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viscosity, but instead it is transferred to smaller eddies, which in turn transfer their
energy to yet smaller eddies, and so on.
A major drawback of the von Kármán spectral densities is that they are not rational
functions of W, which greatly complicates analysis and computations for any
application. To overcome this problem, the simplified Dryden spectral density functions
were introduced:
Sugug (W) = 2su
2Lu
1
1 + (WLu)2 (4.11)
Svgvg (W) = sv
2Lv
1 + 3(WLv)2
(1 + (WLv)2)2 (4.12)
Swgwg (W) = sw
2Lw
1 + 3(WLw)2
(1 + (WLw)2)2 (4.13)
In figure 4.3 typical Dryden spectral density values have been compared with the
Von Kármán functions. The most obvious difference is the asymptotic behaviour
at large values of the spatial frequency, the former having a slope of −53
and the
latter a slope of −2. Although the Von Kármán form seems to fit the experimental
data somewhat better, both representations yield much the same results for aircraft
responses.
4.2.3 Filter design for atmospheric turbulence
For simulation purposes it would be practical to model atmospheric turbulence as
white noise passing through a linear, rational ‘forming filter’, as shown in figure 4.4.
The relationship between the auto-spectral density of the output signal y and the
auto-spectral density of the input signal u of a linear filter can be written as:
Syy(w) = |Hyu(w)|2Suu(w) (4.14)
where |Hyu(w)| denotes the amplitude response of the filter. If the input signal u is
white noise, its spectral density satisfies:
Suu(w) = 1 (4.15)
so for white noise, relation (4.14) simplifies to:
Syy(w) = |Hyu(w)|2 (4.16)
Unfiltered
 

white noise

Turbulence velocity

Linear Filter (‘coloured noise')

(Dryden)
Figure 4.4: Modelling atmospheric turbulence as filtered white noise
50 Chapter 4. External atmospheric disturbances
To apply these relations, the spatial spectral density functions of the turbulence velocities
must be transformed to functions of the circular frequency w. This can be
done, because Taylor’s hypothesis implies that the circular frequency w [rad s−1] encountered
at the aircraft’s center of gravity is related directly to the spatial frequency
W [rad m−1]:
w = WV (4.17)
The resulting transformation is given by:
S(w) = 1
V
S (W) (4.18)
The extra term 1/V in the spectral density function is due to the fact that we now
integrate over the time instead of over the distance in the Fourier transform.
The Dryden spectra were developed to approximate the von Kármán turbulence
spectra by means of rational functions. This makes it possible to apply relation (4.18)
for the generation of turbulence velocity components from white noise inputs. From
the definitions of the Dryden spectra in equations (4.11) to (4.13) and relation (4.18)
the following expressions are found:
|Hugw1 (w)|2 = 2su
2 Lu
V
1
1 +
􀀀
Lu
w
V

2 (4.19)
|Hvgw2 (w)|2 = sv
2 Lv
V
1 + 3
􀀀
Lv
w
V

2

1 +
􀀀
Lv
w
V

2
2 (4.20)
|Hwgw3 (w)|2 = sw
2 Lw
V
1 + 3
􀀀
Lw
w
V

2

1 +
􀀀
Lw
w
V

2
2 (4.21)
Solving equations (4.19) to (4.21) yields the following candidate functions for the
frequency responses of the forming filters:
Hugw1 (w) = su
r
2Lu
V
1
1 ± Lu
V jw
(4.22)
Hvgw2 (w) = sv
r
Lv
V
1 ±
p
3 Lv
V jw

1 ± Lv
V jw
2 (4.23)
Hwgw3 (w) = sw
r
Lw
V
1 ±
p
3 Lw
V jw

1 ± Lw
V jw
2 (4.24)
In these equations w1, w2, and w3 are independent white noise signals. Choosing the
minus sign in the denominators would lead to unstable filters and hence should be
rejected for physical reasons. And choosing the minus sign in the numerators leads
to non-minimum phase systems, so we shall use positive signs in both the numerator
and denominator [27].
The Laplace transforms H(s) are obtained by replacing the imaginary variable iw
by the more general complex variable s, and the resulting transfer functions can subsequently
be converted to state-space systems using the technique from section 6.4.2.
4.2. Stochastic disturbances 51
　
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