DESCRIPTION OF THE PROGRAM

Transcription

1 1 DESCRIPTION OF THE PROGRAM TwoDoFSim_BB VERSION 1. May 213.

2 2 TABLE OF CONTENTS TABLE OF CONTENTS2 1. INTRODUCTION3 1.1 ABBREVIATIONS:... ERROR! BOOKMARK NOT DEFINED. 1.2 GENERAL INFORMATION ABOUT PROGRAM... 3 Purpose and Possiility of the Program... 3 Short Description... 3 Limitations... 4 Standard Conditions INPUT DATA AND EXECUTION OF THE PROGRAM5 2.1 STRUCTURE AND REVIEW OF INPUT DATA... 5 Projectile data file (PMMdata_.m) MODELING INPUT DATA FOR ARTILLERY ROCKETS... 8 Total Impulse and Burning Time... 8 Aerodynamic Drag Coefficient EXECUTION OF THE PROGRAMS... 9 Execution of the Main program TwoDoFsim_BB.m... 9 Execution of the Program Geom_InternalBurningTueWithSlots.m RESULTS NUMERICAL RESULTS GRAPHS DIAGRAMS Graphs Generated y Grain Geometry Data File Graphs Generated y Main program... 16

3 3 1. INTRODUCTION 1.1 GENERAL INFORMATION ABOUT PROGRAM Purpose and Possiility of the Program Capaility:Calculation of projectile trajectory and firing elements ased on Euler equations of motion for point mass (two degrees of freedom). Method:Numerical integration of Euler differential equations of motion with two degrees of freedom of projectile and integration of differential equation of urning the propellant. Purpose of program is quick calculation of trajectory of unguided projectile including firing elements (firing prolem solution). It can also e used for: the determination of rocket motor thrust characteristics (total impulse, urning time) for the design of artillery rockets, the determination of rocket motor thrust characteristics (total impulse, urning time, optimal ignition time) for the extension of range of classical artillery projectiles and mortar shells, the design of the ase-leed unit for the extension of range of classical artillery projectiles, Motion of projectile is modeled y four differential equations of the first order. Progression of urning area is descried y first order differential equation. Equations are solved numerically y fourth order Runge-Kutta method. Burning area with urning thickness of ase-leed fuel can e entered through specified separate file, or calculated in the program for common geometry. Program is written in Matla. It automatically prints all characteristic functions of the process in time and produces graphs. Short Description Program numerically solves (integrates) four differential equations which define flight of projectile for a given initial conditions dv K =- g sin q (1.1) dt

4 4 do g =- coso (1.2) dt V K dx = VK coso dt (1.3) dh =- VK sin q dt (1.4) x a x = L (1.5) d dt and differential equation of urning the fuel (propagation of urning area in time) dy J r x dt = (1.6) In the aove system of equations independent variale is time t. The five dependent variales are V (or simple V ) projectile velocity, q( = g) path angle angle from horizontal line to the K velocity vector, x range, h height from ground, x x spin rate, L a aerodynamic rolling moment, J x axial moment of inertia, and y urning thickness of fuel. Burning rate r depends on pressure in the chamer p c, which itself depends on projectile ase pressure, and consequently on amient atmospheric pressure. Pressure in chamer is determined y solving nonlinear equation which defines quasi steady mass flow rate through BB unit opening. For the meaning of other quantities see lecture supplement and nomenclature in this manual. Initial conditions are defined at launcher muzzle at initial time which is usually taken to e zero: t = : V = V, q= q, x = x =, h = h =, y =. Terminal condition for process in ase-leed unit is defined y maximal urning thickness: y = y : t = t y. max.max Terminal condition for trajectory is defined y minimum height (impact to ground): h : t = t. For the integration of system of differential equations the fourth order Runge-Kutta solver is used. Limitations Main limitations of the program are: Trajectory is in vertical plane derivation and deviation due to disturances cannot e calculated. Earth is flat and stationary (Coriolis force is not taken into account). e

5 5 Only one propellant with homogeneous characteristics can e considered. Burning law of ase-leed fuel is exponential (no free term). Standard Conditions Standard conditions in this software are defined y: Standard gravity gn ms 2 Air temperature on MSL T 15 C Atmospheric pressure on MSL pa ar Temperature of rocket motor propellant T. 2 C Temperature of ase-leed fuel T. 2 C a pf pn 2. INPUT DATA AND EXECUTION OF THE PROGRAM 2.1 STRUCTURE AND REVIEW OF INPUT DATA Basically input data are supplied through two input files: A. Grain geometry data file FileName.mat (default file name is yavaplp.mat) B. Projectile data file FileName.m (default file name is PMMdata_.m). Both of them are Matla files (generated y Matla). The relations with notation in theory is given in the following tale. Tale 1 Grain geometry data file (yavaplp.mat) yavaplp.mat No. Quantity Symol in file in theory Unit Description 1. y y [m] Burnt or urning thickness of propellant 2. A A [m 2 ] Burning area BurntVol V BurntMass m V [m 3 ] Volume of urnt mass m [kg] Burnt mass 5. Ap A p [m 2 ] Port area 6. lp l p [m] Port perimeter

6 6 User can generate that file either y using analytical formulas or y any 3D solid modeler CAD program (SolidWorks, AutoCAD, CATIA...) and transfer to Matla and make grain geometry data file. In oth cases finally it should e made in Matla in order to e written in.mat format. Projectile data file (PMMdata_.m) File contains necessary data to calculate trajectory and process in ase-leed unit. No. Quantity Symol in file in theory Unit Tale 2 Projectile data file (PMMdata_.m) PMMdata_.m Description 1. clc; Matla statement clears Matla work space 2. close all; Matla statement Closes all Matla opened figures 3. Title Identifier of weapon/projectile. It serves as information purposes not significant for calculation. It is printed in work space. 4. dref d [m] Reference length - calire 5. Sref S [m 2 ] Reference area. Usually S d mass m [kg] Projectile initial mass 7. twistd j [deg] Barrel rifling twist angle 8. Jx m [kgm 2 ] Projectile initial moment of inertia Drag form factor - correction factor for drag coefficient. For the example in this report (Rdz_Fr_155mm) i et 1, ecause coded drag coefficient is calculated exactly for the projectile considered in 9. iet i et [-] the example. For any other shape of projectile, or any other coded drag coefficient drag form coefficient should e determined to achieve range otained y experiment. See also section Coding the Aerodynamic Coefficients in this report and corresponding LS. 1. IdentRM [-] RM operation On/Off identifier. For IdentRM= operation of rocket motor is off.. For IdentRM=1 operation of rocket motor is on. 11. Itot IT, Itot [Ns] Total impulse of RM 12. propmass m p [kg] Propellant mass of RM 13. turn t [s] Burning time of RM 14. tign t ign [s] Ignition time of RM 15. IdentBB [-] 16. tignbb t i.bb [s] Ignition time of BB BB operation On/Off identifier. For IdentBB= operation of rocket motor is off.. For IdentBB=1 operation of rocket motor is on. 17. dase d [m] Base diameter of projectile ase diameter of BB unit 18. Sase S [m 2 ] Base area, S d 2 4

7 7 19. rhof r f [kg/m 3 ] Density of BB fuel 2. Lp L p [m] Length of BB grain of BB grain 21. Ns N s [-] Numer of slots of BB grain 22. Dp D p [m] Diameter (outer) of grain of BB fuel 23. Di D i [m] Inner diameter of BB grain 24. s s [m] Slot width of BB grain Diameter of cylindrical part with nozzle. This diameter exists when 25. dm d [m] BB unit is around the RM nozzle. For the design case with no RM m dm. 26. Ae A e m 2 Opening (exit) area 27. Geom_File() Name of file which generates file yavaplp.mat which contains geometrical characteristics of BB grain and urning geometry with respect to urning thickness. In this example file name is Geom_InternalBurningTueWithSlots (); See Grain Geometry Data File section. The file can e disaled y inserting character % efore the file name. Than file yavaplp.mat is not generated y running the file, ut it may e generated y earlier run of this file, or it can e generated upon the running any other program. 28. Rgas R Gas (specific) constant of BB urning product. In this program it is [J/kgK] assumed that it is independent of pressure in the chamer. 29. Tflame T f [K] 3. kappa k 31. etath h q 32. cetad z d 33. r [m/s] 34. nr n 35. krp k 1 Adiaatic flame temperature of BB fuel. In this program it is assumed that it is independent of pressure in the chamer. Ratio of specific heats. In this program it is assumed that it is independent of pressure in the chamer. Coefficient of thermal reduction. It is assumed that mean temperature in chamer is less than flame temperature y amount defined y this coefficient, so that Tc Tf. Practical values for the coefficient are The coefficient is not implemented in this version it will e implemented in the next version. Coefficient of discharge efficiency. It is assumed that discharge mass flow rate is less that theoretical one y amount defined y this coefficient, so that m e dm e. theor. Practical values for the coefficient are The coefficient is not implemented in this version it will e implemented in the next version. n Coefficient in urning rate law defined y r pc for standard BB fuel temperature Tpf. 21 C and stationary BB unit (nonaccelerated, non-spinning propellant). Note that value of coefficient is for pressure is in Pascals. n Exponent in urning rate law defined y r pc. The value ranges from to 1 depending of propellant composition. Note that higher values than 1 generate unstale urning process in chamer. Coefficient of the influence of spin rate on urning rate r k 1 p of BB fuel. It depends on spin rate and of fuel composition. The value ranges from 1.1 to 1.2 for typical spin stailized projectiles, n c

8 8 36. etar [1/K] 37. Tac Ta [ C] Ground air temperature 38. pa pa 39. Windx W x ut user can enter experimentally determined value for corresponding spin rate. Note that spin vs. time is calculated in the program and can e used for coding function k1 f x, x eing spin rate (pspin). Coefficient of the influence of propellant temperature on urning p pn rate T T. r n e k1 pc. The value ranges from.12 to.16 depending of propellant composition. The coefficient is not implemented in this version it will e implemented in the next version. [mar] Ground air pressure [m/s] 4. V V [m/s] 41. ithetad q i [deg] Initial elevation angle, i Ground longitudinal wind. Positive value is in the direction of flight (tail wind). Negative value is in opposite direction of flight (head wind). Muzzle velocity - Projectile initial velocity with respect to ground. It should e greater than zero. 42. dthetad dq [deg] Increment of elevation angle 43. NThetad n q [-] Numer of elevation angles 44. Ts T s 45. jprint 46. PrintToFileID [s] Integration step for solving system of differential equations which define process in RM. To large integration step can produce instaility in integration. It is recommended to choose 1ms at start and carefully increase the value preserving desired accuracy Printing step ratio ratio of the printing step and integration step. It is always greater than unity. For example, to print each tenth point otain y integration, put jprint=1 Output file Identifier. PrintToFileID= Results printed to Matla Workspace; PrintToFileID=1 Results printed to file PPMResults.txt; 2.2 MODELING INPUT DATA FOR ARTILLERY ROCKETS Total Impulse and Burning Time Program does not calculate motion of projectile in arrel or on the launcher. The initial velocity should e greater than zero. For artillery rocket user ought to enter muzzle velocity as for the artillery shells. For this reason total impulse of the rocket motor for the remaining of flight should e reduce for the amount consumed to oost rocket to muzzle velocity. Aerodynamic Drag Coefficient Aerodynamic drag coefficient is coded in the program (see section Coding the Aerodynamic

9 9 Coefficients ). For spin stailized projectiles it is suggested to use C D.43 etalon coefficient. For fin stailized projectiles CD. 58 etalon coefficient is more appropriate (see LS Basis of External Ballistics - Point Mass Model ). Based on used chosen drag coefficient, drag form factor is to e determined from firing, or from firing tales. 2.1 EXECUTION OF THE PROGRAMS Execution of the Main program TwoDoFsim_BB.m To run the main program doule click on TwoDoFsim_BB.m. The main program window appears. Or, alternatively run Matla program first and open the file TwoDoFsim_BB.m. In oth cases Matla environment should e opened. Figure Matla environment and main program. Upon the running of the program the following dialog appears for opening/running file with projectile data,

10 1 Figure Dialog for opening file with rocket motor data. And, after that, dialog for opening/running file with grain geometry data appears. User can search any directory and choose any appropriate data. Note that default offered file yavaplp.mat is generated y program Geom_File (Geom_InternalBurningTueWithSlots.m in this example) called in PMMdata_.m file (PMMdata_Rdz_FR_155mm.m in this example). Figure Dialog for opening file with grain geometry data. Execution of the Program Geom_InternalBurningTueWithSlots.m The Program (function) Geom_InternalBurningTueWithSlots(Lp,Ns,Dp,Di,s,rhop,PrintID,PlotID); which is used for generating the geometry data, can e called on two ways: with arguments (like aove) and without arguments geometrical parameters. When it is called with arguments it uses arguments which are passed to function. When it is called without

11 11 arguments it uses geometrical parameters coded in the ody of function. For example calling y statement (no arguments) Geom_InternalBurningTueWithSlots(); effects in using the parameters coded in function ody %============================================================================== function Geom_InternalBurningTueWithSlots(Lp,Ns,Dp,Di,s,rhop,PrintID,PlotID); % calculation of urning area for internal-burning Tue with slots % Side surfaces and outer surface are inhiited. % Note: Dimensions in [m] clc; %clear all; %close all; % if nargin == % If numer of input arguments is zero use the following % values for grain characteristics Lp = 1.e-3; % m Length Ns = 3; % - Numer of slots Dp = 116.7e-3; % m Outer diameter of the grain Di = 6.e-3; % m Inner diameter of the grain s = 3.e-3; % m Slot width we =.5 * (Dp - Di); % m We rhop = 1598; % kg/m³ Density PrintID = 1; % Printing identifier; % PrintID= - No printing, % PrintID=1 - Printed in work space PlotID = 1; % Ploting identifier; % PlotID= - No ploting, % PlotID=1 - Graphs are plotted end; % %

12 12 4. RESULTS 4.1 NUMERICAL RESULTS Upon the successful run of the program results are printed in workspace. Here is an example of eight trajectories (ithetad=42.5, NThetad=8, dthetad=2.5) for one hypothetical 155mm projectile with rocket motor and ase-leed unit. Maximal range is for elevation 56.1deg. TRAJECTORIES FOR SEARCHING MAXIMAL RANGE Theta Range El TOF Vertex Vterm FallAng [deg] [m] [mils] [s] [m] [m/s] [deg] Data at maximal range After the calculation of these set of trajectories program calculates and prints results for maximal range.

13 13 No. Quantity Symol in file in theory Unit Tale 3 Data printed in Work space and in Graphs IBRMResults.txt 1. Theta q [deg] Initial elevation Description Standard set of data printed always 2. Range x e [m] Range 3. El QE [mil] Initial elevation in mils Quadrant elevation, (division 64) 4. TOF t e [s] Time of flight 5. Vertex MaxOrd h s [m] Maximal ordinate 6. Vterm V e [m/s] Terminal speed 7. FallAng q e [deg] Falling angle 8. Time t [s] Time 9. Height h [m] Height aove ground 1. VK V K [m/s] Velocity with respect to ground 11. Theta q [deg] Elevation 12. spin x x, p [rad/s] Spin rate, axial angular speed 13. mass m [kg] Mass 14. pa p a 15. p p 16. pc p c [mar] Amient atmospheric pressure [mar] Base pressure pressure on the ase of projectile [mar] Chamer pressure 17. Cred C red [-] Drag reduction coefficient 18. Cp C p [-] Base drag coefficient without influence of BB 19. Cp C p [-] Base drag coefficient with influence of BB 2. Ma Ma [-] Flight Mach numer 21. mp, mf mf ( t ) [kg] Current mass of BB fuel 22. mdot mfdot m [kg/s] Mass flow rate 23. A A [cm²] Burning area 24. Inject I [-] Base injection parameter 25. Mj M j [-] Mach numer in BB jet at exit area 26. CD CD [-] Drag coefficient without influence of BB 27. CDBB CD. BB [-] Drag coefficient with influence of BB 28. CD CD [-] Base drag coefficient without influence of BB 29. ddrag D D [-] Increment of drag due to influence of BB

14 14 Example of the output data is shown elow. Mass of BB fuel = 1.2 kg Mass of RM prop. = 3.8 kg BASIC QUANTITIES OF TRAJECTORY Time Range Height VK Theta spin mass pa p pc Cred Cp Cp Ma mp mdot A Inject Mj Time [s] [m] [m] [m/s] [deg] [rad/s] [kg] [mar] [mar] [mar] [-] [-] [-] [-] [kg] [g/s] [cm^2] [1^3] [-] [s]

15 GRAPHS DIAGRAMS Graphs Generated y Grain Geometry Data File 45 Variation of urning area with urnt thickness 5 Variation of urning area with urnt mass A [cm²] y [cm] A [cm²] m [kg] Variation of port area with urnt thickness 4 Variation of port perimeter with urnt thickness Ap [cm 2 ] y [cm] lp [cm] y [cm]

16 16 Graphs Generated y Main program Trajectories for Searching Maximal Range 2.5 x Trajectory 14 2 Height, [m] Range, [m] x 1 4 Range, [m] 5.1 x 14 Range vs. theta Theta, [deg]

17 17 Quantities on Trajectory with Maximal Range 9 Velocity 8 Velocity, [m/s] Time, [s] 2 Drag and Spin rate Drag, [N] 15 1 Drag DragBB ddrag Spin rate, [rad/s] Time, [s]

18 18 CD, CDBB, [-] Drag coefficients vs. Mach CD CDBB CD Mach, [-] 12 1 Pressure: amient, in chamer, on ase pa p pc Pressure, [mar] Time, [s]

19 19 2 Trajectory Height, [km] Velocity, [m/s] Time, [s] Range, [km]

20 2 1 Base-leed Characteristics mp, [kg] mdot, [kg/s] A, [cm 2 ] Mj, [-] Time, [s]

21 21 -Cp, [-] Cp, [-] Cred, [-].5 Ma, [-] Time, [s] Time, [s]