Longitudinal Derivatives

These derivatives are related to the sensitivities of the forces in axial and vertical (normal) directions and the pitching moment with respect to changes in forward speed, normal speed, pitch rate, and elevator/elevon control surface deflections.

Effect of forward/ axial speed u:

(Xu, Zu, Mu)

When there is an increase in the axial velocity the drag and lift forces generally increase. Also, the pitching moment changes. Let us consider the effect of change in forward speed of the aircraft on the lift, drag, and pitching moment.

Xu is the change in forward force (often known as the axial force) due to a change in the forward speed (also known as the axial speed). It is called the speed-damping derivative. By definition, following Equation 4.7 (for force derivatives mass m is

TABLE 4.3 Longitudinal ACDs Nondimensional Dimensional Form Unit Form Unit Remarks Cd = drag force/(^S) — Coefficient of drag force. Important for performance C U &Cd — Mach number/aeroelastic effect. Du 2 du Appropriate in the range 0.8 < Mach < 1.2 — pSU (Cd + Cdu ) m 1/s X 1 dX m du 1/s Negative in sign. Affects damping of Phugoid mode C @CD Cd = ia" 1 /rad Positive in sign. Often neglected/but can be important at low speeds (high AOA) Cl = lift force/(#S) — Basic lift force coefficient rSU (Cl — CDa) 1/s X 1 dX m dw 1/s Xa UXw C @Cd Cds dS 1 /rad Normally negligible. Could be +ve or — ve. For tailless a/c it is relatively large pSU2 (—cds ) m X = 1 @x S m dS m (rad s2) (rad s2) C U dCL Lu 2 du — Arises from Mach and aeroelastic effect. Sign can vary. Negligibly small for low Mach maneuvers — P^ (Cl + Clu ) m 1/s Z 1 dZ m du 1 dL Zu = 7Г" m du 1/s Positive downward C dCL Cl“ da 1 /rad Lift-curve slope. Mostly +ve/could be very small or —ve (implying “stall”) pSU im(C-+cd) 1/s 1 dZ ‘Zw = "7 m dw 1/s C @Cl La d(ac/2U) 1 /rad Models the unsteady effects due to the lag in downwash on the horizontal tail pSC ( —CLa ) — Z 1 dZ m dw — Often neglected Xa UXw 1/rad

_ dCL d(qc/2U)

Positive in sign for rigid body a/c and in low speed flights. In high-speed flights sign could be +/—ve

control surface deflection.

1 @M _ pSU2c dCm

Ty me – 2iy ~we

pSU2cC

2Iy mge

Cmg derivative can be considered as a primary design parameter and it specifies longitudinal axis control power. Important and routinely used longitudinal DADs and NDADs (stability and control derivatives) are given in Table 4.3 in a compre­hensive format. Some important longitudinal derivatives are explained next.

(a) CL is defined as a change in lift coefficient for a unit change in AOA. The lift force can be easily given by

L = CLqS

CL represents the lift-curve slope with respect to AOA. This is a very important derivative because it almost directly determines the contribution to the lift force. It also signifies the fact that as AOA increases, the lift force increases proportionally in the linear region up to a certain AOA. Beyond this AOA the lift would remain somewhat constant and would even decrease further. This condition is called ‘‘wing stall’’ (Appendix A). This means that the aircraft loses some lift after it stalls.

(b) Cm is the basic static stability derivative and is also referred to as the pitch stiffness parameter [2]. A negative value of Cm indicates that the aircraft is statically stable, i. e., if the AOA increases then the pitching moment becomes more negative, thereby decreasing the AOA and hence restoring the stability. Cm derivative is proportional to the distance between the aerodynamic center (better referred to and used as neutral point, NP) of the aircraft and the CG. This distance is related to the static margin as

Static margin = (NP distance — CG)/MAC

The distances (of NP and CG) are from some reference point in the front of the aircraft on the x-axis. If the static margin is positive, then the aircraft is said to have static stability. This means that the NP distance is larger than the CG distance from the reference point. If the CG is continuously moved (by some means) rearwards, then at one point the aircraft will become neutrally stable (neutrally unstable!). This point is called the NP of the aircraft or this CG position/location is called the NP. The NP must always be behind the CG location for guaranteed static stability. A slightly more rearward movement of the CG will render the aircraft statically unstable. If the aircraft is dynamically stable then it must have been statically stable. For dynamic stability the static stability is a must, but the converse is not true, meaning that if the aircraft is statically stable it could be dynamically unstable. This derivative is of primary importance to the longitudinal stability of atmospheric vehicles. If the aircraft is inherently statically unstable (by design or for gaining certain benefits in case of a relaxed static stability aircraft like FBW and many modern-day high – performance fighter aircraft), then artificial stabilization is required for the aircraft to fly. The level of instability is dictated by (1) available state-of – the-art technology (computers, actuators, sensors, and control law design tools), (2) possible reduction in the size of the aircraft and subsequent decrease in weight, (3) increase in weight due to additional hardware (redundancy, computer, etc.), (4) obtainable performance, (5) maneuver­ability, and (6) agility. This is the subject of design and development of flight control laws.

(c) Cm signifies a change in pitching moment coefficient due to a small change in pitch rate q. Being (angular) a rate-related derivative it implies, from the control theory point of view, that it must have something to do with damping-in pitch. As such it contributes to the damping-in pitch. Usually more negative values of Cm signify increased damping. The sign is gener­ally negative for both stable and unstable configurations. Since the statically unstable configurations will have stability augmentation control laws, the lower values of this derivative (less —ve values) are acceptable. The aircraft flight-control system (AFCS) also provides some artificial damping. This derivative often has higher prediction uncertainty levels (scatter in esti­mates); however, this is not a problem since AFCS generally tolerates these uncertainties.

(d) Longitudinal control effectiveness derivative Cm is the elevator control effectiveness. In conventional sense, more negative value means more control effectiveness. It also helps determine the sizing of the control surface. More elevator (elevons for FBW delta-wing aircraft configuration) control power means more effective control in generating the control moment for the aircraft. This also applies to horizontal tails, canard, or a combination of these surface movements. The knowledge of the available elevator/elevon power at all the flight conditions (Chapter 7) in the flight envelope is extremely important. The unstable configurations demand higher control power than the stable ones. Numerical values of certain longitudinal derivatives are given in Table 4.4.

Important lateral-directional aerodynamic derivatives are collected in Table 4.5 in a comprehensive and compact manner, along with brief explanations and indication of influence of certain derivatives on aircraft modes.