This is nice, interesting, and helpful, but the question that lingers for me is that, then, how do you calculate the natural frequencies and mode shapes of a damped MDOF system? Or the natural frequencies and mode shapes are for undamped systems only, which would mean that the natural frequencies and mode shapes of the damped system are those of its undamped version? Thank you.
Very interesting question AJ! Under general circumstances, the natural frequencies and natural modes will both change due to damping. In general they cannot be computed analytically. When the damping is proportional (i.e. D is a linear combination of M and K), the natural modes are the same, but the natural frequencies become damped: Like with a SDOF system omega_d = omega_n * sqrt(1-zeta^2). In case the damping is not proportional, also the natural modes will be complex numbers, which basically means that the system's motion will not be synchronous: not all generalized coordinates pass through their equilibrium value at the same time. Analysis should be carried out further in the state space, where both the original eigenvalue problem as well as the so-called adjoint eigenvalue problem should be performed. I think you will find more information when you are searching for "damped vibration modes".
This is nice, interesting, and helpful, but the question that lingers for me is that, then, how do you calculate the natural frequencies and mode shapes of a damped MDOF system? Or the natural frequencies and mode shapes are for undamped systems only, which would mean that the natural frequencies and mode shapes of the damped system are those of its undamped version? Thank you.
Very interesting question AJ! Under general circumstances, the natural frequencies and natural modes will both change due to damping. In general they cannot be computed analytically. When the damping is proportional (i.e. D is a linear combination of M and K), the natural modes are the same, but the natural frequencies become damped: Like with a SDOF system omega_d = omega_n * sqrt(1-zeta^2).
In case the damping is not proportional, also the natural modes will be complex numbers, which basically means that the system's motion will not be synchronous: not all generalized coordinates pass through their equilibrium value at the same time. Analysis should be carried out further in the state space, where both the original eigenvalue problem as well as the so-called adjoint eigenvalue problem should be performed. I think you will find more information when you are searching for "damped vibration modes".