solve step by step this question with other formulas "
این مثال مربوط به یک هواپیمای پروانه‌ای است با ویژگی‌های تراست شکل مسافت و زمان برخاست را برای شرایط بدون باد و بدون شیب باند محاسبه کنید. فرض کنید که شرایط استاندارد در سطح دریا برقرار است. داده‌های زیر برای این هواپیما در دسترس هستند:

W\ =\ 4,600\ lbs
\left(\frac{t}{C}\right){\mathrm{ave}}=0.14
V_w=0
\ \ \phi=0
\delta f={15}^\circ
S=175\mathrm{\ }{\mathrm{ft}}^2
b=35\ ft
\ \ A=7
\mu_g=0.03
C{L_{\mathrm{max}}}=1.69
\ e=0.80\ in\ free\ air
حل:
داده‌های داده‌شده:

وزن W : 4,600 lbs
نسبت میانگین Thrust به Weight (\left(T/W\right)_{\mathrm{ave}}): 0.14
سرعت باد (V_w):\ 0 (بدون باد)
شیب باند (\phi): 0° (باند صاف)
انحراف Flap (\delta_f): 15°
مساحت بال (S): 175 ft²
طول بال (b): 35 ft
نسبت Aspect (A): 7
ضریب اصطکاک زمین mu_g : 0.03
ضریب Lift حداکثر C_{L_{max}} : 1.69
فاکتور کارایی Oswald\ (e) : 0.80
مراحل حل:

تعیین ضریب بالابری در لحظه برخاست (C_{L_{\mathrm{OF}}})

C_{L_{\mathrm{OF}}}=\frac{C_{L_{\mathrm{max}}}}{\left(\frac{V_{\mathrm{LOF}}}{V_S}\right)^2}=\frac{1.69}{{1.15}^2}=1.28

ضریب بالابری در طول حرکت بر روی زمین (خارج از اثر زمین)

[C_{L\left(\mathrm{out\ of\ ground\ effect}\right)}=0.83]ارائه شده

ضریب بالابری در طول حرکت بر روی زمین (در اثر زمین)

C_{L_g}=0.89
C_{L_g}=\frac{C_L}{1+\frac{b}{2h}}
ضریب درگ در طول حرکت بر روی زمین (خارج از اثر زمین)

C_D=0.0620+\frac{C_{L\left(\mathrm{out\ of\ ground\ effect}\right)}^2}{\pi A e}=0.0620+\frac{{0.83}^2}{\pi\times7\times0.80}=0.1012

محاسبات اثر زمین

ارتفاع از خط میانه کورد هندسی، (\ h\ =\ 3.6\ ) فوت

\frac{h}{b}=\frac{3.6}{35}=0.103
[\ \sigma=\ 0.48\ ] (عامل اثر زمین از شکل)

تغییر در ضریب درگ به علت اثر زمین،

\Delta C_{D_i}=-\sigma\frac{C_{L_g}^2}{\pi A}=-0.48\frac{{0.89}^2}{\pi\times7}=-0.0150

ضریب درگ در اثر زمین،

C_{D_g}=C_D\left(\mathrm{out\ of\ ground\ effect}\right)+\Delta C_{D_i}=0.1012-0.0150=0.0862

تعیین (C_D-\mu_gC_L)

[C_D-\mu_gC_L=0.0862-0.03\times0.89=0.0595]

سرعت استال \left(V_S\right)

V_S=\sqrt{\frac{2W}{\rho S C_{L_{\mathrm{max}}}}}=114.4\mathrm{\ فوت بر ثانیه=67.8 گره

سرعت برخاست (V_{\mathrm{LOF}})

[V_{\mathrm{LOF}}=1.15V_S=1.15\times114.4=131.6\mathrm{\ فوت بر ثانیه=77.9 گره]

فشارهای دینامیکی

فشار دینامیکی در لحظه برخاست،

[q_{\mathrm{LOF}}=0.5\rho V_{\mathrm{LOF}}^2=20.6\mathrm{\ پوند بر فوت مربع]

سرعت چرخش،

[V_R=1.10V_S=125.8\mathrm{\ فوت بر ثانیه=74.5 گره]

فشار دینامیکی در سرعت چرخش،

[q_R=0.5\rho V_R^2=18.8\mathrm{\ پوند بر فوت مربع]

نیروی پیشران هنگام برخاست

نیروی پیشران در (V=0)،(TV=0=2,000 پوند)

نیروی پیشران در (V=V_R)،(TV=VR=1,200 پوند)

شتاب‌های برخاست

شتاب در طول حرکت بر روی زمین (شرایط نیروی پیشران اولیه)،

[a_{g_{V=0}}=g\left[\frac{T}{W}-\mu_g\right]-\frac{C_{D_0}-\mu_gC_L}{q_0S/W}=12.9\mathrm{\ فوت بر ثانیه2]

شتاب در طول حرکت بر روی زمین (شرایط نیروی پیشران هنگام چرخش)،

[a_{g_{V=V_R}}=g\left[\frac{T}{W}-\mu_g\right]-\frac{C_D-\mu_gC_L}{q_RS/W}=6.1\mathrm{\ فوت بر ثانیه2]

فاصله رول روی زمین \left(S_{\mathrm{NGR}}\right)

نسبت شتاب‌ها،

[k=\frac{a_{g_{V=V_R}}}{a_{g_{V=0}}}=0.7]

شتاب میانگین،

[a_{g_{\mathrm{ave}}}=a_{g_{V=0}}\times\frac{1+k}{2}=9.1\mathrm{\ فوت بر ثانیه2]

فاصله رول روی زمین،

[S_{\mathrm{NGR}}=\frac{V_R^2}{2a_{g_{\mathrm{ave}}}}=870\mathrm{\ فوت]

فاصله چرخش (S_R)

[S_R=V_R\times t_{\mathrm{rotate}}=125.8\times1=129\mathrm{\ فوت]بافرض(trotate=1 ثانیه)

محاسبات انتقال

تغییر در ضریب بالابری در طول انتقال،

[\Delta C_L=\frac{1}{2}\left[\left(\frac{V_{\mathrm{LOF}}}{V_S}\right)^2-1\right]\left[C_{L_{{\mathrm{max}}{\mathrm{TO}}}}\left(\frac{V_S}{V{\mathrm{LOF}}}\right)^2-0.53+0.38\right]]

شعاع انتقال،

[R_{TR}=2\left(\frac{W}{S}\right)\frac{1}{\rho g\Delta C_L}]

محاسبات صعود

ارتفاع صعود در طول انتقال،

[h_{TR}=R_{TR}\times\left(1-\cos{\theta_{CL}}\right)=47\mathrm{\ فوت]

ارتفاع صفحه نمایش،

[S_{CL}=\frac{h_{\mathrm{screen}}-h_{TR}}{\tan{\theta_{CL}}}=23\mathrm{\ فوت]

فاصله کل برخاست \left(S_{\mathrm{TO}}\right)

[S_{\mathrm{TO}}=S_{\mathrm{NGR}}+S_R+S_{TR}+S_{CL}=870+129+732+23=1,754\mathrm{\ فوت]

محاسبات زمان برخاست

زمان برای رول روی زمین،

[t_{\mathrm{NGR}}=\frac{V_R}{a_{g_{\mathrm{ave}}}}=15.4\mathrm{\ ثانیه]

زمان برای چرخش،

[t_R=1\mathrm{\ ثانیه]ارائهشده

زمان برای انتقال و صعود به ارتفاع صفحه نمایش،
t_{TR}=\frac{S_{TR}}{V_{LOF}}
t_{CL}=\frac{2S_{CL}}{V_{LOF}+V_2}\emsp

[t_{TR}=5.6\mathrm{\ ثانیه]
[t_{CL}=0.2\mathrm{\ ثانیه]
زمان کل برخاست،

[t_{\mathrm{TO}}=t_{\mathrm{NGR}}+t_R+t_{TR}+t_{CL}=15.4+1+5.6+0.2=22.2\mathrm{\ ثانیه]
" solve again with this foemulas "The determination of the RTO distance on a wet runway is determined in the same way as that for a dry runway, with one notable difference: credit for the use of reverse thrust is permitted.
9.3 Forces Acting on the Airplane During the Ground Run
0.0.1. Vector Diagram and Overview of the Forces Acting on an Airplane
The forces that can act on an airplane during takeoff are shown in Figure 9.5. The runway is assumed to have an uphill slope \ ^4 of angle \gamma_G and to be clear of standing water or snow or icy slush that would impede the airplane's acceleration. (The special case of airplanes operating from contaminated runways is described in Section 9.9.) Resistance to forward motion also comes from the aerodynamic drag, rolling friction of the tires on the runway, and the component of the airplane's weight acting parallel to the runway. Opposing these forces is the thrust of the engines \left(F_N\right), which will cause the airplane to accelerate along the runway.
Before a general equation for the takeoff run can be determined, it is necessary to first identify and describe the forces that act on the airplane. The following factors are important: wind (Section 9.3.2), engine thrust (Section 9.3.3), lift and drag forces (Section 9.3.4), airplane configuration (Section 9.3.5), ground effect (Section 9.3.6), and rolling friction (Section 9.3.7).

Figure 9.5 Forces acting on an airplane during the ground run.
0.0.2. Influence of Wind
It is usual that the takeoff direction is into the prevailing wind (i.e., giving a headwind rather than a tailwind) unless the winds are slack and are of little consequence. It is customary to resolve the prevailing wind vector (at the height of the airplane) into crosswind and headwind/tailwind components (see Section 4.5.2). The crosswind component has a negligible influence on the takeoff distance (despite the fact that the pilot may have to use the rudder and ailerons to counter the tendency for the airplane to turn into the wind). The component of wind acting along the runway is herein designated as V_w, which is defined as positive for a headwind and negative for a tailwind. \ ^5

Figure 9.6 Effect of wind on ground speed.
The ground speed \left(V_G\right), which is the time rate of change of ground distance (s), is illustrated in Figure 9.6. It is defined as follows:
VG=ds dt=V-Vw#(9.1)
where V is the true airspeed (typical units: m/s,ft/s ); and
V_W is the wind speed along the runway, defined as positive for a headwind
(typical units: m/s,ft/s ).
0.0.3. Engine Thrust
In general, an engine's thrust depends on the engine setting, bleed air demand, atmospheric conditions (air temperature and pressure), and airspeed (see Section 8.3). For a given thrust setting (e.g., maximum takeoff thrust) and atmospheric conditions, the thrust will decrease during the ground run as the airspeed increases; this thrust-speed relationship depends on the powerplant type. The following expression, which follows directly from Equation 8.42, has been shown [8] to provide a reasonably accurate representation of jet engine thrust at low speeds:
FNFN0=1-k1V+k2V2#(9.2)
where F_N is the thrust (typical units: N,lb );
F_{N_0} is the static thrust (typical units: N,lb ); and
k_1 and k_2 are positive empirical constants selected to match the engine data.
Although Equation 9.2 provides an acceptable representation of the takeoff thrust, it, unfortunately, leads to a mathematical description of the takeoff ground run for which there is no closed-form analytical solution. To facilitate the development of analytical expressions for the ground run, it is necessary that the thrust be described by simpler expressions, as discussed later in Section 9.4.
0.0.4. Lift and Drag Forces
From the definitions of the lift and drag coefficients, given by Equations 7.1 and 7.5, respectively, the lift and drag forces can be expressed as follows:
L=qSCL=12ρSCLV2#(9.3)D=qSCDg=12ρSCDgV2#(9.4)
where L and D are the lift and drag forces, respectively (typical units: N,lb ); q is the dynamic pressure (typical units: N/m^2,lb/{\rm ft}^2 );
C_L and C_{D_g} are the lift and drag coefficients, respectively (in ground effect); \rho is the ambient air density (typical units: kg/m^3, slug /{\rm ft}^3 ); and
S is the wing reference area (typical units: m^2,{\rm ft}^2 ).
When the airplane accelerates down the runway during takeoff, it does so at a nearly constant angle of attack up to the point of rotation. Changes in the angle of attack can only result from differential extension or contraction of the nose and main landing gear legs. The landing gear will extend a little as the lift builds up over the wing and this may result in an angle of attack change, but the magnitude of this change is negligible. For this reason, C_L can, for all practical purposes, be considered constant during the takeoff, up to the point of rotation.
The drag coefficient will also be essentially constant. This is evident from Equation 7.14: C_D is a function of C_L (which is constant), and neither C_{D_0} nor K will change during a typical ground run (as the airplane's configuration does not change). During the ground run, it is generally advantageous to have a low angle of attack for as long as possible. Although the resistance due to the rolling coefficient of friction would decrease if the angle of attack were increased (generating more lift), this effect is outweighed by the increase in lift-induced drag.
The lift and drag forces acting on the airplane thus increase as linear functions of airspeedsquared during the ground run, up until the point of rotation. From the rotation speed \left(V_R\right) to the liftoff speed \left(V_{LOF}\right) the angle of attack will change, and hence C_L and C_D will also changeboth C_L and C_D will increase during this short segment of the ground run.
0.0.5. Airplane Configuration
It is important that the drag polar used to evaluate the takeoff run includes appropriate correction factors to account for the airplane's configuration (see Section 7.5.2). With the slats and flaps set for takeoff and the landing gear extended, the zero-lift drag coefficient can be determined from the following expression:
CD0=CD0clean +ΔCD0slats +ΔCD0flaps +ΔCD0gear #(9.5)
where C_{D_0} is the zero-lift drag coefficient;
\left(C_{D_0}\right){\mathrm{clean\ }} is the clean airplane zero-lift drag coefficient; and
\Delta C{D_0}^0 describes the drag coefficient increments due to the extension of the slats, flaps, and landing gear.
Furthermore, the deployment of the flaps will change the spanwise lift distribution and this will alter the lift-dependent drag coefficient.
0.0.6. Ground Effect
During the ground run, the airplane's lift-dependent drag is reduced due to ground effect (discussed in Section 7.5.6). In the absence of substantive test or wind-tunnel data, a ground effect factor (\lambda) can be applied to the out-of-ground-effect (OGE) drag polar to establish an approximate drag polar for the airplane on the ground. The magnitude of the ground effect factor depends on the airplane type (see also Equations 7.33 and 7.34). The drag coefficient in ground effect (IGE) is given by Equation 7.31, which can be expressed as follows:
CDg=CD0+λKCL2#(9.6)
where C_{D_g} is the drag coefficient of the airplane in ground effect;
C_{D_0} is the zero-lift drag coefficient (OGE);
\lambda is the ground effect factor; and
K is the lift-dependent drag factor (OGE).
0.0.7. Rolling Friction
The coefficient of friction (or friction coefficient), which is usually assigned the Greek letter mu(\mu), is defined as the ratio of the tire friction force (which acts in a direction parallel to the runway surface) to the force acting on the tire normal to the runway surface (i.e., the reaction force):
\mu=\frac{F_{fr}}{R}\bigmwhere \mu is the coefficient of friction (dimensionless);
F_{fr} is the friction force (typical units: N,lb ); and R is the reaction force acting on the tire (typical units: N,lb ).
The total friction force that acts on the airplane can be expressed as follows:
Ffr=μRMW+μRNW#(9.8)
where the subscripts MW and NW designate the main landing gear and nose gear, respectively.
A very good approximation of the total friction force acting on all the tires (nose and main landing gear), up to the point of rotation on an inclined runway (see Figure 9.5), is given by the following expression:
Ffr=μWcos⁡γG-L#(9.9)
where \gamma_G is the "slope" angle of the runway (see Figure 9.5), defined as positive for an uphill runway (typical units: degrees, radians).
The coefficient of friction depends on the nature of the tire-runway surface interface and, critically, on the amount of slip that occurs between the tires and the runway. Slip describes the relative motion between a tire and the runway surface; a slip condition exists when a tire's rotational speed is less than or greater than the free-rolling speed. During takeoff, when the aircraft's tires are rolling freely, they will experience zero slip, but, obviously, this is not the case when the brakes are applied (wheel braking is discussed in Section 9.8).
The coefficient of friction for a free rolling tire is usually taken as a measure of the total resistance to the rolling motion of the wheel; this includes the internal friction of the bearings in the wheel and the energy associated with the lateral flexing of the walls of the tire under the airplane's weight. The tire type and internal pressure thus influence the frictional force. The value of \mu changes a little during the takeoff as the speed increases; however, this has a negligible influence, and for the purpose of computing the takeoff distance, a mean value can be used. The coefficient of friction is approximately 0.015 for a jet transport airplane under rolling conditions on a dry, concrete runway [9].
To accurately measure the effective coefficient of friction of a rolling airplane tire is a surprisingly difficult task. A widely adopted approach utilizes airplane acceleration data-measured during takeoff under known conditions-with knowledge of the other forces that act on the airplane to determine the coefficient of friction. In other words, the friction coefficient is treated as a fudge factor rather than an independently evaluated parameter (such as ambient temperature), and adjusted within a mathematical model to give the measured distance. The resulting model is then used to determine the takeoff distances for other conditions.
0.0.8. Net Acceleration Force
For the sake of brevity, the net acceleration force acting on the airplane in the direction of motion is herein denoted as \sum F, where
F=FN-D-Ffr-Wsin⁡γG#(9.10)
or {\ }\sum F=F_N-D-\mu\left(W\cos\funcapply\gamma_G-L\right)-W\sin\funcapply\gamma_G{\ } (from Equation 9.9)
As the runway gradient is a very small quantity, \ ^6 the small angle approximations \cos\funcapply\gamma_G\cong1 and \sin\funcapply\gamma_G\cong\gamma_G (for \gamma_G measured in radians) can be used without loss of accuracy.
Thus \sum F=F_N-D-\mu(W-L)-W\gamma_G
Substituting Equations 9.3 and 9.4 into Equation 9.12 results in the required expression for the net acceleration force:
F=FN-12ρV2SCDg-μCL-μ+γGW#(9.13)
where C_{D_g} can be approximated by Equation 9.6.
The group of terms C_{D_g}-\mu C_L does not change appreciably during the ground run and a constant value can be used to evaluate the takeoff distance. This is usually determined from tests conducted for a particular airplane type for each required flap setting.
With knowledge of the net force acting on the airplane (in the direction of motion), the acceleration can be determined for a given airplane weight, which can then be used to calculate the ground run.
0.1. Evaluation of the Takeoff Distance from Brake Release to Rotation
0.1.1. Basic Equation for the Takeoff Distance s_R
The distance (s) that the airplane will travel (see Figure 9.6) in a given period of time depends on its ground speed \left(V_G\right) and its acceleration (a). The airplane's acceleration (with respect to the ground) can be expressed as follows:
a=ddtds dt=ddtV-Vw=dV dt if dVw/dt=0#(9.14)
In the development of Equation 9.14, the wind speed is considered invariant with time. Equation 9.14 can now be written in a manner that facilitates the development of an integral expression for the ground distance (s), that is,
a=dV dt=dV dsds dt=dV dsV-Vw#(9.15)
In the analysis that follows, the ground distance (s) is expressed as a function of airspeed (not ground speed). At the start of the ground run (i.e., brake release), the airplane is stationary, but the presence of wind means that the airspeed is not zero, but is equal in magnitude to the headwind V_w. This sets the lower limit to the integral expression. The upper limit is the rotation speed, which is independent of the wind speed (the lift required for takeoff depends on the airspeed and not the ground speed). The ground distance measured from the start of the takeoff to the point of rotation \left(s_R\right) can be determined from the following expression:
sR=start end   ds=VwVR  V-Vwa dV#(9.16)
At any instant during the ground run, the airplane's acceleration can be obtained from Newton's second law (see Section 2.4.5):
a= Fm=g FW#(9.17)
where a is the instantaneous acceleration (typical units: m/s^2,ft/s^2 );
\sum F is the net acceleration force, given by Equation 9.13 (typical units: N , lb);
m is the airplane's mass (typical units: kg , slug);
g is the acceleration due to gravity (typical units: m/s^2,ft/s^2 ); and
W is the airplane's weight (typical units: N,lb ).
Substituting Equation 9.17 into Equation 9.16 yields the required equation:
sR=VwVR  WV-Vwg F dV=VwVR  WVg F dV-VwVR  WVwg F dV#(9.18)
where the net accelerating force is given by Equation 9.13 (and where C_{D_g} can be approximated by Equation 9.6).
In order to integrate Equation 9.18, it is necessary to describe the variations of thrust, lift, drag, weight, and friction with speed. The airplane's weight changes very little during the short time taken for the takeoff; thus, for "hand" calculations (i.e., analytical solutions), weight may be regarded as constant. In detailed computational evaluations (as conducted by aircraft manufacturers, for example), the very small influence of reducing airplane weight due to fuel burn
is taken into account. The rolling coefficient of friction can also be regarded as constant (see Section 9.3.7). The values of C_L and C_{D_g} are dictated by the airframe geometry, and therefore the lift and drag forces can be expressed as functions of V^2. Both lift and drag are also functions of air density-hence altitude and ambient temperature have a major effect on the takeoff.
In general, the thrust will vary as a function of velocity, air temperature, and pressure (see Section 9.3.3). During the ground run, the thrust will reduce as the speed builds up. The exact manner by which the thrust will reduce depends on the powerplant type. The integration of Equation 9.18 can be performed analytically for simple functions of F_N. In Section 9.4.2, this is undertaken by assuming that F_N is constant. The resulting expression is useful for obtaining analytical estimates of the takeoff distance. The solution is also useful for developing an understanding of the impact of the various parameters on the takeoff distance (or for conducting sensitivity analyses, for example).
0.1.2. Determination of the Distance s_R Assuming F_N is Constant
In this section, a mathematical expression for the distance s_R is determined for the general case of an airplane taking off on a dry or wet (i.e., non-contaminated) runway with a gradient \gamma_G and with a headwind V_w. The problem is simplified by assuming that the engine thrust is equal to FN, a mean constant value. This value is not the arithmetic mean of the thrust values at the start and end of the ground run; instead, it is evaluated at a particular speed selected to give an accurate approximation of the takeoff distance. The most frequently used approach assumes that acceleration varies linearly with V^2 (this is discussed further in Section 9.4.3). The mean thrust FN is calculated at a reference speed, herein denoted as V^\ast, which is determined as follows:
V2=12VR-Vw2#(9.19) or V=12VR-Vw=0.707VR-Vw#(9.20)
An expression for the net accelerating force ( \sum F ) can be obtained by substituting Equation 9.6 into Equation 9.13, that is,
F=FN-12ρV2SCD0+λKCL2-μCL-μ+γGW#(9.21)
The required distance \left(s_R\right) is obtained by a summation of the two integrals that are given in Equation 9.18. For the sake of convenience, the two integral expressions will be evaluated separately, where
1  =VwVR  WVg F dV and 2  =VwVR  WVwg F dV#(9.22)
For the first integral, the following variable substitution is made:
f=V2#(9.23)
thus df=2V\mathrm{\ }dV and dV=\frac{1}{2V}\mathrm{\ }df
hence \int_1\hairsp=\int_{V_w}^{V_R}\hairsp\frac{WV}{g\sum F}\mathrm{\ }dV=\int_{\sqrt{f_w}}^{\sqrt{f_R}}\hairsp\frac{WV}{g\sum F}\left(\frac{1}{2V}\right)df=\frac{1}{2g}\int_{\sqrt{f_w}}^{\sqrt{f_R}}\hairsp\frac{W}{\sum F}\mathrm{\ }df
Now substitute \sum F from Equation 9.21 into Equation 9.24:
1  =12gfwfR  1FNW-μ-γG-ρS2WCD0+λKCL2-μCLf df#(9.25)
To simplify the analysis, arbitrary constants A_2,B_2, and C_2 are now introduced, where
A_2=FNW-μ-γG and B2=ρS2WCD0+λKCL2-μCL
and {\ }C_2=\sqrt{\frac{A_2}{B_2}}
Thus \int_1\hairsp=\frac{1}{2g}\int_{\sqrt{f_w}}^{\sqrt{f_R}}\hairsp\frac{1}{A_2-B_2f}\mathrm{\ }df=\frac{-1}{2gB_2}\left[\ln\funcapply\left(A_2-B_2f\right)\right]{\sqrt{f_w}}^{\sqrt{f_R}}
1  =-12gB2ln⁡A2-B2V2VwVR=12gB2ln⁡A2-B2Vw2A2-B2VR2#(9.27)
The second integral is now considered:
\begin{matrix}\ &\ \int{2}\hairsp\hairsp=\frac{V_w}{g}\int_{V_w}^{V_R}\hairsp\hairsp\frac{1}{A_2-B_2V^2}\mathrm{\ } dV=\frac{V_W}{gB_2}\int_{V_w}^{V_R}\hairsp\hairsp\frac{1}{C_2^2-V^2}\mathrm{\ } dV\\end{matrix}
(\int_{2}\ =\frac{V_w}{2gB_2C_2}\left[\ln{\left(\frac{C_2+V}{C_2-V}\right)}\right]_{V_w}^{V_R}=\frac{V_w}{2gB_2C_2}\ln{\left{\frac{\left(C_2+V_R\right)\left(C_2-V_w\right)}{\left(C_2-V_R\right)\left(C_2+V_w\right)}\right}})
The two results are now added to give the distance s_R :
sR=12gB2ln⁡A2-B2Vw2A2-B2VR2-VwC2ln⁡C2+VRC2-VwC2-VRC2+Vw#(9.29)
where A_2,B_2, and C_2 are defined in Equation 9.26 .
By selecting an appropriate speed at which the mean thrust is calculated (see Equation 9.20), an accurate estimate of the required distance can be obtained by this method.
It is evident that an analytical solution can be found for the ground distance s_R if a constant thrust value is assumed. Equation 8.42 (see Section 8.6.4) provides a reasonably accurate representation of a turbofan engine's thrust variation with speed. Unfortunately, it is not possible to obtain a closed-form solution for the ground distance s_R using Equation 8.42, and a numerical technique-such as the one described later in Section 9.4.4-must be employed if greater accuracy is needed.
0.1.3. Determination of the Distance s_R Using a Mean Acceleration
A popular, and relatively simple, method to estimate the ground distance s_R is based on the use of a mean acceleration ( a). The estimate for s_R is then given by the equation for

Figure 9.7 Instantaneous and mean accelerations for a medium-sized twinjet plotted as a function of airspeed squared (note: the same airplane data were used to generate Figures 9.8 and 9.9).
uniform acceleration:
sR=1aVwVR  V-VwdV=1aV22-VVwVwVR=12aVR-Vw2#(9.30)
The influence of the wind can be easily seen from Equation 9.30: a headwind will directly reduce the takeoff ground run and a tailwind will increase the distance.
The acceleration of a jet airplane during the ground run reduces in an almost linear manner with V^2-this is illustrated in Figure 9.7. The mean value a can therefore be calculated at the reference speed given by Equation 9.20 .
The method for calculating the takeoff distance s_R based on a mean acceleration is summarized below:
(1) Determine the reference speed V^\ast using Equation 9.20 .
(2) Determine the thrust FN at V^\ast from thrust versus velocity data or from an appropriate empirical model, such as Equation 9.2.
(3) Calculate a for V=V^\ast and F_N=FN using Equations 9.17 and 9.13 .
(4) Calculate the takeoff distance s_R using Equation 9.30.
0.1.4. Numerical Evaluation of the Distance s_R
It was shown in Section 9.4.2 that an analytical integration of Equation 9.18, to yield the distance s_R, is possible if the thrust is expressed as a mean constant value. An alternative approach is to conduct a numerical integration of the governing equation (this is preferential when actual engine thrust data are available). By substituting the acceleration term in Equation 9.16 using Equation 9.17, the following integral expression is obtained:
sR=VwVR  V-Vwa dV=WgVwVR  V-Vw F dV#(9.31)
where \sum F is given by Equation 9.13.
Table 9.1 Example of takeoff distance calculation
V_G
(kt) V_G
(\mathrm{\ }m/s) V
(kt) V
(\mathrm{\ }m/s) q
(\mathrm{\ }Pa) F_N
(kN) \left(C_{D_g}-\mu C_L\right)qS
(\mathbf{kN}) \mu W
(kN) \gamma_GW
(kN) a
\left(\mathrm{\ }m/s^2\right) \Delta s
(\mathrm{\ }m)
0 0.0 10 5.1 16.2 312 0.24 17.6 10.7 2.60
20 10.3 30 15.4 146 303 2.12 17.6 10.7 2.50 20.8
40 20.6 50 25.7 405 294 5.88 17.6 10.7 2.39 64.9
60 30.9 70 36.0 794 286 11.5 17.6 10.7 2.26 114
80 41.2 90 46.3 1313 279 19.0 17.6 10.7 2.12 169
100 51.4 110 56.6 1961 271 28.4 17.6 10.7 1.97 233
120 61.7 130 66.9 2739 265 39.7 17.6 10.7 1.81 308
140 72.0 150 77.2 3647 258 52.9 17.6 10.7 1.63 401
150 77.2 160 82.3 4150 255 60.2 17.6 10.7 1.53 243

Total =1552
Notes:
(a) Airplane: medium-size twinjet, with a takeoff weight of 1067 kN ( 240000 lb).
(b) Conditions: 1% runway gradient (uphill), 10 kt headwind, dry runway surface ( \mu=0.0165 ), and ISA conditions.
To integrate Equation 9.31 numerically, it is possible to divide the takeoff distance s_R into n segments (or velocity intervals) and to use the trapezoidal rule for incremental values of V :
sR=W2gi=0n  (V)i+1-Vw Fi+1+(V)i-Vw FiΔV#(9.32)
The intervals do not have to be very small for an acceptable result to be obtained ( 10 kt intervals, for example, are adequate for most applications).
The distance s_R can also be calculated by a more direct approach using a spreadsheet-this is illustrated in Table 9.1 for a medium-size twinjet. For the sake of illustration, 20 kt speed intervals have been used. It is convenient to start the calculation with the ground speed (columns 1 and 2 ) and then determine the true airspeed (V) and dynamic pressure ( q ). When evaluating the terms on the right-hand side of Equation 9.13, note that the thrust \left(F_N\right) is a function of V not V_G, the term C_{D_g}-\mu C_L can be assumed constant, and that the resisting forces due to friction and runway slope are both functions of the airplane's weight. The instantaneous acceleration (a), given in column 10, is determined using Equations 9.17 and 9.13. Each entry (or spreadsheet cell) in the table for columns 3 to 10 contains point (or instantaneous) values evaluated for the corresponding ground speed. The interval distance (\Delta s), given in column 11 , is calculated using an average acceleration for the speed interval. The calculation is repeated for each speed interval until the rotation speed is reached. Finally, the interval distances are summed.
Figures 9.8 and 9.9 illustrate the takeoff variables for the medium-size twinjet considered in the previous paragraph; the takeoff conditions are unchanged. In Figure 9.8, the components of force given by Equation 9.12 are plotted against the ground speed. The resultant acceleration, which is shown on the second Y axis, was calculated using the numerical approach illustrated in Table 9.1. In Figure 9.9, the resulting acceleration is repeated (again on the second Y axis), but this time it is plotted against the takeoff distance. The airspeed is seen to increase rapidly at

Figure 9.8 Example of takeoff calculations: components of force and airplane acceleration plotted against ground speed.

Figure 9.9 Example of takeoff calculations: airspeed and acceleration plotted against takeoff distance.
the start of the takeoff when the acceleration is greatest. The rotation speed (TAS) was assumed to be 160 kt .
9.4.5 Estimation of the Time to Reach (V_R)

The time that the airplane takes to accelerate to the rotation speed (\left(V_R\right)) may be determined using similar approaches to those presented for the distance calculation (Sections 9.4.1 to 9.4.4). The time-velocity relationship is given by
[
\mathrm{d}t=\frac{1}{a}\mathrm{d}V\emsp\left(\mathrm{\ based\ on\ Equation\ }9.15\right)
]

Thus\ t=\int_{\mathrm{start\ }}^{\mathrm{end\ }}\mathrm{d}t=\int_{V_w}^{V_R}\frac{1}{a}\mathrm{d}V)
where the acceleration(\left(a\right)) is given by Equation 9.17
The development of an analytical solution to the integral expression of Equation 9.33 is complex. A simplified approach-which assumes a mean, constant acceleration-will usually provide a satisfactory result, though. With a constant acceleration, it follows that the time for the ground run is
t_R=\frac{1}{\bar{a}}\int_{V_w}^{V_R}\mathrm{d}V=\frac{V_R-V_w}{\bar{a}}
In Section 9.4.3, the distance (s_R) was calculated using a mean acceleration determined at a reference velocity (V^) given by Equation 9.20 . The rationale behind this approach is that the acceleration is seen to be almost a linear function of (V^2); however, the time expression is a function of both (V) and (V^2). It is reported by reference [9] that a better estimate of the time for the ground run can be obtained by determining a mean acceleration at the reference speed (V^=0.6 V_R).

The method can be summarized as follows:
(1) Determine the reference speed by assuming that (V^\ast=0.6V_R).
(2) Determine the thrust (\bar{F}_N) at (V=V^) from thrust versus velocity data.
(3) Calculate (\bar{a}) for (V=V^) and (F_N=\bar{F}_N) from Equation 9.17 .
(4) Calculate the time for the ground run (\left(t_R\right)) from Equation 9.34.
9.5 Rotation and Climb-Out to Clear the Screen Height
9.5.1 Rotation and Flare

At the point of rotation, the pilot will pull the stick/yoke back, which will have the following effect: the elevator will be deflected upwards (creating a negatively cambered airfoil shape), which will produce an aerodynamic load acting downwards on the horizontal tail and this, in turn, will raise the nose of the airplane as it pivots around the main wheels. The angle of attack will thus increase, resulting in a progressive increase in lift. The time that it takes the airplane to rotate (typically about 2 to 4 seconds) depends on several factors, including the airplane type, CG position, speed, airplane weight, and the rate that the pilot pulls the stick/yoke back. The typical rate of rotation for jet transport airplanes is reported to be 2.5 to 3.0 degrees per second ([10]).

If a pilot executes an incorrect takeoff technique by rotating too early and/or by pulling the stick/yoke back too quickly, this can result in too high a rate of rotation, which can cause the rear part of the fuselage to strike the runway. This is called a tailstrike and can occur on airplanes that are geometrically limited. ({ }^7) Such events occur infrequently; they tend to be costly mistakes resulting in damage to the aft fuselage.

After the liftoff, there will be a short phase in which the flight path is curved-called the flare ({ }^8)-when the lift is greater than the weight. During this phase the airplane will continue to
accelerate, and there may be a small change in (C_L). After the flare, the airplane will climb steadily at a climb angle ((\gamma)) that is approximately constant. The screen height, as mentioned earlier in Section 9.2 , is 35 ft for a dry runway and 15 ft for a wet runway. The point at which the screen height is reached may be either before or after the end of the flare.

The distance from the point of rotation to the point where the airplane clears the screen is difficult to calculate accurately using a mechanics approach due to the variation of the governing parameters and the influence of varying pilot technique. The most straightforward approach is based on experimentally determined average times for the rotation and flare segments.
9.5.2 Estimation of the Rotation Distance

The distance from rotation to liftoff (\left(s_{R L}\right)) is difficult to estimate accurately-this is largely due to the changes in (C_L). The distance, however, is small in comparison to (s_R). Flight-test data are usually used in performance analyses. An estimate of the rotation distance may be obtained by multiplying a typical rotation time by the average ground speed:
[
s_{RL}=\Delta t_{RL}\left(\frac{V_R+V_{LOF}}{2}-V_w\right)
]
where (s_{R L}) is the distance from rotation to liftoff (typical units: (\mathrm{m}, \mathrm{ft}) );
(\Delta t_{R L}) is the time from rotation to liftoff (typical unit: s);
(V_R) is the rotation speed (typical units: (\mathrm{m} / \mathrm{s}, \mathrm{ft} / \mathrm{s}) );
(V_{L O F}) is the liftoff speed (typical units: (\mathrm{m} / \mathrm{s}, \mathrm{ft} / \mathrm{s}) ); and
(V_w) is the headwind speed (typical units: (\mathrm{m} / \mathrm{s}, \mathrm{ft} / \mathrm{s}) ).
An estimate of the liftoff speed can be obtained by noting that the airworthiness regulations FAR/CS (25.105[3,4]) require (V_{L O F}) to be greater than 1.1 times the minimum unstick speed, (V_{M U}) (defined in Section 10.2.2).

Once the rotation distance is known, the total ground run (\left(s_g\right)) can be determined by summation of the two ground segments:
[
s_g=s_R+s_{RL}
]
9.5.3 Energy Method to Estimate the Distance from Liftoff to (35 \mathbf{f t})

The takeoff distance from liftoff to clear the 35 ft screen can be determined by considering the work done in accelerating the airplane from its liftoff speed to the speed at the 35 ft screen and simultaneously raising it 35 ft above the runway surface. This approximation ignores the fact that the flight path distance is slightly greater than the runway distance and also assumes zero runway slope and zero wind.

The work done is given by the following approximate expression:
[
U\cong\int\left(F_N-D\right)\mathrm{d} s=\left(F_N-D\right){\mathrm{avg\ }}s{L35}
]
where (U) is the work done (typical units: (\mathrm{J}, \mathrm{ft} \mathrm{lb}) ); (\left(F_N-D\right){\text {avg }}) is the average excess thrust (typical units: (\mathrm{N}, \mathrm{lb}) ); and (s{L 35}) is the runway distance from liftoff to the 35 ft screen (typical units: m , ft).

The work done can be equated to the sum of the changes in kinetic and potential energies (see Equations 2.42 and 2.44), that is,
U=\frac{W}{2g}\left(V_{35}^2-V_{LOF}^2\right)+h_{sc}W
where(V_{35})isthespeedatthe35ftscreen\left(typicalunits:(\mathrm{m}/\mathrm{s},\mathrm{ft}/\mathrm{s})\right);and(h_{sc})isthescreenheight\left(i.e.,35ftor10.7m\right).

By combining Equations 9.37 and 9.38, the following expression for the distance from liftoff to the 35 ft point is obtained (for zero wind):
[
s_{L35}=\left(\frac{W}{\left(F_N-D\right){avg}}\right)\left(\frac{V{35}^2-V_{LOF}^2}{2g}+h_{sc}\right)
]
9.5.4 Time-Based Method to Estimate the Distance from Liftoff to (\mathbf{3 5} \mathbf{~ f t})

An alternative-and relatively straightforward-method to estimate the distance from liftoff to the 35 ft point is based on the product of the average time and the average ground speed for this segment of the takeoff:
[
s_{L35}=\Delta t_{L35}\left(\frac{V_{LOF}+V_{35}}{2}-V_W\right)
]
where (\Delta t_{L 35}) is the time from liftoff to clear the 35 ft screen (unit: s ).
The time (\Delta t_{L 35}) is largely a function of the thrust-to-weight ratio of the airplane (the influence of the thrust-to-weight ratio can be seen in Equation 9.39). The time (\Delta t_{L 35}), which is best determined from experimental data, typically varies between 2 and 5 seconds (see Figure 9.10).

Figure 9.10 Illustration of how the time to clear the 35 ft threshold from rotation or liftoff varies with thrust-to-weight ratio. Source: Adapted from [7].
9.6 Empirical Estimation of Takeoff Distances
0.0.1. Introduction
A simple, graphical method \ ^9 to estimate the effect of thrust setting, weight, and air density on the takeoff distance-from brake release to the 35 ft screen height-is presented in this section. The method, which is based on the use of a takeoff parameter (TOP), only applies to nil wind operations from a runway with zero slope. This parameter is used to characterize the actual measured performance (from flight test) so that the effects of changes in thrust or weight on the takeoff distance, for example, can then be approximated. A number of different takeoff parameters can be found in the literature, although they all have a similar format. In Section 9.6.2, an all-engines-operating ( AEO ) parameter ( {\rm TOP}{AEO} ) is derived; the development is repeated in Section 9.6.3 for one-engine-inoperative (OEI) conditions. Note that the correlation method relies on the assumption that the TOP is proportional to the takeoff distance.
0.0.2. Takeoff Parameter for All-Engines-Operating Condition
This takeoff parameter \left({\rm TOP}{AEO}\right) is derived assuming that the airplane has a constant acceleration ( a ). This is a mean acceleration, which would result in the same distance as that obtained during the actual conditions where the acceleration reduces as airspeed increases. From Equation 9.30, the takeoff distance can be expressed as
s35=V3522a#(9.41)
where s_{35} is the distance from brake release (V=0) to the 35 ft screen height (typical units: m,ft ); and
V_{35} is the TAS at 35 ft (equal to the ground speed as V_w=0 ) (typical units: m/s,ft/s ).
As the greatest part of the takeoff distance will be the ground run, Equation 9.21 can be used to derive a relationship for the mean acceleration. Assuming zero slope (i.e., \gamma_G=0 ), it follows that
a= Fm=gWFN-12ρV2SCD-μRCL-μRW#(9.42)
Equation 9.42 can be simplified by noting that the acceleration depends primarily on the mean engine thrust FN and airplane weight (W). The rolling friction and aerodynamic drag terms together account for approximately 10% to 20% of the net acceleration force applied on the airplane. If this fraction is denoted by a fixed constant k, then the equation can be written as
a=gFNW(1-k)#(9.43)
Implicit in this derivation is the assumption that, like acceleration, a constant mean thrust can be used to replace the actual thrust (which reduces with airspeed). It was shown earlier (see Sections 9.4 .2 and 9.4.3) that the appropriate reference speed at which to calculate FN for the ground run (in nil wind) is 0.707V_R. Extending the estimated distance to the 35 ft screen height requires account to be taken of the small increase in potential energy and the
fact that the airplane must accelerate to V_{35}. The assumption is made that these influences are negligible. The mean thrust has been determined at 70% of V_{35}, which is consistent with the approach taken in ESDU 76011 [12]. (The precise value is not critical as it is only used to determine a correlation parameter.) Substituting Equation 9.43 into Equation 9.41 yields a simplified expression for the takeoff distance:
s_{35}=\frac{V_{35}^2W}{2g(1-k)F_{N_{0.7}}}\bigmwhere F_{N_{0.7}} is the thrust evaluated at 0.7V_{35} (typical units: N,lb ).
The speed term in Equation 9.44 is now eliminated using the lift coefficient at the screen height \left(C_{L_{35}}\right), that is,
CL35=2Lσρ0V352S=2nzWσρ0V352S#(9.45) where$\sigma$istherelativeairdensity(dimensionless);$\rho_0$istheISAsea-leveldatumairdensity(typicalunits:$kg/m^3$,slug$/{\rm ft}^3$);$V_{35}$istheTASat35ft(itcanbeassumedthatthisisequaltothetakeoffsafetyspeed,$V_2$)(typicalunits:$m/s,ft/s$);$S$isthewingreferencearea(typicalunits:$m^2,{\rm ft}^2$);$n_z$istheloadfactornormaltotheflightpath(dimensionless);and$W$istheairplaneweight(typicalunits:$N,lb$).Thus$s_{35}=\frac{n_z}{g\rho_0(1-k)}\left[\frac{W^2}{\sigma C_{L_{35}}SF_{N_{0.7}}}\right]$TheTOPcanbedefined,asfollows,byassumingthattheloadfactor$\left(n_z\right)$isconstant: s35∝W2σCL35SFN0.7=TOPAEO#(9.47)
where {\rm TOP}{\mathrm{AEO\ }} is the all-engine-operating takeoff parameter (typical units: \left.N/m^2,lb/{\rm ft}^2\right).
If the distance to the screen height \left(s{35}\right) is measured for a series of actual takeoffs at different combinations of thrust, weight, and ambient conditions, then a correlation curve can be established by plotting s_{35} against TOP \ {\mathrm{AEO\ }}. Subsequently, s{35} can be estimated for an untested combination of thrust, weight, and ambient conditions, simply by calculating the appropriate value for the takeoff parameter and then reading off the corresponding distance from the chart.
0.0.3. Takeoff Parameter for One Engine Inoperative
As it is a requirement to consider the possibility of an engine failure when planning a "regulated" takeoff (see Section 9.2.4), it is necessary to obtain correlations of s_{35} for this case. In deriving a suitable takeoff parameter, it is assumed that the aircraft accelerates normally from rest to V_1 (see Section 10.2.3), while covering a distance s_{0,1}. The aircraft then suffers an instantaneous total failure of one engine, but continues the takeoff with the remaining engine(s) operating
normally, and clears the 35 ft screen height after covering a farther distance s_{1,35}. The takeoff distance is the sum of two segments, that is,
s35=s0,1+s1,35#(9.48)
As before (see Equation 9.41), mean acceleration is used to determine the takeoff distance segments. This time, however, two values are needed:
s35=V122a0,1+V352-V122a1,35=12V12a0,1+V352-V12a1,35#(9.49)
The mean accelerations are now related to a mean thrust, F_{N_{0.7}}, which is evaluated at 0.7V_{35}. The reduction in mean acceleration following the engine failure is assumed to be solely a function of the number of live engines (i.e., N-1 ); no account is taken of the increase in drag from engine windmilling and/or yaw correction. The acceleration for the two segments has a similar form to that of Equation 9.43, that is,
a0,1=gFN0.7(1-k)W#(9.50) and a1,35=gFN0.7(1-k)WN-1N#(9.51)
where N is the number of engines installed.
Substituting Equations 9.50 and 9.51 into Equation 9.49 yields the following expression:
\begin{matrix}s_{35}&\ =\frac{WV_1^2}{2gF_{N_{0.7}}(1-k)}+\frac{W\left(V_{35}^2-V_1^2\right)}{2gF_{N_{0.7}}(1-k)}\left(\frac{N}{N-1}\right)\\end{matrix}
The lift coefficient, as expressed by Equation 9.45, is now introduced:
\begin{matrix}s_{35}=\frac{n_zW^2}{g\rho_0\sigma C_{L_{35}}SF_{N_{0.7}}(1-k)}\left{N-\left(\frac{V_1}{V_{35}}\right)^2\right}\left(\frac{1}{N-1}\right)\\mathrm{\ Thus\ }s_{35}\propto\left(\frac{W^2}{\sigma C_{L_{35}}SF_{N_{0.7}}}\right)\left{N-\left(\frac{V_1}{V_{35}}\right)^2\right}\left(\frac{1}{N-1}\right)={\rm TOP}{OEI}\\end{matrix}
where {\rm TOP}{OEI} is the one-engine-inoperative takeoff parameter (typical units: \left.N/m^2,lb/{\rm ft}^2\right).
This approach to estimate the takeoff distance is illustrated in Figure 9.11. A trend line has been drawn through data taken from ESDU 76011 [12]. The takeoff distance to 35 ft , following a single engine failure at V_1, is shown as a function of {\rm TOP}_{OEI}. In this case, the speed at screen height has been taken as the takeoff safety speed, V_2 (see Section 10.2.2), and the mean thrust evaluated at 0.7V_2.

Figure 9.11 Approximate takeoff distance to 35 ft versus TOP \ {\mathrm{OEI\ }} following an engine failure at V_1. Source: Adapted from [12].
0.1. Evaluation of Rejected Takeoff Runway Distances
0.1.1. Regulations Regarding the Rejected Takeoff (RTO)
The certification regulations for the determination of the accelerate-stop (i.e., rejected takeoff) distance have undergone several revisions (amendments) over the past few decades. \ ^{10} Consequently, the rules that applied at the time of certification need to be considered in evaluating the RTO performance of a particular airplane type. The description presented herein is in accordance with FAR 25 Amendment 25-92.
Accelerate-stop distances are defined in FAR/CS 25.109 [3, 4] for both dry and wet runways. As mentioned in Section 9.2.5, the additional deceleration available from the use of reverse thrust is not included when determining the required accelerate-stop distance for dry runway operations. For wet runway operations, however, a different procedure is adopted as the runway surface friction is addressed directly within the calculations. Reverse thrust, in this case, can be included provided it is safe, reliable, and its selection does not require exceptional skill to operate the airplane [6].
0.1.2. Requirements for the Accelerate-Stop Distance on a Dry Runway
The required accelerate-stop distance is the greater of the one-engine-inoperative (OEI) RTO distance and the all-engines-operating (AEO) RTO distance. The one-engine-inoperative RTO distance on a dry runway is the sum of the distances necessary to:
(1) accelerate the airplane from a standing start with all engines operating to V{EF};
(2) allow the airplane to accelerate from V_{EF} to the highest speed reached during the rejected takeoff, assuming the critical engine fails at V_{EF} and the pilot takes the first action to reject the takeoff at V_1; and
(3) come to a full stop; plus
(4) a distance equivalent to two seconds at the V_1 speed.
The time allowed for the first action by the pilot following the engine failure (e.g., to apply the brakes) is the demonstrated time during testing or one second, whichever is the greater [4,14]. The time required for the second and third actions (e.g., to retard the thrust levers and extend the ground spoilers) is the demonstrated time during testing, which is typically less than one second. (On most modern transport airplanes, retarding the thrust levers in this situation engages the RTO autobrake function, which is described in Section 11.3.8.)
The principle for calculating one-engine-failure RTO distances, as required for standard flight operations, is based on an expansion of certification test data-this is explained in Section 9.7.4. The all-engines-operating RTO distance is determined in an identical manner. The only differences are that all engines operate until the crew takes the first action at the V_1 speed to reject the takeoff, and that the airplane must come to a full stop with all engines still operating.
0.1.3. Requirements for the Accelerate-Stop Distance on a Wet Runway
The accelerate-stop distance on a wet runway is determined in accordance with the procedures for the dry runway, except that:
(1) the wet runway values of V_{EF} and V_1 are used;
(2) the wet runway braking coefficient of friction (as defined in FAR/CS 25.109[3,4] ) applies;
(3) the maximum tire-to-ground wet runway braking coefficient of friction must be adjusted to take into account the efficiency of the anti-skid system on a wet runway; and
(4) available thrust reversers can be used (i.e., for the engine failure RTO, it must be assumed that the corresponding thrust reverser is inoperative).
0.1.4. Evaluation of the Accelerate-Stop Distance
Figure 9.12 illustrates the process for determining an acceptable accelerate-stop distance for flight operations based on an expansion of certification test data, considering either (1) a failure of the critical engine, or (2) another, unspecified, abort event with AEO. The accelerate-stop distance for an RTO is the sum of the following distances (which are defined in Figure 9.12):
s=sAEO+sE1+sV1+sT+sB#(9.55)
The distance s_{AEO} corresponds to the AEO acceleration from a standing start to V_{EF}, the speed at which the critical engine is assumed to fail or, alternatively, to V_{EV}, the speed at which the abort event is assumed to occur. The distance can be determined using the techniques described in Section 9.4-in particular, the numerical approach (see Section 9.4.4) is appropriate. The distance s_{E1} in the case of the engine-failure RTO is the OEI acceleration from V_{EF} to V_1. The minimum time interval between engine failure (at V_{EF} ) and initiation of the first pilot action to stop the airplane (at V_1 ), permitted by the regulatory authorities, is one second [4,14]. In the case of the event-abort RTO, the distance s_{E1} is the AEO acceleration distance from V_{EV} to V_1. For either case, the distance can be determined using the method described in Section 9.4.4, taking into account the appropriate engine thrust and drag changes. The distance s_{V1} is determined by multiplying the speed V_1 by the mandatory two second time allowance. The distance s_T, usually called the transition distance, is the distance covered after the first action of the pilot until the airplane is fully configured for braking (i.e., wheel brakes fully efficient and ground spoilers fully deployed). The corresponding transition time depends on the airplane type; data for six airplane types indicated transition times between 1.3 and 3.6 seconds [13]. The distance s_B is the

Figure 9.12 Determination of accelerate-stop (rejected takeoff) distance.
braking distance. The analytical evaluation of this distance is non-trivial due to the complexities associated with analyzing the wheel braking force-this is discussed in Section 9.8.
Accurate determination of the RTO distances requires knowledge of the residual thrust generated by the airplane's engines as they spin down (spool down). In the case of a failed engine, a conservative assumption can be made that the thrust decays to zero in a few seconds in a manner that mimics a sudden stoppage of the fuel supply. However, when the pilot moves the thrust levers to the idle position (sometimes called a "throttle chop"), the residual thrust from a "good" engine will decay to the idle-thrust level over a slightly longer period (typically about six seconds). It can thus be assumed for theoretical analyses that a "good" engine will produce a little more thrust during the spool-down than a failed engine, and that the idle thrust will be maintained for the entire deceleration period.
The time-dependent nature of the events after V_{EF} or V_{EV} requires a different approach for the numerical evaluation of the accelerate-stop distance to that described earlier for the accelerate-go distance (see Section 9.4.4). To determine s_{AEO}, a numerical method based on velocity increments (as described earlier) can be used. Following an engine failure or closure of the thrust levers, however, it is necessary to determine the distance traveled based on time increments (of, say, one second). The procedure for the first time interval (i.e., t=0\mathrm{\ }s to t=1\mathrm{\ }s ) is as follows:
(1) At t=0\mathrm{\ }s (corresponding to an engine failure or closure of the thrust levers), calculate the instantaneous acceleration ( a_{t=0} ) based on the net accelerating force (see Section 9.3.8).
(2) At t=1\mathrm{\ }s, calculate the airplane speed \left(V_{t=1}\right) based on the assumption that the acceleration determined at the start of the interval (i.e., a_{t=0} ) applies for the entire interval.
(3) At the speed V_{t=1}, calculate the instantaneous acceleration \left(a_{t=1}\right).
(4) Determine the interval mean acceleration, i.e., a1=at=0+at=1/2.
(5) In a repeat of step 2, calculate a revised estimate of the airplane speed \left(V_{t=1}\right) at t=1\mathrm{\ }s based on the interval mean acceleration a1.
(6) Determine the distance covered from t=0\mathrm{\ }s to t=1\mathrm{\ }s (i.e., s_1 ) based on the interval mean acceleration a1.
Having established the distance covered during the first time interval and the end-of-interval speed, it is now possible to move on to the second time interval, which requires steps 1-6 to be repeated. The process is then repeated for each subsequent time interval; the interval distances are evaluated and summed.
0.2. Wheel Braking
0.2.1. Brake System Design
Braking that is accomplished by brakes fitted to the airplane's wheels is often called mechanical braking (to distinguish this feature from aerodynamic braking or reverse thrust). A brake unit, which is fitted to each wheel on the main landing gear, has an array of disks that comprise alternating rotor disks (which rotate with the wheel) and stator disks (which are stationary with respect to the landing gear). The stack of rotor disks is securely attached to the wheel such that their axes of rotation are aligned with the wheel axle. Stator disks, which separate each rotor disk, are installed such that when hydraulic pressure \ ^{11} within the brake system increasesas controlled by the pilot pressing the brake pedal or commanded by the autobrake systempistons press the stator disks against the rotor disks. The friction between the disks produces a braking torque that acts on the wheel, reducing its rotational velocity. The friction wears the contacting surfaces down, necessitating periodic replacement of worn disks.
The heat that is generated by the friction between the rotor and stator disks can affect the performance of steel brakes. At extreme temperatures, the steel disks start to melt, reducing the braking force that can be generated-this is known as brake fade. The standard method for quantifying brake fade is to conduct a series of high-speed tests with increasing kinetic energy at the point of brake application (the brakes are allowed to cool after each test). The maximum braking force that can be achieved is seen to be initially independent of kinetic energy; however, as the kinetic energy increases beyond a limiting value, the maximum braking force tends to reduce. Modern brake disks manufactured from a carbon-fiber-ceramic composite material, however, have a much higher melting temperature and do not fade in this manner. (Brake energy limits, which apply to an RTO, are discussed in Section 10.3.7.)
0.2.2. Airplane Braking Coefficient
The coefficient of friction for a wheel (\mu) moving over a hard surface is defined by Equation 9.7 as the ratio of the tire friction force to the force acting on the tire normal to the runway surface (i.e., the reaction force on the tire). When braking takes place, the friction force increases considerably compared to the rolling friction case (described in Section 9.3.7).
It is customary for the evaluation of airplane stopping distances to use an airplane braking coefficient, which is defined as follows:
μB=FbrWcos⁡γG-L#(9.56)
where \mu_B is the airplane braking coefficient (dimensionless);
F_{br} is the braking force (typical units: N,lb );
W is the airplane weight (typical units: N,lb );
\gamma_G is the runway slope angle (uphill is positive) (typical unit: rad); and
L is the mean lift (typical units: N,lb ).
The braking coefficient \left(\mu_B\right), as defined by Equation 9.56, is not exactly the same as a coefficient of friction (\mu). This braking coefficient is based on the airplane's entire weight and not just the reaction force on the main landing gear (where the brakes are installed). Furthermore, braking coefficients are usually determined by experiment and based on a mean reaction force corresponding to the test duration. In such evaluations, the lift is taken as a mean value.
Unlike the rolling coefficient of friction, the braking coefficient depends significantly on the airplane's speed. As the airplane slows down, the braking coefficient increases. When the runway is dry, the increase is small-permitting a constant value to be used to determine stopping distances. When the runway is wet, however, there is a large non-linear increase as the airplane slows down. Other factors that influence the braking coefficient include:
(1) the amount of slip that occurs between the tire and the runway (see Section 9.8.3);
(2) the maximum brake torque (see Section 9.8.4); and
(3) the nature of the tire-runway-surface interface, which depends on (i) whether the surface is dry, wet, or contaminated; (ii) the type of tire and its pressure and condition (i.e., new or worn); and (iii) the runway surface profile and condition (see Sections 9.8.5 and 9.8.6).
0.2.3. Slip and Anti-Skid Systems
Slip is the difference between the airplane's ground speed and the tire speed; it is usually expressed as a percentage of the airplane's ground speed-for example: an airplane traveling at 120 kt , with a tire speed of 90 kt (based on the rotational speed of the tire), experiences a slip of 30 kt or 25%. A free rolling tire (i.e., zero slip) has the same speed as the airplane, and provides no braking force-just a rolling resistance to the motion. The friction force increases as the amount of slip increases; it reaches a maximum at about 12% slip [15] and then starts to decrease (Figure 9.13). In the extreme condition, 100% slip (i.e., a locked wheel)

Figure 9.13 Friction force versus slip ratio.
will produce a skid, resulting in a substantially reduced friction force (and a danger of a loss of directional control).
Anti-skid systems-employed on all modern jet airplanes-work by comparing the speed of each main gear wheel (obtained from tachometer readings) with the airplane's forward speed. When the wheel speed drops below a predetermined percentage of the airplane's speed (say \sim87% ), the system instructs the brakes to momentarily release, thus maintaining the optimum brake slip ratio (and protecting the wheels from locking).
The braking coefficient \left(\mu_B\right) can be expressed as the product of an anti-skid braking efficiency factor \left(\eta_B\right) and the maximum tire-to-ground braking coefficient \left(\mu_{\mathrm{max\ }}\right) :
μB=ηBμmax#(9.57)
The value of \eta_B depends on the type of anti-skid system installed and is best determined experimentally for the airplane type. The following data are provided in FAR/CS 25.109[3,4] for three different systems: for an on-off system, \eta_B=0.3; for a quasi-modulating system, \eta_B=0.5; and for a fully modulating system, \eta_B=0.8.
0.2.4. Maximum Brake Torque Limit
The determination of the braking force due to the wheel brakes is further complicated by the fact that the theoretical braking force determined by the product of the braking coefficient and the reaction force (see Equation 9.56) may not be achievable in practice because of aircraft brake system limitations. The "internal" braking torque, which is due to the friction that is generated between the stator and rotor disks, opposes the "external" torque, which is the product of the tire friction force (which acts at the tire-runway-surface interface) and the tire radius.
Accelerate-stop tests are conducted with varying airplane weight and configuration. The CG is required to be in the most forward position allowed for takeoff (this conservatism is introduced as it results in the lowest percentage of the airplane's weight being carried by the tires on the main landing gear). The test data are used to establish a range of braking coefficient values for different flap settings and runway conditions. The maximum braking force values achieved in such tests can be used to generate a graph of the form illustrated in Figure 9.14. The indicated limits establish a region of viable braking conditions. At low airplane weights, the anti-skid system limits the braking force-the slope of this line represents the maximum braking coefficient that the system can deliver. At higher weights, a limit is imposed by the maximum torque that the brake system can produce. In the latter case, the anti-skid system plays no part in limiting the braking action.

For a given:
airplane type/model
runway surface (e.g., smooth or grooved)
(1) Limit imposed by anti-skid system
(2) Limit imposed by the maximum torque that the brake system can produce
Figure 9.14 Braking force versus mean reaction force (W-L) for a dry runway.
0.2.5. Runway Pavement Construction
The braking performance of an airplane depends on the micro- and macrotexture of the runway pavement surface. Microtexture, which is important at low speeds, is the fine-scale roughness that would be apparent to the touch. Macrotexture, which is important at high speeds when the runway is wet, is a larger-scale visible roughness of the pavement surface. The purpose of the macrotexture is to provide paths for water to be removed from beneath the airplane's tires, and thus improve braking under wet conditions.
There are two main types of runway surface treatment currently used to produce the desired macrotexture: grooved and Porous Friction Course (PFC). Grooved runways that comply with FAA AC 150-5320-12 [16] have transverse grooves of 6 mm ( 0.25 inch) deep by 6 mm ( 0.25 inch) wide at 38 mm ( 1.5 inch) spacing, which are created either by forming the concrete before it is fully hardened or by saw-cutting the surface. A PFC construction has a top layer, known as a course, of an open-graded hot-mix asphalt matrix, which is 19-38\mathrm{\ }mm(0.75-1.5 inch) thick [16]. Grooved and PFC runway surfaces can achieve airplane braking coefficients under wet conditions in excess of 70% of that which is achievable under dry conditions.
Another aspect that can affect braking is runway maintenance. Localized deposits of rubberleft on the surface when the tires on landing airplanes spin-up on touchdown-reduce braking effectiveness and have to be removed during routine runway maintenance.
0.2.6. Wet Runway Conditions
Water on the runway leads to a loss of contact between the tire and the runway surface, resulting in a reduction in friction compared to a dry runway. As the tire rolls over the surface, the water tends to be squeezed out from beneath the tire; however, this is not always fully achieved. At high speeds, there is insufficient time for the water to be completely squeezed out, resulting in reduced friction-the braking coefficient thus reduces as the airplane's speed increases.
The braking coefficient on wet runways at high speed is relatively low, but it increases substantially with decreasing speed. FAR/CS 25.109[3,4] provides data on the maximum braking coefficients that can be used for braking evaluations on wet runways. \ ^{12} This information is provided for two categories of runway surface: smooth runways (Figure 9.15) and runways that have a grooved or PFC surface (Figure 9.16). It is evident that at high speed the maximum tire-to-ground braking coefficient can be as low as about 0.15 , increasing to about 0.6 to 0.9 (depending on the tire pressure) as the airplane's ground speed drops. These maximum tireto-ground braking coefficient values ( \mu_{max} ) must be multiplied by the braking efficiency factor \left(\eta_B\right) to account for the anti-skid system installed in the airplane (see Section 9.8.3) to give an appropriate airplane braking coefficient.