AE Stability and Control of Aerospace Vehicles

Transcription

1 AE 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 Handling quality of an airplane Oscillations growing exponentially 1

2 Dynamic Stability Airplane Modes of Motion Longitudinal (symmetric) Long period (phugoid) Exchange of KE and PE Easily controlled by pilot (usually) Lightly damped Short period Usually heavily damped Higher frequency than phugoid Lateral-directional (asymmetric) Spiral mode (aperiodic bank angle divergence) Roll mode (aperiodic roll rate convergence) Dutch roll mode Moderately damped Moderate frequency 2

3 Vector Analysis A scalar quantity is one which has only magnitude, whereas a vector quantity has both magnitude and direction. From physical point of view, when a mathematical vector is used to express a physical element, such as force acting on an object, velocity of a mass point, the third factor of location needs to be accounted for. As a result, the vector quantities can be classified into three types: a free vector, such as wind speed, 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 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 x e, y e, z e, and a moving system x b, y b, z b, which is rigidly fixed in the body and participates in its motion. Rigid body equation of motion are obtained from Newton s second law 3

4 Body and inertial axis systems r δ m CM Body frame v c Fixed frame inertial axis Velocity and acceleration of differential mass respect to inertial reference system a,v referred to an absolute reference system (inertial) Relative velocity of δm respect to CM dr v = vc + = vc + ω r 2 d r a = ac + = a c ω r ω ω r CM ω ( ) Center of mass of the airplane Angular velocity ω = pi+ qj+ rk ( t) = ( t) + ( t) P c r Pδ m r c CM o v c y x e e Fixed frame z e inertial axis 4

5 Newton s second law Summation of all external forces acting on a body is equal to the time rate of change of the momentum of the body d F = ( m v ) d M = H F = F d xi+ Fyj + Fzk F = ( mu) ; F = d ( mv) ; F = d ( mw) x y z Summation of all external moments acting on a body is equal to the time rate of change of the moment of the momentum (angular momentum) M = M i+ M j + M k = Li+ Mj+ Nk d x y x; d z L = H M = Hy; N = d Hz 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 Force Equation F = d ( m ) v Resulting force acting on an element of mass (second Newton s law) dr dv δf = δm v = vc + Total external force acting on the airplane d dr d dr δf = F = + δm= δm+ δm vc vc Assuming constant mass: 2 dvc d dr dvc d F = m + δ m= m + δ m 2 r r δ m = 0 r measured from the center of mass dv Force equation: F = m c 5

6 Moment Equation d M = H Resulting moment acting on an element of mass δ H = r v δ δ d d M = ( ) m δ H = r v δ dr v = vc + = vc + ω r Total angular momentum acting on the airplane H = H = ( r vc ) m+ r ( ω r) δ δ δm v c constant with respect to the summation H = rδ m vc + r ( ω r) δ m r δ m = 0 r measured from the center of mass H = r ω r δ ( ) m ( ) m Moment Equation H = r ω r δ ( ) m Angular velocity ω = pi + qj+ rk r = xi+ yj + zk Vector equation for the angular momentum ( ) ( ) H = pi+ qj+ rk x + y + z δ m ( i j k)( ) x + y + z px + qy + rz δ m x 2 2 ( ) 2 ( 2) H = p y + z δ m q xyδm r xzδm H = p xyδ m+ q x + z δm r yzδm z y Position vector 2 2 ( ) H = p xzδ m q yzδm+ r x + y δm I x Propriety of Cross Product r ( ω r) ( ) ( ) r r ω r ω r 6

7 Moment Equation Mass moments and products of inertia x y z ( 2 2) ( 2 2) ( 2 2) I = y + z δm I = xyδm I = x + z δm I = xzδm I = x + y δm I = yzδm xy xz yz The larger moment of inertia, the greater will be the resistance to rotation Moment Equation Scalar equations for the angular momentum Hx = pix qixy rixz Hy = pixy + qiy riyz Hz = pixz qiyz + riz 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

8 v and H referred to the rotating body frame Relationship inertia frame and rotating body frame da I da = + ω A B then = dvc d m m m = vc F + ω v I B ( ) c = dh d = H M + ω H I B Scalar equations of motion for reference axis fixed to the airplane v = ui+ vj+ wk c Force equations ( ); ( ); ( ) F = m u+ qw rv F = m v+ ru pw F = m w+ pv qu x y z Moment equations dh L = ( + ω H) i B L = H + qh rh ; M = H + rh ph ; N = H + ph qh x z y y x z z y x xz plane of symmetry Iyz = Ixy = 0 Moment equations: dv c Fx = m + m( ω v ) c i B ( ) L = I p I r + qr I I I pq x xz z y xz 2 2 ( ) ( ) M = I q + rp I I + I p r y x z xz ( ) N = I p + I r + pq I I + I qr xz z y x xz 8

9 Orientation and position of the airplane (respect to a fixed frame) At t = 0 the axis system fixed to the airplane and the one of a fixed frame coincide 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 = u1cosψ v1sinψ dy = u1sinψ + v1cosψ dz = w1 u, v, w = f( u, v, w ) u, v, w = g( u, v, w) u 1 w 1 v 1 9

10 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 Absolute velocity components along the fixed frame dx dx dy dz ; ; u dy = [ C ] v w uvw ; ; Velocity components dz along the body axes = = = NOTE: Use of Quarternions is sometime better: see Kinematic equations for the Euler angles 10

11 Relationship between body angular velocities (in the body frame) and the Euler rates p 1 0 Sθ Φ q = 0 CΦ Cθ S Φ θ r 0 SΦ Cθ CΦ ψ Φ 1 SΦ tanθ CΦ tanθ p θ 0 CΦ S Φ q = ψ 0 SΦsecθ CΦsecθ r F = Faero + Fgrav + Fprop Gravitational Forces Along the body axes Fgrav = mg F F F x grav y grav z grav = mgsinθ = mgcosθ sin Φ = mgcosθ cos Φ 11

12 Force and Moment due to propulsion system F prop Trust forces F F F L M N x y z prop prop prop prop prop prop = X = Y T = Z = L T T T = M = N T T Summary xz plane of symmetry I = I = 0 yz xy 12

13 Summary 12 equations, 12 unknowns/variables: x, y, z; ψ, φ, θ; u, v, w; p, q, r Nonlinear Equation of Motion 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

14 Linearization of equations of motion 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. This approach was extremely important in the early days of simulation when high speed computers were not available to solve the fully nonlinear system. 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 Much of the mathematics of control system design was developed based on linear models. The theory of linear quadratic regulator design (LQR) and most other optimal control law synthesis techniques are based on a linear system model. 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

15 Linearized Aerodynamics: Stability Derivatives 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

16 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 t 0 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 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. 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

17 Stability Derivative Control 17

18 We obtain the following linearized equations (taking first order approximations), 18

19 Assume the reference flight condition to be symmetric, unaccelerated, steady, and with no angular velocity, therefore Linearized longitudinal and lateral equations 19

20 Linearized longitudinal and lateral equations Small-Disturbance Theory Small deviations about the steady-flight: u = u +Δ u; v = v +Δ v; w= w +Δw; p = p +Δ p; q = q +Δ q; r = r +Δr; X = X +Δ X; Y = Y +Δ Y; Z = Z +ΔZ; L = L +Δ L; M = M +Δ M; N = N +ΔN; δ = δ +Δδ; 0 Symmetric flight condition and constant propulsive forces v0 = p0 = q0 = r0 =Φ 0 = ψ 0 = 0 (x-axis in the direction of the velocity vector) w 0 = 0 20

21 X Force Equation ( ) X mgsinθ = m u+ qw rv 0 0 ( θ θ) X +ΔX mgsin +Δ = d = m ( u0 +Δ u) + ( q0 +Δ q)( w0 +Δw) ( r0 +Δ r)( v0 +Δv) Derivation of the linearized small-disturbance longitudinal and lateral rigid body equation of motion Longitudinal equation for the X force equation (,,, ) ΔX mgδ θ cosθ = mδ u X X X X X u w δ δ u w δ Δ = Δ + Δ + Δ + Δ δ 0 e T e T u w e T d X X X X m u w ( mgcos 0 ) e T u Δ w Δ + θ Δ θ = Δ δ + Δδ e T d Xu Δu ( Xw) Δ w+ ( gcosθ0 ) Δ θ = ( Xδ ) Δ δ ( ) e e + Xδ Δδ T T X u 1 X = m u 21

22 Aerodynamic force and moment representation Expressed by mean of a Taylor series in the term of perturbation variables about the reference equilibrium condition X X X X Δ X = Δ u+ Δ w+ Δ δe + ΔδT u w X Stability derivative (evaluated at the reference flight condition) Stability coefficient x (dimensionless) = CQS X Cx u C 1 1 = QS = QS = C QS u u u u u u u e 0 x xu T Bryan, 1904 X,M (aero) Expressed as function of the instantaneous values of the perturbation variables Change in the force in x direction and change in the pitching moment (in terms of perturbation variables) (,,,,, δe, δe) Δ X u u w w = X X X X Δ X = Δ u+ Δ u + + Δ δe + Δ δe + H.O.T. u u (,,,,,,,,, δ, δ, δ ) Δ M uvwuvwpqr = e a e r M M M M Δ M = Δ u+ Δ v+ Δ w+ + Δ p+ u v w p e 22

23 Most important aerodynamic derivative X X X X Δ X = Δ u+ Δ w+ Δ δe + ΔδT u w Y Y Y Y Δ Y = Δ v+ Δ p+ Δ r+ Δδr v p r e Z Z Z Z Z Z Δ Z = Δ u+ Δ w+ Δ w + Δ q+ Δ δe + ΔδT u w w q L L L L L Δ L = Δ v+ Δ p+ Δ r+ Δ δr + Δδa v p r M M M M M M Δ M = Δ u+ Δ w+ Δ w + Δ q+ Δ δe + ΔδT u w w q N N N N N Δ N = Δ v+ Δ p+ Δ r+ Δ δr + Δδa v p r r r r T e a a e T T Linearized small-disturbance longitudinal and lateral rigid body equation of motion Longitudinal Equations d Xu u Xw w ( gcos 0 ) X e e X T T Δ Δ + θ Δ θ = δ Δ δ + δ Δδ d d Z Δ u+ ( 1 Z ) Z w ( u Z ) gsinθ θ Z Δ + Δ = Δ δ + Z Δδ u w w 0 q 0 δe e δt T 2 d d d MuΔu Mw + Mw w M 2 q θ Mδ δ e e Mδ δ T T Δ + Δ = Δ + Δ Lateral Equations d Yv Δv YpΔ p+ ( u0 Yr) Δr ( gcosθ0) Δ φ = Yδ Δδ r r d I d L Δ v+ L Δp + L Δ r = L Δ + L Δ xz v p r a a r r δ δ δ δ I x I N Δv d + N d Δ p+ N Δ r = N Δ + N Δ xz v p r δ δ a a δ δ r r Iz 23

24 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 X D T Δ X = Δ u = Δ u+ Δu u u u X D T 1 = + = Cx u QS u u u u0 X ρs 2 CD T = u0 + 2u0CD + 2 C 0 u u u Xu ux = QS 0 u CDu Tu D,T in the x direction changes in the x Force 24

25 1 ρs 2 CD T Cx = u0 u0 2u u 0CD0 QS = u u C 1 ρs C T = u + C + = ( C + C ) + C QS 2 u u u u Du C = = uu 2 D 0 2 D0 Du 2 D0 Tu 0 0 C D T ; CTu 0 uu0 CD CDu = M ; M = Mach number M ( 2 ) C = C + C + C x Du D0 Tu u C = 0 C Tu = C Tu D0 Gliders and jet powered a/c (constant trust cruise) Piston Engine powered a/c and variable pitch propeller Change in the Z force Z 1 = ρsu 0 CL + 2C u L0 u 2 CZ = C 2 u L C + u L0 CL M = 0 CL = Pranl-Glauert Formula 2 1 M CL M = 2 C L 0 M 1 M M = CL u0 CL CL M CL = = = M = C u uu a ua M 1 M L M = 0 25

26 Change in the pitching moment M Δ M = Δu u M = C m ρ u Scu u Cm = Cm M u M 0 C zq C mq Derivative due to the Pitching Velocity, q Δ L = C Δα Q S Z t Lα t t t t ql Δ Z = Δ L = C Q S C Z = QS t t Lα t t t u0 ql Q S ql S Δ C = C = C t t t t t Z L L η αt α u t 0 QS u0 S (for the tail) C 2u C l S C C C V Z 0 Z t t Z = = 2 2 q L η = L η α H qc 2u t α t 0 c q c S 26

27 Derivative due to the Pitching Velocity, q ql Δ M = l Δ L = l C Q S C t cgq t t t Lα t t t u0 cg m ηvhcl q Δ = = cg q C m q ΔM QSc Cm 2u0 C = qc 2u c q C lt m 2 C q L η V α H t c 0 m α t ql u t 0 (for the tail) (for the complete aircraft) 1.1 C ; 1.1C Z q m q Due to the lag in the wing downwash getting to the tail Cz α Derivative due to the Time Rate of Change of the AOA, α Cm α Δ t = lt u0 dε dε dα dε Δ αt = Δ t = Δ t = αδt dα dα Δt : Increment in time that it takes to the change in circulation imparted to the trailing vortex wake to reach the tail dε Δ = t αt α dα u0 Lag in the angle of attack at the tail l l t Changes the downwash at the tail 27

28 Derivative due to the Time Rate of Change of the AOA, α t 0 ; dε Δ t = l u Δ αt = Δt Δ dε lt dε dα lt dε lt αt = u = α dα u = dα u Δ L = C Δα Q S t L t t t αt ΔL S dε l S Δ C = = C Δ = C C t t t t z L αη t L α η α QS t α S t dα u0 S C 2u C dε αc u c α dα z 0 z z = = = 2ηVHC α Lα ( /2 t 0 ) V H η = Q t Q lt St = c S Derivative due to the Time Rate of Change of the AOA, α Δ M = l Δ L = lc Δα Q S C cg t t t L t t t Δ = η αt dε l t α C m V cg H C Lα t d α u m 0 m t m = = = 2ηVHC α Lα ( αc /2u t 0 ) c α dα c 0 C 2u C dε l (for the complete aircraft) 1.1 C ; 1.1C Z α m α 28

29 Derivative due to the Rolling Rate, p Cyp, Cnp, Clp (roll rate) ΔLift=C lα py Δ α = u dl = ΔLift y 0 ΔαQcdy CL CL pb 2u0 Derivative due to the Yawing Rate, r Y = C ΔβQ S C C C Yr = Lα v v v Y Yr C Y ( rb /2u ) N = C ΔβQ S l C C rl Δ β = u 0 0 Lα v v v v n nr v 29