Example7 Added

Author: Mansourt

Animator7.m

@@ -0,0 +1,121 @@
+function Animator7(Th, tt,Param)
+% Author: Mansour Torabi
+% Email: smtoraabi@ymail.com
+%%
+try
+    filename = 'Pic/Anim7.gif';
+    Hf = figure;
+    set(Hf,'color',[1 1 1]);
+    
+    th0 = Th(1,1); ths = Th(1,2); dx = Th(1,3);
+    R0 = Param(1); r0 = Param(2); l0 = Param(3);
+
+    RotFcn = @(X,th)[cos(th), -sin(th);  sin(th), cos(th)]*X;
+
+    %% Main Param
+
+    R = 3.5; r = r0*R/R0; l = l0*R/R0;
+    R_od = R-r;
+
+    xRo = 5; yRo = 7;
+
+    %% SemiCircle and Disc
+
+    aa = linspace(-pi,0,100);
+    xR = R*cos(aa) + xRo;
+    yR = R*sin(aa) + yRo;
+    plot(xR, yR, 'LineWidth',2, 'Color',[0,0.6,0]);
+    hold on;
+
+    P_disc = [xRo, yRo] + R_od*[sin(th0), -cos(th0)];
+    Hod = plot(P_disc(1), P_disc(2), 'ro');
+    
+    aa = linspace(-pi,pi,100);
+    xr = r*cos(aa) + P_disc(1);
+    yr = r*sin(aa) + P_disc(2);
+
+    Hd = plot(xr, yr, 'LineWidth',5, 'Color','r');
+    
+    % Line inside disc
+    xld = [0, 0]; yld = [0, -r];
+    XXld = [xld;yld];
+    alpha = R_od * th0 / r;
+    Xld = RotFcn(XXld, alpha) + repmat([P_disc(1); P_disc(2)],1,length(xld)); 
+
+    Hld = plot(Xld(1,:), Xld(2,:), 'r','LineWidth',2);
+    %% Spring and Mass
+    Ls = l + dx;
+
+    a = 0.1; t0 = linspace(0,Ls,1e3);
+    x = a*sin(100/Ls*t0); y = - t0;
+    XX = [x;y];    
+    Xrs = RotFcn(XX, ths) + repmat([P_disc(1); P_disc(2)],1,length(t0)); 
+    Hs = line([P_disc(1), Xrs(1,:)],[P_disc(2), Xrs(2,:)],'linewidth',1.2);
+
+    Pm = R_od*[sin(th0), -cos(th0)] + Ls*[sin(ths), -cos(ths)] + [xRo, yRo];
+    Hm = plot(Pm(1), Pm(2), 'O', 'markersize', 12, ...
+        'linewidth', 2, 'color', 'r', 'markerfacecolor', 'r');
+    %% Axis
+
+    box off; axis equal
+    set(gca,'xlim',[0 10],'ylim',[0 9],'xtick',[], 'ytick', [])
+
+    %% Text
+    Txt = sprintf('Time: %0.2f sec', tt(1));
+    Htxt = text(3.5, 9, Txt);
+    set(Htxt, 'fontsize', 16, 'fontweight', 'bold');
+
+    frame = getframe(Hf); im = frame2im(frame); [imind,cm] = rgb2ind(im,256);
+    imwrite(imind,cm,filename,'gif', 'Loopcount',inf, 'DelayTime',0.05);
+
+    for i = 1:size(Th,1)
+        pause(0.00001)
+        th0 = Th(i,1); ths = Th(i,2); dx = Th(i,3);
+        %% SemiCircle and Disc
+        
+        P_disc = [xRo, yRo] + R_od*[sin(th0), -cos(th0)];
+        set(Hod, 'xdata',P_disc(1), 'ydata', P_disc(2))
+        
+        aa = linspace(-pi,pi,100);
+        xr = r*cos(aa) + P_disc(1);
+        yr = r*sin(aa) + P_disc(2);
+        
+        set(Hd, 'xdata',xr, 'ydata', yr);
+        
+        % Line inside disc
+        xld = [0, 0]; yld = [0, -r];
+        XXld = [xld;yld];
+        alpha = R_od * th0 / r;
+        Xld = RotFcn(XXld, alpha) + repmat([P_disc(1); P_disc(2)],1,length(xld)); 
+        set(Hld, 'xdata', Xld(1,:), 'ydata', Xld(2,:));
+        
+        %% Spring and Mass
+        Ls = l + dx;
+
+        a = 0.1; t0 = linspace(0,Ls,1e3);
+        x = a*sin(100/Ls*t0); y = - t0;
+        XX = [x;y];
+        Xrs = RotFcn(XX, ths) + repmat([P_disc(1); P_disc(2)],1,length(t0));
+
+        set(Hs, 'xdata', Xrs(1,:), 'ydata', Xrs(2,:));
+
+        Pm = R_od*[sin(th0), -cos(th0)] + Ls*[sin(ths), -cos(ths)]  + [xRo, yRo];
+        set(Hm,'xdata', Pm(1), 'ydata', Pm(2));
+
+        %%
+
+        Txt = sprintf('Time: %0.2f sec', tt(i));
+        set(Htxt, 'string', Txt);
+
+
+        frame = getframe(Hf); im = frame2im(frame); [imind,cm] = rgb2ind(im,256);
+        imwrite(imind,cm,filename,'gif', 'WriteMode','append', 'DelayTime',0.05);
+
+    end
+catch
+    fprintf('You Forced to stop Animation!!\n\n')
+end
+
+
+
+

EVAL6.m

@@ -48,7 +48,7 @@
 set(H,'interpreter','latex','fontsize',18,'location','SouthWest');
 
 hx = xlabel('Time (sec)');  set(hx, 'fontsize', 18);
-hy = ylabel('Angles (rad)- Lenghth (m)'); set(hy, 'fontsize', 18);
+hy = ylabel('Angles (rad)'); set(hy, 'fontsize', 18);
 set(gca, 'fontsize', 18);
 saveas(gcf, 'Pic/Ex6.png')
 %%

EVAL7.m

@@ -4,54 +4,49 @@
 %%
 clc, clear
 
-syms th0 th x Dth0 Dth Dx
-syms k M m l R r g 
+syms th0 ths x Dth0 Dths Dx
+syms R r M J m k l g 
 
 %% Kinetic and Potential Energy
 
-v1x = l1*Dth1*cos(th1) ;
-v1y = -l1*Dth1*sin(th1);
+VoM = (R-r)*[cos(th0), sin(th0)];
+Wd  = -(R-r)*Dth0/r;
 
-v2x = l2*Dth2*cos(th2) ;
-v2y = -l2*Dth2*sin(th2);
+Vm = (R-r)*Dth0*[cos(th0), sin(th0)] + (l+x)*Dths*[cos(ths), sin(ths)] + Dx*[sin(ths), -cos(ths)];
 
-v1t = v1x^2 + v1y^2; 
-v2t = v2x^2 + v2y^2; 
+yM = -(R-r)*cos(th0);
+ym = yM - (l+x)*cos(ths);
 
-T = 1/2*m1*v1t + 1/2*m2*v2t;
+T = 1/2*M*(VoM*VoM.') + 1/2*m*(Vm*Vm.') + 1/2*J*Wd^2;
 
-dXX = l0 + l2*sin(th2) - l1*sin(th1);
-dYY = l1*cos(th1) - l2*cos(th2);
-dx = (dXX^2 + dYY^2)^0.5 - l3;
-
-V1 = -m1*g*(l1*cos(th1)) + 1/2*k*dx^2;
-V2 = -m2*g*(l2*cos(th2));
-V = V1 + V2;
+V = M*g*yM + m*g*ym + 1/2*k*x^2;
 
 L = T - V;
 %%
-q  = [th1, th2];
-Dq = [Dth1, Dth2];
+q  = [th0, ths, x];
+Dq = [Dth0, Dths, Dx];
 tt = linspace(0, 20, 500);
 Eq = LagrangeDynamicEqDeriver(L, q, Dq);
-l0n = 2; l1n = 1; l2n = 1.5; l3n = 2;
-[SS, xx] = DynamicEqSolver(Eq, q, Dq, [k m1 m2 l0 l1 l2 l3 g],...
-                           [20,1,3, l0n, l1n, l2n, l3n, 9.81], tt, [pi/6, pi/2.5, 0, 0]);
+R0 = 5; r0 = 1; l0 = 2; 
+[SS, xx] = DynamicEqSolver(Eq, q, Dq, [R r M J m k l g],...
+                           [R0, r0, 1, 2, 3, 30, l0, 9.81], tt, [pi/3, pi/2, 0, 0, 0, 0]);
 %%             
 figure; 
 plot(tt,xx(:,1),'color',[0,0.6,0],'linewidth',2); 
 hold on; 
 plot(tt,xx(:,2),'b','linewidth',2); 
+plot(tt,xx(:,3),'r','linewidth',2); 
 
-S1 = sprintf('$ \\theta_1$'); 
-S2 = sprintf('$ \\theta_2$');
-H = legend(S1, S2); 
+S1 = sprintf('$ \\theta_0$'); 
+S2 = sprintf('$ \\theta_s$');
+S3 = sprintf('$x_s$');
+H = legend(S1, S2, S3); 
 set(H,'interpreter','latex','fontsize',18,'location','SouthWest');
 
 hx = xlabel('Time (sec)');  set(hx, 'fontsize', 18);
-hy = ylabel('Angles (rad)- Lenghth (m)'); set(hy, 'fontsize', 18);
+hy = ylabel('Angles (rad) - Length (m)'); set(hy, 'fontsize', 18);
 set(gca, 'fontsize', 18);
-saveas(gcf, 'Pic/Ex6.png')
+saveas(gcf, 'Pic/Ex7.png')
 %%
-Animator6(xx(:,1:2), tt, [l0n, l1n, l2n])
+Animator7(xx(:,1:3), tt, [R0, r0, l0])
 

Pic/Anim6.gif

@@ -3,14 +3,15 @@
 
 Using the above library, one can derive differential equations for any dynamic systems and solve response of the system for a given conditions.
 
-Functinality of the library has been Illustrated by the following examples:
+Functionality of the library has been illustrated by the following examples:
 
 1. Double Pendulum
 2. Spring Pendulum
 3. Pendulum with Spring-loaded support
 4. Double Pendulum with free support
 5. Double Spring Pendulum 
-6. Couple Pendulum
+6. Coupled Pendulum
+7. Spring Pendulum with Rolling base inside a semicircle
 
 ## Example 1: Double Pendulum
 
@@ -383,5 +384,68 @@ Slider Position, Pendulum Anlges:
   </tr>
 </table>
 
+## Example 7: Spring Pendulum with Rolling base inside a semicircle 
+
+**Problem Definition:**
+<table style="width:100%">
+  <tr>
+		<td width="10%"> </td>
+		<td width="80%">
+			<img src="../master/Pic/Ex7A.png" />
+		</td>
+		<td width="10%"> </td>
+  </tr>
+</table>
+
+**How to solve:**
+
+Just run the ```EVAL7.m``` to **derive equations** and solve intial condition problem:
+
+### Code Usage:
+``` MATLAB
+syms th0 ths x Dth0 Dths Dx
+syms R r M J m k l g 
+
+%% Kinetic and Potential Energy
+
+VoM = (R-r)*[cos(th0), sin(th0)];
+Wd  = -(R-r)*Dth0/r;
+
+Vm = (R-r)*Dth0*[cos(th0), sin(th0)] + (l+x)*Dths*[cos(ths), sin(ths)] + Dx*[sin(ths), -cos(ths)];
+
+yM = -(R-r)*cos(th0);
+ym = yM - (l+x)*cos(ths);
+
+T = 1/2*M*(VoM*VoM.') + 1/2*m*(Vm*Vm.') + 1/2*J*Wd^2;
+
+V = M*g*yM + m*g*ym + 1/2*k*x^2;
+
+L = T - V;
+%%
+q  = [th0, ths, x];
+Dq = [Dth0, Dths, Dx];
+tt = linspace(0, 20, 500);
+Eq = LagrangeDynamicEqDeriver(L, q, Dq);
+R0 = 5; r0 = 1; l0 = 2; 
+[SS, xx] = DynamicEqSolver(Eq, q, Dq, [R r M J m k l g],...
+                           [R0, r0, 1, 2, 3, 30, l0, 9.81], tt, [pi/3, pi/2, 0, 0, 0, 0]);
+
+```
+Slider Position, Pendulum Anlges:
+<table style="width:100%">
+   <tr>
+        <th>Angles of spring length:</th>
+		<th>Animated Response:</th>
+  </tr>
+  <tr>
+		<td width="50%">
+			<img src="../master/Pic/Ex7.png" />
+		</td>
+		<td width="50%">
+			<img src="../master/Pic/Anim7.gif" />
+		</td>
+  </tr>
+</table>
+
 # Contact
 Email: smtoraabi@ymail.com