Thesis - Beaver simulation

Chapter 2. Dynamic models of the 'Beaver', atmospheric turbulence, and navigation signals.

2.1 Introduction.

This chapter gives a survey of the mathematical models which will be used within the SIMULINKsimulation structure. These models include the aircraft dynamics, force and moment models, wind, atmospheric turbulence, and navigation signals (ILS approach and VOR navigation). The aircraft equations of motion are treated in detail in appendix A. Appendix B contains more information about atmospheric turbulence, and appendix C gives a more detailed description about the ILS approach system. The equations of motion are very general, but the forces and moments which act upon the aircraft depend on the characteristics of the aircraft itself. In this report, the models of the De Havilland DHC-2 'Beaver' will be used. In table 2-1 some basic information about this aircraft is summarized. See also figures 2-1 and 2-2.

2.2 Aircraft equations of motion.

It is possible to express the aircraft dynamics as a set of nonlinear ordinary differential equations (ODES).The state equations presented here are valid for rigid bodies. They express the motions of the aircraft in terms of external forces and moments, which can be subdivided in a number of categories. Only contributions to the forces and moments from aerodynamics, engine, gravity, and non-steady atmosphere will be considered here. In this section, the equations of motion will be presented along with all relevant force and moment equations and a large number of output equations of which some are needed to calculate these forces and moments.

2.2.1 General nonlinear equations of motion.

The derivation of the state equations is included in appendix A. There are six force and moment equations and six equations which determine the aircraft's attitude and position with respect to the earth. The translational equations are expressed in terms of true airspeed V, angle of attack a, and sideslip angle in stead of the body axes velocity components u, u , and w (see appendix A). The state equations for V, a, p, p, q, and r are valid only when the following restrictive assumptions are made:

1- the airframe is assumed to be a rigid body in the motion under consideration,

2 - the airplane's mass is assumed to be constant during the time interval in which its motions are studied,

3 - the earth is assumed to be fixed in space, i.e. its rotation is neglected, 4 - the curvature of the earth is neglected.

Notice that the aerodynamic forces in the V, a, and f3-equations are expressed in terms of aerodynamic lift, drag, and sideforce in stead of body-axes aerodynamic forces. All other forces and all moments are given in their body-axes

Table 2-1. General aircraft data of the DHC-2 'Beaver', PH-VTH.

Figure 2-1. The De Havilland DHC-2 'Beaver' laboratory aircraft.

reasonable assumption for the 'Beaver'). Also scale effects are not taken into account, since the effect of Reynolds number variations is considered to be negligible a t the relatively high Reynolds numbers that occur in flight. The flight condition for which the model has been determined is listed in table 2-2. Corrections are necessary if a different position of the centre of gravity is used,

Table 2-2. Aircraft data on which the aerodynamic model is based.

The aerodynamic model of the 'Beaver' can be written in terms of dimensionless body-axes force and moment coefficients [24]:

The values of the coefficients are listed in table 2-3. The dimensionless force and moment coefficients can be made non-dimensionless using the following equations:

xa = C, *'pv2s

a 2

Ya = c y a2 * l p v 2 s

za = cza*Lpv2s

2

and:

CY

Table 2-3 (i). Coefficients in the nonlinear aerodynamic model of the DHC-2 'Beaver', valid within the 35-55 m/s TAS-range

(continued on next page).

Table 2-3 (ii). Coefficients in the nonlinear aerodynamic model of the DHC-2 'Beaver' aircraft, valid within the 35-55 m/s TAS-range.

Often, aerodynamic forces are expressed in terms of lift, drag, and sideforce, rather than the body-axes components:

If this conversion is not made, the lift, drag, and sideforce in the state equations (2-1) need to be exchanged for the body-axes aerodynamic forces, using the inverse transform.

Engine forces & moments model for the DHC-2 'Beaver'.

The contributions from the engine to the external forces and moments, and the influence of changes in airspeed, can be expressed in terms of changes of

in front of the propellermand the total pressure behind the propeller. For the

The values of the coeficients are listed in table 2-4. Notice the absence of dynamics of the powerplant itself within these equations! Again we need to

to obtain the actual values of the engine forces and moments; hence:

(2-1l a )

and:

(2-1lb)

parameter value parameter value parameter value

Table 2-4. Coefficients in the nonlinear engine model of the DHC-2 'Beaver' aircraft, valid within the 35-55 mls TAS-range.