MATLAB in Dynamics

Jordan Tonchev Jordanov (view profile)

22 Nov 2004 (Updated )

Companion Software for "MATLAB with Applications in Engineering Research"

dtxy
```                      function dtxy
% *****************************************************************
%                     P r o g r a m    DTXY
% *****************************************************************
%
%  PURPOSE:
%    Resolve numerically differential equations of plane motion
%    of a particle   m*d2x/dt2 = Fx(t,x,y,xt,yt);
%                    m*d2y/dt2 = Fy(t,x,y,xt,yt),
%     plots trajectory and graphics of coordinates and velocity.
%     If possible, the program could solve the problem analytically.
%
%  INPUT DATA:
%     m    - mass of the body ;
%     Fx   - sum of projections of forces on axis x ;
%     Fy   - sum of projections of forces on axis y ;
%     x0   - initial value of the coordinate x ;
%     y0   - initial value of the coordinate y ;
%     v0   - initial value of velocity ;
%     alfa - angle between v0 and horizontal plane;
%     Tend - upper bound of the integration ;
%     eps  - precision of the integration ;
%     np   - number of parameters .
%     P{1}, P{2}, ..., P{np} - names of the parameters (array of cells);
%
%  NOTES:
%   1. The coordinates are designed by the symbols 'x', 'y'
%      and there first derivatives by 'xt', 'yt';
%   2. The physical names of the parameters are assigned to the
%      cells of the array P like this: P{1}='m', P{2}='c',...;
%   3. For analytical solution the values of Tend, eps, np and P are not
%      needed.
%   4. Initial values x0, y0, v0, alfa must to be entered as strings,
%      even though they represent numbers!
%   5. All the data can be input from file or in interactive mode.
%
%  EXAMPLE of DATA FILE:
% % Dynamics of a projectile, lanched
% % with initial velocity v0 under
% % angle "alfa'
%   m    =  'm';
%   Fx   = '-k*xt';
%   Fy   = '-k*yt - m*9.81';
%   x0   = '0';
%   y0   = '0';
%   v0   = 'v0';
%   alfa = 'alfa';
%   Tend = 10;
%   eps  = 1.e-10;
%   np   = 2;
%   P{1} = 'm';
%   P{2} = 'k';

% ---------------------------------------------------------
%                DATA INPUT OF THE PROBLEM
% =========================================================

clear
disp(' ');
disp(' How will you input the data ?    ');
disp('     1. From a data file;         ');
disp('     2. In interactive mode.      ');
ans = input(' Number of Your choice : '  );
flag = 0;
if ans == 1
while 1
disp(' ');
indat = input(' Input the name of the data file :', 's');
if exist([cd,'\',indat]) % Search only in current directory
eval(indat);
flag = 1; break  % Successful
else               % Unsuccessful
disp(' ');
disp([' File ',indat,' not exist!'])
disp(' You have to:')
disp(' 1. Enter another DATA file name, or')
disp(' 2. Input the DATA interactively !')
if ans2 == 2, break , end
end
end
end
if flag == 0
%Input of the data in-line mode
m    = input(' Mass of the body m : ','s'             );
Fx   = input(' Expression of the projection Fx : ','s');
Fy   = input(' Expression of the projection Fy : ','s');
x0   = input(' Initial coordinate x0 : '              );
y0   = input(' Initial coordinate y0 : '              );
v0   = input(' Initial velocity v0 : '                );
alfa = input(' Angle alfa : '                         );
Tend = input(' Upper bound of the integration Tend : ');
eps  = input(' Precision of the calculations eps : '  );
np   = input(' Number of parameters  np : '           );
% Asigning names of the parameters
if np > 0
disp(' ');
disp(' Enter names of the parameters:')
for i = 1:np
ii = num2str(i);
P{i} = input([' Name of parameter P',ii,': '],'s');
end
end
end

% ---------------------------------------------------------
%            Differential Equations of Motion
% =========================================================

syms t x xt Dx D2x y yt Dy D2y
Fx = subs(Fx, {xt,yt}, {Dx,Dy});
Fy = subs(Fy, {xt,yt}, {Dx,Dy});
deq1 = m*D2x - Fx;
deq2 = m*D2y - Fy;

% ---------------------------------------------------------
%                  ANALYTICAL SOLUTION
% =========================================================

disp(' ');
ans = input(' Would you like analytical solution? (Y/N): ','s');
if ans =='Y' | ans == 'y'
if ~isstr(x0), x0 = num2str(x0); end
if ~isstr(y0), y0 = num2str(y0); end
if ~isstr(v0), v0 = num2str(v0); end
if ~isstr(alfa), alfa = num2str(alfa); end
inicond = ['x(0)=',x0,',Dx(0)=','v0*cos(alfa),',...
'y(0)=',y0,',Dy(0)=','v0*sin(alfa)'];
[x,y] = dsolve(char(deq1), char(deq2), inicond, 't');
if ~isempty(x) & ~isempty(y)
x = simple(x); y = simple(y);
disp(' ');
disp('      L o w   o f   M o t i o n   ' );
disp('   *******************************' );
disp(' '); disp('x = '); pretty(x)
disp(' '); disp('y = '); pretty(y)
disp(' ');
fname = input(' Name of file to write solution: ','s');
save(fname, 'x','y');
end
disp(' ');
ans = input(' Would you like numerical solution? (Y/N): ','s');
if ans == 'N' | ans == 'n', return, end
x = 'x'; y = 'y'; % Clear contents of x and y
end

% ---------------------------------------------------------
%                  NUMERICAL SOLUTION
% =========================================================

% Input the name of the file-function
disp(' ');
fname = input(' Name of the File-function to be generated: ','s');
flag1 = 'Y';
if exist([cd,'\',fname]) % Search only in current directory!
disp(' ');
disp([' A file-function with name ',fname,' already exist !'])
flag1 = input(' Overwrite it ? (Y/N): ', 's');
end

% ---------------------------------------------------------
%              GENERATING THE FILE-FUNCTION
% ---------------------------------------------------------

if ( flag1 == 'Y' | flag1 == 'y' )
Fx = subs(Fx,{x,y,Dx,Dy},{'Y(1)','Y(2)','Y(3)','Y(4)'});
Fy = subs(Fy,{x,y,Dx,Dy},{'Y(1)','Y(2)','Y(3)','Y(4)'});
% Opening the file for writing file-function
[Fid,mes] = fopen([fname,'.m'],'wt');
% Generating the string with physical parameters: m, c ...
strpar = '';
for j = 1:np
strpar = [strpar,',',P{j}];
end
disp(' ');
titl = input(' Denomination of the Problem: ','s');
%       Writing the headline of the File-function
fprintf(Fid,['function yt = ',fname,'(t,Y',strpar,')\n']);
fprintf(Fid,['%% ',titl]);
%       Writing the first derivatives
fprintf(Fid,'\n%% The first derivatives\n');
fprintf(Fid, '  yt(1) = Y(3); \n');
fprintf(Fid, '  yt(2) = Y(4); \n');
fprintf(Fid,['  yt(3) = ',char(Fx/m),'; \n']);
fprintf(Fid,['  yt(4) = ',char(Fy/m),'; \n']);
fprintf(Fid,'  yt = yt'';\n');
fprintf(Fid,['%% *** End of File-function ',fname,' ***']);
fclose(Fid);
edit(fname)
end

% ---------------------------------------------------------
%        INTEGRATION AND VISUALIZATION OF THE REZULTS
% ---------------------------------------------------------

flag2 = 0;
% Initial entering values of the parameters and generating
% the string with parameters 'P{1}, P{2}, ..., P{np}' to be
% passed to the File-function as actual arguments
if np > 0
PP = P; % Saving the physical names of the parameters in PP
parameters = ' ';
disp(' ');
disp(' Input the numerical values of the parameters: ')
for i = 1:np
i = num2str(i);
eval(['P{',i,'}=input([''   '',P{',i,'},'' = '']);']);
parameters = [parameters,',P{',i,'}'];
end
else
parameters = [];
end
% Check-up the type of x0, y0, v0 and alfa
% and correct it if needed
if ischar(x0)
x0 = str2num(x0);
if isempty(x0), x0 = input(' x0 = '); end
end
if ischar(y0)
y0 = str2num(y0);
if isempty(y0), y0 = input(' y0 = '); end
end
if ischar(v0)
v0 = str2num(v0);
if isempty(v0), v0 = input(' v0 = '); end
end
if ischar(alfa)
alfa = str2num(alfa);
if isempty(alfa), alfa = input(' alfa = '); end
end
while 1
if flag2 == 1
disp(' ');
eps  = input(' Precision of the computations eps: '  );
Tend = input(' Upper bound of the integration Tend: ');
x0   = input(' Initial coordinate x0 : '             );
y0   = input(' Initial coordinate y0 : '             );
v0   = input(' Initial velocity v0 : '               );
alfa = input(' Angle alfa : '                        );
if np > 0
P = PP; % Restoring the names of the parameters !
disp(' ');
disp(' Input the numerical values of the parameters: ')
for i = 1:np
i = num2str(i);
eval(['P{',i,'}=input([''   '',P{',i,'},'' = '']);']);
end
end
end
inic = [x0 y0 v0*cos(alfa) v0*sin(alfa)]; % initial conditions
options = odeset('AbsTol',eps,'RelTol',100*eps);
% Choosing of the Solver
disp('                                        ');
disp('      Choose the proper Solver:         ');
disp('  -------------------------------       ');
disp(' A. Non stiff differential equations    ');
disp('   1. ode45   - middle precision;       ');
disp('   2. ode23   - low precision;          ');
disp('   3. ode113  - from low to upper.      ');
disp('                                        ');
disp(' B. Stiff differential equations        ');
disp('   1. ode15s  - from low to upper;      ');
disp('   2. ode23s  - low precision;          ');
disp('   3. ode23t  - middle precision;       ');
disp('   4. ode23tb - low precision.          ');
disp('                                        ');
solver = input(' The name of the Solver: ','s');

% Integration of the Differential Equations

eval(['[t,Y] = feval(solver,eval([''@'',fname]),',...
'[0 Tend],inic,options',parameters,');']);

% Plotting graphs

tmin  = min(t); tmax = max(t);
y1min = min(Y(:,1));
y1max = max(Y(:,1));
y2min = min(Y(:,2));
y2max = max(Y(:,2));
dy1   = y1max - y1min;
dy2   = y2max - y2min;
xmin  = y1min - 0.1*dy1;
xmax  = y1max + 0.1*dy1;
ymin  = y2min - 0.1*dy2;
ymax  = y2max + 0.1*dy2;
figure % 1
% Coordinate x
plot(t, Y(:,1), [tmin,tmax], [0,0], 'k');
axis([tmin, tmax, xmin, xmax]);
xlabel('{\itt}'); ylabel('{\itx}'); grid;
title('Coordinate {\itx} = {\itx}({\itt})'); pause;
figure % 2
% Coordinate y
plot(t, Y(:,2), [tmin,tmax], [0,0], 'k');
axis([tmin, tmax, ymin, ymax]);
xlabel('{\itt}'); ylabel('{\ity}'); grid;
title('Coordinate {\ity} = {\ity}({\itt})'); pause;
figure % 3
% Trajectory
comet(Y(:,1), Y(:,2))
plot(Y(:,1), Y(:,2), [xmin, xmax], [0,0],'k',...
[0,0], [ymin, ymax], 'k');
axis([xmin, xmax, ymin, ymax]);
xlabel('{\itx}'); ylabel('{\ity}'); grid;
title('Trajectory of the particle'); pause;
figure % 4
% Velocity
v = sqrt(Y(:,3).^2 + Y(:,4).^2);
vmin = min(v); vmax = max(v);
dv = vmax - vmin;
comet(t, v)
plot (t, v, [tmin,tmax], [0,0], 'k');
axis([tmin, tmax, vmin - 0.1*dv, vmax + 0.1*dv]);
xlabel('{\itt}'); ylabel('{\itv}'); grid;
title('Velocity {\itv} = {\itv}({\itt})'); pause; close;
flag2 = 1;
close all
disp(' ');
ans = input(' Would you like to continue? (Y/N): ','s');
if ans == 'n' | ans == 'N', break, end
end

%  ********************* End of Program DTXY **********************
```