《航空器的稳定与控制》（英文版）Lecture 14 Equations of Motion

16.333: Lecture #14 Equations of Motion in a Nonuniform Atmosphere Gusts and Winds

16.333: Lecture # 14 Equations of Motion in a Nonuniform Atmosphere Gusts and Winds 1

Fa2004 16.33312-2 Equations of Motion Analysis to date has assumed that the atmosphere is calm and fixed Rarely true since we must contend with gusts and winds Need to understand how these air motions impact our modeling of the aircraft Must modify aircraft equations of motion since the aerodynamic forces and moments are functions of the relative motion between the aircraft and the atmosphere and not of the inertial velocities Thus the LHS of the dynamics equations(f= ma)must be written in terms of the velocities relative to the atmosphere If u is the aircraft perturbation velocity(X direction ), and ug is the gust velocity in that direction, then the aircraft velocity with respect to the atmosphere is Now rewrite aerodynamic forces and moments in terms of aircraft velocity with respect to the atmosphere(see 4-11) OX OX OX △X a-a aU )+am(0-y)+ OX OX 0X9 (q-qg)+…+ 0+..+-6+△XC 0Q The gravity terms ae and control terms △Xe=X66e+X820p stay the same

Fall 2004 16.333 12–2 Equations of Motion • Analysis to date has assumed that the atmosphere is calm and fixed – Rarely true since we must contend with gusts and winds – Need to understand how these air motions impact our modeling of the aircraft. • Must modify aircraft equations of motion since the aerodynamic forces and moments are functions of the relative motion between the aircraft and the atmosphere, and not of the inertial velocities. – Thus the LHS of the dynamics equations (F� = m�a) must be written in terms of the velocities relative to the atmosphere. – If u is the aircraft perturbation velocity (X direction), and ug is the gust velocity in that direction, then the aircraft velocity with respect to the atmosphere is ua = u − ug • Now rewrite aerodynamic forces and moments in terms of aircraft velocity with respect to the atmosphere (see 4–11) ∂X ∂X ∂X ΔX = (u − ug) + (w − wg) + ∂ ˙ ( ˙ w˙ g) ∂U ∂W W w − ∂X ∂X ∂Xg + (q − qg) + . . . + θ + . . . + θ + ΔXc ∂Q ∂Θ ∂Θ – The gravity terms ∂Xg and control terms ∂Θ ΔXc = Xδeδe + Xδpδp stay the same

Fa2004 16.33312-3 The rotation gusts pg, g, and rg are caused by spatial variations in the gust components = rotary gusts are related to gradients of the vertical gust field O Pg du and gg ax Ng Gust field creating an effective rolling gust. Equivalent distribution by a pitching motion ed Figure 1: Gust Field creating an effective pitching gust

Fall 2004 16.333 12–3 • The rotation gusts pg, qg, and rg are caused by spatial variations in the gust components ⇒ rotary gusts are related to gradients of the vertical gust field ∂wg ∂wg pg = and qg = ∂y ∂x g g +wg +wg y Gust field creating gust. -w -w an effective rolling Equivalent distribution Equivalent to velocity created by a pitching motion Figure 1: Gust Field creating an effective pitching gust

Fa2004 16.33312-4 The next step is to include these new forces and moments in the equations of motion 00 X mg cos eo 0 z;00 Zu zu la t mUo -mg sin eo 0 Mu Mw Mq q 016 00 0 Zs Zi 00 X1-X20 z2 -M -Mu -M 0 0 →Ei=Ax+Bu+B Multiply through by e to get new state space model A r+B, u+B, w which has both control u and disturbance w inputs A similar operation can be performed for the lateral dynamics in terms of the disturbance inputs vg, Pg, and rg Can now compute the response to specific types of gusts, such as a step or sinusoidal function, but usually are far more interested in the response to a stochastic gust field

� � Fall 2004 16.333 12–4 • The next step is to include these new forces and moments in the equations of motion ⎡ ⎤ ⎡⎤⎡ ⎤ ⎤⎡ m 0 0 0 u˙ Xu Xw 0 −mg cos Θ0 u ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ 0 m − Zw˙ 0 0 0 −Mw˙ Iyy 0 0 0 0 1 w˙ q˙ Zu Zw Zq + mU0 −mg sin Θ0 Mu Mw Mq 0 w q = θ ˙ 0 0 1 0 θ ⎡ ⎤ Xδe Xδp Zδe Zδp Mδe Mδp ⎥ ⎥ ⎥ ⎦ δe δt ⎢ ⎢ ⎢ ⎣ + 0 0 −Xu −Xw 0 ⎡ ⎤ ⎡ ⎤ ⎢ ⎢ ⎢ ⎣ ⎣ ⎥ ⎥ ⎥ ⎦ ug −Zu −Zw 0 −Mu −Mw −Mq + w ⎦g qg 0 0 0 Ex ˆ Buu + ˆ ⇒ ˙ = Ax + ˆ Bww • Multiply through by E−1 to get new state space model x˙ = Ax + Buu + Bww which has both control u and disturbance w inputs. – A similar operation can be performed for the lateral dynamics in terms of the disturbance inputs vg, pg, and rg. • Can now compute the response to specific types of gusts, such as a step or sinusoidal function, but usually are far more interested in the response to a stochastic gust field ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣

Fa2004 16.33312-5 Atmospheric Turbulence Can develop the best insight to how aircraft will behave with gust disturbances if we treat the disturbances as random processes What is a random process? Something(signal) that is random so that a deterministic description is not practical But we can often describe the basic features of the process(e. g mean value. how much it varies about the mean Atmospheric turbulence is a random process, and the magnitude of the gust can only be described in terms of statistical properties For a random process f(t), talk about the mean square 尸2(t)=lim T f(t)dt as a measure of the disturbance intensity (how strong it is) Signal f(t) can be decomposed into its Fourier components, so can use that to develop a frequency domain measure of disturbance strength (w)a that portion of f2(t)that occurs in the frequency band +d (w) is called the power spectral density Bottom line: For a linear system y=G(s w,then 重(u)=a(u)G(j) Given an input disturbance spectral density(e. g gusts), quite si ple to predict expected output (e.g. ride comfort, wing loadia m-

Fall 2004 16.333 12–5 Atmospheric Turbulence • Can develop the best insight to how aircraft will behave with gust disturbances if we treat the disturbances as random processes. – What is a random process? Something (signal) that is random so that a deterministic description is not practical!! – But we can often describe the basic features of the process (e.g. mean value, how much it varies about the mean). • Atmospheric turbulence is a random process, and the magnitude of the gust can only be described in terms of statistical properties. – For a random process f(t), talk about the mean square � T 1 f 2(t) = lim f 2 (t) dt T 0 T→∞ as a measure of the disturbance intensity (how strong it is). • Signal f(t) can be decomposed into its Fourier components, so can use that to develop a frequency domain measure of disturbance strength – Φ(ω) ≈ that portion of f 2(t) that occurs in the frequency band ω ω → + dω – Φ(ω) is called the power spectral density • Bottom line: For a linear system y = G(s)w, then Φy(ω) = Φw(ω)|G(jω) 2 | ⇒ Given an input disturbance spectral density (e.g. gusts), quite sim￾ple to predict expected output (e.g. ride comfort, wing loading)

Fa2004 16.33312-6 mplementing a PSD in Matlab . Simulink has a Band-Limited White noise block that can be used for continuous systems Primary difference from the Random Number block is that this block produces output at a specific sample rate, which is related to the correlation time of the noise Continuous white noise has a correlation time of 0= flat power spectral density(PSD), and a covariance of infinity Non-physical, but a useful approximation when the noise distur- bance has a correlation time that is small relative to the natural bandwidth of the system Can simulate effect of white noise by using a random sequence with a correlation time much smaller than the shortest time con stant of the system Band-Limited White Noise block produces such a sequence where the correlation time of the noise is the sample rate of the block For accurate simulations, use a correlation time t much smaller han the fastest dynamics fmax of the system 12丌 100 fmax

Fall 2004 16.333 12–6 Implementing a PSD in Matlab • Simulink has a Band­Limited White Noise block that can be used for continuous systems. – Primary difference from the Random Number block is that this block produces output at a specific sample rate, which is related to the correlation time of the noise. • Continuous white noise has a correlation time of 0 ⇒ flat power spectral density (PSD), and a covariance of infinity. – Non­physical, but a useful approximation when the noise distur￾bance has a correlation time that is small relative to the natural bandwidth of the system. – Can simulate effect of white noise by using a random sequence with a correlation time much smaller than the shortest time con￾stant of the system. • Band­Limited White Noise block produces such a sequence where the correlation time of the noise is the sample rate of the block. – For accurate simulations, use a correlation time tc much smaller than the fastest dynamics fmax of the system. 1 2π tc ≈ 100 fmax

Fa2004 16.33312-7 Power spectral densities for Von Karman model (see MIL-F-8785C) 2(2)=a2[1+(1.3929-36 there ou is the intensity measure of the disturbance, Q is the spatial frequency variable and ln is a length scale of the disturbance All scale parameters depend on day, altitude and turbulence type Different models available for each direction Scale length parameter L (in feet ) varies with altitude-MIL-F 8785C model valid up to 1000 feet h 0.177+0.00023)12 Turbulence intensity for low altitude flight MIL-F-8785C as 0.1W20 (0.17+0.00823h)04 o where w20 is the wind speed as measured at 20 ft o W2045 knots is "heavy

� Fall 2004 16.333 12–7 • Power spectral densities for Von Karman model (see MIL­F­8785C) −5/6 Φug (Ω) = σ2 2Lu � 1 + (1.339LuΩ)2 u π where σu is the intensity measure of the disturbance, Ω is the spatial frequency variable, and Lu is a length scale of the disturbance. – All scale parameters depend on day, altitude, and turbulence type. – Different models available for each direction. – Scale length parameter L (in feet) varies with altitude – MIL­F­ 8785C model valid up to 1000 feet h Lu = (0.177 + 0.000823h)1.2 – Turbulence intensity for low altitude flight MIL­F­8785C as 0.1W20 σu = (0.177 + 0.000823h)0.4 3 where W20 is the wind speed as measured at 20 ft 3 W20 45 knots is “heavy

Fa2004 16.33312-8 Standard approach: assume turbulence field fixed(frozen)in space aircraft is just flying into it Then can relate turbulence spatial frequency 2 to the temporal frequency w that the aircraft would feel Used to model how the aircraft will behave in bumpy air Atmospheric gust ofi Vertical gust sec 300 ft on radio Touchdown Figure 2: Wind gust examples

Fall 2004 16.333 12–8 – Standard approach: assume turbulence field fixed (frozen) in space ⇒ aircraft is just flying into it. 3 Then can relate turbulence spatial frequency Ω to the temporal frequency ω that the aircraft would feel ω Ω ≡ ω = ΩU0 U0 ⇒ – Used to model how the aircraft will behave in bumpy air. profiles 10 ft/sec altimeter 10 ft/sec Longitudinal gust Atmospheric gust Touchdown 10 ft/sec 300 ft on radio 5 sec Vertical gust Lateral gust Figure 2: Wind gust examples

Fa2004 16.33312-9 Wind shear Wind shear is of special interest because it involves significant loca changes in the vertical and horizontal velocity (e nowara ft Particularly important near airports during landings Simple analysis -consider the case shown, where the change in(hor izontal) wind velocity is represented by du a)△h where Ah corresponds to changes in altitude(what we previously just Value du/dh gives magnitude of wind shear, which is 0.08-0.155- for moderate and 0. 15-0.2s for strong Mean wind Flight path Glide slope Flare Figure 3: Aircraft descending into a horizontal windshear But can write the changes in altitude( use h= Ah/At)as h=U0(6-a)

� � Fall 2004 16.333 12–9 Wind Shear • Wind shear is of special interest because it involves significant local changes in the vertical and horizontal velocity (e.g. downdraft) – Particularly important near airports during landings. • Simple analysis – consider the case shown, where the change in (hor￾izontal) wind velocity is represented by du ug = Δh dh where – Δh corresponds to changes in altitude (what we previously just called h) – Value du/dh gives magnitude of wind shear, which is 0.08–0.15s−1 for moderate and 0.15–0.2s−1 for strong. Glide slope Flight path Flare Mean wind Figure 3: Aircraft descending into a horizontal windshear • But can write the changes in altitude (use h˙ = Δh/Δt) as h˙ = U0(θ − α)

Fa2004 16.3331210 Now have a coupled situation that is quite interesting Forces on the aircraft change with altitude(h)because height langes the gust velocity Changes in the forces on the aircraft impact the height of the airplane since both 0 and a will change Analyze the coupling by looking at the longitudinal equations Lu w q h T h=[00001]x=Ch with controls fixed(zero and input u = AT+Bu( 1)ug= Ar+ Bu d A c +mBu(, 1ChI d A+Bu(, 1Cr dynamics have been modified because of the coupling between the change in altitude and the change in forces(with ug) Closed this loop on the b747 dynamics to obtain du Phugoid poles 0-0.0033士0.0672i 0.08-0.0014±0.1150 0.1500002±01442i 020.0014±01619 Clearly this coupling is not good, and an unstable phugoid mode is to be avoided during landing operations

� � � Fall 2004 16.333 12–10 • Now have a coupled situation that is quite interesting – Forces on the aircraft change with altitude (h) because height changes the gust velocity – Changes in the forces on the aircraft impact the height of the airplane since both θ and α will change. • Analyze the coupling by looking at the longitudinal equations � �T x = u w q θ h h = 0 0 0 0 1 x = Chx with controls fixed (zero) and input ug x˙ = Ax + ˜ ˜ Bw(:, 1) ˜ Bw(:, 1)ug = Ax + ˜ duh dh du = Ax˜ ˜ + Bw(:, 1)Chx � dh du = A˜ + B˜w(:, 1)Ch x dh ⇒ dynamics have been modified because of the coupling between the change in altitude and the change in forces (with ug). • Closed this loop on the B747 dynamics to obtain du Phugoid Poles dh 0 0.08 0.15 0.2 ­0.0033 ± 0.0672i ­0.0014 ± 0.1150i 0.0002 ± 0.1442i 0.0014 ± 0.1619i – Clearly this coupling is not good, and an unstable Phugoid mode is to be avoided during landing operations