3. Fundamentals of Aircraft Dynamics Analysis This chapter deals with fundamentals of aircraft dynamics including common terminology, nomenclature, definitions, and the development of equations of motion. 3.1 Notation 3.2 Equations of motion 3.3 Nondimensionalization 3.4 Linearization 3.5 Static stability 3.6 Dynamic Stability

3.1 Notation Before describing the equations of motion that we will solve to determine vehicle response, we establish some conventions and notation that will be used in these notes. There exist a number of different ways of expressing vehicle position, velocity, orientation, and the forces and moments that act on the vehicle. The conventions adopted here are similar to those in Etkin (1972).

3.1.1 Axis Systems One of the first items that must be agreed upon, and one of the things that is often not agreed upon, is the selection of coordinate systems in which to express the vehicle state. Even the names and direction of the different axes depend on who is doing the work. Wind tunnel studies have, historically used a coordinate system in which x points downstream, y to the right when looking forward into the flow, and z upward. It is sometimes said that dynamicists are pessimists, taking z as positive downward. (x is positive forward and y again out the right wing). Computational aerodynamicists often followed wind tunnel conventions, although recently CFD researchers have used x and y for 2-D airfoil analysis so y points upward. When applied to 3D wings the z axis is added as the lateral coordinate, confusing many of us. Here we will use right-handed conventions of dynamics in which x is roughly downstream, y axis points to the right, and z is roughly downward. More specific definitions are given in the next sections. Body Axes Throughout most of these notes we will deal with a coordinate system fixed to the aircraft as shown in the figure below. The x-axis coincides with some reference axis used to define the geometry, often a fuselage centerline. The z axis lies in the plane of symmetry (or some reference plane in the case of asymmetric shapes), pointing 'downward'. The y-axis is perpendicular to these axes forming a right-handed coordinate system. Positive y, thus points to the right when looking forward. It is often convenient to choose these coordinates in such a way that they are aligned with a set of reference coordinates defining the aircraft geometry: z is perpendicular to the fuselage reference plane and y to the symmetry plane. Sometimes we choose the body axes to be aligned with the vehicle principle axes. The origin is generally taken at the vehicle c.g. or at a fixed reference location relative to the geometry.

Stability Axes Stability axes are similar to body axes except that they are rotated by an angle of attack, α, as shown in the figure below. The y-axis is still perpendicular to the plane of symmetry. Wind Axes Wind axes are often used to express wind tunnel data. The axes are further rotated by a sideslip angle, β, so that the x-axis is parallel to the freestream flow.

This means that if we express the velocity in the body axes components {U,V,W} and Vt2 = U2+V2+W2, then: tan α = W/U and sin β = V/ Vt With the origin at the same location, transformations between these systems, can be easily derived:

3.1.2 Euler Angles In addition to these coordinate systems, several other systems are used in the simulation of aircraft and spacecraft motion. One may want to express the vehicle position and orientation in an inertial reference frame, although the forces and moments are most easily expressed in one of the three body-centered frames above. Throughout these note we will use a simplified reference frame that is based on the assumption that we are not interested in variations in gravity with altitude the effects of the earth's motion. Coordinates in this "flat-earth" reference frame (sometimes called a geographic frame) are related to those in the body-

fixed axis system through a series of rotations. The earth-fixed coordinates must be rotated as follows in order to become aligned with the body axes: 1. Rotate the coordinate system through the angle Ψ about the z axis (positive nose right). 2. Rotate about the new y axis by the angle Θ (positive nose up) 3. Rotate about the new x-axis by Φ (positive roll to the right) The coordinate transformation can therefore be written as the product of three rotation matrices involving these Euler angles, Θ, Φ, Ψ, allowing us to describe the attitude of the vehicle in a physically meaningful way. We can also relate the rotation rates in the body-fixed axis system to these angles. If the body-axis rotation rates in roll, pitch, and yaw are written as: ωx = P, ωy = Q, ωz = R, then the relation with the Euler angle rates is: P = dΦ/dt - dΨ/dt sinΘ Q = dΘ/dt cosΦ + dΨ/dt sinΦ cosΘ R = -dΘ/dt sinΦ + dΨ/dt cosΦ cosΘ Conversely, the Euler angle rates can be written in terms of the body axis rates as follows: dΦ/dt = P + Q sinΦ tanΘ + R cosΦ tanΘ dΘ/dt = Q cosΦ - R sinΦ dΨ/dt = Q sinΦ secΘ + R cosΦ secΘ

3.1.3 Alternative Descriptions: Quaternions Although the equations above can be used to infer the vehicle attitude, given the time history of the bodyaxis rates, there are some problems in doing this. First, some of the terms involve trigonometric functions that go to infinity under certain conditions (e.g. the expression above for dΦ/dt when Θ = 90deg). While this can be tested and accommodated, it does sometimes lead to numerical errors or complications when used in aircraft simulations that involve extreme attitude maneuvers. Second, the equations lead to angles that may lie outside the usual 0 to 360 deg range. This is not a big problem, but one must be careful to handle this ambiguity in a numerically acceptable way (e.g. writing Θ = Θ mod(2π) can lead to discontinuities.) Third, the equations are linear in P,Q, and R, but are nonlinear in Θ, Φ, and Ψ, involving some time-consuming numerics that may be undesirable in some applications. An alternative approach involves the use of four parameters to describe the vehicle orientation. The quaternion approach is described in the book by Stevens and Lewis (1992). The basic idea is to find an alternative way of describing the general rotation of one 3D coordinate system to another. This can be done with the 3 Euler angles or with a general rotation matrix, B, that may be written in terms of four quaternion parameters, q0, q1, q2, q3. Stevens and Lewis, following Robinson (1958) and Shoemaker (1985), show that the matrix B may be written: B = q02+q12-q22-q32 2(q1 q2 + q0 q3) 2 (q1 q3 - q0 q2) 2(q1 q2 - q0 q3) q02-q12+q22-q32 2(q2 q3 + q0 q1) 2(q1 q3 + q0 q2) 2(q2 q3 - q0 q1) q02-q12 - q22 + q32 where: q02+q12-q22-q32 = 1 The quaternion parameters may be related to the Euler angles through the following relations, which are useful to set the initial conditions: q0 = cos Φ/2 cos Θ/2 cos Ψ/2 + sin Φ/2 sin Θ/2 sin Ψ/2 q1 = sin Φ/2 cos Θ/2 cos Ψ/2 - cos Φ/2 sin Θ/2 sin Ψ/2 q2 = cos Φ/2 sin Θ/2 cos Ψ/2 + sin Φ/2 cos Θ/2 sin Ψ/2 q3 = cos Φ/2 cos Θ/2 sin Ψ/2 - sin Φ/2 sin Θ/2 cos Ψ/2

With these definitions, the angular rates may be related to the quaternion parameters as: P = 2 ( q0 dq1/dt + q3 dq2/dt - q2 dq3/dt - q1 dq0/dt) Q = 2 (-q3 dq1/dt + q0 dq2/dt + q1 dq3/dt - q2 dq0/dt) R = 2 ( q2 dq1/dt - q1 dq2/dt + q0 dq3/dt - q3 dq0/dt)

Manipulation of these expressions leads to a state equation that can be combined with the vehicle equations of motion: d/dt {q} = -1/2 [Ω] {q} where: 0 P Q Ω = -P 0 -R -Q R 0 -R -Q P

R Q -P 0

3.1.4 Forces and moments, definition of terms: Most of these note will deal with the prediction of vehicle forces and moments. At this point we just define the notation that will be used to represent these quantities. In body axes, the forces and moments about the axes shown in the figure above may just be written: Fx, Fy, Fz, Mx, My, Mz. A more conventional notation is to write the forces as X, Y, and Z and the moments L,M, and N, which leads to all sorts of confusion, especially with rolling moment, L, and lift. For this reason, Ashley (1974) uses Lr for rolling moment and Mp for pitching moment. Usually the aerodynamic forces are expressed in wind axes or stability axes with lift and drag, L and D, replacing the X and Z forces. While X,Y, and Z are taken as positive forward, right, and down respectively, lift is defined positive upward in a direction perpendicular to the freestream. Drag is positive in the direction of the freestream. Moments are taken as positive in the right-handed sense about the corresponding axes so positive pitching moment is nose-up; positive roll is a roll to the right, and positive yawing moment yaws the airplane to the right. Throughout these notes, vector quantities will be expressed in one of two forms: as boldface variables as in M, or in brackets as in {M}. Matrices will be indicated with square brackets, [I], or as individual components Kij. V = {U, V, W} = x, y, z body axis components of velocity ω = {P, Q, R} = components of angular velocity {Θ, Φ, Ψ} pitch, roll, yaw Euler angles F = {X, Y, Z} body axis Fx, Fy, Fz

3.2 Equations of motion 3.2.1 Derivation The equations of motion for a rigid flight vehicle, expressed as the translation of its center of mass and rotation about the center of mass, decouple into the following two vector equations: (3.1) F = m dV/dt (3.2) M = dh/dt where F is the applied force and M is the moment. (Recall bold quantities are vectors.) V is the velocity vector: V = {U, V, W} and h is the angular momentum. We can write h as the product of the inertia tensor and the angular velocity: (3.3) h = [I] ω, where ω = {P,Q,R} and [I] is the inertia tensor which is defined as follows:

The individual components of the inertia tensor are defined as:

and similarly for the other components. The components of h are therefore: (3.4a) hx = Ixx P - Ixy Q - Ixz R (3.4b) hy = -Iyx P + Iyy Q - Iyz R (3.4c) hz = -Izx P - Izy Q + Izz R Since we want to write the equations of motion in a reference frame that is fixed to the body, we can write the time derivatives in the inertial frame in terms of those in body axes as follows: (3.5) dp/dt = dp/dt + ω x p where the italicized d/dt represents a derivative in the rotating frame. (Recall that a x b = {ay bz - az by, az bx - ax bz, ax by - ay bx} ) Using (3.5) we can rewrite the equations of motion in (2.1) as follows: (3.6) F = m dV/dt + m ω x V or as components: Fx = m [ dU/dt + QW - RV] Fy = m [ dV/dt + RU - PW] Fz = m [ dW/dt + PV - QU]

The moment equations (2.2) become: M = dh/dt + ω x h or: Mx = dhx/dt + Q hz - R hy My = dhy/dt + R hx - P hz Mz = dhz/dt +P hy - Q hx Using the definition of h in 3.4: M = dh/dt + ω x h or (3.7) M = [I] dω/dt + ω x ([I]ω) Equations 3.6 and 3.7 may be rewritten for the purposes of simulation or control as: dV/dt = F / m - ω x V dω/dt = [I]-1M + [I]-1 (ω x [I]ω) These nonlinear, coupled differential equations can be integrated in time, or linearized for use in control system design. The problem then becomes estimating the force and moment vectors which are themselves complex functions of the vehicle state. We note that the above equations are simplified insofar as they ignore the angular momentum that may arise from rotating propellers, for example. We have assumed six degrees of freedom here and actual problems may involve additional states associated with control surface motion, structural deflections, or propulsion system dynamics. These additional degrees of freedom may be added to form a more general dynamical equation that may be written in state vector form as: dX/dt = f(X)

2.2.2 Expanded Versions of 6DOF EOM's in Component Form Writing X = {U, V, W, P, Q, R, Φ, Θ, Ψ}, we can write the complete equations of motion out in component form as: m du/dt = Fx - m Q W + m R V m dv/dt = Fy - m R U + m P W m dw/dt = Fz - m P V + m Q U Ixx dP/dt - Ixz dR/dt - Ixy dQ/dt = L + Iyz (Q2 - R2) + Ixz PQ - Ixy RP + (Iyy-Izz) QR Iyy dQ/dt - Ixy dP/dt - Iyz dR/dt = M + Ixz (R2 - P2) + Ixy QR - Iyz PQ + (Izz-Ixx) RP Izz dR/dt - Iyz dQ/dt - Ixz dP/dt = N + Ixy (P2 - Q2) + Iyz RP - Ixz QR + (Ixx-Iyy) PQ

dΦ/dt = P + Q sinΦ tanΘ + R cosΦ tanΘ dΘ/dt = Q cosΦ - R sinΦ dΨ/dt = Q sinΦ secΘ + R cosΦ secΘ Or in matrix form: [A] X = f(X) Where: [A] =

And f(X) Fx - m Q Fy - m R Fz - m P

m 0 0 0 0 0 0 0 0

0 m 0 0 0 0 0 0 0 = W U V

0 0 m 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 Ixx -Ixy -Ixz -Ixy Iyy -Iyz -Ixz -Iyz Izz 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

{ + m R V, + m P W, + m Q U,

L + Iyz (Q2 - R2) + Ixz PQ - Ixy RP + (Iyy-Izz) QR, M + Ixz (R2 - P2) + Ixy QR - Iyz PQ + (Izz-Ixx) RP, N + Ixy (P2 - Q2) + Iyz RP - Ixz QR + (Ixx-Iyy) PQ, P + Q sinΦ tanΘ + R cosΦ tanΘ, Q cosΦ - R sinΦ, Q sinΦ secΘ + R cosΦ secΘ }

3.3 Nondimensionalization 3.3.1 Forces and Moments Our general nonlinear equations of motion require that we compute the force as a function of the state vector. In the simple six DOF case, this means that we need to represent each of the forces and moments in terms of the velocity and angular rate components. One of the main reasons for doing this is that the aerodynamic forces vary mostly with the square of the speed and angle of attack. If we represent the lift, for example in terms of the six dimensional states, we find that the lift varies almost proportionally to tan-1W/U and to U2+V2+W2 -- a rather messy nonlinear relation. Defining Vt2 = U2+V2+W2 and tan α = W/U allows us to represent the lift in a much simpler form. This nondimensionalization also produces much smaller variations in the dimensionless parameters allowing an analyst to quickly spot mistakes and compare different aircraft. It also makes the results scale independent so we can quickly translate from model scale to full-scale, for example. The dimensionless force and moment coefficients are defined as follows: L = 1/2 ρ Vt2 Sref CL (lift) D = 1/2 ρ Vt2 Sref CD (drag) Y = 1/2 ρ Vt2 Sref CY (side force) Lr = 1/2 ρ Vt2 Sref b Cl (roll) M = 1/2 ρ Vt2 Sref cref Cm (pitch) N = 1/2 ρ Vt2 Sref b Cn (yaw) Where: ρ = density of air Vt = airspeed as defined above: Vt2 = U2+V2+W2 Sref= reference wing area b = wing span cref= reference wing chord

3.3.2 State Variables The reference values are chosen by convention, although conventions vary from company to company and text to text. In these notes we will try to be explicit about the values of these reference parameters for each case we consider. The velocities themselves may be nondimensionalized. We will generally refer to the total velocity, Vt, and the angles of attack and sideslip defined below, rather than the individual component velocities. tan α = W/U sin β = V/ Vt We can similarly nondimensionalize the angular rates, P, Q, and R. This is also done to reduce the dependence of the aerodynamic forces and moments to simpler forms and to restrict the range of typical numerical values of these parameters. The angular rates are usually expressed as velocity ratios as defined below:

We again stress that while we might just retain the dimensional versions of these parameters, the dimensionless versions are very useful. A dimensionless roll rate of 0.1, for example means that the tip of the airplane develops a normal velocity of 10% of the freestream speed (a change of more than 11 degrees in local angle of attack between the left and right tips).

3.3.3 Mass Properties The mass properties may also be nondimensionalized, although this is perhaps less useful and more confusing than it is worth. It is, however, useful in establishing certain scaling relationships for use in dynamic scaling. The dimensionless mass (sometimes called reduced mass) is: µ=2m/ρSc Each of the moments of inertia may also be scaled. For example, iyy = 8 Iyy / ρ S c3 ixx = 8 Ixx / ρ S b3

3.3.4 Dynamic Scaling If all of the dimensionless aerodynamic quantities may be expressed as a function of these dimensionless state variables, and the dimensionless mass and inertias are fixed, then the dynamic response of vehicles of different sizes, at different speeds, and altitudes will be the same, as measured in a dimensionless time variable (e.g. t' = 2 Vt t / c). This is to say that if we played back a movie of the flight, we would see the same thing, as long as we slowed down the film in proportion to the time scale above. This allows tests of stability and control characteristics using smaller scale vehicles that represent larger scale, more expensive, designs. The development of such vehicles is not straightforward however, as the requirements for dynamic scaling are often incompatible with the operation of such vehicles. For example, if we reduce the size of an airplane by a factor λ, then if we wish to fly at the same altitude, the mass must scale with λ3 in order to maintain the same relative mass. This mean that the wing loading, W/S is reduced by the factor λ as well. A recent 6% scale model of a 800,000 lb airplane was built and flown. For dynamic scaling we would require a 172 lb model. The wing loading would be only 6% of the full scale wing loading, or 6 lb/ft2 rather than 100 lb/ft2. If the full scale airplane operated at a CL of 1.0 in this flight condition, it would be flying at 172 kts, while the model would fly at 42 kts. (Recall for level flight L = W = .5 ρ Vt2 S CL). This is rather convenient as long as we do not worry about Mach number or Reynolds number effects on the aerodynamics. The aerodynamic coefficients really do change with Mach number and Reynolds number, so very accurate dynamic testing requires that these dimensionless parameters be matched as well, something that is not easy to do.

3.3.5 Dimensionless Equations of Motion We could now rewrite the equations of motion in terms of these dimensionless quantities. However, because the nondimensionalization involves Vt, which varies in time, the expressions get very messy. It is usually better to keep the nonlinear equations in their dimensional form, using the dimensionless expressions for the aerodynamic forces to then assemble the overall forces and moments in dimensional terms, and finally plug them into the full dimensional nonlinear EOMs. When we linearize the equations about a constant reference condition, then the nondimensionalization of the equations becomes more useful. This is discussed in the following section. Nonetheless, for scaling purposes, it is sometimes useful to see the fully nonlinear dimensionless equations. These are written below just by substituting the expressions for the dimensionless forces and rates into the general equations of motion. du'/dt' = Cx/2µ - q' tan α + r' c/b sin β dv'/dt' = Cy/2µ - r' c/b u' + p' c/b tan α dw'/dt' = Cz/2µ - p' c/b sin β + q' u' Ixx /qSb dP/dt - Ixz dR/dt - Ixy dQ/dt = Cl + Iyz (Q2 - R2) + Ixz PQ - Ixy 8/rSb^3 r' p' + (Iyy-Izz) QR iyy dq'/dt' - Ixy dP/dt - Iyz dR/dt = Cm + ixz b/c (r'2 - p'2) + Ixy 8/rScbc q' r' - Iyz 8/rScbc p' q' + (izz-ixx) r' p' b/c Izz dR/dt - Iyz dQ/dt - Ixz dP/dt = N + Ixy (P2 - Q2) + Iyz RP - Ixz QR + (Ixx-Iyy) PQ with t' = t 2Vt/c (Note that authors sometimes define separate nondimensional times for lateral and longitudinal dynamics. This ambiguity is not introduced here, but one must be careful when dealing with dimensionless time measures. cf. Ashley, 1974.)

3.4 Linearization 3.4.1 Why Linearize? The nonlinear equations of motion given in section 3.3 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. 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. 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 managable.

3.4.2 How Linearization is Accomplished There are several ways in which the equations or parts of the equations can be linearized. One first starts with the specification of a reference flight condition. This, by the way, permits a straight-forward nondimensionalization of the equations of motion based on parameters such as the density at the reference altitude and the reference speed. One may then do the nondimensionalization and linearization algebraically. As shown in the example later in this section, this is straightforward if the reference condition is simple, but gets rather messy in the general case (e.g. perturbations from a steady but rapid turn). Alternatively, one may linearize the equations numerically by specifying a reference condition and constructing a specific linear fit to the computed nonlinear results nearby the reference condition. One must be careful to construct accurate estimates using this numerical finite difference technique. Good methods involve checking results at several difference intervals and using a central differencing technique. One of the advantages of numerical linearization is that one may use a general nonlinear aerodynamic database or analysis code, in fact the same code that is used in the nonlinear simulation, perhaps. Still, a great deal of insight may be obtained by expressing the linearized equations explicitly and our approach here is to do both. Keeping the forces explicit and simply linearizing the dynamic equations of motion about a simple steady symmetric flight condition (P,Q, R, V = 0; W = W0; U = U0) for an airplane with no asymmetric cross moments of inertia (Ixy = Iyz = 0) leads to the following equations of motion: m du/dt = Fx - m Q W0 m dv/dt = Fy - m R U0 + m P W0 m dw/dt = Fz + m Q U0 Ixx dP/dt - Ixz dR/dt = L Iyy dQ/dt = M Izz dR/dt - Ixz dP/dt = N

Even simpler equations are useful for an aircraft with no cross-products of inertia in the case where W0 = 0: m du/dt = Fx m dv/dt = Fy - m R U0 m dw/dt = Fz + m Q U0 Ixx dP/dt = L Iyy dQ/dt = M Izz dR/dt = N Note that the v - equation involves R and the w equation involves Q, but otherwise the equations are coupled only through the force terms.

3.4.3 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 types of models are discussed in the following chapter in some detail, but for our present discussion it is useful to note that 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 NavierStokes solvers in much the same way as they do with wind tunnel data. If we need linear models for simulation and control design, we must construct them explicitly. Such 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. Effects of altitude may also be important, although as discussed below this can often be accommodated rather simply. 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 is discussed later in these notes, but 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. The notation for these dimensionless derivatives is illustrated by the moment coefficient: Cmu' = dCm/du' Cmβ = dCm/dβ Cmα = dCm/dα Cmp' = dCm/dp' Cmq' = dCm/dq' Cmr' = dCm/dr' where the perturbation velocities {u, v, w} are written in terms of their dimensionless parameters u', β, and α, while the perturbation rates {p, q, r} are nondimensionalized as described in section 2.3 as {p', q', r'}. Although we require 36 or more derivatives to form a linear model, most airplanes require far fewer. This is because for symmetric airplanes the longitudinal forces and moments must not change to first order with the lateral state variables as long as the reference condition is also symmetric. (See Ashley's text for a proof of this.) Similarly a symmetric airplane at a symmetric flight condition must remain symmetric, so the

derivatives of the lateral quantities with respect to the longitudinal states are zero in this case. So the usual aircraft linear stability model involves only the 3 longitudinal forces and moments with respect to the 3 longitudinal states plus the unsteady terms (the dα/dt or alpha-dot derivatives) and the 3 lateral forces and moments differentiated with respect to 3 dimensionless lateral states. This leaves only 12+9 or 21 derivatives that must usually be specified.

3.4.4 The Decoupled Linear State Equations If the equations of motion and the forces and moments are linearized we may write the linear equations of motion in terms of the derivatives and the reference state explicitly. This is particularly convenient when the reference state involves symmetric flight in a straight line (no angular rates). In that case the equations decouple into a set of longitudinal equations and a set of lateral equations shown below. All time derivatives are in the body axis frame and are with respect to a dimensionless time parameter: d/dt' = c / 2Vtref d/dt. p,q, and r are the dimensionless angular rates while u',β, and α are dimensionless perturbation velocities (i.e. U = U0 + u, and u' = u/U0). The reduced mass µ = m / ρSc, while the dimensionless moments of inertia are defined as in section 2.3). Longitudinal Equations 2 µ du/dt' - 2 CLref tan θref u - Cxu u - Cxα α + CLref θ = 0 2 CLref u - Czu u + (2µ - Czαdot) dα/dt' - Czα α - (2µ + Czq)dθ/dt' + θ CLref tanθref = 0 -Cmu u - Cmαdot dα/dt' - Cmα α + iyy dq/dt' - Cmq q = 0 dθ/dt' = q Lateral Equations 2 µ dβ/dt' - Cyβ β - Cyp p + (2µ c/b - Cyr) r - CLref φ = 0 -Clβ β + ixx b/c dp/dt' - Clp p - ixz b/c dr/dt' - Clr r = 0 -Cnβ β - ixz b/c dp/dt' - Cnp p + izz b/c dr/dt' - Cnr r = 0 b/c dφ/dt' = p + r tanθref The Cx and Cz force derivatives can also be expressed in terms of the more conventional lift and drag coefficients through the rotation from stability to body axes. The linearized result is: Cxα = CLref - CDα Czα = -CDref - CLα In the equations above b is the span and c is the reference chord. These are used to scale certain dimensionless parameters so that we do not have to introduce two different time scales and two different reduced masses.

3.4.5 Eigenvalues and Eigenvectors Once the equations have been linearized, one may obtain solutions to the characteristic equation easily. By assuming a solution of the form x = x0 e-st we can solve for the values of s (complex) that satisfy the equation. The values of s for which the equations are satisfied are the characteristic eigenvalues of the system. They are often written in terms of the frequency and damping of the system: s = n + i ω = -ζ ωn + i ωn (1-ζ2)1/2

So, given the real and imaginary parts of the eigenvalues one can tell a great deal about the characteristic motions of the system, including the natural frequency ωn and the damping ratio ζ, which are directly related to handling qualities specifications. The characteristic period, τ, and time to 1/2 amplitude, T1/2, are given by: τ (sec) = 2 π / ω T1/2 = -0.69315 / n

Note that if one solves the dimensionless equations above, the dimensionless eigenvalue s', must be multiplied by 2Vtref / c to obtain dimensional values of ω and n.

One can obtain the eigenvalues of the longitudinal equations by hand, especially when they are simplified to consider only a single mode, or one can use a compute program to solve the more general problem. Such a program is provided in the course notes.

3.5 Longitudinal Static Stability 3.5.1 Stability and Trim In designing an airplane we would compute eigenvalues and vectors (modes and frequencies) and time histories, etc. But we don't need to do that at the beginning when we don't know the moments of inertia or unsteady aero terms very accurately. So we start with static stability. If we displace the wing or airplane from its equilibrium flight condition to a higher angle of attack and higher lift coefficient:

we would like it to return to the lower lift coefficient. This requires that the pitching moment about the rotation point, Cm, become negative as we increase CL:

Note that: where x is the distance from the system's aerodynamic center to the c.g..

So, If x were 0, the system would be neutrally stable. x/c represents the margin of static stability and is thus called the static margin. Typical values for stable airplanes range from 5% to 40%. The airplane may therefore be made as stable as desired by moving the c.g. forward (by putting lead in the nose) or moving the wing back. One needs no tail for stability then, only the right position of the c.g..

Although this configuration is stable, it will tend to nose down whenever any lift is produced. In addition to stability we require that the airplane be trimmed (in moment equilibrium) at the desired CL.

This implies that: With a single wing, generating a sufficient Cm at zero lift to trim with a reasonable static margin and CL is not so easy. (Most airfoils have negative values of Cmo.) Although tailless aircraft can generate sufficiently

positive Cmo to trim, the more conventional solution is to add an additional lifting surface such as an aft-tail or canard. The following sections deal with some of the considerations in the design of each of these configurations.

3.5.2 Pitching Moment Curves If we are given a plot of pitching moment vs. CL or angle of attack, we can say a great deal about the airplane's characteristics.

For some aircraft, the actual variation of Cm with alpha is more complex. This is especially true at and beyond the stalling angle of attack. The figure below shows the pitching characteristics of an early design version of what became the DC-9. Note the contributions from the various components and the highly nonlinear post-stall characteristics.

3.5.3 Equations for Static Stability and Trim

The analysis of longitudinal stability and trim begins with expressions for the pitching moment about the airplane c.g..

Where: xc.g. = distance from wing aerodynamic center back to the c.g. = xw c = reference chord CLw = wing lift coefficient lh = distance from c.g. back to tail a.c. = xt Sh = horizontal tail reference area Sw = wing reference area CLh = tail lift coefficient

Cmacw = wing pitching moment coefficient about wing a.c. = Cmow Cmc.g.body = pitching moment about c.g. of body, nacelles, and other components The change in pitching moment with angle of attack, Cmα, is called the pitch stiffness. The change in pitching moment with CL of the wing is given by:

Note that:

when

The position of the c.g. which makes dCm/dCL = 0 is called the neutral point. The distance from the neutral point to the actual c.g. position is then:

This distance (in units of the reference chord) is called the static margin. We can see from the previous equation that:

(A note to interested readers: This is approximate because the static margin is really the derivative of Cmc.g. with respect to CLA, the lift coefficient of the entire airplane. Try doing this correctly. The algebra is just a bit more difficult but you will find expressions similar to those above. In most cases, the answers are very nearly the same.)

We consider the expression for static margin in more detail:

The tail lift curve slope, CLαh, is affected by the presence of the wing and the fuselage. In particular, the wing and fuselage produce downwash on the tail and the fuselage boundary layer and contraction reduce

the local velocity of flow over the tail. Thus we write:

where: CLαh0 is the isolated tail lift curve slope. The isolated wing and tail lift curve slopes may be determined from experiments, simple codes such as the wing analysis program in these notes, or even from analytical expressions such as the DATCOM formula:

where the oft-used constant η accounts for the difference between the theoretical section lift curve slope of 2π and the actual value. A typical value is 0.97. In the expression for pitching moment, ηh is called the tail efficiency and accounts for reduced velocity at the tail due to the fuselage. It may be assumed to be 0.9 for low tails and 1.0 for T-Tails. The value of the downwash at the tail is affected by fuselage geometry, flap angle wing planform, and tail position. It is best determined by measurement in a wind tunnel, but lacking that, lifting surface computer programs do an acceptable job. For advanced design purposes it is often possible to approximate the downwash at the tail by the downwash far behind an elliptically-loaded wing:

We have now most of the pieces required to predict the airplane stability. The last, and important, factor is the fuselage contribution. The fuselage produces a pitching moment about the c.g. which depends on the angle of attack. It is influenced by the fuselage shape and interference of the wing on the local flow. Additionally, the fuselage affects the flow over the wing. Thus, the destabilizing effect of the fuselage depends on: Lf, the fuselage length, wf, the fuselage width, the wing sweep, aspect ratio, and location on the fuselage. Gilruth (NACA TR711) developed an empirically-based method for estimating the effect of the fuselage:

where: CLαw is the wing lift curve slope per radian Lf is the fuselage length wf is the maximum width of the fuselage Kf is an empirical factor discussed in NACA TR711 and developed from an extensive test of wing-fuselage combinations in NACA TR540. Kf is found to depend strongly on the position of the quarter chord of the wing root on the fuselage. In this form of the equation, the wing lift curve slope is expressed in rad-1 and Kf is given below. (Note that this is not the same as the method described in Perkins and Hage.) The data shown below were taken from TR540 and Aerodynamics of the Airplane by Schlichting and Truckenbrodt:

Position of 1/4 root chord on body as fraction of body length

Kf

.1

.115

.2

.172

.3

.344

.4

.487

.5

.688

.6

.888

.7

1.146

Finally, nacelles and pylons produce a change in static margin. On their own nacelles and pylons produce a small destabilizing moment when mounted on the wing and a small stabilizing moment when mounted on the aft fuselage. With these methods for estimating the various terms in the expression for pitching moment, we can satisfy the stability and trim conditions. Trim can be achieved by setting the incidence of the tail surface (which adjusts its CL) to make Cm = 0:

Stability can simultaneously be assured by appropriate location of the c.g.:

Thus, given a stability constraint and a trim requirement, we can determine where the c.g. must be located and can adjust the tail lift to trim. We then know the lifts on each interfering surface and can compute the combined drag of the system.

3.6 Dynamic Stability 3.6.1 Introduction The evaluation of static stability provides some measure of the airplane dynamics, but only a rather crude one. Of greater relevance, especially for lateral motion, is the dynamic response of the aircraft. As seen below, it is possible for an airplane to be statically stable, yet dynamically unstable, resulting in unacceptable characteristics.

Just what constitutes acceptable characteristics is often not obvious, and several attempts have been made to quantify pilot opinion on acceptable handling qualities. Subjective flying qualities evaluations such as Cooper-Harper ratings are used to distinguish between "good-flying" and difficult-to-fly aircraft. New aircraft designs can be simulated to determine whether they are acceptable. Such real-time, pilot-in-theloop simulations are expensive and require a great deal of information about the aircraft. Earlier in the design process, flying qualities estimate may be made on the basis of various dynamic characteristics. One can correlate pilot ratings to the frequencies and damping ratios of certain types of motion as in done in the U.S. Military Specifications governing airplane flying qualities. The figure below shows how the shortfrequency longitudinal motion of an airplane and the load factor per radian of angle of attack are used to establish a flying qualities estimate. In Mil Spec 8785C, level 1 handling is considered "clearly adequate" while level 3 suggests that the airplane can be safely controlled, but that the pilot workload is excessive or the mission effectiveness is inadequate.

Rather than solve the relevant equations of motion, we describe here some of the simplified results obtained when this is done using linearized equations of motion. When the motions are small and the aerodynamics can be assumed linear, many useful, simple results can be derived from the 6 degree-of-freedom equations of motion. The first simplification is the decoupling between symmetric, longitudinal motion, and lateral motion. (This requires that the airplane be left/right symmetric, a situation that is often very closely achieved.) Other decoupling is also observed, with 5 decoupled modes required to describe the general motion. The stability of each of these modes is often used to describe the airplane dynamic stability. Modes are often described by their characteristic frequency and damping ratio. If the motion is of the form: x = A e (n + i ω) t, then the period, T, is given by: T = 2π / ω, while the time to double or halve the amplitude of a disturbance is: tdouble or thalf = 0.693 / |n|. Other parameters that are often used to describe these modes are the undamped circular frequency: ωn = (ω2 + n2)1/2 and the damping ratio, ζ = -n / ωn.

3.6.2 Longitudinal Dynamic Modes When the aircraft is not perturbed about the roll or yaw axis, only the longitudinal modes are required to describe the motion. These modes usually are divided into two distinct types of motion.

Short-Period The first, short period, motion involves rapid changes to the angle of attack and pitch attitude at roughly constant airspeed. This mode is usually highly damped; its frequency and damping are very important in the assessment of aircraft handling. For a 747, the frequency of the short-period mode is about 7 seconds, while the time to halve the amplitude of a disturbance is only 1.86 seconds. The short period frequency is strongly related to the airplane's static margin, in the simple case of straight line motion, the frequency is proportional to the square root of Cmα / CL.

Phugoid The long-perioid of phugoid mode involves a trade between kinetic and potential energy. In this mode, the aircraft, at nearly constant angle of attack, climbs and slows, then dives, losing altitude while picking up speed. The motion is usually of such a long period (about 93 seconds for a 747) that it need not be highly damped for piloted aircraft. This mode was studied (and named) by Lanchester in 1908. He showed that if one assumed constant angle of attack and thrust=drag, the period of the phugoid could be written as: T = 1.414 π U/g (= 0.138 U, with U in ft/sec). That is, the period is independent of the airplane characteristics and altitude, and depends only on the trimmed airspeed. With similarly strong assumptions, it can be shown that the damping varies as ζ = .7071 / L/D.

3.6.3 Lateral Dynamic Modes Three dynamic modes describe the lateral motion of aircraft. These include the relatively uninteresting roll subsidence mode, the Dutch-roll mode, and the spiral mode. The roll mode consists of almost pure rolling motion and is generally a non-oscillatory motion showing how rolling motion is damped. Of somewhat greater interest is the spiral mode. Like the phugoid motion, the spiral mode is usually very slow and often not of critical importance for piloted aircraft. A 747 has a non-oscillatory spiral mode that damps to half amplitude in 95 seconds under typical conditions, while many airplanes have unstable spiral modes that require pilot input from time to time to maintain heading. The Dutch-roll mode is a coupled roll and yaw motion that is often not sufficiently damped for good handling. Transport aircraft often require active yaw dampers to suppress this motion. High directional stability (Cnβ) tends to stabilize the Dutch-roll mode while reducing the stability of the spiral mode. Conversely large effective dihedral (rolling moment due to sideslip, Clβ) stabilizes the spiral mode while destabilizing the Dutch-roll motion. Because sweep produces effective dihedral and because low wing airplanes often have excessive dihedral to improve ground clearance, Dutch-roll motions are often poorly damped on swept-wing aircraft.