Having set out a basic aerodynamic analysis with XFLR5, it is then possible to use the code to more fully investigate the stability of the design by locating the eight natural flight modes of oscillation on a root locus plot. These can be compared with the tabulated acceptable values provided by the U.S. military. In a linearized analysis, these modes consist of four longitudinal modes, namely a pair of symmetric phugoid modes and a pair of symmetric short-period modes, and four lateral modes, namely one spiral mode, one roll damping mode, and a pair of Dutch roll modes (free directional oscillations): that is, five distinct types of behavior. XFLR5 does this by using finite differencing in the six degrees of freedom of flight to calculate the first-order stability derivatives, which can then be used to calculate the various eigen- modes (changes are made in wind velocities by 0.01 m/s and angular velocities by 0.001 rad/s). Because the code is so fast, these additional runs are not too significant an overhead in the calculation. In general, the results are conservative because XFLR5 tends to underestimate the various damping terms in the analysis because it tends to underestimate drag.

XFLR5 follows the analysis presented by Etkin and Reid [23] and, in particular, solves equations (4.9,18 and 4.9,19) presented there. To do this, it derives the linearized stability derivatives by finite differencing, and these are then used to solve and store the eigenmodes and 18 nondimensional derivatives. The derivatives are returned as CXu, CLu, Cmu, CXa, CLa, Cma, CXq, CLq, Cmq, CYb, Clb, Cnb, CYp, Clp, Cnp, CYr, Clr, and Cnr. The first nine of these are of horizontal and vertical forces and roll moment with respect to changes in forward speed, of AoA at constant pitch, and of varying pitch at constant AoA (note these last two are different since the direction of instantaneous flight may not be horizontal). They are defined as follows:

The last nine are the side force derivative, the dihedral effect, the weathercock stability, the side force due to rolling, the damping in roll, the yaw due to roll cross derivative, the side force due to yaw, the roll due to yaw, and the damping in yaw. They are defined as follows:

CYb = Y_{v}u0/(qS) = C_{Y}0;

Clb = L_{v}u_{0}/(qSb) = Cip;

Cnb = N_{v}u_{0}/(qSb) = C_{n}p;

CYp = Y_{p}(2u_{0})/(qSb) = Cy_{p};

Clp = Lp(2u0/b)/(qSb) = Cp,

Cnp = Np(2uo/b)/(qSb) = C_{n}p,

CYr = Y_{r}(2u_{0})/(qSb) = C_{Yr};

Clr = L_{r}(2u_{0}/b)/(qSb) = C_{l}r;

Cnr = N_{r} (2u_{0}/b)/(qSb) = C_{nr}.

Here, p is the air density, u_{0} is the reference flight speed, 8_{0} is the reference angle of climb, S is the wing planform area, c is the mean aerodynamic chord, and b is the span. The various terms such as X_{u} and N_{r} are the derivatives, that is, the rates of changes of forces and moments due to changes in velocities and angular velocities.

Phugoid oscillation is a macroscopic mode of exchange between kinetic and potential energies (forward and vertical velocities) and is normally slow, lightly damped, and may be stable or unstable. A very simple estimate of the phugoid frequency is given by Lanchester’s approximation as 9.81/(У2жУ_{0}), where V_{0} is the aircraft’s speed in m/s. An unstable or divergent phugoid is mainly caused by a large difference between the incidence angles of the wing and the elevator. A stable, decreasing phugoid can be attained by adopting a smaller elevator on a longer tail, or, at the expense of pitch and yaw static stability, by shifting the CoG to the rear. As the phugoid frequency is generally very low, it is quite possible to fly an aircraft with a divergent phugoid mode, although this is probably not desirable. Its damping is inversely proportional to the lift/drag ratio, so that more aerodynamically efficient designs tend to have less phugoid damping. Since the lift/drag ratio varies with speed, so also does phugoid damping.

The short-period mode is primarily vertical movement and pitch rate in the same phase and is usually of high frequency, stable, and well damped. An aircraft with a low short-period natural frequency will seem initially unresponsive to control input. If the natural frequency is too high, the aircraft will feel too sensitive in maneuvering and too responsive to turbulence. Aircraft with low short-period damping ratios tend to be easily excited by control inputs and turbulence, and the resulting oscillations take longer to disappear. Aircraft with high short-period damping can be slow to respond and sluggish; thus a compromise in damping is necessary. FAA regulations require that the short-period oscillation must be “heavily damped” and that “Any ... phugoid oscillation ... must not be so unstable as to increase the pilot’s workload or otherwise endanger the aircraft.”

The spiral mode is primarily in-heading, nonoscillatory, slow, and generally unstable, requiring pilot input to prevent divergence (pilots commonly do this without noticing on well-designed aircraft). Roll damping is usually stable because positive dihedral, swept wings, high wings, or a low CoG will have been adopted. Dutch roll is a combination of yaw and roll, phased at 90 , and is usually lightly damped; it is largely controlled by the size of the tail fin, which should have a sensible value if an appropriate volume coefficient has already been used. Larger fins increase both the Dutch roll frequency and its damping. Dutch roll is not an aircraft deficiency but the inevitable result of having directional stability, that is, the fundamental tendency of a stable aircraft to roll away from, but yaw towards, the velocity vector. Aircraft with greater directional than lateral stability tend to Dutch-roll less, but also tend to be spirally unstable. Normally, the design compromise between Dutch roll and spiral instability is to reduce Dutch roll at the expense of spiral instability because spiral dives begin slowly and are more easily controlled than Dutch roll.

To help decide what values for the various dynamic stability modes are acceptable, the U.S. military have developed a specification for the flying qualities of piloted airplanes: MIL-F-8785C. This defines three levels of flying qualities (1-3, with 1 being the best), three phases of flight (A-C, with A being the most demanding and including air-to-air combat and ground attack), and four classes of aircraft (I, light aircraft; II, medium-weight, medium-maneuverability aircraft; III, heavy, low-maneuverability aircraft; and IV, high-maneuverability fighter types). Then for each combination it gives guidance on the acceptable damped natural frequencies and damping ratios. For the purposes of most small UAV designs, class I and flight phases B or C should be assumed. If the aircraft is to be manually flown, a flying quality of level 1 should be aimed at, but if full autopilot control is assumed, then a quality of level 3 can be acceptable as pilot workload is then not an issue. For the five different modes normally encountered, the specification gives the following guidance for class I aircraft:

• Short-period mode. Damping ratio limits for level 1,0.35-1.3 in phases A and C and 0.3-2.0 in phase B (these can be relaxed to 0.25-2.0 and 0.2-2.0 for level 2 in phases A and C and phase B, respectively, and to just lower limits of 0.15 in any phase for level 3);

• Phugoid mode. Damping ratio for level 1 to be at least 0.04 (this can be relaxed to zero for level 2 and to time for the amplitude to double (T_{2}) of 55 s for level 3, that is, divergence is then acceptable);

• Roll mode. Maximum time constant, t_{r},^{[1]} 1 s for level 1, phase A or C and 1.4 s for phase B (these can be relaxed to 1.4 and 3.0 s for level 2, respectively, and to 10 s in any phase for level 3);

• Dutch-roll mode. Minimum damping ratio of 0.19 for level 1, phase A and 0.08 for phase B or C (these can be relaxed to 0.02 in any phase for level 2 and zero for level 3); minimum damped natural frequency of 1 rad/s (0.159 Hz) for level 1, phase A or C and 0.4 rad/s (0.064 Hz) for phase B (these can be relaxed to 0.4 rad/s in any phase for levels 2 and 3);

• Spiral mode. Time for amplitude to double (T_{2}) for level 1 to be at least 12 s in phases A and C or 20 s in phase B (these can be relaxed to 8 s in any phase for level 2, and 4 s for level 3).

Phugoid, short-period, and Dutch-roll modes being oscillatory give complex pairs of roots in the root locus plot, while roll damping and spiral modes lie on the horizontal axis. These quantities can then be converted into frequencies, damping ratios, time constants, and times for the amplitude to double. Apart from the aerodynamic quantities of the aircraft, the stability behavior is controlled by the location of the CoG and moments of inertia; all must be input into XFLR5 to carry out a stability analysis (xy and zy cross-moments are ignored, as the airframe is assumed port/starboard symmetric). If one is unsure of the moments of inertia, these can be estimated by assuming that the radius of gyration for Ixx is a fraction of the main semispan, say 22%; that for Iyy is a fraction of half the overall length of the aircraft, say 35%; and that for Izz is a fraction of the average of semispan and half length, say 38% (Raymer [11] gives values for these nondimensional radii of 22-34%, 29-38%, and 38-52%, respectively; our values are based on the UAVs we have built and flown). The cross-moment Ixz is typically small, and can generally be set as zero until better information is available. These may then be used with the overall mass to compute the required inertias. For Decode-1 at 15 kg maximum take-off weight (MTOW), these estimates give Ixx as 1.6 kg m^{2}, Iyy as 1.2 kg m^{2}, and Izz as 2.7 kg m^{2}. Later on in the design, when a full CAD model is being created, these values can be checked and updated.

Table 13.2 shows the eigenvalues for the stability analysis of Decode-1 with final wing and elevator setting angles, using these inertia values and the previously calculated XFLR5 stability derivative data at 30m/s. The results suggest that the aircraft has quality 1 dynamic stability except in the phugoid mode where it lies on the quality 1/2 boundary. The phugoid prediction is pessimistic because the parasitic drag has not been included in XFLR5; when this is included, clear quality 1 is obtained (at the time of writing, XFLR5 did not include parasitic drag in its internal stability analysis but this can be added in separately using the formulae provided by Phillips [24,25], although these formulae include a few simplifications not needed

Table 13.2 Decode-1 eigenvalues as calculated from XFLR5 stability derivatives using the formulae provided by Phillips [24, 25] and the estimated inertia properties for a flight speed of 30m/s and MTOWof 15kg.

Note that the undamped natural frequency in hertz (Hz) is given by the magnitude of the eigenvalue divided by 2k, while the damping ratio is the negative of the ratio of the real part divided by the imaginary part; T_{2} = log_{e}(2)/eigenvalue (for positive eigenvalues) and t_{r} = log_{e}(1 — 0.632)/eigenvalue (for negative eigenvalues).

“0.151 if parasitic drag is included and thus clearly quality 1.

by the full eigen analysis used in XFLR5). Note also that Dutch roll damping is increased by the drag caused when the fuselage elements are not aligned with the direction of flight, and these aspects are not generally captured by XFLR5 either (it is possible to approximate the fuselage as a further lifting surface, but clearly most fuselages are not really wing-shaped!). Thus the XFLR5 results are essentially worst case values. This analysis has been confirmed by flight trials; the aircraft is responsive and benign when flown manually, and the phugoid and Dutch-roll modes are easily controlled. Figure13.27 illustrates the behavior of the modes listed in Table 13.2 (with the added parasitic drag) in the time domain.

[1] The roll mode time constant is the time it takes to achieve 63.2% of the final roll rate following a step input.