


Under Damped Systems For underdamped systems, ζ<1, and so ω is imaginary, so p_{1} and p_{2} are both complex. The general solution is then where α = ωζ and γ = . This consists of a decaying exponential multiplied by an oscillation at the damped frequency γ. If we place the initial conditions of x(0)=1 and x'(0)=0, then the solution will be
A plot of the first 5 seconds of a system with ζ = 0.1 and ω = 2π then looks like Note that the damping factor is never negative. When it is zero, the system continues to oscillate undamped indefinitely, and when 0 < ζ < 1 the system oscillates with an exponentially decreasing amplitude, as shown above. 