Classical Damping, Non-Classical Damping and Complex Modes

with initial conditions r(0) = do and ṙ(0) = vo. If the system is un-damped (C = 0N×N), the free response of the system will not decay with time, and a suitable trial solution to the differential equation (1) is r(t) = r̄ sin(ωnt), where r̄ is a constant vector of dimension N . Differentiating r(t) twice, r̈(t) = −ω2 nr̄ sin(ωnt), and substituting the trial solution into equation (1) we obtain − ω2 nMr̄ sin(ωnt) + Kr̄ sin(ωnt) = 0. (2)