# MATLAB in Dynamics

### Jordan Tonchev Jordanov (view profile)

22 Nov 2004 (Updated )

Companion Software for "MATLAB with Applications in Engineering Research"

lagre1
                    function lagre1
% ***************************************************************
%                   P r o g r a m   LAGRE1
% ***************************************************************
%
%  PURPOSE:
%    Derive differential equation of motion of a mechanical system
%    with one degree of freedom by means of Lagrange equation
%          d/dt(dL/dqt) - dL/dq = QN(t,q,qt)
%    Resolve the equation numerically and plots the graphics
%    of the coordinate, velocity and phase plane.
%    If possible, the program could solve the problem analytically.
%
%  INPUT DATA:
%    L    - expression of the Lagrangian L = L(t,q,qt);
%    QN   - generalized non potential force QN = QN(t,q,qt);
%    q0   - initial value of the coordinate;
%    qt0  - initial value of the velocity;
%    Tend - upper bound of the integration;
%    eps  - precision of the calculations;
%    np   - number of parameters .
%    P{1}, P{2}, ..., P{np} - names of the parameters (array of cells);
%
%  NOTES:
%   1. The coordinate is designed by the symbol 'q' and velocity by 'qt';
%   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 q0, qt0 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:
%   % Data for problem ...
%    L    = '1/2*a*qt^2 + 9.81*l*cos(q)';
%    QN   = '-k*qt';
%    q0   = '0.5'; ( or q0 = 'q0';)
%    qt0  = '10';  ( or qt0 = 'qt0';)
%    Tend = 20;
%    eps  = 1.e-7;
%    np   = 3;
%    P{1} = 'a';
%    P{2} = 'l';
%    P{3} = '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
L    = input(' Expression of the Lagrangian  L : ','s' );
QN   = input(' Generalized unpotential force QN : ','s');
q0   = input(' Initial value of the coordinate q0 : '  );
qt0  = input(' Initial value of the velocity qt0 : '   );
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

% ---------------------------------------------------------
%       Deriving Differential Equation of Motion
% =========================================================

syms t q qt qtt Dq D2q
Lqt = diff(L,qt);
deq = diff(Lqt,q)*qt + diff(Lqt,qt)*qtt + ...
diff(Lqt,t) - diff(L,q) - QN;
deq = subs(deq,{qt,qtt},{Dq,D2q});
deq = simple(deq);
disp(' ');
disp('    D i f f e r e n t i a l   E q u a t i o n ')
disp(' ***********************************************')
disp(' ');
disp([' DEq: ', char(deq),' = 0'])
%
% ---------------------------------------------------------
%                  ANALYTICAL SOLUTION
% =========================================================

disp(' ');
ans = input(' Would you like analitical solution? (Y/N): ','s');
if ans =='Y' | ans == 'y'
if ~isstr(q0) , q0  = num2str(q0) ; end % Repairing user
if ~isstr(qt0), qt0 = num2str(qt0); end % input errors!
inicond = ['q(0)=',q0,',Dq(0)=',qt0];
q = dsolve(char(deq), inicond, 't');
if ~isempty(q)
disp(' ');
disp('   Low of Motion   ');
disp(' ***************** ');
disp(' ');
disp('q = '); pretty(q)
disp(' ');
fname = input(' Name of file to write solution: ','s');
save(fname, 'q');
end
disp(' ');
ans = input(' Would you like numerical solution? (Y/N): ','s');
if ans == 'N' | ans == 'n', return, end
q = 'q'; % Clear analytical solution from q !
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 curent 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' )
qtt = solve(deq, 'D2q'); % Second derivative
qtt = subs(qtt, {q,Dq},{'y(1)','y(2)'}); % Canonization
% Opening the file to write 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(2); \n');
%fprintf(Fid,['  yt(2) = ',char(b/a),'; \n']);
fprintf(Fid,['  yt(2) = ',char(qtt),'; \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 type of q0 and qt0 and correct
% it if needed
if ischar(q0)
q0 = str2num(q0);
if isempty(q0), q0 = input(' q0 = '); end
end
if ischar(qt0)
qt0 = str2num(qt0);
if isempty(qt0), qt0 = input(' qt0 = '); end
end
while 1
if flag2 == 1
disp(' ');
eps  = input(' Precision of the computations eps: ');
Tend = input(' Upper bound of the integration Tend: ');
q0   = input(' Initial coordinate q0: ');
qt0  = input(' Initial velocity qt0: ');
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
y0 = [q0 qt0]; % 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],y0,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;
% Coordinate q = q(t)
figure % 1
comet(t,y(:,1))
plot(t,y(:,1),[tmin tmax],[0 0],'k'), grid on
axis([tmin, tmax, xmin, xmax]);
set(gca,'FontName','Arial Cyr','FontSize',12);
title('Low of motion {\itq} = {\itq}({\itt})')
xlabel('{\itt}'); ylabel('{\itq}'); pause
% Velocity qt = qt(t)
figure % 2
comet(t,y(:,2))
plot(t,y(:,2),[tmin tmax],[0 0],'k'), grid on
axis([tmin, tmax, ymin, ymax]);
set(gca,'FontName','Arial','FontSize',12);
title('Velocity {\it qt} = {\it qt}({\itt})')
xlabel('{\itt}'); ylabel('{\it qt}'); pause
% Coordinate and Velocity
figure % 3
subplot(2,1,1), plot(t,y(:,1),[tmin tmax],[0 0],'k')
grid on, axis([tmin, tmax, xmin, xmax]);
set(gca,'FontName','Arial','FontSize',12);
title('Low of motion {\itq} = {\itq}({\itt})')
subplot(2,1,2), plot(t,y(:,2),[tmin tmax],[0 0],'k')
grid on, axis([tmin, tmax, ymin, ymax]);
set(gca,'FontName','Arial','FontSize',12);
title('Velocity {\it qt} = {\it qt}({\itt})'), pause
% Phase Plane
figure % 4
subplot(1,1,1)
comet(y(:,1),y(:,2))
plot(y(:,1),y(:,2), [xmin, xmax],[0 0],'k',...
[0 0],[ymin, ymax],'k'), grid on
axis([xmin, xmax, ymin, ymax]);
set(gca,'FontName','Arial Cyr','FontSize',12);
title(' Phase Plane {\it qt} = {\it qt}({\itq})')
xlabel('{\itq}'), ylabel('{\it qt}'), pause
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 LAGRE1 ******************