Analysis Regarding the Effects of Atmospheric Turbulence on Aircraft Dynamics

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1 Analysis Regarding the Effects of Atmospheric Turbulence on Aircraft Dynamics Gabriela STROE *,1, Irina-Carmen ANDREI 2 *Corresponding author *,1 POLITEHNICA University of Bucharest, Faculty of Aerospace Engineering, Gh. Polizu Street 1-7, Sector 1, Bucharest, , Romania ing.stroe@yahoo.com 2 INCAS National Institute for Aerospace Research Elie Carafoli, B-dul Iuliu Maniu 220, Bucharest , Romania andrei.irina@incas.ro, icandrei28178@gmail.com DOI: / Received: 06 May 2016 / Accepted: 24 May 2016 Copyright Published by INCAS. This is an open access article under the CC BY-NC-ND license ( 4 th International Workshop on Numerical Modelling in Aerospace Sciences, NMAS 2016, May 2016, Bucharest, Romania, (held at INCAS, B-dul Iuliu Maniu 220, sector 6) Section 2 Flight dynamics simulation Abstract: This paper will analyze the Gust Load Alleviation (GLA) systems which can be used to reduce the effects of atmospheric turbulences generated by wind gusts on vertical acceleration of aircraft. Their purpose is to reduce airframe loads and to improve passenger comfort. The dynamic model of the aircraft is more realistic than a rigid-body model, since it includes the structural flexibility; due to its complexity, such model can make feedback control design for gust load alleviation more challenging. The gust is generated with the Dryden power spectral density model. This kind of model lends itself well to frequency-domain performance specifications in the form of the weighting functions. Two classical analytical representations for the power spectral density (PSD) function of atmospheric turbulence as given by Von Kármán and Dryden, were used. The analysis is performed for a set of specified values for flight velocity and altitude (as test cases), with different gust signals that must be generated with the required intensity, scale lengths and PSD functions. Key Words: Gust Load Alleviation (GLA) systems, feedback control design, power spectral density (PSD) function. 1. INTRODUCTION Two classical analytical representations for the power spectral density (PSD) functions of atmospheric turbulence were given by Von Kármán and Dryden, [1], [2]. As the Dryden PSD function has a simpler form than Von Kármán's, then the former will be chosen to be used, [7-9], [12]; it can be written as (1): where: Φ w (ω) = σ w 2 L w πu 0 [1+3(L ω w [1+(L w ω 2 (1), pp ISSN

2 Gabriela STROE, Irina-Carmen ANDREI 124 σ w - is the vertical gust velocity, [m/s] L w - is the turbulence scale length, [m] U 0 - is the aircraft trim velocity, [m/s]. The turbulence scale length L w (2) depends on the aircraft's height h when atmospheric turbulence is encountered, as follows: 580 [m] if h > 580 [m] h > 1750 [ft] L w = { h [m] if h < 580 [m] h < 1750 [ft] For thunderstorms, at any height: L w = 580 [m] L w = 1750 [ft] (2a) σ w = 7 [m/s] σ w = 21 [ft/s] (2b) Power spectral density functions for vertical (3), lateral (4), and longitudinal (5) dynamics [7-9] are given as: Φ u (ω) = 2σ u 2 L u πu 0 Φ v (ω) = σ v 2 L v πu 0 Φ w (ω) = σ w 2 L w πu 0 1 (2) (3) 1+(L u ω U0 )2 [1+3(L ω v [1+(L v ω 2 (4) [1+3(L ω w [1+(L w ω 2 (5) 2. GUST MODEL Gust signals have to be generated with the required intensity, scale lengths and power spectral density - PSD functions for some given flight velocity and height. In order to generate these gust signals, a noise source with power spectral density - PSD function Φ n (ω) = 1 in the frequency band of interest is used to provide the input signal to a linear filter G w (s) - is chosen such that the squared magnitude of its frequency response is the power spectral density - PSD function Φ n (ω), [3-6], [7-9]. Fig. 1. Gust signal generator The Dryden Wind Turbulence Model (Continuous) block uses the Dryden spectral representation to add turbulence to the aerospace model by passing band-limited white noise through appropriate forming filters. This block implements the mathematical representation in the Military Specification MIL-F-8785 C, [7] and Military Handbook MIL-HDBK-1797, [8]. According to the military references [7-8], turbulence is a stochastic process defined by velocity spectra. For an aircraft flying at a speed U 0 through a frozen turbulence field with a spatial frequency of Ω radians per meter, the circular frequency ω is calculated by

3 125 Analysis Regarding the Effects of Atmospheric Turbulence on Aircraft Dynamics multiplying U 0 by Ω, [3-5]. The gust generator setup is shown in Fig. 1; the significance of the notations in Fig. 1 is: n(t) N(0,1) is a Gaussian white noise process of unit intensity and zero mean, w g1 (t) is the random continuous vertical gust, so that, w g1 = G w n, [10]. The power spectral density - PSD of the output signal (6) is related to the PSD of the input signal as follows: Φ w (ω) = G w (jω) 2 Φ N (ω) = G w (jω) 2 (6) To generate a signal with the correct characteristics, a unit variance, band-limited white noise signal is passed through forming filters. The forming filters are derived from the spectral square roots of the spectrum equations. [11-13] An expression for the Dryden filter can be found through spectral factorization of Φ w (ω), which yields relations (7), (8) and (9), [7-9]: G u (s) = σ u 2L u πu ( L u U0 s) (7) G v (s) = σ v L v 1+ 3( Lv U0 s) πu 0 [1+( L v U0 s)2 ] (8) G w (s) = σ w L w 1+ 3( Lw U0 s) πu 0 [1+( L w U0 s)2 ] The Low Altitude Model, [7-9] is defined for an altitude less than or equal to 1000 feet. Three values are defined in this region: turbulence scale length (10), (11), turbulence intensity (12), (13), and turbulence axes orientation. Therefore, the Low-Altitude Model, which is defined for altitude h < 1000 [ft], gives the turbulence scale lengths at low altitudes, with the altitude h [ft], as expressed in [1], [7-9]: L u = L v = (9) L w = h (10) h ( h) 1.2 (11) σ w = 0.1 W 20 (12) σ u 1 = (13) σ w ( h) 0.4 W [ft] (14) The turbulence intensities (12) and (13) are given above, where W 20 represents the wind speed (14) at 20 [ft], that is 6 [m]. Typically for light turbulence, the wind speed at 20 feet is 15 knots; for moderate turbulence, the wind speed is 30 knots; and for severe turbulence, the wind speed is 45 knots. The turbulence axis orientation is defined in case of the low altitude region, as: 1-/ Longitudinal turbulence velocity, along the horizontal relative mean wind vector, and 2-/ Vertical turbulence velocity, aligned with the vertical, [1], [7-9]. The Medium / High Altitude Model, [7-8] is defined for altitude greater than or equal to 2000 feet. Three values are defined in this region: turbulence scale length, turbulence intensity (16), and turbulence axes orientation.

4 Gabriela STROE, Irina-Carmen ANDREI 126 Therefore, the Medium/ High Altitudes Model, for h > 2000 [ft], is defined by turbulence scales (15) and turbulence intensity (16): L u = L v = L w = 1750 [ft] (15) For medium to high altitudes the turbulence scale lengths and intensities are based on the assumption that the turbulence is isotropic, [1], [7-9]. The turbulence intensities are determined from a lookup table that provides the turbulence intensity as a function of altitude and the probability of the turbulence intensity being exceeded, [1], [7-9]. The relationship of the turbulence intensities is represented by equation (16): σ u = σ v = σ w (16) Fig. 2. Medium / High Turbulence Intensities, [8] The Discrete Wind Gust Model block implements a wind gust of the standard 1- cosine shape, [8]. This block implements the mathematical representation in the Military Specification MIL-F-8785 C, [8]. The gust is applied to each axis individually, or simultaneously to all three axes. As input data, there must be specified: 1/ the gust amplitude (the increase in wind speed generated by the gust), 2/ the gust length (length, in meters, over which the gust builds up) and 3/ the gust start time. The discrete gust can be used solely or in multiples to assess airplane response to large wind disturbances. The mathematical representation of the discrete gust (17), [1], [10-12] is: where: V wind = { V m 2 V m is the gust amplitude, d m is the gust length, x is the distance traveled, 0 if x < 0 [1 cos (πx d m )] if 0 x d m V m if x > d m (17)

5 127 Analysis Regarding the Effects of Atmospheric Turbulence on Aircraft Dynamics V wind represents the resultant wind velocity in the body axis frame, [1]. The spectral density definitions of turbulence angular rates are defined in the specifications as three variations, [1] which are displayed in equations (18), (19) and (20): p g = w g y q g = w g r g = v g (18) p g = w g y q g = w g r g = v g (19) p g = w g y q g = w g r g = v g The variations affect only the vertical (q g ) and lateral (r g ) turbulence angular rates, [1]. The longitudinal turbulence angular rate spectrum Φ px (ω) is a rational function. The rational function is derived from curve-fitting a complex algebraic function, not the vertical turbulence velocity spectrum, Φ w (ω), that is multiplied by a scale factor. Because the turbulence angular rate spectra contribute less to the aircraft gust response than the turbulence velocity spectra, it may explain the variations in their definitions, [1]. The variations lead to the following combinations of vertical Φ q (ω) and lateral Φ r (ω) turbulence angular rate spectra: (20) Φ q (ω) Φ r (ω) (21) Φ q (ω) Φ r (ω) (22) Φ q (ω) Φ r (ω) (23) To generate a signal with the correct characteristics, a unit variance, band-limited white noise signal is passed through forming filters. The forming filters are derived from the spectral square roots of the spectrum equations, [1]. At altitudes between 1000 feet and 2000 feet, the turbulence velocities and turbulence angular rates are determined by linearly interpolating between the value from the low altitude model at 1000 feet transformed from mean horizontal wind coordinates to body coordinates and the value from the high altitude model at 2000 feet in body coordinates, [7-9]. The measured wind speed at a height of 6 meters (20 feet) provides the intensity for the low-altitude turbulence model. The measured wind direction at a height of 6 meters (20 feet) results in an angle to aid the transforming of the low-altitude turbulence model into body coordinates. Above 2000 feet, the turbulence intensity is determined from a lookup table that gives the turbulence intensity as a function of altitude and the probability of the turbulence intensity's being exceeded. The turbulence scale length above 2000 feet is assumed constant, and from the military references, a figure of 1750 feet is recommended for the longitudinal turbulence scale length of the Dryden spectra, [7-9]. The frozen turbulence field assumption is valid for the cases of mean-wind velocity and the root-mean-square turbulence velocity, or intensity, is small relative to the aircraft's ground speed, [1]. The turbulence model [15] describes an average of all conditions for clear air turbulence because the following factors are not incorporated into the model: 1. Terrain roughness 2. Lapse rate 3. Wind shears 4. Mean wind magnitude 5. Other meteorological factions (except altitude). [1]

6 Gabriela STROE, Irina-Carmen ANDREI FLEXIBLE AIRCRAFT MODEL The longitudinal dynamics of the flexible aircraft in terms of the state variable representation (24), [4-6], [9], [15] is expressed by the following equations: x = Ax + Bu + B g w g (24.1) y = Cx + Du (24.2) where: x is the state vector, u is the control vector, y represents the vertical acceleration, w g is the vertical gust velocity, [11], [13]. The form (25) of the A matrix in state-space equation (24) shows the couplings between the aircraft's flexible structure and rigid-body dynamics: A RA A = [ A RR A AR ] A AA (25) where: A RR - are the rigid-body terms, A RA - the rigid/aero elastic terms, A AR - the aero elastic/rigid terms, and A AA - the structural flexibility terms. [14], [15]. Input terms associated with the effects of wind gust acting at different stations are also included in equation (24). They appear as different gust signals w gi, i = 1, n acting in the longitudinal dynamic equations. Thus, the gust vector is defined as (26): w g = [w g1 w g2 w g3 w gn ] T (26) Introducing the influence of the gust, expressed by equations (27), (28) and (29), u u + u g (27) α α + α g = α + w g U 0 (28) α g = w g U 0 (29) then equation (24.1) containing the variation of the state vector x with respect to time, becomes (30): u = X u (u + u g ) + X α (α + α g ) gθ α = Z u (u + u g ) + Z α (α + α g ) + (q + q g ) + Z δe δ e (30) q = M u (u + u g ) + M α (α + α g ) + M q q + M δe δ e { θ = q and further (31) as in a matrix form: u + u g α + α x = A [ g ] + Bδ q e (31) θ

7 129 Analysis Regarding the Effects of Atmospheric Turbulence on Aircraft Dynamics The state vector x in eqn. (31) can be connected with the control vector u (32), δ e = u (32) resulting a new equation (33), which can be further developed as (34) and (35): u u + u g α α + α [ q ] = A [ g ] + Bδ q e (33) θ θ X u Z u X α u u α [ q ] = A [ α Z q ] + [ α B] δ M u M e (34) α θ θ 0 0 u u α [ q ] = A [ α q ] + B g δ e (35) θ θ The gust is characterized by the matrix B g (36), which was obtained after extending the matrix [ X u Z u M u 0 X α Z α M α 0 ] with the matrix [B]. B g = [ X u Z u M u X α Z α M α B] (36) 0 0 The turbulence is characterized by the turbulence scale (15) and turbulence intensity (16). σ u = σ v = σ w = 21 [ft/s] (37) Gust response analysis [13] requires a huge amount of calculations, since often a single aircraft has to be certified through aviation regulations. Different types of gusts, ranging from discrete to stochastic, have to be tested on an airplane and for each case the designer must prove that either with or without a proper control system, the airplane is capable of standing critical design gust loads, and grants ride comfort together with equipment functionality. The development of a gust alleviation control system, anyway, requires a good knowledge of both the dynamics of the same aircraft to be controlled, and of the possible interactions of airplane flexible modes with flight mechanics. This last aspect is nowadays increasing its importance. The run for increasing endurance range on jet liners lead to the choice of new productive technologies which allowed to use new materials to provide new shapes to airplanes, consequently changing the typical approach to structural stiffness. The increasing importance of sensor crafts, using very original geometric configurations, slender wings and composite materials, compelled aero elasticity to start looking at methodologies capable of including flight mechanics within aero elastic modeling, in order to have a unified approach to aero elastic issues, [14], [15]. Traditionally, when dealing with conventional aircraft configuration, due to the frequency separation between rigid and vibration mode, specific flight controllers for flight mechanics and vibration control are designed to work in a well separated frequency band.

8 Gabriela STROE, Irina-Carmen ANDREI NUMERICAL SIMULATION AND CONCLUSIONS Turbulence is a stochastic process defined by velocity spectra. The turbulence field is assumed to be visualized as frozen in time and space (i.e.: time variations are statistically equivalent to distance variations in traversing the turbulence field). This assumption implies the turbulence-induced responses of the aircraft result only from the motion of the aircraft relative to the turbulent field. Either the Dryden spectral representation or the Von Kármán spectral representation can be used to generate turbulence by filtering band-limited white noise with an appropriate forming filter derived from the spectral representation. To generate a turbulence signal with the correct characteristics, a unit variance, band-limited white noise signal is passed through or used in the appropriate filters. 350 Step Response Amplitude Time (seconds) Fig. 3 - Step response These filters are ultimately derived from the turbulence spectra. For the Dryden Wind Turbulence Model (Continuous) block, the specifications result in the same transfer function after evaluating the turbulence scale lengths, and the turbulence transfer functions balance each other out. To use the continuous Von Kármán filter properly, the unit variance, bandlimited white noise signal is passed through the Von Kármán forming filters. Deriving forming filters requires that the spectral square roots be obtainable from the spectrum equations. Since the Von Kármán spectra are not spectrally factorable, the Von Kármán spectra must be curve-fitted to a satisfactory degree of approximation with a factorable spectral form for which transfer functions may be obtained. The Von Kármán forming filters are then derived from the spectral square roots of the approximated spectrum equations. To use the continuous Dryden filter properly, the unit variance, band-limited white noise signal is passed through the Dryden forming filters. Deriving forming filters requires that the spectral square roots be obtainable from the spectrum equations. Since the Dryden spectra are spectrally factorable, the Dryden forming filters are then derived from the spectral square roots of the Dryden spectrum equations. To use the discrete Dryden filter properly, the unit variance, band-limited white noise signal is used in the digital filter finite difference equations. To derive the discrete Dryden finite difference equations, the Euler integration method is applied to the continuous Dryden forming filters.

9 131 Analysis Regarding the Effects of Atmospheric Turbulence on Aircraft Dynamics 1.5 Impulse Response Amplitude Time (seconds) Fig. 4 - Impulse Response The turbulence model is divided into two distinct regions, low altitude and medium / high altitude, affecting the turbulence scale lengths and intensities used to generate wind velocities and angular rates. Additionally, the turbulence axes orientation also differs in these regions theta gust time(s) Fig. 5 - Variation of Gust The turbulence intensities are defined in a diagram that provides the turbulence intensity as a function of altitude and the probability of the turbulence intensity being exceeded. The unified wind turbulence block calculates both low altitude turbulence and medium / high altitude turbulence. If the altitude is above the low altitude region then the turbulence is calculated at 1000 ft or if the altitude is below the medium / high altitude region then the turbulence is calculated at 2000 ft. If the altitude lies in neither the low nor the medium /

10 Gabriela STROE, Irina-Carmen ANDREI 132 high altitude region, the interpolation method is implemented. The interpolation method takes turbulence values and linearly interpolates between the value from the low altitude model at 1000 feet transformed body axes and the value from the high altitude model at 2000 feet in body axes. Development of a gust alleviation control system requires a good knowledge either of the dynamics of the same aircraft to be controlled, and the possible interactions of airplane flexible modes with flight mechanics. For the Dryden Wind Turbulence Model (Continuous) block, the specifications result in the same transfer function after evaluating the turbulence scale lengths, and the turbulence transfer functions balance each other out. REFERENCES [1] * * * Inc. Aerospace Blockset. [2] F. Auger, P. Flandrin, P. Goncalves and O. Lemoine, Time-Frequency Toolbox for use with MATLAB, Rice University (USA), October 26, [3] M. Rauw, FDC A SYMULINK Toolbox for Flight Dynamics and Control Analisys, [4] B. Etkin, Dynamics of Flight-Stability and Control, Wiley, New York, USA, 2-nd edition, [5] D. McLean, Automatic Flight Control System, Prentice Hall, Hertfordshire, UK, [6] L. Moysis, M. Tsiaousis, N. Charalampidis, M. Eliadou, I. Kafetzis, An Introduction to Control Theory Applications with MATLAB, 31 August [7] * * * U.S. Military Handbook MIL-HDBK-1797, 19 December [8] * * * U.S. Military Specification MIL-F-8785C, 5 November [9] D. McRuer, I. Ashkenas, D. Graham, Aircraft Dynamics and Automatic Control, Princeton University Press, July [10] J. Yeager, Implementation and Testing of Turbulence Models for the F18-HARV Simulation, NASA CR , Lockheed Martin Engineering & Sciences, March [11] T. Pistner, G. J. Rabadan, N. P. Schmitt and W Rehm, Airborne lidar for automatic feedforward control of turbulent in-flight phenomena, Journal of Aircraft, 47(2): , Mar.-Apr [12] R. Szabolcsi, Mathematical Models for Gust Modeling Applied in Automatic Flight Control Systems' Design, Proceedings of the 5 th International Conference New Challenges in the Field of Military Sciences 2007, Vol. II, pp (95-118), Nov. 2007, Budapest. [13] R. Szabolcsi, G. Mészáros, Computer Aided Simulation of the Random Atmospheric Turbulences, CD-ROM Proceedings of the 6 th International Conference on Crisis Management, pp ( ), May, 2008, Brno, Czech Republic. [14] B. L. Stevens, F. L. Lewis, Aircraft Control and Simulation, John Wiley, USA, [15] D. Sheu, and C. E. Lan, Estimation of Turbulent Vertical Velocity from Nonlinear Simulations of Aircraft Response, Journal of Aircraft, Vol. 48, No. 2, March-April, 2011.