/
Appendix - Mellinger

Appendix - Mellinger

May 11, 2021


Works Cited

Ahmad Y Hassan. The Crank-Connecting Rod System in a Continuously Rotating Machine.


Code

%% Derive Position 1st Chain

syms a b c theta2 theta3 bdot w3


realx = a*cosd(theta2) + b*cosd(theta3) == 0;

imgx = a*sind(theta2) + b*sind(theta3) - c == 0;


[b,theta3] = solve(simplify(realx),simplify(imgx),b,theta3);

%% Derive Velocity 1st Chain

syms a b c theta2 theta3 bdot w3 w2


realdx = -a*w2*sind(theta2)+ bdot*cosd(theta3) - b*w3*sind(theta3) == 0;

imgdx = a*w2*cosd(theta2) + bdot*sind(theta3) + b*w3*cosd(theta3) == 0;


[w3,bdot] = solve(simplify(realdx),simplify(imgdx),bdot,w3);

%% Derive Acceleration 1st Chain

syms ddb a2 a theta2 b c theta2 theta3 bdot w3 w2 a3


realddx = -2*bdot*w3*sind(theta3) + a*w2^2*cosd(theta3) + ddb*cosd(theta3) - b*w3^2*cosd(theta3) - b*a3*sind(theta3) == 0;

imgddx =  bdot*w3*cosd(theta3) + bdot*w3*cosd(theta3) - a*w2^2*sind(theta2) + ddb*sind(theta3) - b*w3^2*sind(theta3) + b*a3*cosd(theta3) == 0;


[ddb,a3] = solve(simplify(realddx),simplify(imgddx),ddb,a3);

%% Derive Position 2nd Chain

f = d*cosd(theta3);

d = e/sind(theta3);

%% Derive Velocity 2nd Chain

syms d w3 theta3 ddot fdot


realdx2 = -d*w3*sind(theta3)+ddot*cosd(theta3)-fdot == 0;

imgdx2 = d*w3*cosd(theta3)+ddot*sind(theta3) == 0;


[ddot,fdot] = solve(simplify(realdx2),simplify(imgdx2),ddot,fdot);

%% Derive Acceleration 2nd Chain

syms theta3 bdot w3 ddf ddot a3 d e f ddd


realddx = -2*ddot*w3*sind(theta3)-d*a3*sind(theta3) - d*w3^2*cosd(theta3)+ddd*cosd(theta3) - ddf == 0;

imgddx = 2*ddot*w3*cosd(theta3)+d*a3*sind(theta3) -d*w3^2*sind(theta3)+ddd*sind(theta3) == 0;


[ddd,ddf] = solve(simplify(realddx),simplify(imgddx),ddd,ddf);

%% Mechanism Kinematic Simulation

% Link Lengths

a = 1;

c = 2.5;

e = 4.5;

L3 = 5;

w2 = 1; % Input angular velocity (constant)

i = 1;

for theta2 = 0:360

    % Position Analysis

    theta3 = atand((a*sind(theta2)+c)/(a*cosd(theta2)));

    if theta3 > 0

        theta3 = theta3 - 180;

    end

    b = -a*cosd(theta2)/cosd(theta3);

    % Velocity

    w3 = a*w2*sin((pi*(theta2 - theta3))/180);

    bdot = -(a*w2*cos((pi*(theta2 - theta3))/180))/b;

    d = e/sind(theta3);

    f = d*cosd(theta3);

    ddot = -(d*w3*cos((pi*theta3)/180))/sin((pi*theta3)/180);

    fdot = (9*29^(1/2)*a*w2*sin((pi*(theta2 - theta3))/180))/(10*sin((pi*theta3)/180));


    MA = abs(w2/w3)*abs(a/d);

   

    % Acceleration

    ddb = b*w3^2 - a*w2^2 + a*w2^2*sin((pi*theta3)/180)^2 + a*w2^2*sin((pi*theta2)/180)*sin((pi*theta3)/180);

    a3 = ((a*w2^2*sin((pi*theta3)/90))/2 - 2*bdot*w3 + a*w2^2*cos((pi*theta3)/180)*sin((pi*theta2)/180))/b;

    ddd = -(- d*sin((pi*theta3)/180)*w3^2 + 2*ddot*cos((pi*theta3)/180)*w3 + a3*d*sin((pi*theta3)/180))/sin((pi*theta3)/180);

    ddf = -(a3*d + 2*ddot*w3 - a3*d*cos((pi*theta3)/180)^2 + a3*d*cos((pi*theta3)/180)*sin((pi*theta3)/180))/sin((pi*theta3)/180);

   

    % Coriolis Input Slider

    coriolis_x = -2*bdot*w3*sind(theta3);

    coriolis_y = 2*bdot*w3*cosd(theta3);

    % Coriolis Output Slider

    coriolis_x2 = -2*ddot*w3*sind(theta3);

    coriolis_y2 = 2*ddot*w3*cosd(theta3);

   

   

    % Output Arrays

    ddf_out(i) = ddf;

    a3_out(i) = a3;

    d_out(i) = d;

    b_out(i) = b;

    MA_out(i) = MA;

    fdot_out(i) = fdot;

    ddot_out(i) = ddot;

    bdot_out(i) = bdot;

    f_out(i) = f;

    d1_out(i) = d;

    theta2_out(i) = theta2;

    theta3_out(i) = theta3;

    w2_out(i) = w2;

    w3_out(i) = w3;

    coriolis_x2_out(i) = coriolis_x2;

    coriolis_y2_out(i) = coriolis_y2;

    coriolis_x_out(i) = coriolis_x;

    coriolis_y_out(i) = coriolis_y;

    i = i + 1;

end


% Plot Position

figure(1)

sgtitle('Position Analysis')


subplot(3,1,1)

plot(theta2_out,theta3_out, 'k','linewidth', 2)

title('Theta 3 Relative Motion')

ylabel('Theta 3 (deg)')


subplot(3,1,2)

plot(theta2_out,f_out, 'k','linewidth', 2)

title('Output Slider Relative Motion')

ylabel('F, X-axis (inches)')


subplot(3,1,3)

plot(theta2_out,b_out, 'b','linewidth', 2)

hold on

plot(theta2_out,-d_out, 'r','linewidth', 2)

title('Slider Positions Along Link 3')

ylabel('Slider Pos. (inches)')

xlabel('Input Angle, Theta 2 (deg)')

legend('Input Slider, b', 'Output Slider, d')


% Plot Velocity

figure(2)

sgtitle('Velocity Analysis')


subplot(3,1,1)

plot(theta2_out,w3_out, 'k','linewidth', 2)

title('Link 3 Rotational Velocity')

ylabel('Omega 3 (deg/s)')


subplot(3,1,2)

plot(theta2_out,fdot_out, 'k','linewidth', 2)

title('Output Slider Velocity')

ylabel('Output Velocity (in/s)')


subplot(3,1,3)

plot(theta2_out,bdot_out, 'b','linewidth', 2)

hold on

plot(theta2_out,ddot_out, 'r','linewidth', 2)

title('Slider Velocity Along Link 3')

ylabel('Slider Vel. (in/s)')

xlabel('Input Angle (deg)')

legend('Input Slider, b', 'Output Slider, d')


% Plot Acceleration

figure(3)

sgtitle('Acceleration Analysis')

subplot(3,1,1)

plot(theta2_out,a3_out, 'k','linewidth', 2)

title('Link 3 Rotational Acceleration')

ylabel('Omega 3 (deg/s^2)')


subplot(3,1,2)

plot(theta2_out,ddf_out, 'k','linewidth', 2)

title('Output Slider Acceleration')

ylabel('Output Velocity (in/s^2)')


subplot(3,1,3)

plot(theta2_out,sqrt(coriolis_x_out.^2 + coriolis_y_out.^2), 'b','linewidth', 2)

hold on

plot(theta2_out,sqrt(coriolis_x2_out.^2 + coriolis_y2_out.^2), 'r','linewidth', 2)

title('Slider Coriolis Acceleration')

ylabel('Slider Acc. Magnitude (in/s^2)')

legend('Input Slider, b', 'Output Slider, d')

xlabel('Input Angle (deg)')



% Animation

v = VideoWriter('animation.avi');

open(v);

for i = 1:length(theta2_out)

    % Robot Coordinates

    O2 = [0 0];

    O3 = [0 -2.5];

    L2x = O2(1) + a*cosd(theta2_out(i));

    L2y = O2(2) + a*sind(theta2_out(i));

    L2 = [L2x L2y];

    L3x = O3(1) - L3*cosd(theta3_out(i));

    L3y = O3(2) - L3*sind(theta3_out(i));

    bx = L2x + b_out(i)*cosd(theta3_out(i));

    by = L2y + b_out(i)*sind(theta3_out(i));

    O4y = ones(1,length(theta2_out))*2;

    figure(4)

    plot([3, 3],[3, 3])

    axis([-3, 3, -3, 3])

    hold on

    inp = line([O2(1),L2x],[O2(2),L2y], 'color', 'b', 'linewidth', 2);

    line([L2x,f_out(i)],[L2y, 2], 'color', 'k', 'linewidth', 2)

    line([O3(1),L3x],[O3(2),L3y], 'color', 'k', 'linewidth', 2)

    cor = quiver(L2x,L2y,coriolis_x_out(i),coriolis_y_out(i), 'color', 'g', 'linewidth', 1); % Coriolis

    quiver(f_out(i),2,coriolis_x2_out(i),coriolis_y2_out(i), 'color', 'g', 'linewidth', 1);

    slider = plot(f_out(i),2,'s','color', 'r', 'linewidth', 4);

    plot(L2x,L2y,'s','color', 'r', 'linewidth', 4)

    pin = plot(O3(1),O3(2),'o', 'color', 'b', 'linewidth', 4);

    plot(O2(1),O2(2),'o', 'color', 'b', 'linewidth', 4)

    outp = plot(f_out(1:i),O4y(1:i), '--k');

    title('Mechanism Animation')

    legend([inp, outp, slider, pin, cor], 'Input','Output', 'Slider', 'Pin','Coriolis Acc.','location', 'southoutside', 'Orientation', 'horizontal')


   

    pause(0.001)

    hold off

    frame = getframe(gcf);

    writeVideo(v,frame);

end

close(v)

Arduino


#include <Stepper.h>


const int stepsPerRevolution = 1600;  // change this to fit the number of steps per revolution

// for your motor



// initialize the stepper library on pins 8 through 11:

Stepper myStepper(stepsPerRevolution, 8, 9, 10, 11);


int stepCount = 0;  // number of steps the motor has taken


void setup() {

  // nothing to do inside the setup

}


void loop() {

  // read the sensor value:

  int sensorReading = analogRead(A0);

  // map it to a range from 0 to 100:

  int motorSpeed = map(sensorReading, 0, 1023, 0, 100);

  // set the motor speed:

  if (motorSpeed > 0) {

    myStepper.setSpeed(9.55);

    // step 1/100 of a revolution:

    myStepper.step(stepsPerRevolution / 100);

  }

}


Confluence Documentation | Web Privacy Policy | Web Accessibility