Lecture D32 : Damped Free Vibration Spring-Dashpot-Mass System Spring Force Fs = ?kx, k > 0 Dashpot Fd = ?c˙x, c > 0 Newton’s Second Law (m¨x =summationtextF) m¨x+c˙x+kx = 0 (Define) Natural Frequency ωn = radicalBig k/m, and Period τ = 2pi/ωn (Define) Damping Factor ζ = c/(2mωn) Equation of motion ¨x+2ζωn˙x+ω2nx = 0 1 Solution Try x(t) = Aeλt Characteristic Polynomial λ2 +2ζωnλ+ω2n = 0 Roots λ1 = ωn(?ζ + radicalBig ζ2?1), λ2 = ωn(?ζ ? radicalBig ζ2?1) General Solution (superposition) x = A1eλ1t +A2eλ2t = A1e(?ζ+ √ ζ2?1)ωnt +A2e(?ζ? √ ζ2?1)ωnt 2 Types of Solutions ? ζ > 1 (Overdamped) radicalBig ζ2?1 > 0 ? λ1,2 < 0 ? ζ = 1 (Critically Damped) λ1 = λ2 = ?ωn < 0 x(t) = (A1 +A2t)e?ωnt 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 -0.2 One zero crossing at most !! 3 Types of Solutions (cont’d) ? ζ < 1 (Underdamped) Damped Natural Frequency (and Period) ωd = ωn radicalBig 1?ζ2, τd = 2piω d Solution x(t) = e?ζωnt(Acosωdt+Bsinωdt) or, x(t) = Ce?ζωntsin(ωdt+φ) -1 -0.5 0 0.5 1 1 2 3 4 4 Damping Factor Needs to be estimated experimentally: Measure the ratio of two (or more) succes- sive amplitudes x1 and x2, x1 x2 = Ce?ζωnt1 Ce?ζωn(t1+τd) = e ζωnτd Let δ = ln(x1/x2). Since ωd = ωn radicalBig 1?ζ2, ζ = δradicalBig (2pi)2 +δ2 For lightly damped systems, ζ ≈ δ2pi 5 Energy Decay For Undamped harmonic oscillator: E = 12m˙x2 + 12kx2 = 12kx2max (constant) For Underdamped oscillator xmax(t) = xmax(0)e?ζωnt Thus, we expect E(t) = E(0)e?2ζωnt ...valid for lightly damped systems This can be used to estimate ζ if Energy (decay) can be measured. 6