The spectacle of a rolling aircraft close to the tarmac invariably draws a standing ovation during

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1 Approximating the rolling aircraft J G Manathara & S Pradeep Department of Aerospace Engineering, Indian Institute of Science, Bangalore A R T I C L E I N F O Unpublished manuscript. A B S T R A C T This paper evaluates the existing literal approximations to roll mode of a conventional aircraft and shows them to be unsatisfactory. The paper proposes a new and simple roll mode approximation that is accurate for a wide range of aircraft in a variety of flight conditions. Keywords: literal approximations, roll mode, aircraft equations The spectacle of a rolling aircraft close to the tarmac invariably draws a standing ovation during air shows. The governing equations of the aircraft motion also show that an airplane is capable of almost pure roll. An aircraft when excited by the disturbances in the atmosphere, either responds with motions purely in the plane of symmetry or with motions out of the plane of symmetry. The former is called longitudinal motion and the later, lateral-directional motion. In longitudinal motion, the response is either a phugoid long period and lightly damped) or a short period short period and highly damped). There are three lateral-directional modes: dutch roll, roll and spiral modes. The governing equation of longitudinal dynamics is quartic. Lateral-directional motion also essentially has a governing equation that is quartic as one of the roots of the actual qunic governing equation is zero which belongs to the neutrally stable yaw mode. A neutrally stable yaw mode implies that the heading direction does not affect the dynamics of an airplane. Exact closed form solutions exist to the governing equations as they are quartic [1]. However, the solution is huge and does not provide a physical insight into the dynamics of aircraft. Same is the case with numerical solutions obtained by solving the governing equations on a computer as they do not bring out the effect of parameters stability derivatives) on the modes. This has led to the quest for approximate solutions to the modes. A good approximation must be accurate and simple and should bring out all the essential physics. Currently, we are looking into the lateral-directional approximations. Flight dynamics texts state that the approximations to the roll and the dutch roll frequency are good and support this statement with an example or two. In our study, we found that a good roll approximation is necessary for a good dutch roll approximation. This led to our reexamination of the roll approximations. We found that it is good as stated in texts, but not good enough if we are to derive dutch roll approximation based on the roll approximation. In this paper we propose a new approximation to the roll.

2 Roll mode approximations 2 1 Governing equations Following the notation of Roskam [2], the stick fixed lateral-directional small perturbation equations for steady, level, one g trim flight condition, are: = Y + Y p p Y r ) r + g cos Θ 1 ϕ ṗ A 1 ṙ = L + L p p + L r r ṙ B 1 ṗ = N + N p p + N r r 1) ψ = r The corresponding lateral-directional characteristic equation is where A = 1 A 1 B 1 ) sas 4 + Bs 3 + Cs 2 + Ds + E) = 0 2) B = Y 1 A 1 B 1 ) L p + N r + A 1 N p + B 1 L r ) C = L p N r L r N p ) + Y L p + N r + A 1 N p + B 1 L r ) Y p L + N A 1 ) + L B 1 + N ) Y r L B 1 + N ) 3) D = Y L p N r L r N p ) + Y p L N r N L r ) g cos Θ 1 L + N A 1 ) + L N p N L p ) Y r L N p N L p ) E = g cos Θ 1 L N r N L r ) Decoupling the angular acceleration equations in the equation set 1), which are coupled through inertial terms, results in, = Y + Y p p Y r ) r + g cos Θ 1 ϕ ṗ = L + A 1 N ) 1 A 1 B 1 + L p + A 1 N p ) 1 A 1 B 1 p + L r + A 1 Nr) 1 A 1 B 1 r ṙ = N + B 1 L ) + N p + B 1 L p ) p + N r + B 1 L r ) r 4) 1 A 1 B 1 1 A 1 B 1 1 A 1 B 1 ψ = r Introducing the notations, L i = L i + A 1 N i 1 A 1 B 1 & N i = N i + B 1 L i 1 A 1 B 1 5)

3 Roll mode approximations 3 the set of equations 4) can be rewritten as, = Y + Y p p Y r ) r + g cos Θ 1 ϕ ṗ = L + L pp + L rr ṙ = N + N pp + N rr 6) ψ = r The characteristic equation can then be written as, where a 3 = B A = Y L p N r ss 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 ) = 0 7) a 2 = C A = L pn r L rn p) + Y L p + N r) Y p L + N Y r N a 1 = D A = Y L pn r L rn p) + Y p L N r N L r) g cos Θ 1 L 8) + L N p N L p) Y r L N p N L p) a 0 = E A = g cos Θ 1 L N r N L r) The roots of the characteristic equation 7) gives the modes of the lateral-directional dynamics which are the spiral mode, the roll mode and the dutch roll mode. The characteristic equation can then be factored as follows, λ 4 + a 3 λ 3 + a 2 λ 2 + a 1 λ + a 0 = λ λ s )λ λ r )λ 2 + 2ζ D ω nd λ + ω 2 n D ) 9) where λ s is the spiral root, λ r is the roll root, ω nd damping. is the dutch roll frequency, and ζ D is the dutch roll 2 Existing roll approximations A lot of distinct approximations for roll mode exists in literature. This section gives an almost exhaustive list of existing roll approximations. 2.1 One degree of freedom approximation The philosophy behind such an approximation is that the roll mode consists largely of rolling motion. Therefore, it can be approximated as having only one degree of freedom. Thus all the variables in equation 1) except p and ϕ are set to zero. For sake of brevity, the kinematic equation ψ = r, which is responsible for the neutrally stable yaw mode, is omitted and will be omitted in all further listings. The equation

4 Roll mode approximations 4 1) then becomes, 0 = 0 ṗ = L p p 0 = 0 10) The characteristic equation of above set of equations is λλ L p ) = 0 11) which gives the rolling approximation as λ r = L p 12) Such an approximation to roll is given by Roskam [2], Nelson [3], Hancock [4], Babister [5], Blakelock [6], and Ananthkrishnan and Unnikrishnan [7]. Etkin and Reid [8], McLean [9], Seckel [10], and Cook [11] follow a similar approach but use equation 6) instead of equation 1) to arrive at an approximation for roll mode as, λ r = L p 13) 2.2 Two degree of freedom approximation In this approximation, as given by Russel [12], the side force equation is set to zero. It is assumed that the yawing moment generated by the derivative N p is balanced by that generated by the sideslip derivative N. Also assuming the yawing velocity to be small, the equation 1) can be written as, This gives the rolling approximation as 0 = 0 ṗ = L + L p p 0 = N + N p p 14) 2.3 Three degree of freedom approximation λ r = L p L N p N 15) The roll mode involves very small sideslip ) motions. Thus in the equation 6), and can be set to zero in the side force equation. Further with assumptions that Y r 0, Θ 1 0, Y p 0, the equation 6) becomes the three degree of freedom equation for the spiral and roll mode, 0 = r + g ϕ ṗ = L + L pp + L rr ṙ = N + N pp + N rr 16)

5 Roll mode approximations 5 The resulting characteristic equation is, λ 2 + L p + L N N p g ) ) ) λ + g L N N r L r = 0 17) Assuming that the spiral root is small, the roll approximation is given as, ) λ r = L p L N p N gu1 18) McRuer [13] and Stevens [14] gives such an approximation to the roll mode. 2.4 Approximation by comparison of coefficients Comparing the coefficients in equation 9) an approximate expression for roll mode root can be obtained as, λ r D C Simplifying the expressions for C and D in equation 3) by neglecting relatively small derivatives, the roll root becomes This approximation is given by Bu Aer Report [15]. 19) λ r = L p + gl N + Y L p N r N 20) 2.5 Kolk s approximation Kolk [16] gives an approximate expression for roll root as λ r = b3 2 + b 0 b b 1 21) where b 2 = B A E [ 1 + EC ] D D 2 b 1 = C A E [ 1 + EC ] D D 2 b 2 22) b 0 = D A E [ 1 + EC ] D D 2 b 1 Such an expression is arrived at by using Lin s Method [17] of approximate factorization to obtain approximate roots of an algebraic equation. 2.6 Livneh s approximation The literal approximation for roll root as given by Livneh [18] is N p B 1 λ r g cos Θ 1 / λ r λ r + L ) 23) 1 A 1 B 1 ) λ 2 r 2ζ D ω nd λ r + ω n 2D

6 Roll mode approximations 6 where 2.7 Mengali and Giuliettis approximation λ r = L p ω n 2 D = N + N ry N Y r 24) 2ζ D ω nd = N r Y Assuming the spiral root to be very small, the remaining characteristic polynomial can be written as s 3 + B A s2 + C A s + D A = s2 + 2ζ D ω nd s + ω n 2 D )s λ r ) 25) Equating the the coefficients on both sides of the equation [ λ r 26)] 27) gives B A = λ r + 2ζ D ω nd 26) C A = ω n 2 D 2ζ D ω nd λ r 27) D A = ω n 2 Dλ r 28) C/A) B/A) λ r = λ r ω nd )λ r + ω nd ) 29) Divide 27) by ω nd and combine add and subtract) with 26) to obtain 30) 31) using 29) gives C/A B/A) = λ r + ω nd )1 2ζ D ) ω nd 30) C/A B/A) = λ r ω nd )1 + 2ζ D ) ω nd 31) B/A) λ r C/A)) 1 4ζD 2 ) = B/A) 2 C/A)2 ω 2 32) nd 1 Noting that ω 2 n = λ r D D/A) and 1 4ζ2 D equation 32) becomes B/A) λ r C/A) = B/A) 2 + C/A)2 D/A) λ r 33) This gives the approximation for roll mode as [ ] B/A) 2 + C/A) λ r = [ B/A) + C/A) 2 / D/A) ] 34) Using equation 8) and simplifying assumptions such as Y N r/u 1 N, N p L N r and Y L r/ the above equation can be reduced as L λ r = L p + N r + Y L p N r Y + ) 2 + N + L p [ N + L p ) N r + Y N r + Y )] 2 / N λc r ) 35)

7 Roll mode approximations 7 Table 1: Airplanes in Roskam s database Aircraft Representative of: Flight Conditions Cessna 172 A small, single piston engine 1) Power approach general aviation airplane Beech M99 1) Power approach B small, twin turboprop 2) Low altitude cruise regional commuter airplane 3) High altitude cruise SIAI Marchetti S211 1) Power approach C small, single jet engine 2) Normal cruise military training airplane 3) High altitude cruise Gates Learjet M24 1) Power approach D twin jet engine 2) Maximum weight cruise corporate airplane 3) Low weight cruise McDonnell Douglas F4C 1) Power approach E twin jet engine 2) Subsonic cruise fighter/attack airplane 3) Supersonic cruise Boeing 747 1) Power approach F large, four jet engine 2) High altitude cruise commercial transport airplane 3) Low altitude cruise where ) λ c r = L p + L N p N gu1 This approximation is given by Mengali and Giulietti [19]. 3 Evaluation of accuracies of existing approximations The accuracy of an approximation should be evaluated for different types of aircraft and different flight conditions to establish the extend of its generality. Such a wide database is given in Appendix C of Roskam [2]. It gives data pertaining to six modern aircrafts in a total of sixteen flight conditions the details of which are given in Table 1. The flight conditions range from power approach at sea level to cruise at low and high altitudes. The database is thus representative of a wide spectrum of airplanes and flight conditions. The metric for the evaluation of the accuracies of various approximations is selected as the percentage error in the root computed through the approximate expression relative to the exact root. The accuracies of various roll mode approximations as discussed above over the chosen database is given in Table 2. For the one degree of freedom approximation, the percentage error of the approximation as given by equation 13) only is presented in the table as it is relatively more accurate than the approximation in equation 12). 4 Comments on existing roll mode approximations As seen from Tables 2, most of the existing roll mode approximations are inaccurate except for Mengali and Giuliettis approximation 35) which performs well except for three cases - A1, B2 and C1. Kolks

8 Roll mode approximations 8 Table 2: Accuracy of existing roll mode approximation Aircraft Flight eqn 13 eqn 15 eqn 18 eqn 20 eqn 21 eqn 23 eqn 35 Phase % Error % Error % Error % Error % Error % Error % Error A B C D E F approximation 21) is good except for the power approach of the aircraft E and high and low altitude cruises of aircraft F. However, this approximation is too huge an expression to receive any worthy appreciation. Approximation as in equation 20) gives reasonably good accuracies except for the power approach cases. Same is the case with Livnehs approximation 23) but in this case, again the expression is large. It has been observed that a good rolling approximation holds the key to an accurate dutch roll frequency and damping approximations. Thus an accurate but simple expression for roll mode approximation is inevitable. The development of a new accurate roll mode approximation is presented in next section. 5 Development of a new roll approximation Russel [12] derives the three degree of freedom dutch roll approximation in the following manner. Neglecting small and cross coupling derivatives except those due to slideslip, the equation 1) can be written as, = Y r + g cos Θ 1 ϕ ṗ = L + L p p ṙ = N + N r r 36) These equations lead to the characteristic equation given by, λ 4 + N r Y ) L p λ 3 Y + N r + N + L p ) Y L p N r + N g cos Θ 1 L )) Y + N r λ ) ) λ + g cos Θ 1 L N r = 0 38)

9 Roll mode approximations 9 This can be factorized approximately into quadratic factors as, [ λ 2 + N r Y ) λ + L 2 p + N + Y + L g cos Θ 1 / L 2 p + N ) λ + [ λ 2 + L p L g cos Θ 1 / L 2 p + N L 2 p L N r g cos Θ 1 / L 2 p + N )] N r )] 0 39) Although Russel does not proceed to do this, an approximate expression for roll mode can be derived from the above equation. The second quadratic factor in the characteristic equation represents combined roll and spiral modes. Assuming spiral root to be negligible in comparison to the roll root, an approximation to roll can be derived as, λ r = L p + L g cos Θ 1 / L 2 p + N 40) Using equation 6) instead of equation 1), assuming that Θ 1 0 and accounting for the cross coupling derivatives which was earlier ignored for the ease of derivation, a new accurate expression for roll mode approximation can be obtained as, λ r = L p + L ) g N p L 2 p + N ) + YL rn p L pn r) L 2 p + N ) 41) Our studies through extensive simulations revealed that the roll modes of at least some airplanes involve the participation of yaw and sideslip which if not accounted for will result in an incomplete and inaccurate representation of the mode. Therefore, any good approximation to roll should respect the participation of yaw and sideslip. The new approximation ensures this through the presence of yaw in roll equation L r ), roll in yaw balance N p ) and the side force term Y ). The accuracy of above approximation over the selected database is given in Table 3. As seen from the table, the new approximation is almost exact except for two cases. It is to be appreciated that the expression is really small and yet captures the whole physics of the roll mode so as to give an accurate result for the wide database chosen. This ensures the generality of the new approximation derived. 6 Conclusions An almost exhaustive survey of existing roll mode approximations was conducted and listed. The accuracies of above approximations were evaluated over a database with data for wide range of aircraft types and different flight conditions thus studying the generality of these approximations. It was shown that no simple yet accurate approximation for roll existed. A new approximation was derived. The new approximation is simple and was shown to be very accurate over the chosen wide database. References [1] Abramowitz, M. and Stegun, I. A. Eds.), Solutions of Quartic Equations, Section 3.8.3, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972, pp

10 Roll mode approximations 10 Table 3: Accuracy of the new roll mode approximation Aircraft Flight eqn 41 Phase % Error A B C D E F [2] Roskam, J., Airplane Flight Dynamics and Automatic Flight Controls, Part I, DAR Corporation, Lawrence, Kansas, [3] Nelson, R.C., Flight Stabiliy and Automatic Control, McGraw-Hill, New York, [4] Hancock, G., An Introduction to the Flight Dynamics of Rigid Aeroplanes, Ellis Horwood, [5] Babister, A., Aircraft Dynamic Stability and Response, Pergamon Press, [6] Blakelock, J., Automatic Control of Aircraft and Missiles, J. Wiley & Sons, [7] Ananthkrishnan, N., and Unnikrishnan, S., Literal Approximations to Aircraft Dynamic Modes, Journal of Guidance, Control and Dynamics, Vol. 24, No. 6, 2001, pp [8] Etkin, B., and Reid, L., Dynamics of Flight: Stability and Control, J. Wiley & Sons, [9] McLean, D., Automatic Flight Control Systems, Prentice Hall, [10] Seckel, E., Stability and Control of Airplanes and Helicopters, Academic Press, [11] Cook, M.V., Flight Dynamics Principles, Arnold, London, [12] Russell, J.B., Performance and Stability of Aircraft, Butterworth-Heinemann, [13] McRuer, D., Ashkanas, and Graham, D., Aircraft Dynamics and Automatic Control, Princeton University Press, [14] Stevens, B., and Lewis, F., Aircraft Control and Simulation, J. Wiley & Sons, 2004.

11 Roll mode approximations 11 [15] Bu Aer Report, AE , April [16] Kolk, R. W., Modern Flight Dynamics, Prentice Hall, [17] Lin, S. N., A Method for Finding Roots of Algebraic Equations, J. Math. and Phys., vol. 22, No. 2, 1943, pp [18] Livneh, R., Improved Literal Approximation for Lateral-Directional Dynamics of Rigid Aircraft, Journal of Guidance, Control and Dynamics, Vol. 18, No. 4, 1995, pp [19] Mengali, G., and Giulietti, F., Unified Algebraic Approach to Approximation of Lateral-Directional Modes and Departure Criteria, Journal of Guidance, Control and Dynamics, Vol. 27, No. 4, 2004, pp