function [v,w,zeta]=vtb4_3(m,d,k)%VTB4_3 VTB4_3(m,d,k)% [u,w,zeta]=VTB4_3(m,d,k)% function VTB4_3 finds the mode shapes and natural frequencies of% a linear second order matrix equation.  % [u,w]=VTB4_3(m,k) finds the mode shapes and natural frequencies % for the undamped case.if nargin==2  k=d;  [v,w]=eig(m\k);  w=sqrt(diag(w));  [d,i]=sort(sqrt(w/2/pi));  w=w(i);  v=real(v(:,i));  vnorms=sqrt(v'*m*v);	  v=real(v/vnorms);endif nargin==3  if norm(d/m*k-k/m*d) < 1e-8*norm(k/m*d)    disp('Damping is proportional, eigenvectors are real.')    [v,w]=eig(m\k);    w=sqrt(diag(w));    [f,i]=sort(sqrt(w/2/pi));    w=w(i);    v=real(v(:,i));    vnorms=sqrt(v'*m*v);	    v=real(v/vnorms);    zeta=diag((v'*m*v)\(v'*d*v)/2/diag(w));   else    disp('Damping is non-proportional, eigenvectors are complex.')    a=[0*k eye(length(k));-m\k -m\d];    [v,w1]=eig(a);    w=abs(w1);    zeta=-real(w1)/w;  endend%if nargout==3;zeta=diag(zeta);end