Statistics and Probability

Several definitions and concepts from statistics, probability theory, and linear algebra are collected in nearly alphabetical order [1-5], all of which might not have been used in this book; however, they will be very useful in general for aerospace science and engineering applications.


If x(t) is a random signal, then it is given as R^j) = E{x(t)x(t +t)}; here, t is the ‘‘time-lag,’’ and E is the expectation operator. For a stationary process, Rxx is not dependent on ‘‘t,’’ and has a maximum value for t = 0. This is then the variance of the signal x. With increasing time if Rxx shrinks, then it means that the nearby values of the signal x are not correlated and hence are not dependent on each other. The autocorrelation of the white noise process is an impulse function. The autocorrelation of discrete-time residuals is given as

1 N—t

Rrr(t) =———- r(k)r(k + t); t = 0, … , tmax (are discrete time lag)

iV 1 k=1


Bias is given as (8) = 8 — E(8), the difference between the true value of the parameter 8 and the expected value of its estimate. The estimates would be biased if the noise were not zero mean. The idea is that for a large amount of data used for estimation of a parameter, an estimate is expected to center closely on the true value, and the estimate is called unbiased if E{8 — 8} = 0. The bias should be very small.


Analytical expressions for temperature, density, and other parameters with altitude were discussed in Section A1. The variations in pressure and temperature with altitude in a standard atmosphere are also available in the form of Tables (see Refs. [1,3]).

The standard atmosphere is thought to be based on the average pressure, temperature, and air density at various altitudes [12]. This information is very useful for engineering calculations for aircraft. It shows what pressures and temperatures are to be expected at various altitudes. The standard atmosphere is based on mathematical formulas that relate the temperature and pressure as altitude is gained or reduced. The results are close to averages of balloon and airplane measurements at these altitudes. Table A2 uses metric units.

A table using U. S. units—altitude in feet, temperature in Fahrenheit, and pressure in inches of mercury is given in Williams (www. usatoday. com/weather/wstdatmo. htm [USATODAY. com]). It gives density in slugs per cubic foot because it uses the American system. People often use pounds per cubic foot as a measure of density in the United States; however, pounds are a measure of force, not mass. Slugs are the correct measure of mass, and one needs to multiply slugs by 32.2 for a rough value in pounds.


Standard Atmosphere Data

Height (m)

Temperature (°C)

Pressure (hPa)

Density (kg/i



















































































































































Once the baseline configuration of an aircraft is selected, the WT experiments are conducted on the (physically) scaled (down) models of the vehicle. WT testing mainly yields the aerodynamic coefficients. Static rigs are used to obtain the steady-state aerodynamic characteristics. The tests are conducted in low-/high – speed WTs. The effectiveness of various control surfaces is tested. To determine the dynamic derivatives, the forced oscillation test rigs are used. The aircraft’s physical model is oscillated and force/moment measurements are made. Rotary rigs are used to determine the characteristics at high AOA and AOSS.

To extend the force and moment coefficient measurements made in the WT for small-scaled models to full scale, it is necessary to match the similarity parameters, e. g., Reynold’s number, Mach number, and Froude number. The Froude number is

related to the ratio of inertia force to gravity force as mass *acceleratlon = YL which is a

mass * g dg

dimensionless quantity.

The matching of Reynold’s number is essential at low-speed tests while Mach number is matched for WT tests carried out at high speeds. Generally, it suffices to match these two similarity parameters for models held fixed within the WT. For a free-flight model, however, it becomes necessary to match the Froude number.


Derivatives from stability to body axes are given as follows:

Cn„ = CLa cos a + CDa sin a + Cc Cc = CD cos a — CL sin a — CN

t-a l-‘ a L^a 1 v

Cma (Cm,)s

Cib = (Ch)) cos a – (C„b)) sin a

Cip = (Clp)s cos2 a + (C„r)s sin2 a – (C^ + Ci)s sin a cos a Cib = (СіД cos a – (c,*) sin a

Cir = (C^ )s cos2 a – (C^ sin2 a – (C„r – Ci^s sin a cos a

Cid = (Cid )) cos a – (Cn )s sin a

Cnb = (C„b)) cos a + (C^ ) sin a

C„r = (Cnr )s cos2 a + (Cip )s sin2 a + (Cnp + Ci^ sin a cos a Cnb = ДпД cos a + (Cib^) sin a

Cnp = (C^ cos2 a – (Cir)s sin2 a – (Cnr – C^ sin a cos a

Cn = (Cn)) cos a + (Ci5)) sin a

Derivatives from body to stability axes are given as

Cl„ = Cn„ cos a – Cca sin a – Cd Cd„ = Cca cos a + Cn„ sin a + Cl

(Cma)) = Cma

(Cib) ) = Cib cos a + Cnb sin a

(Cir)) = Cir cos2 a – Cnp sin2 a + (Cnr – Cip) sin a cos a ДД = Ci^j cos a + Cnbb sin a

(Cip)) = Cip cos2 a + Cnr sin2 a + (Cir + Cnp) sin a cos a (Ci5)) = Ci5 cos a + Cn5 sin a (CnJ) = Cnb cos a – Cib sin a

(Cnr)) = Cnr cos2 a + Cip sin2 a – (Cir – Cnp) sin a cos a (Cnb) = Cnbb cos a – Cib sin a

(CnJ) = Cnp cos2 a – Cir sin2 a + (Cnr – Cip) sin a cos a (Cn5)) = Cn5 cos a – Ci5 sin a


SARAS aircraft (Figure A12), designed and developed by National Aerospace Laboratories, India, is a multirole light transport aircraft ideal for executive travel, air-ambulance, and surveillance. It is an unconventional configuration with clean wing and pusher propellers on aircraft fuselage mounted turboprop engines and is capable of operating from semiprepared and high-altitude runways. It has a T-tail with delta-shaped vertical fins to provide pitch recovery at high AOAs. A single slotted fowler flap is provided to meet the takeoff and landing requirements. Powered with P&WC PT6A-67A turboprop engine, it has a maximum cruise speed of 550 km/h. At present, with two prototypes flying, flight testing is underway to meet FAR-25 airworthiness standards.

HJT-36 (Figure A13) is an Intermediate Jet Trainer (IJT) aircraft under devel­opment by Hindustan Aeronautics Limited (HAL) for the Indian Air Force. It is slated to replace the aging HJT-16 Kiran aircraft, which is currently used by the Indian Air Force. It has a conventional jet trainer design, with a low, swept wing, staggered cockpits, and small air intakes on either side of its fuselage. It has manually operated flight controls with three-axes trim capability using electrical actuators. With an overall length of 11.0 m, a service ceiling of 9 km, and a



FIGURE A14 HANSA two-seater aircraft.

maximum takeoff weight of 4500 kg, the aircraft can achieve a maximum Mach number of 0.75.

HANSA-3 aircraft (Figure A14), designed by National Aerospace Laboratories of India, is an all-composite light trainer aircraft ideal for training, sport, and hobby­flying. It is certified under FAR-23 and is successfully flying in various flying clubs of India. Its main features include a neat cockpit with good visibility, dual controls, and turbo charged engines with a constant speed propeller. It has an overall length of

7.6 m, wing span of 10.47 m, and all-up weight of 750 kg with usable fuel capacity of 85 L. It has a maximum cruise speed of 96 knots.

Aircraft Tejas (Light Combat Aircraft – LCA; Figure A15) is designed and devel­oped by Aeronautical Development Agency (ADA), India, with an aim to replace the aging MiG-21 aircraft as the Indian Air Force’s primary multirole tactical aircraft. It is a single engine tail-less delta wing aircraft designed to be aerodynamically unstable in the longitudinal axis. The wing supports three-segment leading edge slats


and two-segment elevons that span the entire length of the trailing edge. The elevons are the primary control surfaces in pitch and roll. In order to stabilize the airframe and achieve the desired performance over the entire flight envelope, it incorporates a quad redundant full authority fly by wire (FBW) flight-control system (FCS). Its takeoff weight is around 8500 kg and has a maximum speed of 1.8 Mach.


Aircraft have several instruments that aid the pilot in flying. These can be categorized according to their use, e. g., flight instruments, engine instruments, navigational instruments, environmental instruments, electrical system instruments, and so on. Some of the instruments that fall in the category of flight instruments are discussed in Refs. [1,11]. Sensors/transducers are used to measure control positions, pressures, temperatures, and loads. The electrical signal from each transducer is routed through an instrumentation wiring to a signal conditioning circuit or device in the airplane. The signal conditioning would include multiplexing, commutating, digitizing,

ADC/DAC conversions, time-code generation, and pulse code modulation. Optical head-up display allows the pilot to observe the flight instrument information pre­sented in the head-up form as well as the out-of-window view of the world.

Air Data Instruments: These instruments use a pitot-static system to measure ambient atmospheric pressure. Ambient or static pressure is measured through a set of holes provided on the side of a tube projecting into the free stream, typically on both sides of the fuselage (see Figure A1). To minimize the errors in the measured static pressure, the sensor is placed clear of the disturbed airflow. The pitot or the total pressure is sensed by the open end of the pitot-static tube, which is directly facing the oncoming airflow. The pitot pressure is the sum of static and dynamic pressures, the latter caused by the forward movement of the aircraft. Nose boom allows measurements of pressure and flow angles far in front of the fuselage. For flow angles the differential pressures are measured in both the axes: horizontal and vertical relative to the aircraft. These differential pressures along with the dynamic pressure (at the aircraft nose boom) are used and the flow angles are computed. In many situations five-hole probes are employed to measure 3-D flows. These probes can determine pitch and yaw angles also. Interestingly, a simple relation exists between yaw coefficient and measured pressures: Cyaw = PPl_Pp> . . From this coef­ficient yaw angle can be computed. However, in reality algorithms to accurately compute various such quantities would be complicated.

Pressure Altimeter: This is used to measure the aircraft altitude. It is basically a pressure transducer calibrated to read height above a specific pressure datum. It measures ambient pressure and indicates the value of the altitude on the instrument dial. The altimeter readings can become inaccurate at higher altitudes where the pressure change with a change in height is small.

Air Speed Indicator: The air speed indicator measures the dynamic pressure, which is computed as the difference between the pitot (total) and static pressure and converts it to airspeed. As the airspeed indicator is usually calibrated to standard sea – level conditions, the speed indicated is called IAS. The IAS corrected for the instrumentation errors and compressibility effects gives the EAS. The TAS can be obtained from the EAS by accounting for the change in the air density from standard sea-level conditions (see Equation A.11). The compressibility and density correc­tions are normally included in the navigation computer from which CAS and TAS can be found.

Mach Meter: This instrument comprises both the pressure altimeter and the air speed indicator. Mach number is defined as the ratio of TAS to local speed of sound (see Section A1). In Mach meter, the altimeter and ASI capsules are linked to the pointer of the Mach meter through links and gears. The pointer rotates on a dial indicating the airspeed in terms of Mach number.

Vertical Speed Indicator: This is also known as the rate of climb/descent indicator. It indicates the rate of change of height per minute as the aircraft is climbing or descending. It essentially senses the pressure difference between the pressure in the instrument casing and the static pressure inside the capsule, due to a change in height. The capsule is connected to a pointer, which moves on the dial. In level flight, the pressure in the instrument casing is equal to the static pressure in the capsule and the pointer will indicate zero.


FIGURE A12 SARAS transport aircraft.


A few important aspects of flight vehicle performance are mentioned here.

The drag force acting an aircraft is typically a function of the lift coefficient and Mach number.


Cd = f (Cl, M)


The equation for aircraft drag polar, given by Equation A.22, can be expressed as

Cd = CDo + KC2l (A.38)

where K can be estimated from flight data using parameter-estimation techniques. The theoretical value for the drag-due-to-lift factor K at subsonic speeds is given by 1/pAe, while its value at transonic and supersonic speeds is close to 1/Cl.

Assuming thrust T to be inclined at angle sT with the flight path direction, the equations of motion for a level (U = 0) unaccelerated flight can be expressed as


T cos sT = D T sin sT + L = W




For small values of sT, Equation A.39 becomes


T = D L=W




Rearranging Equation A.40, the required thrust for an unaccelerated level flight is given by








Thus, minimum thrust is required where L/D is maximum.

If V is the velocity of the aircraft during level unaccelerated flight, and the lift is equal to weight, we have


I 2W VpScL



V pSC3l





FIGURE A11 Airplane in a steady climb.

Another important parameter in aircraft performance is the rate of climb. Compared to the level unaccelerated flight, the thrust in this case, in addition to overcoming drag, will also be required to compensate for the component of weight (Figure A11).

T = D + W sin U

The vertical velocity component gives the rate of climb, as shown in Figure A11.

Rate of climb = V sin U

The topic of performance is incomplete without the mention of engines used for propulsion. The piston engines with propellers still rule the roost in low-speed flights while gas turbine engines are used for jet propulsion at higher speeds. Turbo jets are equipped with a compressor, a combustion chamber, turbine, and exhaust nozzle. Thrust is provided by reaction of the exhaust gases thrown backward through the nozzle. Another type is the turbofan or the bypass engine, which has a large fan that accelerates the air ahead of the compressor, thus resulting in more thrust with higher efficiency. While turbojet and turbofan engines have rotating parts, there is another type called the ramjet, which uses the ram effect generated from the forward speed to pass the compressed air to the combustion chamber and out of the exhaust nozzle at very high speeds. As is obvious, ramjet will give no thrust at zero forward speeds.


Neutral and maneuver points are important longitudinal stability parameters, which critically determine the aft CG limit of an aircraft. Since these parameters are a function of speed, AOA, external store configuration, control surface deployment (slats), etc., extensive flight tests are conducted to accurately determine these critical stability parameters. Existing methods based on steady-state trim flights turn out to be time-consuming and are error-prone because the results are dependent on air data and aircraft weight. An alternative approach, based on system theoretic concepts, is to estimate aircraft longitudinal static and dynamic stability from flight dynamic maneuvers in terms of neutral and maneuver points. In this procedure, the stability information is extracted from the short-period dynamic response of the aircraft, which leads to substantial reduction in flight test time compared with the conven­tional steady-state flight tests. Since this method does not use air data information or mass/inertia data, the resulting estimates of neutral and maneuver points are gener­ally more accurate [9]. The neutral point is related to short-period static stability parameter Ma and hence the natural frequency (see Chapter 4). We estimate Ma values (using the parameter-estimation method) from short-period maneuver flight data of an aircraft, flying it for different CG positions (minimum three), and then plot with respect to CG, and extend this line to the x-axis. The point on the x-axis when this line passes through ‘‘zero’’ on the y-axis is the neutral point (see Figure A10). The distance between the neutral point NP and the actual CG position is called the static margin and when this margin is zero, the aircraft has neutral stability.

Подпись: 0Подпись: CG, CGПодпись:Подпись: CGПодпись: '2Подпись:image221Ma


If the sum of all the resultant forces and moments acting on the aircraft is zero about its center of gravity (CG), the aircraft is in a state of equilibrium. In this condition, the aircraft is said to be flying at trim condition in a steady uniform flight. In this state, the translational and angular accelerations are normally zero.

An aircraft is said to be stable if, on encountering a disturbance during steady uniform flight, it is able to come back to its initial trim state. There are two types of stability—static and dynamic stability. An aircraft possesses positive static stability if it tends to return to its equilibrium position after disturbance. An example to illustrate this point is shown in Figure A8. The ball in position 1 is stable as it will return to its equilibrium after disturbance. The ball in position 2 is unstable as any disturbance will cause it to roll off. In position 3, the ball possesses neutral stability because, being on a flat surface, any disturbance will cause it to seek a new position. Dynamic stability, on the other hand, deals with the time history of the aircraft motion. After a disturbance, the aircraft may tend to move back because of its static

Position 2


CG >NP (static instability)


FIGURE A9 Longitudinal static stability as affected by CG location.

stability and settle down either monotonically or after a few oscillations. In either case, if it goes back to its equilibrium position on its own over a period of time, it is said to be dynamically stable. It should be noted that a dynamically stable aircraft must always be statically stable. On the other hand, static stability does not auto­matically ensure dynamic stability of the aircraft. Dynamic stability is very important for flight-control system design.

From the above explanation it is clear that if the AOA increases, the aircraft should develop a negative pitching moment so that it has the tendency to return to its equilibrium position. In other words, for an aircraft to possess static longitudinal stability, the slope of the pitching moment curve should be negative, i. e.,

m < 0 or C-a < 0 da

In addition to a negative pitching moment slope, the aircraft must also have a positive pitching moment at zero AOA to achieve trim at positive angles of attack, i. e., Cm0 should be positive (see Figure A9).

The value of Cm depends on the location of the CG. The point for which Cm = 0 defines the aft limit of the CG. At this point, the aircraft possesses neutral stability. This location of CG is called the neutral point. Estimation of neutral point from flight test data is discussed in Section A9.

Controllability is the ability to control the aircraft to maneuver and change the course of the flight path. Controllability depends on the pedal/stick force and the displacement needed to achieve the task. The faster the aircraft response to the force and displacement, the higher is the controllability.

that can degrade the aircraft handling qualities. This interaction between the con­troller and the aircraft rigid body dynamics is known as aeroservoelasticity [8]. This problem can be circumvented by designing filters that eliminate the higher-frequency structural modes from the response. Normally, ground resonance and structural coupling tests are carried out on ground to determine the degree of coupling between airframe and controller dynamics. The mathematical model to study aeroservoelastic effects should account for not only the rigid body dynamics, but also the primary flexible modes of the aircraft as well as the unsteady aerodynamic effects.

Transformation from Stability Axis to Body Axis

The origin of the body and stability axes is the same and the only rotation is the inclination a of that of the XZ plane of the stability axis with the body axis. By putting C = f = 0 and U = a in the elements of matrix T in Equation A.28, one obtains the matrix TSB to transform from stability axis to body axis

cos a


—sin a






sin a


cos a

Transformation from Stability Axis to Wind Axis

This transformation is given by the following matrix

cos b

— sin b



sin b

cos b





Transformation from Body Axis to Wind Axis

The origin of wind and body axes is the same and the two axes system are related through the flow angles a and b – Substituting C = b, U = a, and f = 0 in the elements of matrix T in Equation A.28, one obtains the matrix TWB to transform from body axis to wind axis-

Подпись: cos a cos b sin b sin a cos b TBW = —cos a sin b cos b —sin a sin b —sin a 0 cos a (A.37)

Thus, using the transformation matrix T defined in Equation A.28 and knowing angles C, U, f, one can obtain a transformation from one axis to another.

A6 aerodynamic derivatives—preliminary determination

The preliminary estimates of the derivatives are generally obtained by using methods like data compendium/handbook methods (DATCOM) and computational fluid dynamics (CFD). These methods are valid for low AOA and subsonic/supersonic Mach number regions. At high AOA, the effects of flow separation, boundary layer flow, vortex flow, and shock intensity and location at transonic Mach number complicate the accurate determination of these derivatives. For these conditions, the data from the WT experiments and flights of similar aircraft are used in conjunction with the handbook methods (DATCOM). Subsequently or concurrently, the derivatives are determined from suitable WT experiments. Further refinements can be made using advanced CFD codes. The estimates thus obtained are employed for studies related to the selection of the baseline configurations for aircraft and missiles. These estimates are generally known as predicted values (or initial refer­ence values) of the aerodynamic derivatives or simply the ‘‘predictions.’’ After conducting several WT tests, the preliminary estimates are refined and the vehicle configurations are optimized for obtaining the desired performance and controllabil­ity and used for studies on load estimation. Subsequently, the prototype vehicle is built and preliminary flight tests are conducted. One of the purposes of these flight tests is to estimate the aerodynamic derivatives from the flight data generated by conducting certain specific maneuvers on the vehicle (usually aircraft and rotor – crafts). These aspects are discussed in Chapters 7 and 9. Mathematical models

(specifically the aerodynamic derivatives or stability and control derivatives, which occur in the equations of motion) of aircraft dynamics are generally available from WT experiments and analytical methods before flight tests. Due to extensive WT testing and progress in aerodynamics and system technologies, reasonably accurate mathematical representations (models) are available. As a result of this, satisfactory characteristics, validated through extensive flight simulation, could be designed into the aircraft prior to flight. This has given more weightage to the model verification exercise to be performed along with pilot’s assessment. The vehicle’s dynamical characteristics are described by equations of motion and parameters that have physical meaning. These parameters are to be estimated from flight test data. These estimated parameters are compared with those obtained from WT experiments and analytical methods (e. g., CFD/DATCOM—all put together as prediction methods). Flight-determined stability and control derivatives (FDD) are also used in handling quality criteria (see Chapter 10) to assess the overall pilot-aircraft interactions and performance. Hence, Taylor series (expansion) of aerodynamic coefficients is generally found very useful in representing the stability and control derivatives.