SolSec3.9 - 3-53Problems and Solutions from Section 3.9

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Unformatted text preview: 3-53Problems and Solutions from Section 3.9 (3.50-3.57)3.50*.Numerically integrate and plot the response of an underdamped systemdetermined by m= 100 kg, k= 1000 N/m, and c= 20 kg/s, subject to the initialconditions of x= 0 and v= 0, and the applied force F(t) = 30(t-1). Then plot theexact response as computed by equation (3.17). Compare the plot of the exact solution tothe numerical simulation.Solution: First the solution is presented in Mathcad:The Matlab code to provide similar plots is given next:3-54%Numerical Solutions%Problem #50clcclearclose all%Numerical Solutionx0=[0;0];tspan=[0 15];[t,x]=ode45('prob50a',tspan,x0);figure(1)plot(t,x(:,1));title('Problem #50');xlabel('Time, sec.');ylabel('Displacement, m');hold on%Analytical Solutionm=100;c=20;k=1000;F=30;w=sqrt(k/m);d=c/(2*w*m);wd=w*sqrt(1-d^2);to=1;phi=atan(d/sqrt(1-d^2));%for t<tot=linspace(0,1,3);x=0.*t;plot(t,x,'*');%for t>=tot=linspace(1,15);x=F/k-F/(k*sqrt(1-d^2)).*exp(-d.*w.*(t-to)).*cos(wd.*(t-to)-phi);plot(t,x,'*');legend('Numerical', 'Analytical')%M-file for Prob #50function dx=prob(t,x);[rows, cols]=size(x);dx=zeros(rows, cols);m=100;c=20;k=1000;F=30;if t<1dx==0;elsedx(1)=x(2);dx(2)=-c/m*x(2) - k/m*x(1) + F/m;end3-553.51*.Numerically integrate and plot the response of an underdamped systemdetermined by m= 150 kg, andk= 4000 N/m subject to the initial conditions of x= 0.01m and v= 0.1 m/s, and the applied force F(t) = F(t) = 15(t-1), for various values of thedamping coefficient. Use this program to determine a value of damping that causes thetransient term to die out with in 3 seconds. Try to find the smallest such value ofdamping remembering that added damping is usually expensive.Solution: First the solution is given in Mathcad followed by the equivalent Matlab code.A value of c= 750 kg/s will do the job. This corresponds to = 0.466.3-56%Clay%Vibrations%Numerical Solutions%Problem #51clcclearclose all%Numerical Solutionx0=[0.01;0];tspan=[0 15];[t,x]=ode45('prob51a',tspan,x0);figure(1)plot(t,x(:,1));title('Problem #51');xlabel('Time, sec.');ylabel('Displacement, m');hold on%Analytical Solutionm=150;c=0;k=4000;F=15;w=sqrt(k/m);...
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SolSec3.9 - 3-53Problems and Solutions from Section 3.9

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