Definition of Aerodynamic Derivatives

Essentially these derivatives are sensitivities to changes in flight variables. A change in AOA results in a change in the pitching moment coefficient and an appropriate change in the pitching moment M (see Equation 4.2). How this change in turn alters the aircraft responses is the subject of the EOM (Chapters 3 and 5). There could be cross-coupling terms, i. e., a change in the aerodynamic coefficient of one axis due to a change in the variable of the other axis. Several possibilities exist. The analysis is limited to straightforward effects. The force and moment coefficients of Equations

4.1 and 4.2 are expressed in terms of the aerodynamic derivatives as seen in Equation 4.6. These are often called stability and control derivatives. This is because they directly or implicitly govern the stability and control behavior of the vehicle. They are called derivatives as they specify the variation of aerodynamic forces and moments with respect to a small change in the perturbed variable. A particular derivative could vary with velocity or Mach number, altitude, and AOA. And in turn the coefficients that depend on these derivatives also vary with these and several other variables. However, the variation of these coefficients soon gets translated to variations with time because of the composition of the coefficients in terms of motion variables, which vary with time.

Table 4.1 gives the matrix of relationships in terms of the sensitivities between the forces/moments and the response variables u, v, w, p, q, r, and control surface deflections in terms of the aerodynamic derivatives. Table 4.2 depicts the major functional (effect) relationships between forces/moments and the aircraft response variables and classifies them broadly as mentioned in the table. In other words, we have the following classification: (1) speed derivatives, (2) static derivatives, (3) dynamic derivatives, and (4) control derivatives. The static derivatives are mainly with respect to AOA and AOSS. They govern the static stability

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TABLE 4.1 Relationship Matrix for Forces/Moments and the Independent Response Variables With Respect to With Respect to Angular With Respect to Control Component Velocities Velocities (Rates) Surface Deflections Response Variables^ Se (Eleven/ Forces/Moments J. и (Axial) v (Side) w (Normal) p (Roll) q (Pitch) r (Yaw) Elevator) Sa (Aileron) 8, (Rudder) X (axial) xu Aw X» Xs, Cd, Cd<x Cds, Y (side) Yv YP Yr Ysa Ys, Суs Cyp Су, Cysa Crs, Z (vertical/normal) zu Zw zq Ze, Си cLa Cl, Cls, L (rolling) LP Lp L, LSa Ls,. Cl„ Qp Q, Cm, Cm, M (pitching) Mu Mw Mq Ms, Сш, c Аша Сш, Cm8e N (yawing) Np Np N, NK Ns, Cn„ c ‘-■np c ^nr CnSa Cn8,

TABLE 4.2 Effective Consequential Relationships of Forces and Moments and the Response Variables Linear Velocities Angular Velocities (Rates) Control Surface Deflections Forces Speed derivatives/static derivatives (with respect to AOA, AOSS) Usually unimportant and often neglected Control derivatives Moments Speed derivatives/static stability derivatives (with respect to AOA, AOSS) Dynamic derivatives/damping (related) derivatives Control derivatives

(Appendices A and C) of the vehicle. Dynamic derivatives are with respect to the rotational motion of the vehicle and mainly specify the damping (of certain modes of the vehicle) in the respective axis. The speed derivatives are with respect to linear velocities of the vehicle. The control derivatives specify the control effect­iveness of the control surface movements in changing the forces and moments acting on the vehicle.

We see from Table 4.2 that the static and the dynamic derivatives, as they are termed, imply something definitely more in terms of the behavior of the aircraft from control theory point of view (Appendix C). What is important to know is that these names themselves relate to certain stability and damping properties of the aircraft, which is considered as a dynamic system. This aspect is further explored in Chapter 5.

The derivative effects are considered in isolation assuming that the perturbations occur in isolation. The aerodynamic derivatives are evaluated at steady-state equi­librium condition with a small perturbation around it. Hence, these derivatives are called quasi-static aerodynamic derivatives. Interestingly enough, these derivatives are used in dynamically varying conditions also, mainly for small amplitude man­euver analysis, thereby assuring the linear domain operations and analysis. For large amplitude maneuvers where AOA are high, the unsteady and nonlinear effects need to be incorporated into the analysis for which additional aerodynamic derivatives must be included. For definition of dimensional aerodynamic derivatives (DADs) and nondimensional aerodynamic derivatives (NDADs) we closely follow Ref. [2] because the definitions and procedures therein are consistent and straightforward. The present study is also enhanced by the research presented in Ref. [3,5-9]. Although there might be some differences in the formulae of certain derivatives between various sources [2,3,5-9], the presentation in this book is more uniform and highly standardized as in Ref. [2]. DADs are defined as

The derivatives specify the change in pitching moment/vertical force due to a small change in vertical speed (w) of the aircraft. Next, we have NDADs defined as

since Cm = M/(qSc) and a = w/U (for small alpha), we get

dM 1

d(w/U) qSc U dM qSc dw

Щ

qSC

Finally, we get

Mw = qSc/(IyU)Cma

= pUS-c/(2Iy )Cma (4.9)

A similar procedure can be applied to all the dimensional derivatives to obtain the equivalent nondimensional derivatives. The decoupling between longitudinal and lateral-directional derivatives is presumed. We also assume that during a small perturbation maneuver, the Mach number, Reynolds number, dynamic pressure, velocity, and engine parameters do not change much, so their effects can be neglected.

The important longitudinal aerodynamic derivatives are collected in Table 4.3 in a comprehensive and compact manner along with brief explanations and an indica­tion of the influence of certain derivatives on aircraft modes.