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were found to be of considerable help in designing the fiight control systems.
For a statically unstable airplane, the usual short-period and phugoid modes
do not exist. Instead, we have two exponential modes and an oscillatory mode.
The oscillatory mode is called the third oscillatory mode that has short-period-like
damping and phugoidlike frequency. Because of static instability, at least one of
the two exponential modes wrllbe unstable.
The handling quality requirements form the basis of the design of stabiliLy
augmentation systems and autopilots. The longitudinal flying quality requirements
were presented in terms of damping ratios and frequencies of the short-period and
phugoid modes. Similarly, the lateral-directional fiying qualit/es were presented
in terms of damping ratios and frequency for Dutch-roll mode and in terms of time
for half or double the amplitude for roll-subsidence and spiral modes.
We have illustrated the basic design procedures for stability augmentation sys-
tems and autopilots. It was not possible to cover all types of stability augmentation
systems and autopilots that are in use on modern airplanes. The interested reader
may refer elsewhere5.7 for additional information.
References
I Pro-MATLAB for Srm Workstations, The MathWorks, Natick, MA, Jan. 1990.
zTeper, G. L., "Aircraft Stability and Control Dat.a:' NASA CR-96008, April 1969.
3Roskam, J.,Airplane Flight Dynamics and Automatic Fkght Control, Pari H, Roskam
Aviation and Engineering Corp., LaWfence, KS, 1979.
4Nelson, R. C., Flig/zt Stability and Automatic Control, McGraw-Hill, New York, 1989.
5Stevens, B. L., and Lewis, F. L., Aircraft Control and Simulation, Wiley, New York,
J 994.
6MIL_F_8785C, "Flying Qualities of Piloted Airplanes:' U-S. Department of Defense
Military Speczjicr.Uions, Nov. 5, 1980.
7Blakelock, R. C.,Automatic Control of Arrcra.ft cmd Missiles, Wiley, New York, 1965.
Problems
6.1 The mass properties and aerodynamic characteristics of a business jet air-
plane4 are as folfows.
Mass properties: weight W = 17,347.9 kg, wing area S = 50.3983 m2, wing span
b - 16.383 m, mean aerodynanuc chord c - 3.3315 m, distance between center
ofgravity and aerodynamic centerin terms ofmean aerodynamic chord X cg = 0.25,
Ix = 161,032.43 kg/m2, Iy = 184,211.19 kg/m2, lz = 330,142.72 kg/m2, /A:). =
Iyz = 0, and Ixz = 6861.7104 kg/m2.
Longitudinal aerodynamic characteristics: CL -.0.737, CD -0.095, CLa,=
5.0, CDa - 0.75, Cma - -0.8, CLcr - 0, CDd = 0, Cma - -3.0, CLM - 0, CDM - 0,
624 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Cm M = -0.05, CL8t = 0.4, CD8e = 0, Cm~e = -0.81, CLq = 0, CDq = O, and Cmq =
- 8.0.
Lateral-directional aerodynamic character:istics: Cyp = - 0.720, Cip = -0.103,
Cnp = 0.137, C:.& = O, Cl8a = 0.054, Cn8a = -0.0075, C:-8r = 0.175, Ct& = 0.029,
Cn~r = -0.063, C).p = 0, Ctp = -0.37, C,p = -0.14, C).r - 0, Clr -- O.ll, Cnr -
-0.16, C)/ = O, CW = 0, and Cnp = O.
Note that all der:ivatives are per radian.
Determine the following:
(a) The state-space representations, eigenvalues, free responses, and unit-step
responses for theolvongitudinal andlateral- directional dynamics based on acomplete
system as well as.longitudinal (short-period and phugoid) and lateral-directional
(roll-subsidence, Dutch-roll, and spiral) approximations.
(b) Transfer functions, roots of characteristic equations, steady-state values
for unit-step inputs for longitudinal and lateral directional variables based on a
complete system as well as longitudinal (short-period and phugoid) and lateral-
directional (roll subsidence,Dutch-roll, and spiral) approximations.Cornpare these
results with those obtained in (a).
(c) The longitudinal (elevator input) and lateral-directional (aileron and rudder
inputs) frequency responses based on a complete system.
6.2 The longitudinal characteristic equation of an airplane is given by
s4+1.8s3+3.5s2+0.25s+0.05 - O
Examine the longitudinal stability of the airplane using Routh's criterion.
63 The longitudinal transfer functions of an airplane based on the short-period
approximations are given by
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