AE 430 - Stability and Control of Aerospace Vehicles

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 force acting on an object, velocity of a mass 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 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 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 forces 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 (angular 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 moment of inertia, 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 +×• 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.

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