SolSec3_9 - 3- 72Problems and Solutions from Section 3.9...

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Unformatted text preview: 3- 72Problems and Solutions from Section 3.9 (3.57-3.64) 3.57*.Numerically integrate and plot the response of an underdamped system determined by m= 100 kg, k= 1000 N/m, and c= 20 kg/s, subject to the initial conditions of x= 0 and v= 0, and the applied force F(t) = 30Φ(t-1). Then plot the exact response as computed by equation (3.17). Compare the plot of the exact solution to the numerical simulation. Solution: First the solution is presented in Mathcad: The Matlab code to provide similar plots is given next: 3- 73%Numerical Solutions %Problem #57 clc clear close all %Numerical Solution x0=[0;0]; tspan=[0 15]; [t,x]=ode45('prob57a',tspan,x0); figure(1) plot(t,x(:,1)); title('Problem #57'); xlabel('Time, sec.'); ylabel('Displacement, m'); hold on %Analytical Solution m=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<to t=linspace(0,1,3); x=0.*t; plot(t,x,'*'); %for t>=to t=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 #50 function dx=prob(t,x); [rows, cols]=size(x);dx=zeros(rows, cols); m=100; c=20; k=1000; F=30; if t<1 dx==0; else dx(1)=x(2); dx(2)=-c/m*x(2) - k/m*x(1) + F/m; end 3- 743.58*.Numerically integrate and plot the response of an underdamped system determined by m= 150 kg, andk= 4000 N/m subject to the initial conditions of x= 0.01 m and v= 0.1 m/s, and the applied force F(t) = F(t) = 15Φ(t-1), for various values of the damping coefficient. Use this “program” to determine a value of damping that causes the transient term to die out with in 3 seconds. Try to find the smallest such value of damping 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= 710 kg/s will do the job. 3- 75%Vibrations %Numerical Solutions %Problem #51 clc clear close all %Numerical Solution x0=[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 Solution m=150; c=0; k=4000; F=15;...
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SolSec3_9 - 3- 72Problems and Solutions from Section 3.9...

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