Chapter 4
Dynamic of flights

4.1 Wing geometry
4.2 Summary of main equations
4.3 images and plots collected
4.4 Some strange shaped airplanes
4.5 links
4.6 references

4.1 Wing geometry

\(C_{r}\) 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 stick-fixed static margin \(k_{m}=\left ( h_{n}-h\right ) \bar {c}\) Must be positive for static stability

4.2 Summary of main equations

4.2.1 definitions

This table contain some definitions and equations that can be useful.





\(\begin {array} [c]{lcl}C_{L} & = & \frac {\partial C_{L}}{\partial \alpha }\alpha \\ & = & C_{L_{\alpha }}\alpha \\ & = & a\alpha \end {array} \)

\(C_{L}\) is lift coefficient. \(\alpha \) is angle of attack. \(a\) is slope \(\frac {\partial C_{L}}{\partial \alpha }\) which is the same as \(C_{L_{\alpha }}\)


\(C_{L_{w}}=C_{L_{w_{\alpha }}}\alpha \)

wing lift coefficient


\(C_{D}=C_{D_{\min }}+kC_{L}^{2}\)

drag coefficient


\(C_{m_{w}}=C_{m_{ac_{w}}}+\left ( C_{L_{w}}+C_{D_{\min }}\alpha _{w}\right ) \left ( h-h_{n_{w}}\right ) +\left ( C_{L}\alpha _{w}-C_{D_{w}}\right ) \frac {z}{\bar {c}}\)

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 \(\left ( C_{L}\alpha _{w}-C_{D_{w}}\right ) \ll 1\)


\(C_{m_{w}}=C_{m_{ac_{w}}}+C_{L_{w}}\left ( h-h_{n_{w}}\right ) \)

simplified wing Pitching moment


\(\begin {array} [c]{lll}C_{m_{wb}} & = & C_{m_{ac_{wb}}}+C_{L_{wb}}\left ( h-h_{n_{w}}\right ) \\ & = & C_{m_{ac_{wb}}}+\frac {\partial C_{L_{wb}}}{\partial \alpha _{wb}}\alpha _{wb}\left ( h-h_{n_{w}}\right ) \\ & = & C_{m_{ac_{wb}}}+a_{wb}\alpha _{wb}\left ( h-h_{n_{w}}\right ) \end {array} \)

simplified pitching moment coefficient  due to wing and body about the C.G. of the airplane. \(\alpha _{wb}\) is the angle of attack


\(C_{L_{t}}=\frac {L_{t}}{\frac {1}{2}\rho V^{2}S_{t}}\)

\(C_{L_{t}}\) is the lift coefficient generated by tail. \(S_{t}\) is the tail area. \(V\) is airplane air speed



total lift of airplane. \(L_{wb}\) is lift due to body and wing and \(L_{t}\) is lift due to tail


\(C_{L}=C_{L_{wb}}+\frac {S_{t}}{S}C_{L_{t}}\)

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


\(M_{t}=-l_{t}L_{t}=-l_{t}C_{L_{t}}\frac {1}{2}\rho V^{2}S_{t}\)

pitching moment due to tail about C.G. of airplane


\(C_{m_{t}}=\frac {M_{t}}{\frac {1}{2}\rho V^{2}S_{t}\bar {c}}=-\frac {l_{t}}{\bar {c}}\frac {S_{t}}{S}C_{L_{t}}=-V_{H}C_{L_{t}}\)

pitching moment coefficient due to tail. \(V_{H}=\frac {l_{t}}{\bar {c}}\frac {S_{t}}{S}\) is called tail volume


\(\begin {array} [c]{lcl}V_{H} & = & \frac {l_{t}}{\bar {c}}\frac {S_{t}}{S}\\ \bar {V}_{H} & = & \frac {\bar {l}_{t}}{\bar {c}}\frac {S_{t}}{S}\end {array} \)

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. \(\bar {l}_{t}=l_{t}+\left ( h-h_{n_{wb}}\right ) \bar {c}\)


\(C_{m_{t}}=-\bar {V}_{H}C_{L_{t}}+C_{L_{t}}\frac {S_{t}}{S}\left ( h-h_{n_{wb}}\right ) \)

pitching moment coefficient due to tail expressed using \(\bar {V}_{H}\). This is the one to use.



pitching moment coefficient due to propulsion about airplane C.G.



total airplane pitching moment coefficient about airplane C.G.


\(\begin {array} [c]{lll}C_{m} & = & C_{m_{wb}}+C_{m_{t}}+C_{m_{p}}\\ & = & \left [ C_{m_{ac_{wb}}}+C_{L_{wb}}\left ( h-h_{n_{w}}\right ) \right ] +\left [ -\bar {V}_{H}C_{L_{t}}+C_{L_{t}}\frac {S_{t}}{S}\left ( h-h_{n_{wb}}\right ) \right ] +C_{m_{p}}\\ & = & C_{m_{ac_{wb}}}+\overbrace {\left ( C_{L_{wb}}+C_{L_{t}}\frac {S_{t}}{S}\right ) }^{C_{L}} \left ( h-h_{n_{w}}\right ) -\bar {V}_{H}C_{L_{t}}+C_{m_{p}}\\ & = & C_{m_{ac_{wb}}}+C_{L}\left ( h-h_{n_{w}}\right ) -\bar {V}_{H}C_{L_{t}}+C_{m_{p}}\end {array} \)

simplified total Pitching moment coefficient about airplane C.G.


\(\begin {array} [c]{lcl}\frac {\partial C_{m}}{\partial \alpha } & = & \frac {\partial C_{m_{ac_{wb}}}}{\partial \alpha }+\frac {\partial C_{L}}{\partial \alpha }\left ( h-h_{n_{w}}\right ) -\bar {V}_{H}\frac {\partial C_{L_{t}}}{\partial \alpha }+\frac {\partial C_{m_{p}}}{\partial \alpha }\\ C_{m_{\alpha }} & = & \frac {\partial C_{m_{ac_{wb}}}}{\partial \alpha }+C_{L_{\alpha }}\left ( h-h_{n_{w}}\right ) -\bar {V}_{H}\frac {\partial C_{L_{t}}}{\partial \alpha }+\frac {\partial C_{m_{p}}}{\partial \alpha }\end {array} \)

derivative of total pitching moment coefficient \(C_{m}\) w.r.t airplane angle of attack \(\alpha \)


\(h_{n}=h_{n_{wb}}-\frac {1}{\frac {\partial C_{L}}{\partial \alpha }}\left ( \frac {\partial C_{m_{ac_{wb}}}}{\partial \alpha }-\bar {V}_{H}\frac {\partial C_{L_{t}}}{\partial \alpha }+\frac {\partial C_{m_{p}}}{\partial \alpha }\right ) \)

location of airplane neutral point of airplane found by setting \(C_{m_{\alpha }}=0\) in the above equation


\(\begin {array} [c]{lcl}\frac {\partial C_{m}}{\partial \alpha } & = & \frac {\partial C_{L}}{\partial \alpha }\left ( h-h_{n}\right ) \\ C_{m_{\alpha }} & = & C_{L_{\alpha }}\left ( h-h_{n}\right ) \end {array} \)

rewrite of \(C_{m_{\alpha }}\) in terms of \(h_{n}\). Derived using the above two equations.



static margin. Must be Positive for static stability Writing the equations in linear form

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.





\(\begin {aligned} C_{L_{wb}} &= \frac {\partial C_{L_{wb}}}{\partial \alpha _{wb}}\alpha _{wb}\\ &= a_{wb}\alpha _{wb}\\ C_{L_t} &= a_t \alpha _t \\ C_{mp} &= C_{m_{0}p} + \frac {\partial C_{mp}}{\partial \alpha } \alpha \end {aligned}\)

\(a_{wb}\) 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 \(\alpha \)


\(\begin {aligned} \alpha _{t} &= \alpha _{wb} - i_t - \epsilon \\ \epsilon &= \epsilon _0 + \frac {\partial \epsilon }{\partial \alpha }\alpha _{wb} \end {aligned}\)

main relation that associates \(\alpha _{wb}\) with \(\alpha _{t}\). \(\alpha _{wb}\) is the wing-body angle of attack, \(\epsilon \) is downwash angle at tail, and \(i_{t}\) is tail angle with horizontal reference (see diagram)


\(\begin {aligned} C_{L_{t}} &= a_t \alpha _t \\ &= a_t \left [ \alpha _{wb}\left ( 1-\frac {\partial \epsilon }{\partial \alpha }\right ) -i_{t}-\epsilon _{0}\right ] \end {aligned}\)

Lift due to tail expressed using \(\alpha _{wb}\) and \(\epsilon \) (notice that \(\alpha _{t}\) do not show explicitly)


\(a=a_{wb}\left [ 1+\frac {a_{t}}{a_{wb}}\frac {S_{t}}{S}\left ( 1-\frac {\partial \epsilon }{\partial \alpha }\right ) \right ] \)

\(a\) defined for use with overall lift coefficient


\(\begin {array} [c]{lll}C_{L} & = & \overset {a_{wb}\alpha _{wb}}{\overbrace {C_{L_{wb}}}}+\frac {S_{t}}{S}C_{L_{t}}\\ & = & a_{wb}\alpha _{wb}+\frac {S_{t}}{S}a_{t}\left [ \alpha _{wb}\left ( 1-\frac {\partial \epsilon }{\partial \alpha }\right ) -i_{t}-\epsilon _{0}\right ] \\ & = & a\alpha \\ & = & \left ( C_{L}\right ) _{\alpha _{wb}=0}+a\alpha _{wb}\end {array} \)

overall airplane lift using linear relations


\(\begin {array} [c]{lll}\alpha & = & \alpha _{wb}-\frac {a_{t}}{a}\frac {S_{t}}{S}\left ( i_{t}+\epsilon _{0}\right ) \end {array}\) pict

overall angle of attack \(\alpha \) as function of the wing and body angle of attack \(\alpha _{wb}\) and tail angles


\(\begin {array} [c]{lll}C_{m} & = & C_{m0}+\frac {\partial C_{m}}{\partial \alpha }\alpha =C_{m0}+C_{m_{\alpha }}\alpha \\ C_{m} & = & \bar {C}_{m0}+\frac {\partial C_{m}}{\partial \alpha }\alpha _{wb}=\bar {C}_{m0}+C_{m_{\alpha }}\alpha _{wb}\end {array} \)

overall airplane pitch moment. Two versions one uses \(\alpha _{wb}\) and one uses \(\alpha \)


\(\begin {array} [c]{lll}C_{m_{\alpha }} & = & a\left ( h-h_{n_{wb}}\right ) -a_{t}\bar {V}_{H}\left ( 1-\frac {\partial \epsilon }{\partial \alpha }\right ) +\frac {\partial C_{mp}}{\partial \alpha }\\ C_{m_{\alpha }} & = & a_{wb}\left ( h-h_{n_{wb}}\right ) -a_{t}V_{H}\left ( 1-\frac {\partial \epsilon }{\partial \alpha }\right ) +\frac {\partial C_{mp}}{\partial \alpha }\end {array} \)

Two versions of \(\frac {\partial C_{m}}{\partial \alpha }\) one for \(\alpha _{wb}\) and one one uses \(\alpha \)


\( \begin {array}[c]{lll}C_{m_{0}} & = & C_{m_{ac_{wb}}}+C_{m_{o_{p}}}+a_{t}\bar {V}_{H}\left (\epsilon _{0}+i_{t}\right ) \left [ 1-\frac {a_{t}}{a}\frac {S_{t}}{S}\left (1-\frac {\partial \epsilon }{\partial \alpha }\right ) \right ] \\ \bar {C}_{m_{0}} & = & C_{m_{ac_{wb}}}+\bar {C}_{m_{o_{p}}}+a_{t}V_{H}\left (\epsilon _{0}+i_{t}\right ) \end {array} \)

\(C_{m_{0}}\) 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.


\(\bar {C}_{m_{0_{p}}}= C_{m_{0_{p}}}+(\alpha -\alpha _{wb}) \frac {\partial C_{mp}}{\partial \alpha }\)


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

Used to determine \(h_{n}\)

4.2.1 definitions

  1. 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.

  2. stall from

    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.
  3. Aerodynamics in road vehicle wiki page
  4. some demos relating to airplane control

  6. Lectures Helicopter Aerodynamics and Dynamics by Prof. C. Venkatesan, Department of Aerospace Engineering, IIT Kanpur
  7. has easy to understand definitions airplane geometry. "The MAC is the mean average chord of the wing"
  8. airfoil design software

4.3 images and plots collected

These are diagrams and images collected from different places. References is given next to each image.

This below from

From and


Images from and Flight dynamics principles by Cook, 1997.



From 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

Image from

Image from

Image from FAA pilot handbook and

Image and

4.4 Some strange shaped airplanes


Image from

Image from

Image from "Boeing Pelican ground effect vehicle"


4.6 references

  1. Etkin and Reid, Dynamics of flight, 3rd edition.
  2. Cook, Flight Dynamics principles, third edition.
  3. Lecture notes, EMA 523 flight dynamics and control, University of Wisconsin, Madison by Professor Riccardo Bonazza
  4. Kuethe and Chow, Foundations of Aerodynamics, 4th edition