 20 )β − P0(D + 1)α + P0(nr + Dzα)α

−pα0 + (nrα0 − DQ0 − DP0β0)p = 0 (14)

Apart from α and β, the above equations also contain the variables p and p. If p and p can be expressedin terms of α and β, then the above equations can be cast in second-order vector form.

The vector y for the second-order form is chosen as yT = [α, β]. Four linearized first-order equations, i.e,Eqs. (6), (7), (9) and (10), are now recast in the second-order form:

y + Cy + Ky = 0 (15)

The condition for onset of static instability as given by Routh’s criterion is that the constant termin the characteristic equation must be zero. Using the linearized equations in the second-order form, thecharacteristic equation can be written as

|λ2I + λC + K| = 0

where I is the 2 × 2 identity matrix. It is easy to see that the zero constant term condition corresponds to|K| = 0. The stiffness matrix K for this sytem is given by a 2 × 2 matrix:

K =

[

−(mα − mqzα + CP 20 ) + f11(p, p) −P0(mq + Cyβ) + f12(p, p)

P0(nr + Dzα) + f21(p, p) (nβ + nryβ − DP 20 ) + f22(p, p)

]

(16)

where f11, f12, f21 and f22 are given by:

f11(p, p) = (−mqβ0 − CR0 − CP0α0)∂ps

∂α+ β0

∂ps

∂α(17)

f12(p, p) = (−mqβ0 − CR0 − CP0α0)∂ps

∂β+ β0

∂ps

∂β(18)

f21(p, p) = (nrα0 − DQ0 − DP0β0)∂ps

∂α− α0

∂ps

∂α(19)

f22(p, p) = (nrα0 − DQ0 − DP0β0)∂ps

∂β− α0

∂ps

∂β(20)

The expression for p is obtained by solving the linearized equation (8). In the existing literature, thecontribution of p has been usually neglected, assuming the roll mode to be fast. However, we need to checkfor the existence of a static residual roll rate. This is the component of roll perturbation p that persists evenafter the fast dynamics in roll is “complete”. Stengel? has incorporated the static residual roll rate into afourth-order residualised model.

B. Small Perturbations in Roll

The small perturbation equation (8) in roll can be rewritten in terms of α, α, β, β, and p, by using Eqs. (11)and (12). The roll mode can be represented by the first-order dynamics of the following equation:

p = −ER0α + (lrP0 − EQ0P0 + ER0zα)α + (EQ0 − lr)β + (lβ + lryβ − EQ0yβ − ER0P0)β

+(lp + lrα0 − EQ0α0 − ER0β0)p (21)

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It is next assumed that the sideslip and angle of attack vary as per slow modes, therefore, the rates ofchange of the perturbed sideslip and angle of attack during the roll mode are negligible, i.e., β ≈ 0 and α ≈0. The perturbations in the slow mode variables, however, cannot be taken to be zero. These perturbationsact as static forcing functions for the roll mode. At p = 0, the perturbed roll rate dies down to a static valuegiven by:

ps = −(lrP0 − EQ0P0 + ER0zα)α + (lβ + lryβ − EQ0yβ − ER0P0)β

lp + lrα0 − EQ0α0 − ER0β0(22)

On differentiating Eq. (22), an expression for ps is obtained:

ps = −(lrP0 − EQ0P0 + ER0zα)α + (lβ + lryβ − EQ0yβ − ER0P0)β

lp + lrα0 − EQ0α0 − ER0β0(23)

The static residual value of the perturbed roll rate and its derivative from Eqs. (22) and (23) can besubstituted in Eqs. (17) through (20). For this we need to compute the following partial derivatives:

∂ps

∂α= −

(lrP0 − EQ0P0 + ER0zα)

lp + lrα0 − EQ0α0 − ER0β0(24)

∂ps

∂β= −

(lβ + lryβ − EQ0yβ − ER0P0)

lp + lrα0 − EQ0α0 − ER0β0(25)

∂ps

∂α= 0,

∂ps

∂β= 0 (26)

Thus, we now have the complete expression for the K matrix in Eq. (16). The condition for inception ofinstability, i.e., |K| = 0, now becomes

∣

∣

∣

∣

−(mα − mqzα + CP 20 ) + f11(ps) −P0(mq + Cyβ) + f12(ps)

P0(nr + Dzα) + f21(ps) (nβ + nryβ − DP 20 ) + f22(ps)

∣

∣

∣

∣

= 0 (27)

where

f11(ps) =(mqβ0 + CR0 + CP0α0)(lrP0 − EQ0P0 + ER0zα)

lp + lrα0 − EQ0α0 − ER0β0(28)

f12(ps) =(mqβ0 + CR0 + CP0α0)(lβ + lryβ − EQ0yβ − ER0P0)

lp + lrα0 − EQ0α0 − ER0β0(29)

f21(ps) =(DQ0 + DP0β0 − nrα0)(lrP0 − EQ0P0 + ER0zα)

lp + lrα0 − EQ0α0 − ER0β0(30)

f22(ps) =(DQ0 + DP0β0 − nrα0)(lβ + lryβ − EQ0yβ − ER0P0)

lp + lrα0 − EQ0α0 − ER0β0(31)

The alternative to this exercise is to compute the eigenvalues of the Jacobian of the complete fifth-ordersystem at a given trim condition.4 However, there is no known analytical solution to a quintic equation.Thus, handling a 5 × 5 matrix is a tedious exercise. Though the formula in Eq. (27) looks complicated, itis not computationally demanding. Calculating the determinant of a 2 × 2 matrix is not an involved task.Hence, the technique described here is a significant improvement over existing methods. In fact, it is thefirst known analytical criterion for roll-coupled jump instability.

Calculations have been done to verify the predictions of critical roll rate from Eq. (27) against computedresults obtained from AUTO (Figs. 1 and 2). The value of the trim rolling rate P0 has been taken as inputand all trim states α0, β0, Q0 and R0 have been computed. These values have been used to compute thedeterminant of the stiffness matrix K in Eq. (27) for both Aircraft A and B .

Figure 7 shows the plot of det(K) versus P0 for Aircraft A with δe = 2 deg and δr = 0. det(K) has justtwo roots: 1.70 rad/sec and 3.03 rad/sec. These match perfectly with the saddle-node points predicted inFig. 1 by the AUTO computation.

The plot of det(K) versus P0 for Aircraft B with δe = 2 deg and δr = 0 is given in Fig. 8. det(K) has fourroots, and these too match well with the bifurcation points predicted by the AUTO computation, labeled

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a, b, c, d in Fig. 2. There is a discontinuity in the value of the determinant at four points (for 0 < P0 < 5rad/sec). These points are very close to the values of critical roll rate for pitch/yaw divergence as predictedin Table A, corresponding to the roots of the denominator of the expressions for f11(p), f12(p), f21(p) andf22(p), i.e., where

lp + lrα0 − EQ0α0 − ER0β0 = 0

Thus, it can be concluded that:

1. The determinant of the K matrix goes to zero for all saddle-node bifurcation points. This was verifiedfor two different sets of aircraft data.

2. The determinant does not go to zero at any other point.

Figure 7. det(K) as a function of trim roll rate P0 for Aircraft A.

V. Zero-Sideslip Roll Maneuvers

In practice, the sideslip is never allowed to build up freely. The zero-sideslip maneuver is much morerealistic than its unconstrained counterpart. The prediction of the onset of departure in constrained rollmaneuvers requires the recalculation of the trim values of the aircraft. The departure values predicted bythe criterion described in the previous chapter also need to be verified against computational results.

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Figure 8. det(K) as a function of trim roll rate P0 for Aircraft B.

A. Computation of Trim Values

The trim values for the zero-sideslip roll maneuver can be computed by substituting β = 0 in the PSSequations (Appendix A.2). Leaving aside the rolling equation, the other four equations are linear in α, q, rand the control inputs δe and δr.

α = q + zαα

β = pα − r

q = mqq + mαα + Cpr + mδeδe

r = nrr + npp − Dpq + nδrδr (32)

In order to obtain the PSS solutions, the L.H.S. of Eq. (32) must be set to zero. In vector-matrix form, theresultant fourth-order system can now be written as follows:

A1x + b1u = 0 (33)

where xT = [α, q, r, δr] and uT = δe. The coefficient matrices of Eq. (33) are given by:

A1 =

zα 1 0 0P0 0 −1 0mα mq CP0 00 −DP0 nr nδr

, b1 =

00

mδe

0

, (34)

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Thus, the trim values of α, q and r, and the required rudder deflection δr for the zero-sideslip roll maneuverare given by the following:

x0 = −A−11 [b1u] = −

adj[A1]

det[A1][b1u] (35)

B. Computations using a Continuation Algorithm

PSS solutions for the constrained rolling motion have been computed for Aircraft A, for δe = 2 deg andvarying aileron deflection δa, by using the AUTO2000 continuation algorithm. The bifurcation analysis ofconstrained maneuvers calls for the use of the Extended Bifurcation Analysis (EBA) technique. Thereforeto obtain bifurcation diagrams for the zero-sideslip roll maneuver, the model in Appendix A.2 is appendedwith the zero-sideslip constraint in the following form:7

x = f(x, u, s)

β = 0 (36)

where u is the continuation parameter (δa in this case), and s consists of the other control parameters. Ofthese, δr is freed and δe is kept fixed at 2 deg. In the first step of the EBA, a continuation of the system inEq. (36) is carried out. The rudder schedule required to maintain zero-sideslip is computed as a function ofaileron deflection. The variation of δr as a function of δa is included in the model as follows:

x = f(x, δa, δr(δa), δe) (37)

In the second step of the EBA, a bifurcation analysis of the model in Eq. (37) is carried out, with δa

as the continuation parameter. The PSS solutions, so obtained, are plotted in Fig. 9 (reproduced fromreference9 along with the rudder schedule). The straight-line solution branch starting at p=0, δa=0 is thezero-sideslip branch. This is intersected by another solution branch at δa ≈ -0.15 rad, at which an exchangeof stability takes place. This is known as a transcritical bifurcation point. Beyond this bifurcation value ofδa, the zero-sideslip roll maneuver cannot be sustained, as it is dynamically unstable. Thus, the transcriticalbifurcation point represents the onset of departure from the zero-sideslip maneuver, and is also indicative ofthe maximum roll rate the aircraft can achieve in this maneuver.

C. Simulations

All computations so far have used the approximate equations of motion, the steady-state solutions of whichyield the PSS solutions. To verify the accuracy of the solutions, simulations of the time histories of thecomplete seventh-order model must be carried out.

The sideslip response was evaluated at different values of aileron input for Aircraft A. The response ofthe sideslip to the aileron input is illustrated in Fig. 10. Though the sideslip is close to zero for low valuesof aileron, there is a build-up of sideslip for values of aileron greater than 9 deg, which is just to the left ofthe transcritical bifurcation in Fig.9.

Thus, it can be concluded that the transcritical bifurcation signals the loss of stability and the onset ofdeparture from the zero-sideslip roll maneuver. This corresponds to a specific value of aileron input.

D. Prediction from Analytical Criterion

Calculations have been done to compare the predictions of critical roll rate from Eq. (27) against computedresults obtained from AUTO in Fig. 9. The value of the trim roll rate P0 was taken as an input and trimstates α0, Q0 and R0 were computed (β0 = 0). These values were used to compute the determinant of thestiffness matrix K.

A plot of det(K) versus P0 for Aircraft A with δe = 2 deg is given in Fig. 11. det(K) has two roots:2.1293 rad/sec and 4.4546 rad/sec. The first one matches perfectly with the roll rate at which the transcriticalbifurcation occurs in Fig. 9. The second one is outside the range of values plotted in Fig. 9. Thus, it can beverified that the analytical criterion in Eq. (27) is able to capture the onset of instability at a transcriticalbifurcations in constrained maneuvers as well.

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VI. Conclusion

This paper gives a comprehensive analysis of the roll-coupling problem. It lays down criteria for both yaw-pitch divergence as well as jump instabilities. A new analytical criterion for the prediction of jump phenomenain inertia-coupled roll maneuvers has been obtained in this paper. This criterion involves computing thedeterminant of a 2×2 matrix. This technique is a huge improvement over existing techniques, i.e., evaluatingthe eigenvalues of a fifth-order system, which have no analytical solutions. This criterion has been validatedby matching the predicted onset of departure with the computational results generated by a continuationalgorithm, for both unconstrained and constrained roll maneuvers. The analytical criterion derived here alsopredicts the maximum roll rate attainable in the zero-sideslip roll maneuver.

Also, a shortcoming of the existing literature, which neglects perturbation in roll, has been brought tolight. The two pairs of critical roll rates, which were traditionally interpreted as the points of inception ofinstability, are actually points where all trim states (except roll rate) blow up, leading to yaw-pitch divergence.These points could themselves be stable or unstable. Thus, the inadequacy of the steady rolling assumptionhas been demonstrated. Thus, the steady rolling assumption has been discarded and the influence of thestatic residual of the fast roll mode has been incorporated into the traditional fourth-order model consideredin literature to obtain the aforementioned analytical criterion for prediction of jump phenomena.

Appendix A

A.1. Equations of Motion (in Airplane Principal Axes)

Assuming constant velocity and throttle input, the aircraft dynamics can be modeled by the following set ofseven coupled nonlinear first-order differential equations:9

α = q − pβ + zαα + (g

V) cos θ cosφ

β = pα − r + yββ + (g

V) cos θ sin φ

p = lpp + lββ + lrr − Eqr + lδaδa


r = nrr + nββ + npp − Dpq + nδrδr + nδaδa

φ = p + tan θ(q sin φ + r cosφ)

θ = q cosφ − r sin φ

where the inertia coupling terms are given by

C =Iz − Ix

Iy

D =Iy − Ix

Iz

E =Iz − Iy

Ix

These equations include the assumption that α and δe are measured from trim values for straight and levelflight. Also, the aerodynamics are linear.

A.2. Pseudosteady State (PSS) Equations

The PSS equations are obtained by dropping the θ and φ equations, and consequently the θ and φ terms.The PSS method neglects varying weight components along the axes.

α = q − pβ + zαα

β = pα − r + yββ

p = lpp + lββ + lrr − Eqr + lδa)δa


r = nrr + nββ + npp − Dpq + nδrδr + nδaδa

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A.3. Aircraft Data

For computational and simulation purposes, data for two aircraft have been used. These are shown inTable A.1.

Table A.1. Summary of constants for Aircraft A and Aircraft B

Quantity Unit Aircraft A9 Aircraft B2

C 0.959 0.949

D 0.786 0.716

E 0.705 0.727

zα sec−1 -3.0076 -1.329

yβ sec−1 -0.0558 -0.196

lp sec−1 -21.7529 -3.933

lr sec−1 1.5207 0.126

lβ sec−2 -110.45 -9.99

lδa sec−2 -327.28 -45.83

mq sec−1 -1.815 -0.814

mα sec−2 -13.52504 -23.18

mδe sec−2 -33.268 -28.37

np sec−1 0 0

nr sec−1 -0.2608 -0.235

nβ sec−2 3.716 5.67

nδa sec−2 0 0

References

1Phillips, W. H., “Effect of Steady Rolling on Longitudinal and Directional Stability,” NACA TN 1627, 1948.2Rhoads, D. W., and Schuler J. M, “A Theoretical and Experimental Study of Airplane Dynamics in Large Disturbance

Maneuvers,” Journal of Astronautical Sciences, Vol. XXIV, July 1957, pp. 507-526, 532.3Hacker, T., and Oprisiu, C. A., “A Discussion of the Roll-Coupling Problem,” Progress in Aerospace Sciences, Vol. 15,

1974, pp. 151-180.4Schy, A.A., and Hannah, M. E., “Prediction of Jump Phenomena in Roll-coupled Maneuvers of Airplanes,” Journal of

Aircraft, Vol. 14, No. 4, 1977, pp. 375-382.5Young, J.W., Schy, A.A., and Johnson, K. G.,“Pseudosteady State Analysis of Nonlinear Aircraft Maneuvers,” NASA

TP 1758, 1980.6Ananthkrishnan, N., and Sudhakar, K., “A Strategy to Avoid Jump Phenomena in Rapid Rolling Aircraft,” J. Instn

Engrs (India), Vol. 74, 1993, pp. 16-20.7Ananthkrishnan, N., and Sudhakar, K.,“Inertia-coupled Constrained Roll Maneuvers of Airplanes,” Journal of Aircraft,

Vol. 32, No. 4, 1994, pp. 883-884.8Ananthkrishnan, N., and Sudhakar, K.,“Prevention of Jump in Inertia-coupled Roll Maneuvers of Aircraft,” Journal of

Aircraft, Vol. 31, No. 4, 1994, pp. 981-983.9Ananthkrishnan, N., and Sinha, N. K., “Bifurcation Analysis of Inertia-coupled Roll Manoeuvres of Airplanes,” Proc.

Instn Mech. Engrs. Journal of Aerospace Engineering, Vol. 217, Part G, 2003, pp. 75-85.10Nelson, R.C., Flight Stability and Automatic Control, McGraw-Hill, New York, 1989.11Roskam, J.,Airplane Flight Dynamics and Automatic Flight Controls, Part I, Roskam Aviation and Engineering Corpo-

ration, 1979.12Ananthkrishnan, N., and Unnikrishnan, S.,“Literal Approximations to Aircraft Dynamic Modes,” Journal of Guidance,

Control, and Dynamics, Vol. 24, No. 6, 2001, pp. 1196-1201.13Doedel, E. J., Paffenroth, R. C., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B., and Wang, X.,

“AUTO2000: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont),” Technical Report,California Inst. of Technology, Pasadena, CA, 2001.

14Stengel, R.F., Flight Dynamics, Princeton University Press, New Jersey, 2004.

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Figure 9. ARI (top left) and PSS solutions for a zero-sideslip maneuver as a function of aileron deflection δa, for δe = 2 deg(full lines: stable equilibria; dashed lines: unstable equilibria; empty square: transcritical bifurcation point)9

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Figure 10. Sideslip response of Aircraft A for various aileron inputs with δe = 2 deg for zero-sideslip roll maneuver

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Figure 11. det(K) as a function of trim roll rate P0 for Aircraft A for the zero-sideslip roll maneuver

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