MATLAB (6)

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m=10kg, c=0.4Ns/m, k=4Nm(Mass-Spring-Damper System)

General solution, Natural response, Step response

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syms x y;
u=dsolve('D2y+0.04*Dy+0.4*y=0','x')
U=dsolve('D2y+0.04*Dy+0.4*y=0','y(0)=1','Dy(0)=1','x');
subs_U=subs(U,x,0:0.1:250);
t=0:0.1:250;
plot_U=double(subs_U);
plot(t,plot_U)
hold on

figure
num=[1];
den=[10 0.4 4];
h=tf(num, den);
step(h)

General solution

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>> test
 
u =
 
C5*exp(-x/50)*cos((3*111^(1/2)*x)/50) - C6*exp(-x/50)*sin((3*111^(1/2)*x)/50)

natural-response
step-response

subs plot

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subs_U=subs(U,x,0:0.01:10);
t=0:0.01:10;
plot_U=double(subs_U);
plot(t, plot_U)

pole

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pole(h)

ans =

  -0.0200 + 0.6321i
  -0.0200 - 0.6321i

State-space equation

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[A,B,C,D]=tf2ss(num, den)

A =

   -0.0400   -0.4000
    1.0000         0


B =

     1
     0


C =

         0    0.1000


D =

     0

TF by solve

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syms x y;
u=dsolve('D2y+0.04*Dy+0.4*y=0','x')
U=dsolve('D2y+0.04*Dy+0.4*y=0','y(0)=1','Dy(0)=1','x');
subs_U=subs(U,x,0:0.1:250);
t=0:0.1:250;
plot_U=double(subs_U);
plot(t,plot_U)
hold on

syms s t Y;
f=heaviside(t);
Y1=s*Y;
Y2=s*Y1;
sol=solve(10*Y2+0.4*Y1+4*Y-laplace(f),Y);
pretty(sol)
num=[1];
den=[10 0.4 4];
h=tf(num, den);
figure
step(h)

pole(h)

[A,B,C,D]=tf2ss(num, den)
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>> test
 
u =
 
C3*exp(-x/50)*cos((3*111^(1/2)*x)/50) - C4*exp(-x/50)*sin((3*111^(1/2)*x)/50)
 
          1
---------------------
  /     2   2 s     \
s | 10 s  + --- + 4 |
  \          5      /


ans =

  -0.0200 + 0.6321i
  -0.0200 - 0.6321i


A =

   -0.0400   -0.4000
    1.0000         0


B =

     1
     0


C =

         0    0.1000


D =

     0