exam3 - Doug Rivas ME 218 Exam 3 M5-7 PROBLEM 1 a. Ѳ 1...

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Unformatted text preview: Doug Rivas ME 218 Exam 3 M5-7 PROBLEM 1 a. Ѳ 1 ’= v 1 Ѳ 2 ’= v 2 (-g(2m1+ m2)sin( Ѳ 1 )-m2gsin( Ѳ 1-2 Ѳ 2 )-2sin( Ѳ 1- Ѳ 2 )m2(v 2 2 a 2- v 1 2 a 1 cos( Ѳ 1- Ѳ 2 ) v 1 ’= a 1 (2m1+m2-m2cos(2( Ѳ 1- Ѳ 2 ))) (2sin( Ѳ 1- Ѳ 2 )( v 1 2 a 1 (m1+m2)+g(m1+m2)cos( Ѳ 1 )+v 2 2 a 2 m2cos( Ѳ 1- Ѳ 2 ) v 2 ’= a 2 (2m1+m2-m2cos(2( Ѳ 1- Ѳ 2 ))) b. theta1.m: function dz= theta1(t,z) a1 = 0.25; m1 = 4.0; a2 = 0.5; m2 = 7.0; g = 9.81; v1 = z(1); v2 = z(2); theta1 = z(3)*pi/180; theta2 = z(4)*pi/180; theta1prime = v1; theta2prime = v2; v1prime = (-g*(2*m1 + m2)*sin(theta1) - m2*g*sin(theta1 - 2*theta2) - 2*sin(theta1 - theta2)*m2*(v2^2*a2 - v1^2*a1*cos(theta1 - theta2)))/ (a1*(2*m1+m2-m2*cos(2*(theta1-theta2)))); v2prime = (2*sin(theta1 - theta2)*(v1^2*a1*(m1 + m2) + g*(m1 + m2)*cos(theta1) + v2^2*a2*m2*cos(theta1 - theta2)))/(a2*(2*m1+m2- m2*cos(2*(theta1-theta2)))); dz= [v1prime; v2prime; theta1prime; theta2prime]; prob1.m clear all ; clf tspan = [0 2]; %INITIAL CONDITIONS v10 = 0; v20 = 0; theta10 = 5; theta20 = 0; [t, z] = ode45( 'theta1' , tspan, [v10, v20, theta10, theta20]); theta1 = z(:,3); theta2 = z(:,4); subplot(2,1,1), plot(t, theta1, t, theta2); xlabel( 'Time (seconds)' ); ylabel( 'Theta (degrees)' ); title( 'Doug Rivas- Prob 2- Angles v. Time' ); legend( 'Theta 1' , 'Theta 2' ); tspan2 = [0 10]; [t, z] = ode45( 'theta1' , tspan2, [v10, v20, theta10, theta20]); theta1 = z(:,3); theta2 = z(:,4); subplot(2,1,2), plot(t, theta1, t, theta2); xlabel( 'Time (s)' ); ylabel( 'Angle (degrees)' ); legend( 'Theta 1' , 'Theta 2' ); c. prob1.m clear all ; clf tspan = [0 2]; %INITIAL CONDITIONS v10 = 0; v20 = 0; theta10 = 45; theta20 = 45; [t, z] = ode45( 'theta1' , tspan, [v10, v20, theta10, theta20]); theta1 = z(:,3); theta2 = z(:,4); subplot(2,1,1), plot(t, theta1, 'b.' , t, theta2, 'r.' ); xlabel( 'Time (seconds)' ); ylabel( 'Theta (degrees)' ); title( 'Doug Rivas- Prob 2- Angles v. Time' ); legend( 'Theta 1' , 'Theta 2' ); tspan2 = [0 10]; [t, z] = ode45( 'theta1' , tspan2, [v10, v20, theta10, theta20]); theta1 = z(:,3); theta2 = z(:,4); subplot(2,1,2), plot(t, theta1, 'b.' , t, theta2, 'r.' ); xlabel( 'Time (s)' ); ylabel( 'Angle (degrees)' ); legend( 'Theta 1' , 'Theta 2' ); The chart does indeed show adaptive step sizing when using the zoom in tool. The time between the next successive point varies across the two plots. Below is a zoomed-in version of the two plots above at places where Matlab’s use of adaptive step size is fairly clear. d. prob1.m clear all ; clf tspan = [0 10]; %INITIAL CONDITIONS v10 = 0; v20 = 0; theta10 = 180; theta20 = 0; [t, z] = ode45( 'theta1' , tspan, [v10, v20, theta10, theta20]); theta1 = z(:,3); theta2 = z(:,4); plot(t, theta1, 'b.' , t, theta2, 'r.' ); xlabel( 'Time (seconds)' ); ylabel( 'Theta (degrees)' ); title( 'Doug Rivas- Prob 2- Angles v. Time' ); legend( 'Theta 1' , 'Theta 2' ); There are a few reasons that this problem behaves in this matter considering the initial condition of Ѳ 1 =180 degrees. As the plot illustrates, the pendulum remains motionless. =180 degrees....
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This note was uploaded on 06/23/2011 for the course ME 218 taught by Professor Unknown during the Fall '08 term at University of Texas.

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exam3 - Doug Rivas ME 218 Exam 3 M5-7 PROBLEM 1 a. Ѳ 1...

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