ME3200- HW # 8 SOLUTIONS – FALL 2003
Page 1 of 7
University of Utah
Mechanical Engineering Department
Salt Lake City, Utah 84112
ME3200- HW # 8 SOLUTIONS – FALL 2003
% Rise parameters:
h1=1.5;
theta0=0;
beta1=pi/4;
theta1=theta0+beta1;
s0=1; % Assumed
% Dwell parameters:
beta2=5*pi/6;
theta2=theta1+beta2;
% Fall parameters:
s2=s0+h1;
beta3=pi/2;
theta3=theta2+beta3;
h3=-1.5;
% Dwell parameters:
beta4=5*pi/12;
theta4=theta3+beta4;
incr=360; %number of increments
dtheta=(theta4-theta0)/incr;
theta=theta0;
for j=1:incr+1
if theta<=theta1
h=h1;
beta=beta1;
scyc(j)=s0+(h/beta*(theta-theta0)-h/2/pi*sin(2*pi/beta*(theta-theta0)));
vcyc(j)=(h/beta-h/beta*cos(2*pi/beta*(theta-theta0)))*w;
acyc(j)=2*pi*h/beta^2*sin(2*pi/beta*(theta-theta0))*w^2;
jcyc(j)=4*pi^2*h/beta^3*cos(2*pi/beta*(theta-theta0))*w^3;
% HW #8 Pro. 8.8
% Plot the SVAJ diagram and cam profile using a cycloidal profile
elseif theta<=theta2
scyc(j)=scyc(j-1);
vcyc(j)=0;
acyc(j)=0;
jcyc(j)=0;
% Cam rotational speed (rad/sec)
w=2*pi/6;
elseif theta<=theta3
h=h3;
Page 2 of 7
University of Utah
Mechanical Engineering Department
Salt Lake City, Utah 84112
ME3200- HW # 8 SOLUTIONS – FALL 2003
beta=beta3;
theta0=theta2;
scyc(j)=s2+(h/beta*(theta-theta0)-h/2/pi*sin(2*pi/beta*(theta-theta0)));
vcyc(j)=(h/beta-h/beta*cos(2*pi/beta*(theta-theta0)))*w;
acyc(j)=2*pi*h/beta^2*sin(2*pi/beta*(theta-theta0))*w^2;
jcyc(j)=4*pi^2*h/beta^3*cos(2*pi/beta*(theta-theta0))*w^3;
else
scyc(j)=scyc(j-1);
vcyc(j)=0;
acyc(j)=0;
jcyc(j)=0;
end
th(j)=theta*180/pi;
theta=theta+dtheta;
end
figure(1)
subplot(4,1,1)
plot(th,scyc)
title('Pro.8.8 SVAJ Diagram')
ylabel('S(in)'), grid on
subplot(4,1,2)
plot(th,vcyc)
ylabel('V(in/s)'), grid on
subplot(4,1,3)
plot(th,acyc)
ylabel('A(in/s^2)'), grid on
subplot(4,1,4)
plot(th,jcyc)
ylabel('J(in/s^3)')
xlabel('theta, deg'), grid on
figure(2)
polar(th/180*pi,scyc), hold on
title('Cam profile(S vs theta in polar plot)')
axis equal
Page 3 of 7
University of Utah
Mechanical Engineering Department
Salt Lake City, Utah 84112
ME3200- HW # 8 SOLUTIONS – FALL 2003
Page 4 of 7
University of Utah
Mechanical Engineering Department
Salt Lake City, Utah 84112
ME3200- HW # 8 SOLUTIONS – FALL 2003
% HW #8 Pro. 8.9 ME3200
% Plot the SVAJ diagram and cam profile using Double-Harmonic motion
% Cam rotational speed (rad/sec)
w=2*pi/2;
% Rise parameters:
h=2;
theta0=0;
beta1=pi/3;
theta1=theta0+beta1;
s0=6;
vharm(j)=h/2*(pi/beta)*w*(-sin(pi*(theta-theta0)/beta)-1/2*sin(2*pi*(thetatheta0)/beta));
aharm(j)=h/2*(pi/beta)^2*w^2*(-cos(pi*(theta-theta0)/beta)-cos(2*pi*(thetatheta0)/beta));
jharm(j)=h/2*(pi/beta)^3*w^3*(+sin(pi*(theta-theta0)/beta)+2*sin(2*pi*(thetatheta0)/beta));
elseif theta<=theta3
beta=beta3;
theta0=theta2;
sharm(j)=s0;
vharm(j)=0;
aharm(j)=0;
jharm(j)=0;
% Fall parameters:
beta2=pi/2;
theta2=theta1+beta2;
end
th(j)=theta*180/pi;
theta=theta+dtheta;
end
% Dwell pararameters:
beta3=7*pi/6;
theta3=theta2+beta3;
figure(1)
subplot(4,1,1)
plot(th,sharm)
title('HW #7.2b - SVAJ Diagram, Rb=81mm')
ylabel('S(in)'), grid on
incr=360; %number of increments
dtheta=(theta3-theta0)/incr;
theta=theta0;
for j=1:incr+1
if theta<=theta1
beta=beta1;
sharm(j)=h/2*(1-cos(pi*(theta-theta0)/beta)-1/4*(1-cos(2*pi*(thetatheta0)/beta)))+s0;
vharm(j)=h/2*(pi/beta)*w*(sin(pi*(theta-theta0)/beta)-1/2*sin(2*pi*(thetatheta0)/beta));
aharm(j)=h/2*(pi/beta)^2*w^2*(cos(pi*(theta-theta0)/beta)-cos(2*pi*(thetatheta0)/beta));
jharm(j)=h/2*(pi/beta)^3*w^3*(-sin(pi*(theta-theta0)/beta)+2*sin(2*pi*(thetatheta0)/beta));
elseif theta<=theta2
beta=beta2;
theta0=theta1;
sharm(j)=h/2*(1+cos(pi*(theta-theta0)/beta)-1/4*(1-cos(2*pi*(thetatheta0)/beta)))+s0;
subplot(4,1,2)
plot(th,vharm)
ylabel('V(in/s)'), grid on
subplot(4,1,3)
plot(th,aharm)
ylabel('A(in/s^2)'), grid on
subplot(4,1,4)
plot(th,jharm)
ylabel('J(in/s^3)'), grid on
figure(2)
polar(th/180*pi,sharm), hold on
title('Cam profile(S vs theta in polar plot)')
axis equal
Page 5 of 7
University of Utah
Mechanical Engineering Department
Salt Lake City, Utah 84112
ME3200- HW # 8 SOLUTIONS – FALL 2003
Page 6 of 7
University of Utah
Mechanical Engineering Department
Salt Lake City, Utah 84112
ME3200- HW # 8 SOLUTIONS – FALL 2003
Page 7 of 7
University of Utah
Mechanical Engineering Department
Salt Lake City, Utah 84112