1. Contents: Slide 2: Bu=0.002 H axis is w0*t Slide 3: Bu=0.002 H axis is w/w0 Slide 4: Bu=0.001 H axis is w0*t Slide 5: Bu=0.001 H axis is w/w0 Slide 6: Bu=0.0005 H axis is w0*t Slide 7: Bu=0.0005 H axis is w/w0 Slide 8 - Matlab code (main program) Slide 9 - Matlab code (derivative function) Slide 10 - expected w(t) vs t and theta

2. Bu=0.002. H axis is w0*wt. Peak at w0*t =280. Expect peak near 1/B=500 (see slide 10)

3. Bu = 0.002. H axis is w/w0. Peak at w/w0=0.6. Expect peak near 1

4. Bu = 0.001, H axis is w0*wt. Peak near w0*t = 540. Expect peak near 1/B = 1000 (see slide 10)

5. Bu = 0.001. H axis is w/w0. Peak at 0.55 Expect peak near w/w0=1

6. Bu = 0.0005. H axis is w0*wt. Peak near w0*t ~ 1050. Expect peak near 1/B = 2000

7. Bu = 0.0005. H axis is w/w0. Peak at 0.52 expect peak near w/w0 = 1

8. Main program file (Matlab) global zeta Bu Bu = 0.0005 zeta = 0.01 thetastop=2000 tol1=1.00E-12 minstep=1.00E-15 maxstep=0.05 maxiter= 10000 tol1=1e-12 init_state=[0,0]'; sim_interval = [0,thetastop]; echo off options = odeset('AbsTol',[tol1 tol1]); [theta,state]=ode45('x_dot',sim_interval,init_state,options); figure % plot displacement vs w0*t plot(theta,state(:,1),'r') title(strcat('Displacement vs w0*t with zeta =',num2str(zeta),' and Bu = ',num2str(Bu)),'fontsize',22) pause % plot displacement vs w/w0 w = Bu*theta plot(w,state(:,1),'r') title(strcat('Displacement vs w/w0 with zeta =',num2str(zeta),' and Bu = ',num2str(Bu)),'fontsize',22) pause

9. Derivative Function (Matlab file x_dot.m) function state_dot = x_dot_h4(theta,x); % x_dot.m global zeta Bu % Input of this function is theta=w*t and state variable x % Output of this function is state_dot which is calcualted time derivative of x % equation of motion: % sin(Bu*theta^2) = [xu'' + zeta*xu' + xu] % state variables: % x1 = xu % x2 = d/dtheta (xu) % rewrite equation of motion in terms of state variables % sin(Bu*theta^2) = [x2' + zeta*x2 + x1] % solve for derivative of x2: % d/dtheta(x2) = sin(Bu*theta^2) - zeta*x2 - x1 % initialize state dot as a column vector of dimension 4 state_dot=zeros(2,1); % find dx1/dt state_dot(1) = x(2); % dx1/dt = x2 % Find dx2/dt state_dot(2) = sin(Bu*theta^2) - zeta*x(2) - x(1);

10. Evolution of frequency as a function of t and theta t w(t) = B*w0*t 1/B w0 theta=w0*t w(t) = B*w0*t = B*theta=w0*Bu*theta 1/Bu w0 w(t) = B*theta w(t) = (w0*Bu)*theta w(t)/w0 = Bu * theta