# SolSec3_9 - 3Problems and Solutions from Section...

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3- 72 Problems 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 = 0 and v 0 = 0, and the applied force F( t ) = 30 Φ ( 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:

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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- 74 3.58*. Numerically integrate and plot the response of an underdamped system determined by m = 150 kg, and k = 4000 N/m subject to the initial conditions of x 0 = 0.01 m and v 0 = 0.1 m/s, and the applied force F ( t ) = F( t ) = 15 Φ ( 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.

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