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 simple to predict expected output (e.g. ride comfort, wing loading)