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Research Article
Aircraft Response to Atmospheric Turbulence at Various Types of the Input Excitation

Emad N. Abdulwahab and Chen Hongquan
Based on statistical and power spectral technique, methodology to predict the responses of aircraft in the atmospheric turbulence is presented. Modified longitudinal aircraft equations of motion to reflect gust inputs are solved. The responses of aircraft were tested under two categories of turbulence excitation, including the discrete and random disturbance atmosphere. Five new linear dynamics models of increasing gust excitation complexity are developed to describe the normal load factor throughout an aircraft due to vertical gust. Numerical model constructed for a Convair CV-M880 jet transport is solved to illustrate the results. 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 (2-3.5%) error compared with model 2 which considered all gust penetration effects. Finally, the results show a good agreement with the published work in load factor determination, at different probabilities not exceeding this value (load factor estimation) when encountering a turbulent vertical gust.
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Emad N. Abdulwahab and Chen Hongquan, 2008. Aircraft Response to Atmospheric Turbulence at Various Types of the Input Excitation. Space Research Journal, 1: 17-28.

DOI: 10.3923/srj.2008.17.28



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 sharp-edged step function used in the airworthiness requirements to the currently favored one-minus-cosine. Static gust loadings are still determined by one-minus-cosine 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 qusi-steady 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 CV-880 jet transport aircraft model.


Aerodynamic and Stability Derivatives Model: The Aircraft selected as a model in this research work is the Convair CV-M880 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 CV-M880 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 CV-880M jet transport layout

Fig. 2: One-minus-cosine discrete gust

Table 1: Flight conditions and stability derivatives of Convair CV-M880 jet transport flying at Mach = 0.86 in both conditions

Aircraft Response Model to Discrete Gust: The idealized sharp-edged 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- minus-cosine model (Fig. 2) is more frequently used in the determining gust-induced load factors rather than a ramp rising to a steady peak gust. The aircraft response when interring one-minus-cosine 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):

Δnz (t) = Local load factor,
  λ = Time constant (in seconds)

d = Gradient distance,
W = Aircraft weight,
S = Wing span,
Wg = Vertical gust velocity and
C = 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,
Zα = Normal force due to angle of attack,
V = Aircraft velocity,
q = Pitch rate,
Zq = Normal force due to pitch rate,
Zδ = Normal force due to elevator,
= Pitching moment due to angle of attack rate,
Mu = Longitudinal stability derivative,
Mq = Pitch damping,
δ = Control input and
Mδ = 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 Zq 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 αg (t) and qg (t) in place of control inputs. The longitudinal equations of motion can be written in state space form as follow :


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 time-varying 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 xT (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


Consequently, that a Fourier transformation of XT(t) exists may be expressed as:


Since XT(ω) in Eq. 6 is a complex quantity whereas XT(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)


σw = Root mean square (rms) of stochastic gust.

The transfer function Gwg (S) can be expressed in terms of the Laplace transform variable as follows,



L2 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 αg = wg/V. The state variable form,


αn = 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 Gnwg(S) becomes


Which leads to PSD as:


Table 2: Model assumption
PS: Power Spectral, θo: Pitch angle, (rad), qg: 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 B-757-200 research aircraft (Buck and Newman, 2005). In current study, the rms values of turbulent vertical gust (αw) 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 qg 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.


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 CV-880 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 1-cos gust) as well as the phenomenon described as continuous turbulence. The maximum load factor (Δnz)max = 1.79 g at time = 0.285 sec is found from the time history response to one-minus-gust (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 CV-880 Jet transport at H = 7005 m (23000 ft): Mach = 0.86: Wg = 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 σw (normalized input) and σw (output) were determined by using trapezoidal integral approximation for finite frequency range The normalized gust transfer function Eq. 11 yielded to σw = 0.382 m sec-1 (1 ft sec-1) if ωmax were infinite; however, the frequency truncation results in σW 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 |Gwg (ω)|2 |Gwg (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 σw = 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 σw. The effects of σw (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 σw 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 non-causal transfer function structure resulting from noted assumption. Models 1 and 5 predict the load factor with (2-3.5%) 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 φn (w) for σw = 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): σw = 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 σw values of stochastic vertical gust upon load factor (model 3)

Fig. 10: Peak normal load factor for all suggestion models. Convair CV-880M, At H = 7005 m: Mach = 0.86: σw = 6.1 m sec-1

Fig. 11: Effects of σw on peak normal load factor for all models for CV-880M 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 σw ( rms) of vertical gust at high altitude H = 10661, m: Mach 0.86

Figure 11 introduces the effect of σw values on peak normal load factor. Other test conditions of aircraft at altitude 10661 m (35000 ft) was made at various σw (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.


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 (2-3.5%) 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.
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Etkin, B., 1981. Turbulent wind and its effect on flight. J. Aircraft, 18: 327-345.

Farrell, B.F. and P.J. Ioannou, 1995. Stochastic dynamics of mid latitude atmospheric jet. J. Atmosph. Sci., 52: 1642-1656.
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