Thesis - Beaver simulation

yields :

The static air pressure p, can be found by integrating (2-20), which yields:

which can be written as:

where:

po=air pressure on sea level ( p o = 101325 [ ~ / m ~ ] ) ,

R = specific gas constant (R I R, / M o , with Mo = 28.9644 [kg/kmol] = molecular weight of the air on sea level).

The acceleration of gravity g was held constant during this integration. It is actually necessary to replace the geometrical altitude h by the geopotential altitude H I), but the slight distinction between h and H will be neglected here, in view of the altitudes considered. In other words: we will assume that g = gothroughout the whole flight envelope of the aircraft.

The airdensity p (in [kg/m3]) is calculated fromp, and T using the ideal gas law (2-lg), which yields :

If scale effects are taken into account, the Reynolds number needs to be known. The Reynolds number with respect to c equals:

p v c

R, = -

P

and the Reynolds number per unit length (in [l/m]) is:

The coefficient of the dynamic viscosity p, which appears in the equations (2-31) and (2-32) can be calculated with Sutherland's equation 1271:

For solving the equations of motion for the 'Beaver', the dynamic pressure is needed to calculate the external forces and moments acting upon the airplane. This means that p, T, p, and qdynalways have to be calculated to solve the equations of motion. Inclusion of other variables may be necessary to calculate the external forces and moments for other aircraft. For the implementation within S I ~ L I N Kthese, equations will be logically subdivided in a number of 'atmosphere' and 'airdata'-groups, see appendix F.

2.2.5 Additional output equations.

It is possible to include a number of additional output equations to the state variables, state derivatives, forces and moments, and the atmosphere and airdata variables. Here, a list of normalized accelerations and specific forces will be included, a s well as some flightpath (-related) variables.

Accelerations and specific forces.

It is possible to calculate a number of different accelerations and accelerometer outputs, which are often important in the aircraft control analysis and design problem (consider for example turncoordination by applying feedback of the acceleration along the YB-axis,or manoeuvre load limiting). Here, accelerations in the vehicle's centre of gravity will be considered. The expressions are reproduced from ref. [l11.

The body axis acceleration vector a can be expressed as:

Finally, the azimuth angle x and the bank angle @ are added to the list of outputs, using the following equations:

2.3 Wind and atmospheric turbulence.

For the evaluation of aircraft control systems, it is necessary to include simulations of the aircraft responses to wind and atmospheric turbulence. In this section, only the most important expressions will be described. A more detailed treatment of wind and atmospheric turbulence can be found in refs.[l] and [24] and in appendix B of this report.

Wind and wind shear.

First of all, it is necessary to distinguish between wind and atmospheric turbulence. Wind is the mean or steady-state velocity of the atmosphere with respect to the earth a t a given position. Usually, the mean wind is measured over a certain time interval of several minutes. The remaining fluctuating part of the wind velocity is then defined a s atmospheric turbulence.

The wind velocity and direction with respect to the ground is usually not constant along the flightpath. This wind variation is called wind shear. Since the wind velocity varies most strongly as a function of altitude in the earth's boundary layer (H < approximately 300 m), it is particulary important to consider wind shear during the simulation of approach and landing or take-off and climb. Ref.[l] gives the following idealized equation of the wind velocity as a function of altitude:

0.2545 ( h r300m)

where VW9.is1Sthe wind speed a t 9.15 m altitude. More extreme wind profiles in lower atmosphere have often been measured [I] and have sometimes resulted in accidents. Actual measurements of extreme wind profiles can therefore play a n important role in the assessment of approach and landing or take-off and climb control functions of AACSs.

In equation (2-13), the wind velocity components along the aircraft's bodyaxes were needed. If the direction of the wind, measured relatively to the earth, is denoted a s qw,where ly, = 0° if the wind is directed southwards, the following transformation equations can be derived (see section B.2):

runway centerline in landing direction, ZF points downwards, and Y, points to the right, a s seen from a n approaching aircraft. The origin of the earth-fixed reference frame FE is chosen to coincide with the projection of the point of intersection of the runway treshold and the centerline of the runway (the origin of FF)on the horizontal plane a t sea level. The transformation from FEto FF then becomes:

where xf, y , and zf are the coordinates in the reference frame FF, Hf is the altitude o f t6e aircraft above sea level, HRWis the altitude of the runway above sea level, and qRWis the heading of the runway.

The ILS system provides guidance in two planes. The localizer reference plane is a vertical plane, passing through the runway centerline, which provides guidance in lateral direction. The instruments in the aircraft measure the angle between the line through the projection of the aircraft on the ground and the localizer antenna, and the runway centerline rhc,which is equal to:

where:

and:

Rhc is the ground distance from aircraft to localizer antenna, dbc is the lateral deviation of the aircraft from the localizer reference plane, and xl, is the distance from the runway treshold to the localizer antenna (measured along the XF-axis).

The glideslope reference plane, which is actually a cone, provides guidance in longitudinal direction. The line of intersection of the glideslope reference cone and the localizer reference plane should provide a straight glide path with a reference glide path angle of about 3'. In practice, however, the glideslope antenna is located a t some 300 metres besides the runway centerline, so the glideslope actually looks like a hyperbola. See section C.2 of appendix C for more details.

The angle eg, between the line through the aircraft and glideslope transmitter, and the reference cone is measured. If ygs is the reference glide

The VOR signals.

The VOR system is the standard short-range radio navigation aid. The VOR signal, received by the aircraft, depends linearly on the angle rVoRbetween some reference VOR-radian, called the Course Datum (CD) and the actual

bearing of the aircraft, denoted as QDR (a radio navigation term). The angle rVoRnow equals:

rvoR= CD - QDR

In this case, the coordinates of the aircraft, relative to the earth-fixed reference frame FEare xe ,y e , and ze (He). The coordinates of the VOR station, relative to FEare XVOR 9 YVOR ,and +OR (HVOR1-

The VOR signals have also a limited coverage, and can not be received accurately if the aircraft enters the cone of silence. See section C.3.2 of appendix C. In section C.3.3, some remarks about VOR steady-state errors are made.

2.5 Conclusions.

To simulate an aircraft for the evaluation of automatic control systems, mathematical models for all relevant (sub-) systems are needed. This chapter has presented a survey of these models1), starting with the general nonlinear equations of motion, models for the external forces and moments, and a large number of output equations. Furthermore, a description of wind and atmospheric turbulence has been added, so that it is possible to evaluate control systems in conditions where external disturbances affect the flight path of the aircraft. Finally, a model of VOR navigation and ILS approach signals has been treated. See appendices A, B, and C for more details.

Models for the Flight Control System and engine dynamics of the 'Beaver' will be treated in part I1 of this report.