INTRODUCTION
The effects of atmospheric turbulence on many of the modern sophisticated transport
systems have become an important design parameter from both structural and performance
aspects. Aircraft encounters with turbulence represent a serious safety threat
for airlines. The problem of gusty winds proved itself to be a major obstacle
to successful flight. The history of aviation abounds with incidents and accidents
in which the variability of the wind in space or time played a decisive role.
Loss of control of the altitude or the flight path and even the crashes of jet
aircraft were not uncommon. Aside from the human catastrophes, annual injuries
to passengers and flight crew cost airlines millions in lost work time and medical
expenses (Prince and Robinson, 2001). Turbulence refers to an irregular or disturbed
flow in atmosphere that produces gusts and eddies. The most economic and practical
method to explore innovative concepts and to investigate configuration options
at an early stage is to first conduct an analytical and/or simulation study
using an appropriate engineering mathematical model of the relevant physics
(Buck and Newman, 2005). Often in an aircraft model simulation development,
the gust effects of the atmosphere are neglected for various reasons and removed
in the final form of the equations. Here, gust effects are the key excitation
of interest. The models of the wind have to accommodate both events that are
perceived as discrete (usually described as gusts), as well as the phenomenon
described as continuous turbulence. Discrete events are isolated encounters
with steep gradients (horizontal or vertical) in horizontal or vertical speed
of air. The discrete gust has evolved over the years from the isolated sharpedged
step function used in the airworthiness requirements to the currently favored
oneminuscosine. Static gust loadings are still determined by oneminuscosine
vertical gust velocity shape with the aircraft motion constrained to the plunge
mode only. Haddadpour and Shams (2005) showed that the linear model analysis
technique and linear qusisteady aerodynamic are still used for structure modeling
and aerodynamic modeling, respectively. Random turbulence is a chaotic motion
of air that is described by its statistical properties (Kim et al., 1999).
The main statistical features that need to be considered are: stationary, homogeneity,
probability distribution and correlations and spectra. The power spectral approach
offers a more realistic representation of the continuous nature of atmospheric
turbulent and it allows more rational consideration of design and operational
variations such as configuration changes, mission changes and airplane degrees
of freedom. The main object of this paper is to analyze the response of aircraft
under excitation of various types of turbulence atmosphere, based on statistical
and spectral technique. Five new linear dynamics models are developed to describe
the normal acceleration throughout an aircraft due to vertical gust effect.
To the best of authors knowledge, no attempts have been made to investigate
the effect of atmospheric turbulence on aircraft response with these various
models for the base line Convair CV880 jet transport aircraft model.
MATERIALS AND METHODS
Aerodynamic and Stability Derivatives Model: The Aircraft selected
as a model in this research work is the Convair CVM880 jet transport
operating at Mach = 0.86 and altitudes of 7005 m (23000 ft) and 10661
m (35000 ft).The airframe fixed coordinate system, dimensional aerodynamic
and stability derivatives influence coefficients of aircraft are access
to flight data test from the model original in accord with NASA convention
in USA, 1973 (Schmidt, 1998). The Convair CVM880 jet transport layout
can be shown in Fig. 1. Flight conditions and stability
derivatives of jet transport were illustrated in Table 1.

Fig. 1: 
Convair CV880M jet transport layout 

Fig. 2: 
Oneminuscosine discrete gust 
Table 1: 
Flight conditions and stability derivatives of Convair CVM880
jet transport flying at Mach = 0.86 in both conditions 

Aircraft Response Model to Discrete Gust: The idealized sharpedged
gust is a very severe type of a velocity profile that seldom occurs in
nature. Instead, a discrete gust may be modeled more practically by a
ramp input that reaches a peak value in a distance known as the gradient
distance. The one minuscosine model (Fig. 2) is more
frequently used in the determining gustinduced load factors rather than
a ramp rising to a steady peak gust. The aircraft response when interring
oneminuscosine gust is in the vertical (plunging) degree of freedom
mode only. The load factor for aircraft constrained to the plunging mode
can be obtained from Eq. 1. The full derivation can
be found in (Schmidt, 1998):
Δn_{z} (t) 
= 
Local load factor, 

λ 
= 
Time constant (in seconds) 
d 
= 
Gradient distance, 
W 
= 
Aircraft weight, 
S 
= 
Wing span, 
W_{g} 
= 
Vertical gust velocity and 
C_{lα} 
= 
Lift curve slope. 
The maximum load factor will occur near to the time for peak gust value.
Modified Aircraft Equations of Motion to Reflect the Gust Input: The
use of the short period dynamic model will provide an insight as to import
of increasing the airframe degrees of freedom when representing the airframe
dynamics. The simplified set of short period equations of motion can be
expressed as (Schmidt, 1998):

= 
Normal force due to angle of attack rate, 
Z_{α} 
= 
Normal force due to angle of attack, 
V 
= 
Aircraft velocity, 
q 
= 
Pitch rate, 
Z_{q} 
= 
Normal force due to pitch rate, 
Z_{δ} 
= 
Normal force due to elevator, 

= 
Pitching moment due to angle of attack rate, 
M_{u} 
= 
Longitudinal stability derivative, 
M_{q} 
= 
Pitch damping, 
δ 
= 
Control input and 
M_{δ} 
= 
Pitching moment due to elevator. 
The two coupled linear Eq. 2 will be restated as functions
solely of α and q by the use of algebraic substitutions, i.e.,
A further simplification can be made by recognizing that both
and Z_{q} are nearly zero in magnitude and most assuredly are negligible
when compared to the free stream velocity in the preceding equations. The short
period approximation in a commonly used becomes
In present study the longitudinal model (short period response) modified
to reflect gust inputs of α_{g }(t) and q_{g }(t)
in place of control inputs. The longitudinal equations of motion can be
written in state space form as follow^{ }:
Where,
The normal acceleration output is given by
Power Spectral Technique: The power spectrum represents a frequency
viewpoint for describing the square of random variable that is originally considered
in time domain. The original timevarying random signal or function x (t), shown
in Fig. 3a, is processed (or filtered) through a unit rectangular
filter, shown in Fig. 3b, to yield a truncated signal x_{T
}(t) that is zero when
T as shown in Fig. 3c and this signal is absolutely integrable
because is finite and the function is assumed to be bounded variation (McLean,
1990).

Fig. 3: 
Truncation of a random signal 
Hence
Consequently, that a Fourier transformation of X_{T}(t) exists
may be expressed as:
Since X_{T}(ω) in Eq. 6 is a complex quantity
whereas X_{T}(t) is a real quantity.
From Parseval`s theorem, which can, which can be described in the preceding
notation as:
The mean square expectation can be defined as:
The development of Power Spectral Density (PSD) follows from applying
Parseval`s theorem to Eq. 8 to obtain an alternate form
for the mean square that involves frequency dependent function, i.e.,
The limiting action on the integr and in the preceding expression leads
to the definition of the power spectral density,

Fig. 4: 
Load factor response to a vertical gust input 
Therefore the expectation for mean square may be described statistically
in term of frequency content by
Aircraft Response Models to Random Gust: The aircraft normal load
factor, in response to a turbulent vertical gust may be found by the series
application by Dryden vertical gust model`s transfer function (squared)
to the aircraft transfer function (squared) of normal load factor to vertical
gust input. This statement can be shown in Fig. 4. The
expectation of the normal load factor response is obtained by integrating
the power spectral density. The Dryden vertical gust model may be expressed
in a transfer function format as (Schmidt, 1998)
σ_{w} = Root mean square (rms) of stochastic gust.
The transfer function Gw_{g} (S) can be expressed in terms of
the Laplace transform variable as follows,
Where,
L_{2 } is the scale of vertical turbulence gust.
The aircraft longitudinal response is based on short period approximation
where it is noted that α = w/V and α_{g }=_{ }w_{g}/V.
The state variable form,
α_{n} = Normal acceleration, x = State variables
Where [A], [B], [C] and [D] are in accord with equation 4 whereas {X} = [wq]^{T}.
The transfer function of G_{nwg(S)} becomes
Which leads to PSD as:
Table 2: 
Model assumption 

PS: Power Spectral, θ_{o}:
Pitch angle, (rad), q_{g}: Pitch rate gust, body
axis, (rad sec^{1}) 
and the output PSD as:
Finally, the normal load factor is obtained by the integration of the
output power spectral density
In an attempt to understand the nature of atmospheric turbulent better,
to provide data through which mathematical modeling of turbulence may
be made and an improve means for treating the response of aircraft in
turbulent air, many experimental studies have been made to predict and
measure the vertical gust velocity in various circumstances, using aircraft
probing (such as NASA probe used in flight measurement of turbulence (Houbolt,
1973)^{ } and NASA B757200 research aircraft (Buck and Newman,
2005). In current study, the rms values of turbulent vertical gust (α_{w})
are detected experimentally from (Etkin, 1981).
Five new models with different gust excitation complexities are used in present
work. The assumptions used in each models are presented in Table
2. Models 1 and 5 are the simplest models for short period and Lyapunov
approach (Farrell, 1994). As shown, the value of
(part of aerodynamic damping) and the pitch rate gust signal q_{g }
are zero. Models 3 and 4 are considered the effects of
and ignore the pitch rate gust signal in two approaches while model 2, accounts
all gust penetration effects of the aircraft in short period response. The numerical
simulation model was built by using the MATLAB software.
RESULTS AND DISCUSSION
The maximum load factor determination is the primary purpose of the current
work to predict the aircraft response resulting from flight within a different
turbulence atmospheric environment in degrees of severity (moderate to
severe turbulence, usually the latter is storm related, such as thunderstorm).
The longitudinal equations of motion are modified to include the gust
effect. New models are developed to estimate the mentioned purpose. The
base line aircraft was taken into consideration in this analysis, Convair
CV880 jet transport, when operating at Mach 0.86 at altitude 7005 m (23000
ft) and 10661 m (35000 ft). The response of aircraft tested under two
categories of turbulence excitation, including discrete gusts (usually
1cos gust) as well as the phenomenon described as continuous turbulence.
The maximum load factor (Δn_{z})_{max} = 1.79 g at
time = 0.285 sec is found from the time history response to oneminusgust
(Fig. 5). The effects on gust response of degree of
freedom in present method can appear, with maximum load factor = (1.72
g) occurring at time = 0.252 sec. The addition of pitch angle rotation
to response model results in maximum load factor decreasing by
Fig. 5: 
Effects on gust response of degree of freedom for Convair
CV880 Jet transport at H = 7005 m (23000 ft): Mach = 0.86: W_{g}
= 21 m sec^{1} 
about 0.07 g after the startup transient has occurred. The other type of turbulence
under consideration in this study is a random turbulence, which was modeled
by an appropriate power spectral density. The transfer function approach was
applied here to determine the aircraft gust response based on short period approximation.
The area under the power spectrum curve represented the mean square of load
factor. The numerical estimation for σ_{w} (normalized input) and
σ_{w} (output) were determined by using trapezoidal integral approximation
for finite frequency range
The normalized gust transfer function Eq. 11 yielded to σ_{w}
= 0.382 m sec^{1} (1 ft sec^{1}) if ω_{max} were
infinite; however, the frequency truncation results in σ_{W} estimation,
which corresponds to 0.6% error. Figure 6a is a spectral representation
of Dryden vertical gust model when normalized to unit area. The aircraft normal
load factor transfer function due to vertical gust input shown in Fig.
6b with speak response value occuring near the short period frequency. Figure
6c represents the frequency distribution of aircraft
normal acceleration (product of G_{wg} (ω)^{2 }G_{wg}
(ω)^{2}). The three sigma value for aircraft normal load
factor estimation of 1.1 g with the probability of 99.7% does not exceed
this value when encountering a turbulent vertical gust at variance σ_{w}
= 6.1 m sec^{1} (Fig. 7).
To validate the numerical results, a comparison between the present work
and data in reference (Schmidt, 1998) was made to determine the load factor
for Lockheed jet transport when operating at Mach 0.75 and altitude 6092
m (20000 ft). The results show a good agreement with 1.6% error. These
verification results are shown in Fig. 8. Five new models
(previously discussed) for the aircraft acceleration response are excited
by vertical gust with different values of σ_{w}. The effects
of σ_{w} (rms) values of stochastic vertical gust upon load
factor (model 3 taken as an example for calculation) is illustrated in
Fig. 9. The results show that the load factor and σ_{w}
are directly related at different probabilities. Peak values of normal
acceleration for all models are presented in Fig. 10.
An alternate approach adopted in this study was the application of the
Lyapunov equation, which directly yielded
the mean square of the load factor (Farrell, 1994; Ogata, 1990), resulting
in small error when the variance is estimated. This is noted when the
values of load factor for models 1, 5 and 3, 4 is compared, respectively.
The most energetic responses are models 1 and 5. As expected, these models
exhibit higher frequency content due to the noncausal transfer function
structure resulting from noted assumption. Models 1 and 5 predict the
load factor with (23.5%) error compared with model 2 which considered
all gust penetration effects (Fig. 10).
Fig. 6: 
Aircraft spectral response resulting from a vertical gust:
H = 7005 m: Mach = 0.86, a) Dryden vertical gust input, b) Aircraft transfer
function, c) Aircraft normal load factor φ_{n} (w) for σ_{w}
= 0.328 m sec^{1} (1 ft sec^{1}) 
Fig. 7: 
Normal load factor estimation values for Convair 880M transport
(Model 3): H = 7005 m: Mach = 0.86: at different probabilities not exceeding
these values (load factor): σ_{w }= 6.1 m sec^{1} 
Fig. 8: 
Lockheed jet transport comparison results of peak normal load
factor at different probabilities of not exceeding these values (model 3):
H = 6092 m, Mach = 0.75:1.65% error in the estimation of load factor response 
Fig. 9: 
Effects of σ_{w} values of stochastic vertical
gust upon load factor (model 3) 
Fig. 10: 
Peak normal load factor for all suggestion models. Convair
CV880M, At H = 7005 m: Mach = 0.86: σ_{w} = 6.1 m sec^{1} 
Fig. 11: 
Effects of σ_{w} on peak normal load factor for
all models for CV880M transport with 99.7% probability not exceeding the
value of peak normal load 
Fig. 12: 
The load factor predication (model 3) at different values
of σ_{w} ( rms) of vertical gust at high altitude H = 10661,
m: Mach 0.86 
Figure 11 introduces the effect of σ_{w}
values on peak normal load factor. Other test conditions of aircraft at
altitude 10661 m (35000 ft) was made at various σ_{w} (rms)
according to probability not exceeding the predicted values of aircraft
acceleration. These results are presented in Fig. 12.
Finally, this study provides increased motivation to improve airplane
response in gust turbulence atmospheric by using modern optimal control
methods.
CONCLUSIONS
Methodology to estimate aircraft transient response resulting from flight
within turbulent atmosphere based on statistical and power spectral technique
is presented. Modified longitudinal equations of motion which includes
the effects of atmospheric gust are solved. The responses of aircraft
are tested under two categories of turbulence excitation input (discrete
and continuous random turbulence). Five linear dynamics models are developed
to describe the normal acceleration throughout an aircraft when it encounters
a vertical gust. The following conclusions have been obtained in the present
research:
• 
The numerical results show dependencies on which gust
excitation type and evaluation criteria are considered. 
• 
Models 1 and 5 exhibit higher frequency contents and give a rapid
estimation of normal load factor in case complete data are not readily
available. These models predict the load factor with (23.5%) error
compared with model 2 which considered all gust penetration effects. 
• 
It can be concluded that the agreement between the finite frequency
limit on integration of the spectral distribution and Lyapunov`s results
obtaining an estimate for the output deviation of load factor is well
within the accuracy. 