a very good explanation thank you sir. At task 2 of this series where it is asked to find the free vibration response of the 5 story building with initial conditions ( 5 4 3 2 1 ) when I try to uncouple the system of equations by using the transformation ΦtMΦ the result is not a diagonal matrix. I get non zero values outside of the diagonal. How can I fix this? I used the exact same 5 mode shapes you calculated with matlab.