Thesis - Beaver simulation

S-functions can be connected to eachother in a separate block-diagram, and each (sub-) model can easily be updated without affecting the total simulation structure. The use of different S-functions is particulary recommended for large models, which contain many levels. In the previous section, it was shown that the model of the aircraft dynamics itself is indeed very complicated.

In this report, separate libraries with models of wind, atmospheric turbulence, ILS signals, and VOR signals will be created. All models will be described in detail in appendix F. Similarly to the aircraft model, the equations of the other models will be implemented as SIMULINKFunction blocks, which will be grouped together in 'basic' blocks or subsystems. Figure 4-8 shows how the models of the aircraft dynamics, disturbances, navigation signals, and control loops can be connected together in a modular way. In part I1 of this report, a similar structure, containing the control laws of the 'Beaver' autopilot, the model of the 'Beaver' dynamics, Flight Control System models, and external disturbances, will be treated in more detail.

Contrary to the nonlinear aircraft model of figure 4-2, control laws are often based upon deviations from nominal values of the control variables in stead of the actual values of these signals. Thus, control loops use the difference between the outputvector from the aircraft model and the initial (= trimmed) value of this outputvector, y - yo,in stead of y itself to compute the control signals Au. If the control signals are fed back to the aircraft model, it is necessary to add the initial values of the input signals uoto Au. This is not necessary if a linear small-perturbations model of the aircraft is used in stead of the complete nonlinear aircraft model. In that case, the whole structure presented in section 4.2 can be replaced by one linear state-space block only! The use of trimmed values of the states and inputs for linear and nonlinear aircraft models will be illustrated in chapter 6.

states

Control loops wNavi ation si nals

Figure 4-8. Model structure for simulation of an aircraft, equipped with an automatic controller, and affected by external disturbances.

Chapter 5. Assessment of the SIMULINKimplementation of the 'Beaver' model.

5.1 Introduction

In this chapter, some open-loop results, obtained with the SIMULINKmodels of the 'Beaver' will be analyzed. First, some basic properties of the aerodynamic and engine models will be shown. Section 5.3 gives some linear and nonlinear open-loop responses, which should match similar results from refs.[6] and [18]. In section 5.4, open-loop turbulence responses will be shown, and finally, some results of basic open-loop analysis of the linearized 'Beaver' model will be given. Chapter 6 shows how the results from this chapter were created.

5.2 Results from the aerodynamic and engine models.

Of course, the primary contributions to the aerodynamic forces and moments are functions of the angle of attack a and the sideslip angle p. In ref.[30], graphs in which the aerodynamic forces and moments have been plotted as a

dpt - 0 dpt = 0.5 dpt = 1.0

Figure 5-1. Force and moment coefficients of the nonlinear 'Beaver' model as functions of a and p. Compare with figures 4 and 5 of ref.[30].

options (perturbation levels, linearization algorithm, etc.) give the best performance I). The largest differences can be found in the lateral-longitudinal cross-coupling effects from figures 5-3 and 5-4. The longitudinal-lateral crasscoupling is quite good, especially in figure 5-2. Although there is often some phase shift of the phugoid in linear responses to lateral inputs (compared with the nonlinear results), the lateral responses to lateral inputs match perfectly, as do longitudinal responses to longitudinal inputs.

Note that the time-trajectory of x, is not accurate. This is due to the smallperturbation character of the linear responses: integrating the deviation of the nominal airspeed yields a result which differs a lot from the integral of the actual airspeed! Since the time-trajectory of ye is also not very accurate, the linearized x, and ye-equations shouldn't be used for navigation purposes. For non-turning flights, x, can be approximated by: x, (t) = y (10) + V, *t,where y(10) is the time-trajectory of the tenth output from the state-space model, Ax, (use x, (t) = y (10) + (V, cos a, sin $, ) t if the equilibrium values of a and/or

differ a lot from zero).

The initial condition, which was computed with the trim routine ACTRIM, (see appendix E and section F.2.7 of appendix F) yielded the same results as ref.[30]. This gives us an additional validation of the SIMULINK'Beaver' model. And since the SIMULINKand Fortran versions of the same model have been developed fully independently, this check also raises the confidence in the Fortran programs. It is highly unlikely that the same mistakes are made twice!

Figures 5-2 to 5-7 clearly show the characteristic motions of the 'Beaver'. In figure 5-2, the short-period mode is visible in the a, q, and e plots and also in the f3, p, r, and cp plots, due to longitudinal-lateral cross-coupling. For the 'Beaver', the lateral responses to the longitudinal block-shaped elevator deflection A6, are quite large. One should realize this when using simplified 'Beaver' models, which assume that the longitudinal and lateral dynamics are independent of eachother. The phugoid, a n oscillation with a period of about 23 seconds, is clearly visible in the responses to the elevator deflection, shown in figure 5-2. It is also clearly present in the responses to aileron and rudder deflections, shown in figures 5-3 and 5-4, which again proves the significance of cross-coupling effects for the 'Beaver'.

The Dutch-roll and spiral modes can be distinguished easily in figures 5-3 and 5-4. The 'Beaver' behaves quite well in this respect, since the Dutch-roll mode is strongly damped and the spiral mode is (just) stable. The roll subsidence mode is hardly visible in these plots; only t h e p and cp-responses in figure 5-3 show some influence of this characteristic motion. The influence of the roll subsidence mode would probably have been more clearly in these plots if a step-shaped input to the ailerons would have been applied in stead of a blockshaped deflection of the ailerons.

In figure 5-5, the response of the 'Beaver' to a flap deflection of 3 degrees within 3 seconds is shown. Again, the phugoid is visible in both longitudinal and lateral motion variables. The airspeed varies significantly after application of the flap deflection. The mean value of the airspeed drops slightly, and the

In this report, the linearization blocks and plots with responses of the linear aircraft model have been included to illustrate the principle of linearization in SIMULINK,not to find out which linearization method or options give the best results. However, the differencesbetween linear and nonlinear responses are already quite small!

final value of the angle of attack is larger than without flap deflection. This figure again clearly proves the significance of the longitudinal-lateral crosscoupling for the 'Beaver': due to the flap deflection, the aircraft starts to roll with a roll angle of approximately 10 degrees, and the sideslip angle becomes more negative. The change in cp is due to the negative value of Cl,,t (both a and dpt increase after the flap deflection).

Figures 5-6 and 5-7 show the responses of the 'Beaver' to a n increase in engine speed and manifold pressure, respectively. These two inputs yield almost identical responses, as can be expected, because in both cases the resulting increase in engine power P, and hence, the increase in manifold pressure Apt, causes the changes of the forces and moments. The mean airspeed grows somewhat, the angle of attack becomes smaller. The value of dpt increases due to the larger engine power, even though the airspeed increases too. Again, this yields a larger sideslip angle (more negative), and the aircraft starts to roll.

I

rad

rad

"Hg

RPM

Table 5-1. Initial conditions for V = 45 m/s, H = 6000 ft, obtained with the trim routine ACTRIM (compare with ref.[30]).

The responses illustrate the most important characteristia of the motions of the 'Beaver'. In particular, the strong longitudinal-lateral cross-coupling, and the strong influence of the engine and slipstream effects, are clearly visible. As stated in ref.[l8], the responses obtained with the nonlinear 'Beaver' model do resemble the actual motions of the aircraft quite well.

5.4 Open-loopresponses to atmospheric turbulence.

Figures 5-8 to 5-11 show open-loop responses of the 'Beaver' to atmospheric turbulence. The turbulence has been modelled as white noise, passing through Dryden filters, with Lg = 300 m, and a, = 1m/s. The coefficients of these filters are a function of the airspeed V, but for these plots, the values of the filter coefficients for V = 45 m/s were used. The initial altitude H, was equalled to 2000 ft in stead of the 6000 ft used in section 5.3, but this doesn't really matter for the evaluation of the responses.

It is not possible to compare the results from figures 5-8 to 5-10 rightaway with similar responses presented in ref.[24], because there are some significant differences in the definitions of the variables. Firstly, ref.[24] uses linear aircraft models, which express the motions of the aircraft in terms of small perturbations from an equilibrium condition. Furthermore, in ref.[24] the contributions of the turbulence velocity components to the angle of attack, sideslip angle, and true airspeed are treated as separate state variables (called a,, p, , and u,). Here these contributions are simply included in a, p, and V.

If we now compare figures 5-8 to 5-11 with responses given in ref.[24], we find some differences:

The frequency of the r-response to vertical or longitudinal turbulence (figures 5-8 and 5-10) is larger than the frequency of the p-response, contrary to ref.[24], where the frequency of the p-response was always larger than the frequency of the r-response. However, the responses to lateral turbulence (figure 5-9) are consistent with the results from ref.[24].

The influence of vertical turbulence to the cp-response (figure 5-10) and the amplitude of the high-frequency fluctuations in cp (figure 5-9) are both larger than in ref.[24].

The v, cp, and r-responses to longitudinal, vertical, and lateral turbulence, clearly show the influence of the spiral mode, whereas this influence is not so obvious in the results from ref.[24].

The frequency of the q-response to longitudinal and vertical turbulence is quite large, compared with ref.[24].

Most results, however, are consistent with ref.[24]. The longitudinal variables H and 0 clearly show the influence of the phugoid, which is initiated by longitudinal, vertical, and lateral turbulence. In figures 5-8 to 5-11, as well as in ref.[24], the amplitude of the phugoid tends to grow in time. As can be expected, the phugoid is most strongly excited by vertical turbulence, the longitudinal motion variables vary most strongly for vertical or longitudinal turbulence, and the lateral motion variables vary most strongly for lateral turbulence.