1 16.070 Introduction to Computers and Programming 5 April Recitation 8 Spring 2001 Topics Procedure for the Simulation of Differential Equations (By Example): • Concept • Equations of motion • Continuous State Space • Discrete State Space • Algorithm • Coding • Analysis Procedure for the Simulation of Differential Equations Concept A Mars Lander(/rock/ball) is suspended 500m from the surface of the earth and released. Simulate the dynamics of the problem as the vehicle falls to earth. Only 1-DOF is required: the vertical altitude of the vehicle above the surface of the earth. Assume a constant gravitational acceleration of g=9.81m/s/s. The vehicles mass is 500kg (does this matter?). The vehicle does not thrust and drag is neglected. Write the vehicles state history to a file, in the format: <time> <displacement> <velocity> Figure 1 Lander suspended above the surface of the earth (NO THRUST!) x mg2 Equations of motion Newton’s second law: xmmaF== Initial Conditions: ()()005000==xx Forces acting on the vehicle: mgF−= Complete simplified equation on motion: mgxm−= Continuous State Space The states are similar to the specific of initial conditions that are required: State Vector: ==Χxxxx21 Continuous State Space Representation: )(100010mgm−+Χ=Χ Discrete State Space Approximate dtdx with txxnn∆−+1 then we get the following Discrete State Space result: )(10001021212111mgmxxtxxtxxnnnnnn−+=∆−∆−++ )(0101211211mgmtxxtxxnnnn−∆+∆=++3 Algorithm Variables and constants: mgFtttmtBtAnn−=ΧΧ==∆=∆=∆=+1max02001.000101 Basic Pseudo-code (A flowchart will be added): MAIN MODULE [0] Start [1] Declare all the variables/constants above [2] Open an output file for telemetry [3] Set initial conditions SEPARATE MODULE [4] Write initial states to a file SEPARATE MODULE [5] Increment time (t=t+dt) [6] If t>tmax, goto XX (IMPLEMENTED WITH A WHILE LOOP) [7] Update State (Xn+1=AXn+BF) SEPARATE MODULE [8] Write states to file SEPARATE MODULE [9] Advance the state Xn=Xn+1 [10] Goto [6] [11] Close all open files [12] Exit Code /* Thomas Jones, April 2001 */ /* Recitation 8 */ /* Simulation of a falling lander */ /* (on earth) */ /* -------------------------------- */ /* ALL UNITS ARE STANDARD METRIC */ #define time_step 0.01 #define out_file "rec8_out.txt" #define mass 500.0 #define grav_accel 9.81 #define initial_position 500.0 #define initial_velocity 0.0 #define max_time 10.04 #include <stdio.h> #include <stdlib.h> /* Prototypes */ void Set_Initial_Conditions(double Xn[]); void Write_To_File(double t, double X[],FILE *ofp); void Update_X(double A[][2],double B[],double F,double Xn[],double Xnp1[]); /* Start main function */ int main(void) { /* Variable Declarations and Setup */ /* declare time, mass and gravitational acceleration */ double t=0; /* time */ const double dt=time_step; /* time step */ const double m=mass; /* mass */ const double g=grav_accel; /* gravity acceleration */ /* declare A, B, F, Xn, Xnp1, such that Xnp1 = AXn + BF */ double A[2][2]={1.0, dt, 0.0, 1.0}; double B[2]={0, dt/m}; double Xn[2]={0.0, 0.0}; double Xnp1[2]={0.0,0.0}; double F=-m*g; /* open telemetry output file */ FILE *ofp; printf("Opening Telemetry File...\n"); if ((ofp=fopen(out_file,"w+"))==0) { printf("File open error!\n"); exit(-1); } printf("File Opened...\n"); /* Implementation of Flow Chart /* Main BODY of Code */ /* ---------------------- */ /* !!!!!!!!!!!!!!!!!!!!!! */ printf("Initializing State Variables...\n"); Set_Initial_Conditions(Xn); /* initial conditions */ Write_To_File(t,Xn,ofp); /* save inital telemetry */ printf("Simulating...Please wait...\n"); while(t <= max_time) { t=t+dt; /* increment time */ Update_X(A,B,F,Xn,Xnp1); /* state update */ Write_To_File(t,Xnp1,ofp); /* save telemetry */ Xn[0]=Xnp1[0]; /* advance state */ Xn[1]=Xnp1[1]; } printf("Simulation Complete!\n"); /* !!!!!!!!!!!!!!!!!!!!!! */ /* ---------------------- */ /* close all open files */ printf("Closing Files...\n"); fclose(ofp); printf("Done!\n\n"); return 0; }5 /* ----------------------------------------- */ /* Function sets initial conditions of state */ /* ----------------------------------------- */ void Set_Initial_Conditions(double Xn[]) { Xn[0]=initial_position; Xn[1]=initial_velocity; return; } /* end Set_Initial_Conditions */ /* ------------------------------------------ */ /* Function writes state and time to file ofp */ /* ------------------------------------------ */ void Write_To_File(double t, double X[],FILE *ofp) { fprintf(ofp,"%2.2lf\t%lf\t%lf\n",t,X[0],X[1]); return; } /* end Write_To_File */ /* ------------------------------------------ */ /* Function performs state update */ /* ------------------------------------------ */ /* PS8 requires this state update to be performed by */ /* a general matrix multiplier function, which is quite */ /* different from the code below !!! */ void Update_X(double A[][2],double B[],double F,double Xn[],double Xnp1[]) { Xnp1[0]=A[0][0]*Xn[0]+A[0][1]*Xn[1]+B[0]*F; Xnp1[1]=A[1][0]*Xn[0]+A[1][1]*Xn[1]+B[1]*F; return; } /* end Update_X */6 Analysis An output file called rec8_out.txt contains the telemetry data. Output: Opening Telemetry File... File Opened... Initializing State Variables... Simulating...Please wait... Simulation Complete! Closing Files... Done! Press
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