Chapter III Aerodynamic Modulation3.1 IntroductionThe purpose of this chapter is to determine the mathematical model of the Storm by incorporate the physical and aerodynamic lows in to its mathematical equivalent model so that mathematical laws can be applied to analyse the behaviour and responses of the Storm. In the second order, this model will be define using DATCOM file. 3.2 Aerodynamic forces During the flight, there are four forces that are applied on every aircraft as it is seen in figure 33. This forces are taken about the aircraft’s centre of mass.

Drag D: due to theair resistance. The air molecules move around the aircraft as it moves through the atmosphere. The molecules that cling to the surface of the aircraft create skin friction. The natural texture of the surface of the aircraft is aerodynamically rough and is specified using the Roughness Height Rating (RHR). The RHR is the arithmetic mean of the surface variation in millionths of a millimetre.D= 1/2 .A..v2.CdWhere:D: Drag (N)A: Area (m2): Air density (Kg.m-3)v: velocity (m.s-1)Cd: Drag coefficient The Lift: This force is created by both Bernoulli Lift and Vortex Lift.It is the forces that opposes to the weight force. Its magnitude depends on several factors such as the shape and it is always directed perpendicular to the flight direction. Aero-dynamic coefficients are non-dimensional numbers that are used to determine the aerodynamic characteristics of an aircraft. We will L= 1/2. A..v2.CLWhere;CL: Lift coefficient Weight: Weight is a force that refers basically to all the mass of the airplane and it is always directed towards the centre of the earth.W = m.gWhere ;m: mass (kg)g: the gravity’s acceleration (m.s-2)Thrust: This force is generated due to the propulsion system, and it opposes to the drag force.3.3 Aerodynamic momentsThe aerodynamic moments are applied at the principle axes of theon the aircraft coordinate system. [7] Rolling: about the body x-axis. Pitching: about the body y-axis. Yawing: about the body z-axis.To choose the direction of the moments we should use the rule of the right hand as follows: A moment is noted positively rolling if the pilot is rotated clockwise around the x-axis. A pitching moment is rated positive if the nose of the aircraft moves towards the positive z-axis. The Yawing Moment is rated positive if the pilot is rotated clockwise around the z axis.The rotation rates with respect to the derivative are: the rolling rate p, the pitch rate q and the yaw rate r. We note k a constant that presents the length of the span in the subsonic regime or the average aerodynamic chord with respect to the pitch speed in the supersonic regime. The interest coefficients derived from depreciation are typically: [ 7]Rolling Moment with respect to Roll Rate Clp,Pitching Moment with respect to Pitch Rate Cmq.Yawing Moment with respect to Yaw Rate Cnr , Rolling Moment with respect to Yaw Rate Clr , Yawing Moment with respect to Roll Rate Cnp , Lift Force with respect to Pitch Rate CLr , Side Force with respect to Roll Rate CYP , Side Force with respect to Yaw Rate CY r . There are also derivatives of the force and moment coefficients with respect to the various control surfaces. These are typically: Pitching Moment with respect to Elevator Deflection Angle Cqґele. Lift Force with respect to Elevator Deflection Angle CL ґeleRolling Moment with respect to Aileron Deflection Angle Cl ґail.Yawing Moment with respect to Aileron Deflection Angle Cnґail ,Rolling Moment with respect to Rudder Deflection Angle Cl ґrud Yawing Moment with respect to Rudder Deflection Angle Cnґ rud It should be noted that there are other derivatives for force and moment coefficients as a function of Mach, altitude and thrust. It should also be noted that the aerodynamic forces, moments and derived coefficients are not dimensioned so that the aerodynamic data of an aircraft are scaled from these coefficients.3.4 Aerodynamic ModelThe standard model representing the aerodynamic equations has 6 DOF degrees of freedom. This model is widely used for nonlinear aircraft aerodynamic modelingin the form of this vectorXT = [U V W P Q R] [9]Body axis 6-DOF equations can be given as Force Equations_U= (RV-QW-g0.sin+Fx)/m V=(-RU-PW-g0-sin.cos)/m W= (QU-PV-g0.sin.cos)/m Kinematic Equations=P+tan.(R.cos+Q.sin)=Q.cos- R.sin= (Q.sin+R.cos)/cos Moment EquationsP= (C1.R+C2.P).Q+C3.L+C4.NQ=C5.R.P-C6.(PІ-RІ)+C7.MR=(C2.P-C8.R).Q+C4.L+C9.NWhereC1=({(JX-JZ).JZ-J^2 XZ})/rC2=({(JX-JY+JZ).JXZ})/rC3=JZ/rC4=JXZ/rC5=((JZ-JX))/JYC6=JXZ/JYC7=1/JYC8=({(JX-JY).JX+JІXZ)/rC9=JX/(JX.JZ-JІXZ)3.5 Aerodynamic stability and control coefficient estimationTherefore, based on the geometry data, the aerodynamics stability and control coefficients (ASCC) of a special Piper J-3 Cub model was estimated by two programs, AVL and DATCOM, in the following parts. DATCOM and AVL are better known software for analytical methods. AVL as a main tool to obtain aerodynamics in this thesis and DATCOM as a complement only be used to estimate dynamics derivatives which cannot be estimated by AVL. Except for those data listed in table 3-1, the wing shape is also crucial in determining the aircraft aerodynamics characteristics. According to [17], NACA 2314 type airfoil is selected to the main wing of the aircraft and NACA 0002 is selected for the empennage. Ultra-light aircraft flight envelops is small, it is only used at low height range, so only sea level altitude is selected to get aerodynamics derivatives and control coefficients of the Piper J-3 Cub model. AVL analysis: Athena Vortex Lattice (AVL) was developed by Dr. Drela and his students of Massachusetts Institute of Technology. It can be used to analyze the aerodynamics stability and control characteristics of subsonic aircraft. The aircraft geometry and inertial data should be depicted in three type files in the AVL program, i.e. Geometry Input File, Mass input File and Run-Case File. The Geometry input file describes the vortex lattice geometry and aerodynamics section properties. Mass input file contain the mass and inertia properties of the configuration. It also defines units to be used to run case setup. These units may want to be different from those used to define the geometry. The analysis flight conditions are defined in the Run-Case File. The coordinate system used in these files is X downstream, Y out the right-wing and Z up. The Geometry input file of the Piper J-3 Cub model is given in Appendix A. With the input files, the analyzed results can be obtained by running the AVL program. The most important input files to generate the output files are Geometry Input File and Run-Case File. With these files, aerodynamics output and trim condition can be calculated, and it can also be used to output the aircraft geometry model. The trim condition mentioned here is different from the trim condition to be introduced later, for the AVL program only can calculate the moments balance. The trim in the later section is means aircraft in forces and moments balance. The modelled aircraft in AVL is presented in Figure 3-1.Aerodynamics coefficients and control derivatives can also be obtained by running the input files in AVL. In order to use the software to produce the linearized coefficients, specific flight conditions should be set beforehand. Operating points like Cruise, Climb and Descent are specified in this thesis, and the obtained coefficients are given in Appendix B according to different velocities. DATCOM methods: DATCOM is developed by the U.S. Air Force, adopting a component combination method with modularity ideas, etc. The estimation method it uses is a theory, semi-empirical and Empirical mixed way. It can provide the recipient accurate degree data to different flight conditions to various airplanes. The DATCOM input file includes four parts, which defines flight conditions and geometric characteristics about wings, fuselage, horizontal and vertical stabilizer, aileron, elevator, flap and slat, etc.[18]. Therefore, in order to create an input file to DATCOM, aircraft geometry should be acquired first. Once the input file has been created, then it can be run in DATCOM to get the output files. The output files contain a 3D geometry model, estimated stability and control coefficients and derivatives for the appointed airplane geometry and flight conditions. It shows that in the process of the programming that if the aircraft main body data was not defined in the input file, it means if only wing plus empennage configuration were specified, the DATCOM would not calculate the dynamics derivatives. So the aircraft body data has to be defined in the DATCOM input file which was obtained by estimation based on the picture from some websites. Figure 3-2 gives the 3-D model described by the input file.ConclusionIn this chapter, we determining the mathematical model based on a theoretical study of the different lows, forces and aerodynamic moments. This model of 6 degree of freedom, is present in a matrix. This model will used to determine the aerodynamic coefficients, particularly: drag coefficient, lift coefficient, take-off speed and distance. And to estimate these coefficients, two solutions are proposed: AVL program and DATCOM. We chose the DATCOM because it gives us more precision over then AVL.