AE 430 - Stability and Control of Aerospace Vehicles

Aircraft Equations of Motion

Dynamic Stability

Degree of dynamic stability: time it takes the motion to damp to half or to double the amplitude of its initial amplitude Oscillations growing “Handling exponentially quality of an airplane”

1 Dynamic Stability

Airplane Modes of Motion

z Longitudinal (symmetric) – Long period (phugoid) z Exchange of KE and PE z Easily controlled by pilot (usually) z Lightly damped – Short period z Usually heavily damped z Higher frequency than phugoid z Lateral-directional (asymmetric) – Spiral mode (aperiodic bank angle divergence) – Roll mode (aperiodic roll rate convergence) – Dutch roll mode z Moderately damped z Moderate frequency

2 Vector Analysis

z A scalar quantity is one which has only magnitude, whereas a vector quantity has both magnitude and direction. z From physical point of view, when a mathematical vector is used to express a physical element, such as acting on an object, velocity of a point, the third factor of location needs to be accounted for. z As a result, the vector quantities can be classified into three types: – a free vector, such as wind , is one with a specified slope (direction) and sense (magnitude) but not acting through any particular point ; – a sliding vector, such as the moment acting on the body depends upon the line of action of the force, has definite or specific line of action, but is independent of the precise point of application along that line; – a fixed vector is a vector with specified magnitude, direction, and point of application.

Rigid body z A rigid body is a system of particles in which the distances between the particles do not vary. To describe the motion of a rigid body we use two systems of coordinates, a space-fixed system xe, ye, ze, and a moving system xb, yb, zb, which is rigidly fixed in the body and participates in its motion. z Rigid body equation of motion are obtained from Newton’s second law

3 Body and inertial axis systems

Body frame δ m r vc CM

Fixed frame “inertial axis”

Velocity and acceleration of differential mass respect to inertial reference system z a,v referred to an Pcr(ttt) =+( ) ( ) absolute reference P system (inertial) δ m Relative velocity of δm respect to CM r vc dr vv=+ =+× v ω r CM ccdt 2 d r c aa=+cc =+×+×× a ω r ωω()r dt 2 o

xe CM Center of mass of the airplane ye Fixed frame ω Angular velocity z ω =++pqrijk e “inertial axis”

4 Newton’s second law

Summation of all external acting on a body is equal to the time rate of change of the momentum of the body d Fi= FFFx ++yzj k Fv= m ddd ∑ () F ===()mu;; F() mv F() mw dt xyzdt dt dt Summation of all external moments acting on a body is equal to the time rate of change of the moment of the momentum () Mi= M xyz++MMj ki =++ LMNj k d ddd MH= L ===HM;; HN H ∑ dt dtx dtyz dt The time rate of change of linear and angular momentum are referred to an absolute or inertial reference frame

F,M Forces and Moments due to Aerodynamic, Propulsive and Gravitational forces

d Fv= ()m Force Equation ∑ dt

Resulting force acting on an element of mass (second Newton’s law) dv dr δδF = m vv=+c dt dt Total external force acting on the airplane ddd⎛⎞rr ⎛ d ⎞ ∑∑δδδδFF==⎜⎟ vcc +mmm = ∑ ⎜ v + ⎟ dt⎝⎠ dt dt ⎝ dt ⎠ Assuming constant mass: ddvvddr d2 Fr=+mmmmccδ =+ δ dt dt∑∑ dt dt dt2 ∑rδ m = 0 r measured from the center of mass dv Force equation: F = m c dt

5 d MH= Moment Equation ∑ dt

Resulting moment acting on an element of mass δ Hrv=×( )δ m dd dr δ MHrv==×δδ()m vv= +=+× v ω r dt dt ccdt Total angular momentum acting on the airplane

∑∑δ HH==() rv ×c δδmm + ∑⎣⎡ r ××(ω r)⎦⎤

vc constant with respect to the summation

Hrv=×+××∑∑δ mmc ⎣⎡ r(ω r)⎦⎤δ ∑rδ m = 0 r measured from the center of mass

Hr=××∑ ⎣⎡ (ω r)⎦⎤δ m

Moment Equation Hr=××∑ ⎣⎡ (ω r)⎦⎤δ m

Angular velocity Position vector Propriety of Cross ω =++pi qrj k ri= x ++yzj k Product Vector equation for the angular momentum r ××≡(ω r) Hijk=++()pqr∑( x222 ++ y z)δ m ()(rr•−•ω r ω )r −++++∑()()xijky z px qy rzδ m 22 H x =+−−pyzmqxymrxzm∑∑∑()δ δδ 22 H y =−pxymqxzmryzm∑∑δ +( +)δδ − ∑

22 H z =−p∑∑∑ xzmqδ − yzmrδδ +( x + y) m

I x

6 Moment Equation z Mass moments and products of inertia

22 Ixxy=+∫∫∫()yzmδδ I = ∫∫∫ xym 22 IxzmIxzmyxz=+∫∫∫()δδ = ∫∫∫ 22 Izyz=+∫∫∫()xymδδ I = ∫∫∫ yzm

The larger , the greater will be the resistance to rotation

Moment Equation

Scalar equations for the angular momentum

HpIqIrIx = xxyxz−−

HpIqIrIyxyyyz= −+−

HpIqIrIzxzyzz= −−+

NOTE: If the reference frame is not rotating, then as the airplane rotates the moments and the products of inertia will vary with the time

To simplify the problem we will fix the axis system to the aircraft (body axis system)

7 v and H referred to the rotating body frame

z Relationship inertia frame and rotating body frame ddAA = +×ω A dtIB dt

then

ddvvcc F ==+×mmm()ω vc dtIB dt

ddHH M = =+×ω H dtIB dt

Scalar equations of motion for reference axis fixed to the airplane

⎛⎞dv Fm= c +ו mω vi vijkc =++uvw x ⎜⎟()c ⎝⎠dt B Force equations

FmuqwrvFmvrupwFmwpvquxy=+−( );; =+−( ) z =( +−) dH Moment equations L = ()+×ω Hi • dt B L=+ Hx qHzy − rH;; M =+ H yx rH − pH z N =+ H z pH yx − qH

xz plane of symmetry IIyz= xy = 0 Moment equations: L =−+Ipxxzzyxz I r qrI( −− I) I pq 22 M =+Iqyxzxz rpI() −+ I I( p − r)

NIpIrpqIIIqr=−xz + z +( y − x) + xz

8 Orientation and position of the airplane (respect to a fixed frame)

z At t = 0 the axis system fixed to the airplane and the one of a fixed frame coincide z Orientation of airplane described by three consecutive angular rotation (Euler Angles) – rotation about z (through the yaw angle ψ – rotation about y (through the pitch angle θ – rotation about x (through the roll (bank) angle Φ

Euler Angles

Fixed Reference Frame: dx =−uvcosψψ sin dt 11 dy =+uv11sinψψ cos dt uvw11,, 1= fuvw (,, 2 2 2 ) dz uvw,,= guvw (,,) = w 22 2 dt 1

u1 v w1 1

9 Orientation and position of the airplane (respect to a fixed frame)

The orientation of an airplane, relative to local axes, can be specified by the three sequential rotations about the body axes. Starting with the body axes aligned with the local axes, the first rotation is about the z-axis through an angle Ψ , followed by a rotation about the y-axis through an angle Θ , followed by a rotation about the x-axis through an angle Φ . These angles of rotation are the Euler angles, and can represent any possible orientation of the airplane.

Airplane's direction cosine matrix constructed from the Euler angles

⎡⎤dx dx dy dz Absolute velocity components ;; ⎢⎥dt ⎢⎥ ⎡⎤u dt dt dt along the fixed frame dy ⎢⎥= []Cv⎢⎥ ⎢⎥dt ⎢⎥ uvw;; Velocity components ⎢⎥ ⎣⎦⎢⎥w ⎢⎥dz along the body axes ⎣⎦⎢⎥dt = = Kinematic equations for = the Euler angles NOTE: Use of Quarternions is sometime better: see http://www.aerojockey.com/papers/meng/node19.html

10 Relationship between body angular velocities (in the body frame) and the Euler rates

⎡⎤pS ⎡10 − θ ⎤⎡Φ⎤ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎢⎥qCCS= ⎢0 ΦΦθ ⎥⎢θ ⎥ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎣⎦rSCC ⎣0 − ΦΦθ ⎦⎣ψ ⎦

⎡⎤Φ ⎡1tantanSCΦΦθθ⎤⎡ p ⎤ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎢⎥θ =−⎢0 CSqΦΦ⎥⎢ ⎥ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎣⎦ψθθ⎣0secsecSCΦΦ⎦⎣ r ⎦

FF= aero++ F grav F prop

Gravitational Forces

Along the body axes

Fggrav = m

Fmg=− sinθ x grav Fmg= cosθ sin Φ y grav Fmg= cosθ cos Φ z grav

11 Force and Moment due to propulsion system

Fprop Trust forces F = X xprop T

FyT= Y prop

FzT= Z prop

LLprop = T

M prop = MT

NNprop = T

Summary

xz plane of symmetry IIyz= xy = 0

12 Summary

12 equations, 12 unknowns/variables: x, y, z; ψ, φ, θ; u, v, w; p, q, r

Nonlinear Equation of Motion

z The nonlinear equations of motion given previously may be used to predict the motion of a vehicle assuming the forces and moments can be computed at the flight conditions of interest. The equations are nonlinear because of the quadratic dependence of the inertia forces on the angular rates, the presence of trigonometric functions of the Euler angles and angles of attack and sideslip, and the fact that the forces depend on the state variables in fundamentally nonlinear ways. While the quaternion formulation avoids some of the trigonometric nonlinearities, the equations remain nonlinear.

13 Linearization of equations of motion

z Despite the nonlinear character of the equations, one may consider small variations of motion about some reference condition for which the equations (including the forces and moments) may be approximated by a linear model. z This approach was extremely important in the early days of simulation when high speed computers were not available to solve the fully nonlinear system. z Now, the general set of equations is often maintained for the purposes of simulation, although there are still important reasons to consider linear approximations and many conditions for which the linear approximation of the system is perfectly acceptable.

Reasons to consider linear approximations

z Much of the mathematics of control system design was developed based on linear models. z The theory of linear quadratic regulator design (LQR) and most other optimal control law synthesis techniques are based on a linear system model. z Even many nonlinear simulations, that keep the full equations of motion, rely on linear aerodynamic models (or at least partially linearized aero models) to keep the size of the aerodynamic database more manageable

14 Linearized Aerodynamics: Stability Derivatives

z There are two senses in which we may deal with "linear" aerodynamic models. – To most aerodynamicists, this means that the partial differential equations describing the fluid flow are linearized. These linear models lead to aerodynamic characteristics that are nonlinear in the dynamics state variables (such as angle of attack) due to nonlinearities in the boundary conditions and speed-pressure relations. Thus, dynamicists must deal with the results of potential flow codes, Euler codes, or Navier-Stokes solvers in much the same way as they do with wind tunnel data. – The linearizations lead to aerodynamic models that are comprised of a set of reference values and a set of "stability derivatives" or first order expansions of the actual variations of forces and moments with the state variables of interest. – Because these are first order models, the total force can be conveniently "built-up" as the sum of the individual effects of angle of attack, pitch rate, sideslip, etc. Since the six aerodynamic forces and moments do not depend explicitly on the orientation of the vehicle with respect to inertial coordinates, we expect derivatives only with respect to the 3 relative wind velocity components and the 3 rotation rates.

Linearized Aerodynamics: Stability Derivatives

– This means that there are usually 36 stability derivatives required to describe the first order aerodynamic characteristics of a flight vehicle. However, the applied forces and moments may also vary, not just with the values of the state variables, but also their time derivatives. This can represent a significant complication to the basic concept of stability derivatives. In most cases, however, these effects are small and usually the only terms of much significance are those associated with the rate of change of angle of attack. – These derivatives can be expressed in dimensional form making them just the coefficients in the linear state space model, and assigning some direct physical significance to their numerical values, or in dimensionless form. The latter has the advantage that the values are relatively independent of dynamic pressure and model size and that this is the form that is used in wind tunnel databases and computational aerodynamics models.

15 Small-Disturbance Theory

The equations of motion are frequently linearized for use in stability and control analysis. It is assumed that the motion of the aircraft consists of small deviation from a steady flight condition. The use of small disturbance theory predicts the stability of unaccelerated flight. In most cases, a perturbed fluid-aerodynamic force is a function of perturbed linear and angular velocities and their rates:

Thus the aerodynamic force at time t0 is determined by its series expansion of the right-hand side of this equation:

stability derivatives, or more generally as aerodynamic derivatives.

Small-Disturbance Theory

z For small perturbations, the higher-order terms are dropped. Also, due to the assumed symmetry of the vehicle, derivatives of X, Z, M w.r.t. motions out of the longitudinal plane are zero, thus may be visualized by noting that X, Z, M must be symmetrical w.r.t. lateral perturbations. z In other words, we neglect the symmetric derivatives w.r.t. the asymmetric motion variables, i.e., for aerodynamic force X,

and so on.

16 Stability Derivative Control

17 We obtain the following linearized equations (taking first order approximations),

18 z Assume the reference flight condition to be symmetric, unaccelerated, steady, and with no angular velocity, therefore

Linearized longitudinal and lateral equations

19 Linearized longitudinal and lateral equations

Small-Disturbance Theory

Small deviations about the steady-flight: uu=+Δ=+Δ=00 u;; vv v ww 0 +Δ w ;

pp=+Δ=+Δ=+Δ000 pqq;;; qrr r

XX=+Δ=+Δ=+Δ00 XYY;; YZZ 0 Z ;

LL=+Δ000 L;;; MM = +Δ M NN = +Δ N

δδ=+Δ0 δ;

Symmetric flight condition and constant propulsive forces

vpqr000000====Φ==ψ 0

(x-axis in the direction of the velocity vector) w0 = 0

20 X Force Equation

Xmg−=+−sinθ muqwrv( )

XXmg00+Δ −sin (θθ +Δ) = ⎡⎤d =muuqqwwrrvv⎢⎥()()()()()00000 +Δ + +Δ +Δ − +Δ +Δ ⎣⎦dt

Derivation of the linearized small-disturbance longitudinal and lateral rigid body equation of motion

Longitudinal equation for the X force equation

ΔXm−Δg θ cosθ0 =Δmu

∂∂XX ∂ X ∂ X Δ=Δ+Δ+Δ+ΔXuw(),,δeT ,δδδ u w e T ∂∂∂uwδδeT ∂ ⎛⎞⎛⎞dX∂∂ X⎛⎞ ∂ X⎛⎞ ∂ X ⎜⎟⎜⎟muwmg−Δ−Δ+()cosθ0 Δ=Δ+Δθδ⎜⎟eT⎜⎟ δ ⎝⎠⎝⎠dt∂∂ u w ⎝⎠ ∂δδeT⎝⎠ ∂ ⎛⎞d −Δ−Δ+XuXwgcosθθ Δ=Δ+Δ X δ X δ ⎜⎟uw() (0 ) ()δδeT e() T ⎝⎠dt 1 ∂X X = u mu∂

21 Aerodynamic force and moment representation

z Expressed by mean of a Taylor series in the term of perturbation variables about the reference equilibrium condition ∂∂XX ∂ X ∂ X Δ=Xuw Δ+ Δ+ ΔδeT + Δδ Bryan, 1904 ∂∂∂uwδδeT ∂ X,M (aero) Expressed as Stability derivative (evaluated at the function of the reference flight condition) instantaneous Stability values of the perturbation X = CQS coefficient x (dimensionless) variables ∂X ∂∂Cu C 11 ==xx0 QS QS = C QS xu ∂∂uuuuuu0000 ∂ u

Change in the force in x direction and change in the pitching moment (in terms of perturbation variables)

Δ=Xuuww( ,,,,,,… δδee) ∂∂XX ∂ X ∂ X Δ=Δ+Δ++Δ+Δ+Xuu … δδeeH.O.T. ∂∂uu ∂δδee ∂

Δ=M( uvwuvwpqr,, ,,, , ,,,δδδaer , , ) ∂∂∂MMM ∂ M Δ=Muvw Δ+ Δ+ Δ++…… Δ+ p ∂∂∂uvw ∂ p

22 Most important aerodynamic derivative

∂∂XX ∂ X ∂ X Δ=Xuw Δ+ Δ+ ΔδδeT + Δ ∂∂∂uwδδeT ∂ ∂∂∂YYY ∂ Y Δ=Yvpr Δ+ Δ+ Δ+ Δδr ∂∂vpr ∂∂δ r ∂∂∂∂∂ZZZZZ ∂ Z Δ=Zuwwq Δ+ Δ+ Δ+ Δ+ ΔδδeT + Δ ∂∂uwwq ∂ ∂∂δδeT ∂ ∂∂∂LLL ∂ L ∂ L Δ=Lvpr Δ+ Δ+ Δ+ Δδδra + Δ ∂∂vpr ∂∂δδra ∂ ∂∂∂∂∂MMMMM ∂M ΔM = Δ+ uwwq Δ+ Δ+ Δ+ Δδe + ΔδT ∂∂uwwq ∂ ∂∂δe ∂δT ∂∂∂∂NNNN ∂ N Δ=Nvpr Δ+ Δ+ Δ+ Δδδra + Δ ∂∂vpr ∂∂δδra ∂

Linearized small-disturbance longitudinal and lateral rigid body equation of motion

Longitudinal Equations ⎛⎞d ⎜⎟−Δ−Δ+XuXwguw()cosθθ0 Δ=Δ+Δ Xδδ δ e X δ T ⎝⎠dt eT ⎡⎤⎡⎤dd −Δ+−Zu1sin Z − Z Δ− w u + Z − gθ Δ=θδ Z Δ+ Z Δ δ uwwq⎢⎥⎢⎥() ()00δδeT eT ⎣⎦⎣⎦dt dt ⎛⎞ddd⎛⎞2 −Δ−Mu M + M Δ+ w⎜⎟ − M Δ=Δ+θδ M M Δ δ uww⎜⎟⎜⎟2 qδδeT e T ⎝⎠dt⎝⎠dt dt Lateral Equations ⎛⎞d ⎜⎟−Δ−Δ+−Δ−YvYpuYrgvp()()00 rcosθφ Δ=Δ Yδ δ r ⎝⎠dt r ⎛⎞dd⎛⎞I −Δ+Lv − L Δ− pxz + L Δ= r L Δδδ + L Δ vp⎜⎟⎜⎟ rδδar ar ⎝⎠dt⎝⎠ Ix dt ⎛⎞I dd⎛⎞ −Δ−Nvxz + N Δ+ p − N Δ= rN Δ+δ N Δδ vprar⎜⎟⎜⎟δδar ⎝⎠Idtz ⎝⎠ dt

23 Effect of the Mach number on the Stability Derivatives

Derivatives due to the change in Forward Speed

L,M,D,T all vary with changes in the airplane’s forward speed ∂∂∂XDT Δ=Xu Δ=−Δ+ uu Δ D,T in the x direction ∂∂∂uuu changes in the x Force ∂∂∂XDT 1 =− + =CQS xu ∂∂∂uuuu0

∂∂XSρ ⎛⎞2 ∂CD T =−⎜⎟uuC00 +2 D + ∂∂uu2 ⎝⎠0 ∂ u C T uX Du u C = 0 u Xu QS

24 1 ⎛⎞ρST⎛⎞2 ∂CD ∂ CuxD=−000⎜⎟⎜⎟ u +2 uC += u QS⎝⎠2 ⎝⎠∂∂ u0 u 1 ⎛⎞ρST⎛⎞∂C ∂ =−⎜⎟uC2 D +22 + =−+ CCC + ⎜⎟⎜⎟0 DDDT0 ()uu0 QS⎝⎠2 ⎝⎠∂∂ u u00 u u

∂∂CCD T CCDTuu==; ∂∂uu00 uu

∂CD CMD ==; M Mach number Gliders and jet powered u ∂M CTu = 0 a/c (constant trust – cruise) CCCC=− +2 + Piston Engine powered a/c xu ( Du D0 ) Tu CCTu= − D0 and variable pitch propeller

Change in the Z force

∂Z 1 =−ρSu⎡⎤ C +2 C ∂u 2 0 ⎣⎦LLu 0 CCC=−⎡⎤ +2 ZLLuu⎣⎦0 C L M =0 CL = Prandtl-Glauert Formula 1− M 2

∂CL M = CL ∂M 1− M 2 M =0 ∂∂∂CCCu M 2 CMC==LLL0 = = LLu 2 M =0 ∂∂∂uu0 a ua M 1− M

25 Change in the pitching moment

∂M Δ=M Δu ∂u ∂M = CScuρ ∂u mu 0 ∂C CM= m mu ∂M

C C zq mq Derivative due to the Pitching Velocity, q

(for the tail) Δ=LCtLttt Δα QS αt

qlt Δ=−Δ=−ZLCQStL tt αt u0 Z C = Z QS

qlttt Q S ql t S t Δ=−CCZL =− C Lη ααtt uQS00 u S

∂∂CCZZ2ulS0 tt CCCVZ ≡==−=−22LLHη η q ααtt ∂∂qc2 u0 c q c S

26 Derivative due to the Pitching Velocity, q

qlt Δ=−Δ=−M cglL t t lC t L QS t t (for the tail) q αt u0 ΔM cgq qlt Δ=CVCmHL =−η cgqtα QSc u0 ∂∂CuC2 C ≡=mm0 mq ∂∂qc2 u0 c q

lt CCVmLH≡−2 η q αt c 1.1CC ; 1.1 (for the complete aircraft) Zqqm

Due to the lag in the wing downwash getting to the tail C C zα mα Derivative due to the Time Rate of Change of the AOA, α

lt Δ=tlut 0 ddddεεαε Δα = Δ=ttt Δ=α Δ t dt dαα dt d Δt : Increment in time that it takes to the change in circulation imparted to the trailing vortex wake to reach the tail dε lt Δ=ααt Changes the downwash duα 0 at the tail Lag in the angle of attack at the tail

27 Derivative due to the Time Rate of Change of the AOA, α

dε Δ=tlu; Δα = Δ t tt0 dt ddddεεαεlllttt Δ=ααt = = dt u000 dαα dt u d u

Δ=LCtLttt Δα QS αt

ΔLSttdε lS ttQt Δ=−=−ΔCCzLtLαη =− C α η η = ααtt QS S dα u0 S Q

∂∂CCzz2u0 dε lS CVCzHL===−2η tt αα t VH = ∂∂()αcu/2 0 cαα d cS

Derivative due to the Time Rate of Change of the AOA, α

Δ=−Δ=−ΔMlLlCQScg t t t Lα t t t αt

dε lt Δ=−CVCmHLηα cg αt duα 0

∂∂CuCmm2 0 dε l t CVCmHL===−2η αα t ∂∂()αcu/2 0 cαα dc

1.1CC ; 1.1 (for the complete aircraft) Zαmα

28 CCCyp,, np lp Derivative due to the Rolling Rate, p

ΔLift=C ΔαQcdy lα (roll rate) py Δ=α u0 dL =−ΔLift y

∂C ⇒⇒C L L ⎛⎞pb ∂ ⎜⎟ ⎝⎠2u0

Derivative due to the Yawing Rate, r

YC= −ΔLαvvvYβ QSC ⇒⇒ C Yr

∂CY CYr = ∂ ()rb/2 u0

N = CQSlCCLαvvvvnΔ⇒⇒β nr rl Δ=−β v u0

29