function [v,w,zeta]=vtb9_1(m,d,k)%VTB9_1 VTB9_1(m,d,k)% [v,w,zeta]=VTB9_1(m,d,k)% function VTB9_1 finds the mode shapes and natural frequencies of% a linear second order matrix equation.  % [v,w]=VTB9_1(m,k) finds the mode shapes and natural frequencies % for the undamped case.disp('VTB9_1 has been grandfathered. Please use VTB4_3 in the future.')if nargin==2  k=d;  [v,w]=eig(m\k);  w=sqrt(w);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(w);    zeta=(v'*m*v)\(v'*d*v)/2/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;  endendw=diag(w);zeta=diag(zeta);