Thesis - Beaver simulation

Appendix A

137

where TBE= TBV= T ~ is ~the ' transformation~	matrix from FB to FE (see
definition in section 0.5.2). Usually, the aircraft's	altitude H is used in stead
of the coordinate ze .The following relation holds:
= -ze	(A-58)

A6 Summary of the state equations.

The equations of motion can thus be written as a set of first order ordinary differential equations, which is particulary useful for such purposes as simulation, linearization and steady-state trim [29]. The followingvariables will be used as states: V, a, p, p, q, r, 9 , 8, and cp. The set x , , y e , and H is added to this list, but contrary to the Euler angles 9 , 8, and cp, these variables are not necessary to solve the equations for the other state variables. In this appendix, the following state equations have been derived:

+ q - ( p cosa + r s i n a ) t a n a

(A-59b)

+(Y,+Yt + Y w ) c o s p - ( Z p , + Z t + Z w ) s i n a s i n p ] } + p s i n a - r c o s a

(A-59 ~ )

(J = q sincp	+ r- cos cp
cos e	cos e

x, =	(u,cos 0	+ (v,sincp + w ecos cp) sin 0) cosv	- (v, cos cp - w esincp) sin yl
3, =	(u,cos 0	+ (v,sincp + w ecoscp)sine) s i n v	+ (v,coscp - w esincp) CosV
2, =	-u,sin0	+ (v,sincp + w ecoscp) cos0

(A-62 )

Equations (A-53)and (A-54)express the forces from gravity and winds a s functions of states and (wind-) inputs, respectively. The other forces and moments can also be written in terms of states, inputsignals, and sometimes statederivatives, but contrary to the gravity and wind terms, these forces and moments depend on the aircraft considered. For this reason no expressions for aerodynamic and engine forces and moments are given here.

A7 Conclusions.

In this appendix, the equations of motion for steady and nonsteady atmosphere have been derived. The nonlinear state-space formulation that was used is particulary suited for simulation, linearization, and for trimming nonlinear aircraft models for steady-state flight. Aircraft velocity components along the body-axes have been replaced by the true airspeed, angle of attack, and sideslip angle, because these variables are commonly used within aerodynamic models. The equations in nonsteady atmosphere are very general, but it might be necessary to enhance the aerodynamic model with additional contributions to account for rapid changes of windspeeds, i.e. atmospheric turbulence, which may introduce aerodynamic lags due to the finite dimensions of the aircraft.

Appendix A

139

Appendix B

141

Appendix B. Mathematical models of wind and atmospheric turbulence.

B,1 Introduction,

For the assessment of the flying qualities of a n aircraft, its responses to external disturbances such as wind and atmospheric turbulence should be considered. It is therefore necessary to develop mathematical models of these phenomenae. This appendix will give a short description of wind(-shear) and atmospheric turbulence. See refs.[l] and [24] for more details.

B.2 Wind profiles and wind shear.

In this section, a distinction between wind and atmospheric turbulence will be made. Wind is defined here as 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 a few minutes. The remaining fluctuating part of the wind velocity is then the contribution of atmospheric turbulence.

The velocity and direction of the mean wind with respect to the ground usually is not constant along the flightpath. This variation of mean wind velocity and direction along the flightpath is called wind shear1). The influence of wind shear is particulary important during approach and landing, or take-off and climb. An idealized profile of the mean wind as a function of altitude is shown in figure B-1. More extreme wind profiles in lower atmosphere have been measured [I]and have sometimes resulted in accidents. A very serious type of wind shear is encountered in micmbursts, where large nose winds are followed by large tail winds in just a couple of seconds.

In ref.[l], a couple of idealized wind profiles for different atmospheric conditions are presented. Since this report limits itself to the use of the US Standard Atmosphere (see for instance ref.[27]), the standard temperature

lapse rate h = dT = - 0.0065 will be used. For this lapse rate, the wind pro-

dh

file can be represented by the following expression [I]:

Vwg.15is the wind speed a t 9.15 m altitude. The wind profile that is shown in figure B-1 uses VW9.=151 [m/s].

Sometimes, the local variations of the velocity of the atmophere with respect to the ground, including atmospheric turbulence, are noted a s wind shear. In this report, the term will be reserved for a description of the variations of the mean wind.

Figure B-1. Wind profile for h = -0.0065 [K/m] and V, = 1 [m/s].

9.16

This model of the earth's boundary layer wind profile is inadequate to represent the extreme wind profiles which can occur in the lower atmosphere. Thence, actual measurements of extreme wind profiles may have to be used for the assessment of automatic aircraft control systems.

The aircraft model presented in appendix A uses the wind velocities along the three body-axes. If the vertical wind velocity w, is assumed to be zero, we have the situation that is scetched in figure B-2. The wind direction with respect to the earth is denoted as vwand the wind velocity is denoted as V,. In this figure, the usual convention, with q, = 0' if the wind is directed southwards is used. For the components along the XBand YB-axes,we can now write:

If atmospheric turbulence is considered too, the turbulence velocities must be added to these wind velocity components.

B.3 Modelling atmospheric turbulence.

It is possible to create a mathematical model to describe atmospheric turbulence. For purposes where very high accuracy is required, it might be necessary to use actual measurements of atmospheric turbulence in stead of the models. The theory of stochastic processes provides a convenient means to describe atmospheric turbulence. Auto power density spectra form the basic elements of the resulting mathematical model. A number of alternative sets of these spectra can be found in the literature. They all require the selection of intensity levels and scale lengths, before they can be applied in simulations.

Appendix B

143

Figure B-2.Wind velocity components along the aircraft's body axes.

The following six assumptions concerning stochastic processes are often made when they are applied to atmospheric turbulence (ref.[l]):

Ergodicity, which means that time averages in the process are equal to corresponding ensemble averages. This assumption makes it possible to determine all required statistical properties related to a given set of atmospheric conditions from a single time history of sufficient length.

Stationarity, dealing with temporal properties of turbulence. If the statistical properties of a process are not affected by a shift in the time origin, this process is called stationary.

Homogeneity, dealing with spatial properties of turbulence. The statistical properties of homogeneous turbulence are not affected by a spatial translation of the reference frame.

Isotropy, which means that the statistical properties are not changed by a rotation or a reflection of the frame of reference. Complete isotropy implies homogeneity. Because of isotropy, the three mean-square velocity components are equal:

and the scale lengths for the turbulence velocities along the three body axes satisfy:

5 - Taylor's hypothesis of 'frozen atmosphere', which implies that gust velocities are functions of the position in the atmosphere only. During the short time interval in which the aircraft is under the influence of the velocities a t a certain point in the atmosphere, these velocities are assumed not to change with time. This hypothesis allows spatial correlation functions and frequencies to be related to correlation functions and frequencies in the time domain. The following relations are used:

and:

6.) = szv

Ax	:	distance between two points in space (in [m]),
V	:	velocity of the aircraft relative to the air (in [m/s]),
t	:	time taken by the aircraft to cover the distance Ax (in [s]),
51	:	spatial frequency (in [rad/m]),
(I,	:	frequency (in [rad/s]).

6 - Normality, meaning that the probability density function of each turbulence velocity component is Gaussian. With this assumption, the information of the covariance matrix only suffices for a total statistical description of atmospheric turbulence (ref.[24]).

Experimental data on atmospheric turbulence a t low altitudes, i.e., in the boundary layer a t the earth's surface, do not satisfy all these assumptions (ref.[l]). At low altitudes, due to the proximity of the ground, the assumptions of homogeneity and isotropy are not very valid. Both are affected by terrain roughness and the height above the ground. The assumption of stationarity is satisfied only over the short periods of time during which the meteorological conditions remain reasonably constant. Stationarity is also affected by the shape and roughness of the ground surface below the aircraft.

Taylor's hypothesis seems to be valid as long as the aircraft's velocity is large relative to the encountered turbulence velocities. For this reason it is somewhat doubtful that the hypothesis is fully valid when simulating the final approach and landing of SffTOL aircraft. Finally, measurements have provided mounting evidence that atmospheric turbulence is not perfectly Gaussian. The measured departures from a normal amplitude distribution are small, but pilots seem to be quite sensitive to these effects. Actual atmospheric turbulence possess what is sometimes called a 'patchy structure' [I].

In this report, assumptions 1to 6 will all be maintained, but the turbulence models might have to be enhanced in the future to assure a more accurate description of actual atmospheric turbulence, particularly for the simulation of aircraft approach and landing.

Appendix B

145

B.4 Power spectra of atmospheric turbulence.

B.4.1 The von K i i d n spectra.

Several analytical power spectral density functions have been obtained from measured data. The von Kiirmhn spectra functions yield spectra that seem to best fit the available theoretical and experimental data on atmospheric turbulence, particularly a t higher spatial frequencies [24]. The von E r m a n spectra for the three components of the turbulence velocity are:

The cross spectral density functions are zero in isotropic turbulence a t any point in space. Although this approximation is not very valid a t low altitudes, the cross covariances, and hence, the cross power spectral densities are usually neglected (ref.[l6]). The von Khrmhn spectrum yields a n asymptotic behaviour of S(S2) - RdI3 as R approaches infinity. See figure B-3.

Von Kinnh \

Figure B-3.Von KArmiin and Dryden spectra.

A: longitudinal, B: laterallvertical.

B.4.2 The Dryden spectra.

A major drawback of the von G r m a n spectral densities is that they are not rational functions of 52. For this reason, the Dryden power spectrum model is

often used for flight simulation purposes. The Dryden spectra are:

(B-11)

The von Kiirmiin and Dryden spectra are shown in figure B-3. The most obvious difference is the asymptotic behaviour a t large values of the spatial frequency, the former having a slope of -513 and the latter a slope of -2.

B.5 Filter design for atmospheric turbulence.

B.5.1 Modelling atmospheric turbulence as filtered white noise.

For simulation purposes, it would be practical to model atmospheric turbulence a s white noise passing through a linear, rational filter, see figure B-4. The relationship between the auto-spectral density of the output signal and the auto-spectral density of the input signal of a linear filter can be written as:

(B-13)

If the input signal u is white noise, its spectral density satisfies:

so for white noise relation (B-11) simplifies to:

Turbulence velocity

LlNEAR FlLTER (DRYDEN)

Figure B-4.Modelling atmospheric turbulence as filtered white noise.