%%%  mass- spring-damper system
%%%  evaluate the free evolution and the step response
clear all
close all
clc



%%% define the parameters
b=4 ; % N/m*s
m=1;  % Kg
k=4; %N/m

% with these parameters zita=1 and omega_n=2

% define the ss repres.
% define A, B, C and D
A=[0 1; -k/m -b/m];

B=[0;1/m];

C=[1 0];

D=0;

sys=ss(A,B,C,D);

s=tf('s');

% define the tf
W=1/m/(s^2+ b/m*s + k/m);
%or
[num,den]=ss2tf(A,B,C,D);


% define the intial condition
x0=[1;0];

figure(1)
hold on
grid on
initial(sys,x0)

figure (2)
hold on
grid on
step(W)

%%% reduce b
b1=b/10;
%%% re-define 
A1=[0 1; -k/m -b1/m];
sys1=ss(A1,B,C,D);

W1=1/m/(s^2+ b1/m*s + k/m);

%%% increase b
b2=10*b;
%%% re-define 
A2=[0 1; -k/m -b2/m];
sys2=ss(A2,B,C,D);

W2=1/m/(s^2+ b1/m*s + k/m);


figure(1)
initial(sys1,sys2,x0)
hold on
grid on

figure(2)
hold on
grid on
step(sys1,sys2)