\(C_{r}\) below is the core chord of the wing.
This is a diagram to use to generate equations of longitudinal equilibrium.
This distance is called the stickﬁxed static margin \(k_{m}=\left ( h_{n}h\right ) \bar {c}\) Must be positive for static stability
This table contain some deﬁnitions and equations that can be useful.
# 
equation 
meaning/use 
1 
\(\begin {array} [c]{lcl}C_{L} & = & \frac {\partial C_{L}}{\partial \alpha }\alpha \\ & = & C_{L_{\alpha }}\alpha \\ & = & a\alpha \end {array} \) 
\(C_{L}\) is lift coeﬃcient. \(\alpha \) is angle of attack. \(a\) is slope \(\frac {\partial C_{L}}{\partial \alpha }\) which is the same as \(C_{L_{\alpha }}\) 
2 
\(C_{L_{w}}=C_{L_{w_{\alpha }}}\alpha \) 
wing lift coeﬃcient 
3 
\(C_{D}=C_{D_{\min }}+kC_{L}^{2}\) 
drag coeﬃcient 
4 
\(C_{m_{w}}=C_{m_{ac_{w}}}+\left ( C_{L_{w}}+C_{D_{\min }}\alpha _{w}\right ) \left ( hh_{n_{w}}\right ) +\left ( C_{L}\alpha _{w}C_{D_{w}}\right ) \frac {z}{\bar {c}}\) 
pitching moment coeﬃcient due to wing only about the C.G. of the airplane assuming small \(\alpha _{w}.\) This is simpliﬁed more by assuming \(C_{D_{w}}\alpha _{w}\ll C_{L_{w}}\) and \(\left ( C_{L}\alpha _{w}C_{D_{w}}\right ) \ll 1\) 
5 
\(C_{m_{w}}=C_{m_{ac_{w}}}+C_{L_{w}}\left ( hh_{n_{w}}\right ) \) 
simpliﬁed wing Pitching moment 
6 
\(\begin {array} [c]{lll}C_{m_{wb}} & = & C_{m_{ac_{wb}}}+C_{L_{wb}}\left ( hh_{n_{w}}\right ) \\ & = & C_{m_{ac_{wb}}}+\frac {\partial C_{L_{wb}}}{\partial \alpha _{wb}}\alpha _{wb}\left ( hh_{n_{w}}\right ) \\ & = & C_{m_{ac_{wb}}}+a_{wb}\alpha _{wb}\left ( hh_{n_{w}}\right ) \end {array} \) 
simpliﬁed pitching moment coeﬃcient due to wing and body about the C.G. of the airplane. \(\alpha _{wb}\) is the angle of attack 
7 
\(C_{L_{t}}=\frac {L_{t}}{\frac {1}{2}\rho V^{2}S_{t}}\) 
\(C_{L_{t}}\) is the lift coeﬃcient generated by tail. \(S_{t}\) is the tail area. \(V\) is airplane air speed 
8 
\(L=L_{wb}+L_{t}\) 
total lift of airplane. \(L_{wb}\) is lift due to body and wing and \(L_{t}\) is lift due to tail 
9 
\(C_{L}=C_{L_{wb}}+\frac {S_{t}}{S}C_{L_{t}}\) 
coeﬃcient of total lift of airplane. \(C_{L_{wb}}\) is coeﬃcient of lift due to wing and body. \(C_{L_{t}}\) is lift coeﬃcient due to tail. \(S\) is the total wing area. \(S_{t}\) is tail area 
10 
\(M_{t}=l_{t}L_{t}=l_{t}C_{L_{t}}\frac {1}{2}\rho V^{2}S_{t}\) 
pitching moment due to tail about C.G. of airplane 
11 
\(C_{m_{t}}=\frac {M_{t}}{\frac {1}{2}\rho V^{2}S_{t}\bar {c}}=\frac {l_{t}}{\bar {c}}\frac {S_{t}}{S}C_{L_{t}}=V_{H}C_{L_{t}}\) 
pitching moment coeﬃcient due to tail. \(V_{H}=\frac {l_{t}}{\bar {c}}\frac {S_{t}}{S}\) is called tail volume 
12 
\(\begin {array} [c]{lcl}V_{H} & = & \frac {l_{t}}{\bar {c}}\frac {S_{t}}{S}\\ \bar {V}_{H} & = & \frac {\bar {l}_{t}}{\bar {c}}\frac {S_{t}}{S}\end {array} \) 
introducing \(\bar {V}_{H}\) bar tail volume which is \(V_{H}\) but uses \(\bar {l}_{t}\) instead of \(l_{t}\). Important note. \(V_{H}\) depends on location of C.G., but \(\bar {V}_{H}\) does not. \(\bar {l}_{t}=l_{t}+\left ( hh_{n_{wb}}\right ) \bar {c}\) 
13 
\(C_{m_{t}}=\bar {V}_{H}C_{L_{t}}+C_{L_{t}}\frac {S_{t}}{S}\left ( hh_{n_{wb}}\right ) \) 
pitching moment coeﬃcient due to tail expressed using \(\bar {V}_{H}\). This is the one to use. 
14 
\(C_{m_{p}}\) 
pitching moment coeﬃcient due to propulsion about airplane C.G. 
15 
\(C_{m}=C_{m_{wb}}+C_{m_{t}}+C_{m_{p}}\) 
total airplane pitching moment coeﬃcient about airplane C.G. 
16 
\(\begin {array} [c]{lll}C_{m} & = & C_{m_{wb}}+C_{m_{t}}+C_{m_{p}}\\ & = & \left [ C_{m_{ac_{wb}}}+C_{L_{wb}}\left ( hh_{n_{w}}\right ) \right ] +\left [ \bar {V}_{H}C_{L_{t}}+C_{L_{t}}\frac {S_{t}}{S}\left ( hh_{n_{wb}}\right ) \right ] +C_{m_{p}}\\ & = & C_{m_{ac_{wb}}}+\overbrace {\left ( C_{L_{wb}}+C_{L_{t}}\frac {S_{t}}{S}\right ) }^{C_{L}} \left ( hh_{n_{w}}\right ) \bar {V}_{H}C_{L_{t}}+C_{m_{p}}\\ & = & C_{m_{ac_{wb}}}+C_{L}\left ( hh_{n_{w}}\right ) \bar {V}_{H}C_{L_{t}}+C_{m_{p}}\end {array} \) 
simpliﬁed total Pitching moment coeﬃcient about airplane C.G. 
17 
\(\begin {array} [c]{lcl}\frac {\partial C_{m}}{\partial \alpha } & = & \frac {\partial C_{m_{ac_{wb}}}}{\partial \alpha }+\frac {\partial C_{L}}{\partial \alpha }\left ( hh_{n_{w}}\right ) \bar {V}_{H}\frac {\partial C_{L_{t}}}{\partial \alpha }+\frac {\partial C_{m_{p}}}{\partial \alpha }\\ C_{m_{\alpha }} & = & \frac {\partial C_{m_{ac_{wb}}}}{\partial \alpha }+C_{L_{\alpha }}\left ( hh_{n_{w}}\right ) \bar {V}_{H}\frac {\partial C_{L_{t}}}{\partial \alpha }+\frac {\partial C_{m_{p}}}{\partial \alpha }\end {array} \) 
derivative of total pitching moment coeﬃcient \(C_{m}\) w.r.t airplane angle of attack \(\alpha \) 
18 
\(h_{n}=h_{n_{wb}}\frac {1}{\frac {\partial C_{L}}{\partial \alpha }}\left ( \frac {\partial C_{m_{ac_{wb}}}}{\partial \alpha }\bar {V}_{H}\frac {\partial C_{L_{t}}}{\partial \alpha }+\frac {\partial C_{m_{p}}}{\partial \alpha }\right ) \) 
location of airplane neutral point of airplane found by setting \(C_{m_{\alpha }}=0\) in the above equation 
19 
\(\begin {array} [c]{lcl}\frac {\partial C_{m}}{\partial \alpha } & = & \frac {\partial C_{L}}{\partial \alpha }\left ( hh_{n}\right ) \\ C_{m_{\alpha }} & = & C_{L_{\alpha }}\left ( hh_{n}\right ) \end {array} \) 
rewrite of \(C_{m_{\alpha }}\) in terms of \(h_{n}\). Derived using the above two equations. 
20 
\(k_{n}=h_{n}h\) 
static margin. Must be Positive for static stability 




The following equations are derived from the above set of equation using what is called the linear form. The main point is to bring into the equations the expression for \(C_{L_{t}}\) written in term of \(\alpha _{wb}.\) This is done by expressing the tail angle of attack \(\alpha _{t}\) in terms of \(\alpha _{wb}\) via the downwash angle and the \(i_{t}\) angle. \(\frac {\partial C_{L_{wb}}}{\partial \alpha _{wb}}\)in the above equations are replaced by \(a_{wb}\) and \(\frac {\partial C_{L_{t}}}{\partial \alpha _{t}}\) is replaced by \(a_{t}\). This replacement says that it is a linear relation between \(C_{L}\) and the corresponding angle of attack. The main of this rewrite is to obtain an expression for \(C_{m}\) in terms of \(\alpha _{wb}\) where \(\alpha _{t}\) is expressed in terms of \(\alpha _{wb}\), hence \(\alpha _{t}\) do not show explicitly. The linear form of the equations is what from now on.
# 
equation 
meaning/use 
1 
\(\begin {aligned} C_{L_{wb}} &= \frac {\partial C_{L_{wb}}}{\partial \alpha _{wb}}\alpha _{wb}\\ &= a_{wb}\alpha _{wb}\\ C_{L_t} &= a_t \alpha _t \\ C_{mp} &= C_{m_{0}p} + \frac {\partial C_{mp}}{\partial \alpha } \alpha \end {aligned}\) 
\(a_{wb}\) is constant, represents \(\frac {\partial C_{L_{wb}}}{\partial \alpha _{wb}}\) and \(C_{m_{0p}}\) is propulsion pitching moment coeﬀ. at zero angle of attack \(\alpha \) 
2 
\(\begin {aligned} \alpha _{t} &= \alpha _{wb}  i_t  \epsilon \\ \epsilon &= \epsilon _0 + \frac {\partial \epsilon }{\partial \alpha }\alpha _{wb} \end {aligned}\) 
main relation that associates \(\alpha _{wb}\) with \(\alpha _{t}\). \(\alpha _{wb}\) is the wingbody angle of attack, \(\epsilon \) is downwash angle at tail, and \(i_{t}\) is tail angle with horizontal reference (see diagram) 
3 
\(\begin {aligned} C_{L_{t}} &= a_t \alpha _t \\ &= a_t \left [ \alpha _{wb}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) i_{t}\epsilon _{0}\right ] \end {aligned}\) 
Lift due to tail expressed using \(\alpha _{wb}\) and \(\epsilon \) (notice that \(\alpha _{t}\) do not show explicitly) 
4 
\(a=a_{wb}\left [ 1+\frac {a_{t}}{a_{wb}}\frac {S_{t}}{S}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) \right ] \) 
\(a\) deﬁned for use with overall lift coeﬃcient 
5 
\(\begin {array} [c]{lll}C_{L} & = & \overset {a_{wb}\alpha _{wb}}{\overbrace {C_{L_{wb}}}}+\frac {S_{t}}{S}C_{L_{t}}\\ & = & a_{wb}\alpha _{wb}+\frac {S_{t}}{S}a_{t}\left [ \alpha _{wb}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) i_{t}\epsilon _{0}\right ] \\ & = & a\alpha \\ & = & \left ( C_{L}\right ) _{\alpha _{wb}=0}+a\alpha _{wb}\end {array} \) 
overall airplane lift using linear relations 
6 
\(\begin {array} [c]{lll}\alpha & = & \alpha _{wb}\frac {a_{t}}{a}\frac {S_{t}}{S}\left ( i_{t}+\epsilon _{0}\right ) \end {array}\) 
overall angle of attack \(\alpha \) as function of the wing and body angle of attack \(\alpha _{wb}\) and tail angles 
7 
\(\begin {array} [c]{lll}C_{m} & = & C_{m0}+\frac {\partial C_{m}}{\partial \alpha }\alpha =C_{m0}+C_{m_{\alpha }}\alpha \\ C_{m} & = & \bar {C}_{m0}+\frac {\partial C_{m}}{\partial \alpha }\alpha _{wb}=\bar {C}_{m0}+C_{m_{\alpha }}\alpha _{wb}\end {array} \) 
overall airplane pitch moment. Two versions one uses \(\alpha _{wb}\) and one uses \(\alpha \) 
8 
\(\begin {array} [c]{lll}C_{m_{\alpha }} & = & a\left ( hh_{n_{wb}}\right ) a_{t}\bar {V}_{H}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) +\frac {\partial C_{mp}}{\partial \alpha }\\ C_{m_{\alpha }} & = & a_{wb}\left ( hh_{n_{wb}}\right ) a_{t}V_{H}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) +\frac {\partial C_{mp}}{\partial \alpha }\end {array} \) 
Two versions of \(\frac {\partial C_{m}}{\partial \alpha }\) one for \(\alpha _{wb}\) and one one uses \(\alpha \) 
9 
\( \begin {array}[c]{lll}C_{m_{0}} & = & C_{m_{ac_{wb}}}+C_{m_{o_{p}}}+a_{t}\bar {V}_{H}\left (\epsilon _{0}+i_{t}\right ) \left [ 1\frac {a_{t}}{a}\frac {S_{t}}{S}\left (1\frac {\partial \epsilon }{\partial \alpha }\right ) \right ] \\ \bar {C}_{m_{0}} & = & C_{m_{ac_{wb}}}+\bar {C}_{m_{o_{p}}}+a_{t}V_{H}\left (\epsilon _{0}+i_{t}\right ) \end {array} \) 
\(C_{m_{0}}\) is total pitching moment coef. at zero lift (does not depend on C.G. location) but \(\bar {C}_{m_{0}}\) is total pitching moment coef. at \(\alpha _{wb}=0\) (not at zero lift). This depends on location of C.G. 
10 
\(\bar {C}_{m_{0_{p}}}= C_{m_{0_{p}}}+(\alpha \alpha _{wb}) \frac {\partial C_{mp}}{\partial \alpha }\) 

11 
\(\begin {array} [c]{lll}h_{n} & = & h_{n_{wb}}+\frac {a_{t}}{a}\bar {V}_{H}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) \frac {1}{a}\frac {\partial C_{mp}}{\partial \alpha }\\ & = & h_{n_{wb}}+\frac {a_{t}}{a_{wb}\left [ 1+\frac {a_{t}}{a_{wb}}\frac {S_{t}}{S}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) \right ] }\bar {V}_{H}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) \frac {1}{a_{wb}\left [ 1+\frac {a_{t}}{a_{wb}}\frac {S_{t}}{S}\left ( 1\frac {\partial \epsilon }{\partial \alpha }\right ) \right ] }\frac {\partial C_{mp}}{\partial \alpha }\end {array} \) 
Used to determine \(h_{n}\) 




Remember that for symmetric airfoil, when the chord is parallel to velocity vector,
then the angle of attack is zero, and also the left coeﬃcient is zero. But this
is only for symmetric airfoil. For the common campbell airfoil shape, when the
chord is parallel to the velocity vector, which means the angle of attack is zero,
there will still be lift (small lift, but it is there). What this means, is that the
chord line has to tilt down more to get zero lift. This extra tilting down makes
the angle of attack negative. If we now draw a line from the right edge of the
airfoil parallel to the velocity vector, this line is called the zero lift line (ZLL) see
diagram below.
Just remember, that angle of attack (which is always the angle between the
chord and the velocity vector, the book below calls it the geometrical angle of
attack) is negative for zero lift. This is when the airfoil is not symmetric. For
symmetric airfoil, ZLL and the chord line are the same. This angle is small, \(3^{0}\) or
so. Depending on shape. See Foundations of Aerodynamics, 5th ed, by Chow and
Kuethe, here is the diagram.
stall from http://en.wikipedia.org/wiki/Stall_(flight)
In fluid dynamics, a stall is a reduction in the lift coefficient generated by a foil as angle of attack increases.[1] This occurs when the critical angle of attack of the foil is exceeded. The critical angle of attack is typically about 15 degrees, but it may vary significantly depending on the fluid, foil, and Reynolds number.
some demos relating to airplane control http://demonstrations.wolfram.com/ControllingAirplaneFlight/
These are diagrams and images collected from diﬀerent places. References is given next to each image.
This below from http://www.grc.nasa.gov/WWW/k12/UEET/StudentSite/dynamicsofflight.html
http://www.grc.nasa.gov/WWW/k12/airplane/alr.html
From http://en.wikipedia.org/wiki/Lift_coefficient and http://en.wikipedia.org/wiki/File:Aeroforces.svg
from http://adg.stanford.edu/aa241/drag/sweepncdc.html
Images from http://adamone.rchomepage.com/cg_calc.htm and Flight dynamics principles by Cook, 1997.
From http://chrusion.com/BJ7/SuperCalc7.html
From http://www.willingtons.com/aircraft_center_of_gravity_calcu.html
From http://www.solarcity.net/2010/06/airplanecontrolsurfaces.html nice diagram that shows clearly how the elevator causes the pitching motion (nose up/down). From same page, it says "The purpose of the ﬂaps is to generate more lift at slower airspeed, which enables the airplane to ﬂy at a greatly reduced speed with a lower risk of stalling."
Images from ﬂight dynamics principles, by Cook, 1997.
Images from Performance, stability, dynamics and control of Airplanes. By Pamadi, AIAA press. Page 169. and http://www.americanflyers.net/aviationlibrary/pilots_handbook/chapter_3.htm
Image from http://www.americanflyers.net/aviationlibrary/pilots_handbook/chapter_3.htm
Image from http://www.americanflyers.net/aviationlibrary/pilots_handbook/chapter_3.htm
Image from FAA pilot handbook and http://www.youtube.com/watch?v=8uT55aei1NI
Image http://www.youtube.com/watch?v=8uT55aei1NI and http://www.youtube.com/user/DAMSQAZ?feature=watch
Image http://edition.cnn.com/2014/01/16/travel/insideairbusbeluga/index.html?hpt=ibu_c2
Image from http://edition.cnn.com/2014/01/16/travel/insideairbusbeluga/index.html?hpt=ibu_c2
Image from http://www.nasa.gov/centers/dryden/Features/super_guppy.html
Image from http://www.aerospaceweb.org/question/aerodynamics/q0130.shtml "Boeing Pelican ground eﬀect vehicle"