Chapter 4
Dynamic of flights

Figure 4.1: Dynamic of flights

4.1 Wing geometry

Figure 4.2: Wing geometry

\(C_{r}\) below is the core chord of the wing.

Figure 4.3: 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

Figure 4.4: equations of longitudinal equilibrium

4.2 Summary of main equations

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	\(C_{m_{w}}=C_{m_{ac_{w}}}+\left ( C_{L_{w}}+C_{D_{\min }}\alpha _{w}\right ) \left ( h-h_{n_{w}}\right ) +\left ( C_{L}\alpha _{w}-C_{D_{w}}\right ) \frac {z}{\bar {c}}\)	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 \(\left ( C_{L}\alpha _{w}-C_{D_{w}}\right ) \ll 1\)
5	\(C_{m_{w}}=C_{m_{ac_{w}}}+C_{L_{w}}\left ( h-h_{n_{w}}\right ) \)	simplified wing Pitching moment
6	\(\begin {array} [c]{lll}C_{m_{wb}} & = & C_{m_{ac_{wb}}}+C_{L_{wb}}\left ( h-h_{n_{w}}\right ) \\ & = & C_{m_{ac_{wb}}}+\frac {\partial C_{L_{wb}}}{\partial \alpha _{wb}}\alpha _{wb}\left ( h-h_{n_{w}}\right ) \\ & = & C_{m_{ac_{wb}}}+a_{wb}\alpha _{wb}\left ( h-h_{n_{w}}\right ) \end {array} \)	simplified pitching moment coefficient  due to wing and body about the C.G. of the airplane. \(\alpha _{wb}\) is the angle of attack
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. \(\bar {l}_{t}=l_{t}+\left ( h-h_{n_{wb}}\right ) \bar {c}\)
13	\(C_{m_{t}}=-\bar {V}_{H}C_{L_{t}}+C_{L_{t}}\frac {S_{t}}{S}\left ( h-h_{n_{wb}}\right ) \)	pitching moment coefficient due to tail expressed using \(\bar {V}_{H}\). This is the one to use.
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	\(\begin {array} [c]{lll}C_{m} & = & C_{m_{wb}}+C_{m_{t}}+C_{m_{p}}\\ & = & \left [ C_{m_{ac_{wb}}}+C_{L_{wb}}\left ( h-h_{n_{w}}\right ) \right ] +\left [ -\bar {V}_{H}C_{L_{t}}+C_{L_{t}}\frac {S_{t}}{S}\left ( h-h_{n_{wb}}\right ) \right ] +C_{m_{p}}\\ & = & C_{m_{ac_{wb}}}+\overbrace {\left ( C_{L_{wb}}+C_{L_{t}}\frac {S_{t}}{S}\right ) }^{C_{L}} \left ( h-h_{n_{w}}\right ) -\bar {V}_{H}C_{L_{t}}+C_{m_{p}}\\ & = & C_{m_{ac_{wb}}}+C_{L}\left ( h-h_{n_{w}}\right ) -\bar {V}_{H}C_{L_{t}}+C_{m_{p}}\end {array} \)	simplified total Pitching moment coefficient about airplane C.G.
17	\(\begin {array} [c]{lcl}\frac {\partial C_{m}}{\partial \alpha } & = & \frac {\partial C_{m_{ac_{wb}}}}{\partial \alpha }+\frac {\partial C_{L}}{\partial \alpha }\left ( h-h_{n_{w}}\right ) -\bar {V}_{H}\frac {\partial C_{L_{t}}}{\partial \alpha }+\frac {\partial C_{m_{p}}}{\partial \alpha }\\ C_{m_{\alpha }} & = & \frac {\partial C_{m_{ac_{wb}}}}{\partial \alpha }+C_{L_{\alpha }}\left ( h-h_{n_{w}}\right ) -\bar {V}_{H}\frac {\partial C_{L_{t}}}{\partial \alpha }+\frac {\partial C_{m_{p}}}{\partial \alpha }\end {array} \)	derivative of total pitching moment coefficient \(C_{m}\) w.r.t airplane angle of attack \(\alpha \)
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	\(\begin {array} [c]{lcl}\frac {\partial C_{m}}{\partial \alpha } & = & \frac {\partial C_{L}}{\partial \alpha }\left ( h-h_{n}\right ) \\ C_{m_{\alpha }} & = & C_{L_{\alpha }}\left ( h-h_{n}\right ) \end {array} \)	rewrite of \(C_{m_{\alpha }}\) in terms of \(h_{n}\). Derived using the above two equations.
20	\(k_{n}=h_{n}-h\)	static margin. Must be Positive for static stability

4.2.0.1 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.

#	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 wing-body 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	\(\begin {array} [c]{lll}C_{m_{\alpha }} & = & a\left ( h-h_{n_{wb}}\right ) -a_{t}\bar {V}_{H}\left ( 1-\frac {\partial \epsilon }{\partial \alpha }\right ) +\frac {\partial C_{mp}}{\partial \alpha }\\ C_{m_{\alpha }} & = & a_{wb}\left ( h-h_{n_{wb}}\right ) -a_{t}V_{H}\left ( 1-\frac {\partial \epsilon }{\partial \alpha }\right ) +\frac {\partial C_{mp}}{\partial \alpha }\end {array} \)	Two versions of \(\frac {\partial C_{m}}{\partial \alpha }\) one for \(\alpha _{wb}\) and one one uses \(\alpha \)
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 \(h_{n}\)

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.

   

   Figure 4.5: diagram from 5th ed, by Chow and Kuethe

2. 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.
    
   

3. Aerodynamics in road vehicle wiki page

4. some demos relating to airplane control http://demonstrations.wolfram.com/ControllingAirplaneFlight/

   http://demonstrations.wolfram.com/ThePhysicsOfFlight/

5. http://www.americanflyers.net/aviationlibrary/pilots_handbook/chapter_3.htm
6. Lectures Helicopter Aerodynamics and Dynamics by Prof. C. Venkatesan, Department of Aerospace Engineering, IIT Kanpur http://www.youtube.com/watch?v=DKWj2WzYXtQ&list=PLAE677E56C97A7C7D
7. http://avstop.com/ac/apgeneral/terminology.html has easy to understand definitions airplane geometry. "The MAC is the mean average chord of the wing"
8. http://www.tdmsoftware.com/afd/afd.html 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.

Figure 4.6: Main forces on airplane

Figure 4.7: Local stability

Figure 4.8: diherdal angle

Figure 4.9: Definitions

This below from http://www.grc.nasa.gov/WWW/k-12/UEET/StudentSite/dynamicsofflight.html

Figure 4.10: Drag coefficient

http://www.grc.nasa.gov/WWW/k-12/airplane/alr.html

Figure 4.11: Roll, Yaw and Pitch

From http://en.wikipedia.org/wiki/Lift_coefficient and http://en.wikipedia.org/wiki/File:Aeroforces.svg

Figure 4.12: Forces diagram

from http://adg.stanford.edu/aa241/drag/sweepncdc.html

Figure 4.13: From sweepncdc website

Images from http://adamone.rchomepage.com/cg_calc.htm and Flight dynamics principles by Cook, 1997.

Figure 4.14: From Flight dynamics principles by Cook

From http://chrusion.com/BJ7/SuperCalc7.html

Figure 4.15: From chrusion site

From http://www.willingtons.com/aircraft_center_of_gravity_calcu.html

Figure 4.16: center of gravity

From http://www.solar-city.net/2010/06/airplane-control-surfaces.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."

Figure 4.17: airplane control surfaces

Images from flight dynamics principles, by Cook, 1997.

Figure 4.18: from flight dynamics principles, by Cook

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

Figure 4.19: from Performance, stability, dynamics and control

Image from http://www.americanflyers.net/aviationlibrary/pilots_handbook/chapter_3.htm

Figure 4.20: from pilots handbook

Image from http://www.americanflyers.net/aviationlibrary/pilots_handbook/chapter_3.htm

Figure 4.21: from pilots handbook (2)

Image from FAA pilot handbook and http://www.youtube.com/watch?v=8uT55aei1NI

Figure 4.22: from FAA pilot handbook

Image http://www.youtube.com/watch?v=8uT55aei1NI and http://www.youtube.com/user/DAMSQAZ?feature=watch

Figure 4.23: from youtube

Figure 4.24: from youtube (2)

4.4 Some strange shaped airplanes

Image http://edition.cnn.com/2014/01/16/travel/inside-airbus-beluga/index.html?hpt=ibu_c2

Figure 4.25: airbus beluga 1

Figure 4.26: Concorde

Image from http://edition.cnn.com/2014/01/16/travel/inside-airbus-beluga/index.html?hpt=ibu_c2

Figure 4.27: airbus beluga (2)

Image from http://www.nasa.gov/centers/dryden/Features/super_guppy.html

Figure 4.28: NASA SGT super guppy

Image from http://www.aerospaceweb.org/question/aerodynamics/q0130.shtml "Boeing Pelican ground effect vehicle"

Figure 4.29: pelican 01

4.5 links

1. https://3dwarehouse.sketchup.com/search.html?redirect=1&tags=airplane

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