Appendix (MATLAB Code).

Owned by Jonathon G Vandyke
May 10, 2021
2 min read

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%% inputs %%

a=1.5;

b=7+3/8;

c=4+3/8;

theta1=0;

g1=4+3/16;

g2y=1.5;


speed=1;

final=500;

%%for 1 cycle uncomment below%%

speed=pi/10;

final=200;


%% angular position

for i=1:final

    omega2(i)=speed;

    theta2(i)=speed*i/10;

    theta3(i)=atan2(-g1*sin(theta1)+a*sin(theta2(i)),-g1*cos(theta1)+a*cos(theta2(i)));

    b1(i)=(-g1*cos(theta1)+a*cos(theta2(i)))/cos(theta3(i));

    b2(i)=b-b1(i);

    theta4(i)=asin((-b2(i)*sin(theta3(i))+g2y)/c);

    theta4(i)=pi-theta4(i);

    g2x(i)=-c*cos(theta4(i))-b2(i)*cos(theta3(i));

end


%% angular velocity %%

for i=1:final

    omega3(i)=(a*omega2(i)*cos(theta2(i))+tan(theta3(i))*a*omega2(i)*sin(theta2(i)))/...

        (b1(i)*cos(theta3(i))+b1(i)*sin(theta3(i))*tan(theta3(i)));

    b1dot(i)=(b1(i)*omega3(i)*sin(theta3(i))-a*omega2(i)*sin(theta2(i)))/cos(theta3(i));

   

    b2dot(i)=-b1dot(i);

   

    omega4(i)=(-b2(i)*omega3(i)*cos(theta3(i))-b2dot(i)*sin(theta3(i)))/(c*cos(theta4(i)));

    g2xdot(i)=c*omega4(i)*cos(theta4(i))+b2(i)*omega3(i)*sin(theta3(i))+b2dot(i)*cos(theta3(i));

end


%% angular acceleration %%

for i=1:final

    alpha2(i)=0;

    alpha3(i)=(tan(theta3(i))*(-omega2(i)^2*a*cos(theta2(i))+omega3(i)^2*b1(i)*cos(theta3(i))...

        +2*omega3(i)*sin(theta3(i)))+omega2(i)^2*a*sin(theta2(i))-omega3(i)^2*b1(i)*sin(theta3(i))...

        +2*omega3(i)*b1dot(i)*cos(theta3(i)))/(b1(i)*cos(theta3(i))+tan(theta3(i))*b1(i)*sin(theta3(i)));  

    b1dot2(i)=(-omega2(i)^2*a*cos(theta2(i))+omega3(i)^2*b1(i)*cos(theta3(i))+alpha3(i)...

        *b1(i)*sin(theta3(i))+2*omega3(i)*b1dot(i)*sin(theta3(i)))/cos(theta3(i));

    b2dot2(i)=-b1dot(i);

   

    alpha4(i)=(omega3(i)^2*b2(i)*sin(theta3(i))-b2dot2(i)*sin(theta3(i))+c*omega4(i)^2*sin(theta4(i))...

        -b2(i)*alpha3(i)*cos(theta3(i))-2*b2dot(i)*omega3(i)*cos(theta3(i)))/(c*cos(theta4(i)));

    gx2dot2(i)=c*alpha4(i)*sin(theta4(i))+omega3(i)^2*b2(i)*cos(theta3(i))-b2dot2(i)*cos(theta3(i))...

        +c*omega4(i)^2*cos(theta4(i))+b2(i)*alpha3(i)*sin(theta3(i))+2*b2dot(i)*omega3(i)*cos(theta3(i));

   

%     s_coriolis(i)=2*b2dot(i)*omega


     g_coriolis(i)=2*omega3(i)*b1dot(i);

end


%% point kinematics %%

for i=1:final

  %position

  p1x(i)=0;

  p1y(i)=0;

  p2x(i)=a*cos(theta2(i));

  p2y(i)=a*sin(theta2(i));

  p3x(i)=g1;

  p3y(i)=0;

  p4x(i)=p3x(i)-b2(i)*cos(theta3(i));

  p4y(i)=p3y(i)-b2(i)*sin(theta3(i));

  p5x(i)=p4x(i)-c*cos(theta4(i));

  p5y(i)=p4y(i)-c*sin(theta4(i));


  %velocity

  v1x(i)=0;

  v1y(i)=0;

  v2x(i)=-omega2(i)*a*sin(theta2(i));

  v2y(i)=omega2(i)*a*cos(theta2(i));

  v3x(i)=0;

  v3y(i)=0;

  v4x(i)=omega3(i)*b2(i)*sin(theta3(i))-b2dot(i)*cos(theta3(i));

  v4y(i)=-omega3(i)*b2(i)*cos(theta3(i))-b2dot(i)*sin(theta3(i));

  v5x(i)=omega4(i)*c*sin(theta4(i))+v4x(i);

  v5y(i)=0;


  %acceleration

  a1x(i)=0;

  a1y(i)=0;

  a2x(i)=-alpha2(i)*a*sin(theta2(i))-omega2(i)^2*a*cos(theta2(i));

  a2y(i)=alpha2(i)*a*cos(theta2(i))-omega2(i)^2*a*sin(theta2(i));

  a3x(i)=0;

  a3y(i)=0;

  a4x(i)=alpha3(i)*b2(i)*sin(theta3(i))+omega3(i)^2*b2(i)*cos(theta3(i))...

      +2*omega3(i)*b2dot(i)*sin(theta3(i))-b2dot2(i)*cos(theta3(i));

  a4y(i)=-alpha3(i)*b2(i)*cos(theta3(i))+omega3(i)^2*b2(i)*sin(theta3(i))...

      -2*omega3(i)*b2dot(i)*cos(theta3(i))-b2dot2(i)*sin(theta3(i));

  a5x(i)=a4x(i)+alpha4(i)*c*sin(theta4(i))+omega4(i)^2*c*cos(theta4(i));

  a5y(i)=0;

end


%% animation %%

axis(gca,'equal')

axis([-3 15 -3 10]);

P1=[0 0];

title('Slotted 5 bar with slider')

for i=1:final

    theta2a=speed*i/10;

   

    theta3a=theta3(i);

    theta4a=theta4(i);

    b1a=b1(i);

    b2a=b2(i);

    g2xa=g2x(i);


    P2=[a*cos(theta2a) a*sin(theta2a)];

    P3=[P2(1)-b1a*cos(theta3a) P2(2)-b1a*sin(theta3a)];

    P4=[P3(1)-b2a*cos(theta3a) P3(2)-b2a*sin(theta3a)];

    P5=[P4(1)-c*cos(theta4a) P4(2)-c*sin(theta4a)];

  

    P6=[5 P5(2)];

    P7=[12 P5(2)];

   

    if i>1

        if g2x(i)>g2x(i-1)

            direction=1;

        else

            direction=-1;

        end

    else

        direction=1;

    end

    P8=[P5(1)+direction*2 P5(2)];

   

    if i>1

        if b1(i)>b1(i-1)

            direction2=1;

        else

            direction2=-1;

        end

    else

        direction2=1;

    end

    P9=[P3(1)+abs(g_coriolis(i))*cos(theta3a+direction2*pi/2), P3(2)+abs(g_coriolis(i))*sin(theta3a+direction2*pi/2)];

   

    crank=line([P1(1),P2(1)],[P1(2),P2(2)]);

    slot1=line([P2(1) P3(1)],[P2(2),P3(2)]);

    slot2=line([P3(1) P4(1)],[P3(2),P4(2)]);

    coupler=line([P4(1) P5(1)],[P4(2),P5(2)]);

%     vertical=line([P6(1) P7(1)],[P6(2) P7(2)]);

%     slider_coriolis=line([P5(1) P8(1)],[P5(2) P8(2)]);

    ground_coriolis=line([P3(1) P9(1)],[P3(2) P9(2)]);

   

    P1_circ=viscircles(P1,0.1);

    P2_circ=viscircles(P2,0.1);

    P3_circ=viscircles(P3,0.1);

    P4_circ=viscircles(P4,0.1);

    P5_circ=viscircles(P5,0.1);


  

    pause(.001);

   

    delete(crank)

    delete(slot1)

    delete(slot2)

    delete(coupler)

    delete(P1_circ)

    delete(P2_circ)

    delete(P3_circ)

    delete(P4_circ)

    delete(P5_circ)

    delete(ground_coriolis)

end


%% Plots for points %%

figure (2)

plot(theta2,v1x)

hold on

plot(theta2,v2x)

hold on

plot(theta2,v3x)

hold on

plot(theta2,v4x)

hold on

plot(theta2,v5x)

grid on

legend('Point 1', 'Point 2', 'Point 3', 'Point 4', 'Point 5');

title('Velocity in x direction')

ylabel('Velocity (in/s)')

xlabel('Theta2 (rad)')


figure (3)

plot(theta2,v1y)

hold on

plot(theta2,v2y)

hold on

plot(theta2,v3y)

hold on

plot(theta2,v4y)

hold on

grid on

plot(theta2,v5y)

legend('Point 1', 'Point 2', 'Point 3', 'Point 4', 'Point 5');

title('Velocity in y direction')

xlabel('Theta2 (rad)')

ylabel('Velocity (in/s)')


figure (4)

plot(theta2,a1x)

hold on

plot(theta2,a2x)

hold on

plot(theta2,a3x)

hold on

plot(theta2,a4x)

hold on

grid on

plot(theta2,a5x)

legend('Point 1', 'Point 2', 'Point 3', 'Point 4', 'Point 5');

title('Acceleration in x direction')

xlabel('Theta2 (rad)')

ylabel('Acceleration (in/s^2)')


figure (5)

plot(theta2,a1y)

hold on

plot(theta2,a2y)

hold on

plot(theta2,a3y)

hold on

plot(theta2,a4y)

hold on

grid on

plot(theta2,a5y)

legend('Point 1', 'Point 2', 'Point 3', 'Point 4', 'Point 5');

title('Acceleration in y direction')

xlabel('Theta2 (rad)')

ylabel('Acceleration (in/s^2)')


figure (6)

plot(theta2,p1x)

hold on

plot(theta2,p2x)

hold on

plot(theta2,p3x)

hold on

plot(theta2,p4x)

hold on

grid on

plot(theta2,p5x)

legend('Point 1', 'Point 2', 'Point 3', 'Point 4', 'Point 5');

title('Position along x axis')

xlabel('Theta2 (rad)')

ylabel('Position (in)')


figure(7)

plot(theta2,p1y)

hold on

plot(theta2,p2y)

hold on

plot(theta2,p3y)

hold on

plot(theta2,p4y)

hold on

grid on

plot(theta2,p5y)

legend('Point 1', 'Point 2', 'Point 3', 'Point 4', 'Point 5');

title('Position along y axis')

xlabel('Theta2 (rad)')

ylabel('Position (in)')


%% plots for output links %%

figure(8)

plot(theta2,theta3)

hold on

plot(theta2,theta4)

grid on

legend('Slotted Link','Coupler')

title('Angular position of linkages')

xlabel('Theta2 (rad)')

ylabel('Angular position (rad)')


figure(9)

plot(theta2,omega3)

hold on

plot(theta2,omega4)

grid on

legend('Slotted Link','Coupler')

title('Angular Velocity of linkages')

xlabel('Theta2 (rad)')

ylabel('Angular velocity (rad/s)')


figure(10)

plot(theta2,alpha3)

hold on

plot(theta2,alpha4)

grid on

legend('Slotted Link','Coupler')

title('Angular acceleration of linkages')

xlabel('Theta2 (rad)')

ylabel('Angular acceleration (rad/s^2)')



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