We discuss in detail all of the required aerodynamic and aerothermal information that is needed in order to launch a 200g, 1kg, and 5kg payload into low earth




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Mach number – comes directly from trajectory’s output. Any assumptions trajectory makes in their code can also be regarded as assumptions for the aero code.


Cd (historical) - The coefficient of drag with respect to Mach number and angle of attack for a multitude of historically successful launch vehicles was analyzed and compressed into a single equation for each region of flight, subsonic, transonic, and supersonic. These equations became the basis for the historical coefficient of drag code.


Normal – the normal force acting on the launch vehicle was found by assuming linear supersonic flow. The launch vehicle was broken up into sections so the coefficient of normal force could be calculated for each section. Once each section had been found, their values were added together to output the final coefficient of normal force, which is then multiplied by the dynamic pressure.


Pitching moment – the pitching moment acting on the launch vehicle was found in the same manner as the normal force stated above. Linear supersonic flow was also assumed.


Xcp – the location of center of pressure is measured in meters from the tip of the nose. The same assumptions used in solving CN and CM are also assumed in the case of solving Xcp as the location of the center of pressure is simply the ratio of the pitching moment coefficient to the normal coefficient.


Shear – The shear force acting on the launch vehicle is found by integrating the normal force over a specified length of the launch vehicle.


Bending Moment – the moment is computed by measuring the center of pressure for a specified length of the launch vehicle and then multiplying the distance between that center and the stage by the integrated normal force.


Axial – The axial force acting on the launch vehicle is found by integrating the pressure distribution around the launch vehicle and multiplying it by the radius of the launch vehicle at each measured location. These values are then added up in much the same manner as the CM and CN were.


CD (dimensional) – Calculated using the axial force coefficient, normal force coefficient, and angle of attack. Assumptions used in finding those values hold here as well.

Important Notes:

In the final design, trajectory’s code uses Cd (historical), and the angle of attack is considered to be 0 degrees at all times.


Input Section:

The call line of the function is:


[M, Cd, Normal, Pitching_Moment, Xcp, Shear_Force, Bending_Moment, Axial, CD] = call_aerodynamics(V,r,a,D)

All of the variables that are passed into the function are described below:


Variable Name

Description

V


Velocity – can be input in the form of an array, or as a single variable [UNITLESS]


a


Angle of Attack – measure of the angle between the free stream velocity and the centerline of the launch vehicle. Can be input in the form of an array, or as a single variable. It must, however, have the same length as M.

It would be wise to check: length(a) = length(M)? [DEGREES]


r

Radius – measure of distance from surface of the Earth to the rocket. This value cannot exceed 290,000 ft, as this is considered the end of the atmosphere. It also happens to be the location at which atmosphere4.m falls apart.

D


Rocket Dimensions – Must be input as a matrix comprised in the following manner:


D = [N,L,D,T]

N - the location of the geometry in order starting from the nose of the launch vehicle

- example: nose = 1, shoulder = 2, stage 2 = 3,…

L - length of specified portion of launch vehicle [METERS]

-example: nose = 3, shoulder = 4, stage 2 = 6,…


D - diameter of specified portion of launch vehicle [METERS]

-example: nose = 1, shoulder = 2, stage 2 = 2,…

T - Type of geometry used in the vehicle

1 specifies conical nose (example: nose cone)

2 specifies cylinder (example: all stages)

3 specifies Shoulder or Boattail (example: skirt)

This is an example of what the Vanguard launch vehicle's "D" matrix would look like:

D = [N, L D T]

D = [ 1,86.4,.84, 1

2,159-86.43,.84,2;

3,358-159,.84,2;

4,57.638,1.14,3;

5,448.19,1.14,2; ]



Output Section:


Variable Name

Description

M


Mach number [UNITLESS]


Cd


Coefficient of drag (historical)[UNITLESS]


Normal


Normal Force [N]


Pitching Moment

Pitching moment [N-METERS]

Xcp

Location of Center of Pressure – measured from tip of nose cone [METERS]

Shear

Shear force – shear force acting on each stage [N]

Bending Moment

Moment – moment force acting on each stage [N-METERS]

Axial

Axial Force – axial force acting on the launch vehicle [N]

CD

Coefficient of Drag (dimensional) [UNITLESS]



Sample Output:

The variables named in the output section will print to the screen. Outputs occur at intervals of 1 second. Nothing other information will be output. Suggestion: write all output variables to an excel file using the xlswrite(‘filename’, variable) command in MATLAB.


User’s Guide for wing_fin.m

Written by Brian Budzinski

Revision 2.0 – 3/29/08

Description:

The purpose of the code ‘wing_fin.m’ is to output all aerodynamic coefficients and loads needed for the addition of a wing and/or fins for an aircraft launch. These outputs include lift coefficient, drag coefficient, normal coefficient, axial coefficient, pitching moment coefficient, and shear coefficient. All of these aerodynamic coefficients and loads are output with respect to Mach number, and angle of attack.


Assumptions:

Several assumptions are made within the ‘wing_fin.m’ function. The assumptions can be further explained as follows:


Mach number – The Mach number is assumed to range from Mach 1.0 to Mach 15.0. This is assumed because for an aircraft launch the initial Mach would be near 1.0 and the wing would remain attached up until near Mach 15.0.


Angle of Attack – The angle of attack is assumed to range from an initial horizontal configuration (AoA = 0°) to a final vertical configuration (AoA = 90°).


Normal – The normal force was found assuming the Newtonian Method. Though the Newtonian Method is geared towards hypersonic flow, the launch vehicle will quickly enter the hypersonic regime with an aircraft launch. Therefore, this remains a reasonable assumption.


Axial – Similarly, the axial force was found assuming the Newtonian Method.


Shear – The shear force acting on the launch vehicle from the wing and/or fin is assumed to be equivalent to the normal force acting on the wing and/or fin itself.


Pitching Moment – The pitching moment is found by calculating the normal and axial forces and multiplying them by their corresponding distances from the center of mass.


Input Section:

All of the variables that are passed into the code are described below:

Variable Name

Description

h

Altitude – measure of the height above ground level, and is initially chosen to begin at 12,000 m for an aircraft launch. [m]


rho

Air Density – an array initially chosen from an altitude of 12,000 m to a final altitude of 205,000 m. [kg/m3]


T


Variable Name

Air Temperature – an array initially chosen from an altitude of 12,000 m to a final altitude of 205,000 m. [K]

Description

Mach

Mach Number – an array initially chosen to range from Mach 1.0 to Mach 15.0. [Non dimensional]


alpha

Angle of Attack – measure of the angle between the free stream velocity and the centerline of the launch vehicle. Initially chosen to range from 0 to 90 degrees for an aircraft launch. [Degrees]


taper_ratio

Taper Ratio – measure of the taper ratio of the wing attached to the rocket. [Non dimensional]


c

Chord – measure of the chord length of the wing. [m]


sweep

Wing Sweep – the angle at which the wings are swept backwards. [Radians]


lambda_eff

Effective Sweep – the effective sweep angle [Radians]


S, S_w, S_LS

Reference Wing Area [m2]


S_fin

Reference Fin Area [m2]


length_LE

Leading Edge Length [m]


length_fin

Fin Length [m]


Rn

Nose Radius – Radius of the nose of the launch vehicle. [m]


RLE

Leading Edge Radius – Radius of the leading edge of the wing. [m]


Rf

Fin Radius – Radius of the leading edge of the fin(s). [m]


Xn

Axial distance from nose to center of mass. [m]


Zn

Radial distance from nose to center of mass. [m]


XLE

Axial distance from wing leading edge to center of mass. [m]


ZLE

Radial distance from wing leading edge to center of mass. [m]


XLS

Axial distance from wing lower surface to center of mass. [m]


ZLS

Radial distance from wing lower surface to center of mass. [m]


Xfin

Axial distance from fin to center of mass. [m]


Zfin

Radial distance from fin to center of mass. [m]


Output Section:

All of the variables that are passed out of the code are described below:

Variable Name

Description

Cn

Normal Force Coefficient [Non dimensional]


Ca

Axial Force Coefficient [Non dimensional]


CL

Lift Coefficient [Non dimensional]


CD

Drag Coefficient [Non dimensional]

CM

Pitching Moment Coefficient [Non dimensional]

q

Shear Force Coefficient [Non dimensional]


Sample Output:

The variables listed in the output section will be plotted as functions of angle of attack.



Author: Brian Budzinski
1   2   3   4   5   6   7

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