Analysis Of Engineering Systems - Homework 2

David Clark MAE 488 Homework #1

3.36) Input program: %2(dx/dt)+x=f(t) %f(t)=m1*t*us(t)-m1(t-D1)*us(t-D1)-m2*(t-D2)*us(t-D2)+m2*(t-D3)*us*(t-D3) %F(s)=m1/s^2-(m1*e^(-D1*s))/s^2-(m2*e^(-D2*s))/s^2+(m2*e^(-D3*s))/s^2 %X(s)=(1/(2*s-1))*(F(s)) %c=14; %D1=10; %D2=30; %D3=40; %x(t)=(7/10)[t*t-4+4*exp(-t/2)-[2*(t-10)-4+4*exp(-(t-10)/2)]us(t-10)]-(7/10)[[2*(t30)-4+4*exp(-(t-30)/2)]us*(t-30)]+(7/10)[[2*(t-40)-4+4*exp(-(t-40)/2)]*us(t-40)]

3.39) Input program: %3.39 conv( [1 4], [1,4]) conv( ans, [1 1]) [r,p,K]=residue([5], ans) disp('x(t)=0.5556*exp(-4*t)-1.6667*exp(-4*t)+0.5556*exp(-1*t)') conv( [1 6 34], [1 4 20] ) [r,p,K]=residue([4 9], ans) disp('x(t)=2*exp(-3*t)*(-0.1159*cos(5*t)-0.1073*sin(5*t)+2*exp(2*t)*(0.1159*cos(4*t)+0.1052*sin(4*t)))') Output: ans = 1 8 16 ans = 1 9 24 r= -0.5556 -1.6667 0.5556 p= -4.0000 -4.0000 -1.0000 K= []

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David Clark MAE 488 Homework #1

x(t)=0.5556*exp(-4*t)-1.6667*exp(-4*t)+0.5556*exp(-1*t) ans = 1 10 78 256 680 r= -0.1159 + 0.1073i -0.1159 - 0.1073i 0.1159 - 0.1052i 0.1159 + 0.1052i p= -3.0000 + 5.0000i -3.0000 - 5.0000i -2.0000 + 4.0000i -2.0000 - 4.0000i K= [] x(t)=2*exp(-3*t)*(-0.1159*cos(5*t)-0.1073*sin(5*t)+2*exp(2*t)*(0.1159*cos(4*t)+0.1052*sin(4*t)))

3.40) Input Program: sys1=tf(1, [3, 21, 30]); [y, t] = step(sys1); plot(t, 20*y), xlabel('t'), ylabel('x(t)') sys1=tf(1, [5, 20, 65]); [y, t] = step(sys1); plot(t, 20*y), xlabel('t'), ylabel('x(t)') sys1=tf([3,2], [4, 32, 60]); [y, t] = step(sys1); plot(t, 20*y), xlabel('t'), ylabel('x(t)')

David Clark MAE 488 Homework #1

Output:

David Clark MAE 488 Homework #1

3.45 Input t = linspace(0, 2, 300); f=6*cos(3*t); sys1 = tf(1, [3, 21, 30]); [y, t] = lsim(sys1, f, t); plot(t, y, t, f),xlabel('t'),ylabel('x(t) and f(t)') t = linspace(0, 2, 300); f=6*cos(3*t); sys2 = tf(1, [5, 20, 65]); [y, t] = lsim(sys2, f, t); plot(t, y, t, f),xlabel('t'),ylabel('x(t) and f(t)') t = linspace(0, 2, 300); f=6*cos(3*t); sys3 = tf([3, 2], [4, 32, 60]); [y, t] = lsim(sys3, f, t); plot(t, y, t, f),xlabel('t'),ylabel('x(t) and f(t)')

David Clark MAE 488 Homework #1

Output

David Clark MAE 488 Homework #1