SDOF Results

function [zi,wn,A,phi,s] = fr1dof(t,r,a,f)

%fr1dof SDOF System Paremeters
%   Finds the parameters of a second order 1 DOF system
%   and reconstructs the governing equation.

%   Input Arguments
%   t    :  time vector
%   r    :  free response
%   a    :  instant amplitude of the free response
%   f    :  intant frequency (Hz) of the free response

%   Output Arguments
%   zi   :  time vector
%   wn   :  free response
%   A    :  instant amplitude of the free response
%   phi  :  intant frequency (rad) of the free response

wd = 2*pi*mean(f); % damped frequency

s = find(t == 3); e = find(t == 7); % indices to avoid border effects

v = zeros(1,size(r,1));
for i = 1:size(r,1)
    amp = polyfit(t(s:e),log(a(i,s:e)),1);
    ziwn(i) = -amp(1); % dampened natural frequency
    % Initial Conditions
    x0 = r(i,1); v(i,:) = diff(r(i,:))./diff(t); v0 = v(i,1);
end

wn = sqrt(wd.^2 + ziwn.^2); % natural frequency

zi = ziwn./wn; % damping ratio

A = sqrt(x0.^2 + ((v0+zi.*wn.*x0)./wd).^2); % amplitude
phi = atan((v0 + zi.*wn.*x0) ./ (wd.*x0)); % phase

s = A.*exp(-ziwn.*t).*sin(wd.*t-phi); % reconstructed signal

end