EPILOGUE

Since the direct use of EOMs in their full form is usually avoided, except in the full dynamics flight simulation, simplified models are useful in various applications: linear models for control law design, quick analysis to understand the dynamic characteristics of the vehicle, mathematical modeling from flight data, and related handling qualities analysis. This evokes the need of the system’s approach in understating and use of flight mechanics aspects in various aerospace engineering applications. In Ref. [20], the authors give the literal approximate factors to obtain approximate relations for TF poles and zeros, in terms of the stability and control derivatives of the atmospheric vehicle.

EXERCISES

5.1 The aerodynamic derivatives for a transport aircraft are given as

Lo = 0.0095, Lp = -2.0374, Lr = 0.876, LK = -5.933, LSr = 1.034,

N0 = 0.03, Np = -0.177, Nr = -0.57, NSa = -0.292, NSr = -1.749

Write an appropriate mathematical model and obtain various TFs and fre­quency responses of the lateral-directional model using MATLAB functions. Determine the frequency and damping ratio of the mode.

5.2 What is the significance of the natural frequency, say of the SP mode, from the pitching moment point of view?

5.3 Recall Equations 3.23 and 5.32, say for rolling moment and yawing moment. Assume the neutral static condition and neglect cross-coupling inertia terms and obtain the relation between lateral-directional static stability and dihedral (static stability) derivatives in terms of control effectiveness derivatives (see Chapter 4).

5.4 The aerodynamic derivatives of a transport aircraft in phugoid mode are given as

Xu = -0.011, Zu = -0.1583, u0 = 111 m/s

Use the phugoid model of Equation 5.20 and obtain the frequency responses and other characteristics.

5.5 For an airplane with the following characteristics compute (1) the lateral-direc­tional dimensional derivatives.

Сур = -1.125; Cyr = 0.8; С„э = 0.27; CHr = -0.5; = 0.28;

Cnsr = -0.16; Cyp = 0.17; Ckr = 0.0697; Ch = -0.133;

C1p = -0.96; C1r = 0.42; Cnp = -0.1; Ckr = -0.2496

Other data are

Ix = 105,500 kgm[3];	Iy = 250,000 kgm2;	Iz	= 340,000 kgm2;
Ixz = 11,500 kgm2	V = 72.114;	r	= 0.72851 kg/m[4]
b = 21.5 m;	S = 65 m2;	mass = 5,000 kg;
u0 = 140 knots;	H = 5100 m

5.9 If you want to add another state variable in the phugoid model, which one would you add?

5.10 If the moment-speed derivative in Equation 5.22a becomes sufficiently nega­tive, what happens to the phugoid mode characteristic?

5.11 In SP mode/model, which aerodynamic derivative would have the largest influence on the natural frequency?

5.12 In SP mode which aerodynamic derivatives would have influence on the damping ratio?

5.13 What is inertial and aerodynamic ‘‘damping’’ in SP mode?

5.14 In phugoid mode which is the most influential aerodynamic derivative and in what way?

5.15 Which aerodynamic derivatives have the most significant influence on the DR frequency?

5.16 Which are the most effective aerodynamic derivatives in spiral and roll sub­sidence modes?

5.17 Which is the most effective DR damping aerodynamic derivative?

5.18 Explain the name spiral divergence.

5.19 Derive the condition for spiral stability.

5.20 Using small disturbance theory and assuming wings-level symmetric flight condition, show that the expression W = —PV + QU+ g cos Q cos Ф + Az reduces to w = qu0 — gU sin U0 + az.