Lecture D33 : Forced Vibration Spring Force Fs = ?kx, k> 0 Dashpot Fd = ?c˙x, c> 0 Forcing Fext =F0sinωt Newton’s Second Law (m¨x = summationtextF) m¨x+c˙x+kx =F0sinωt ωn = radicalBig k/m, ζ = c/(2mωn), Equation of motion ¨x+2ζωn˙x+ω2nx = F0m sinωt 1 Undamped Forced Vibration ¨x+ω2nx= F0m sinωt (1) General Solution x(t) = xc(t)+xp(t) ? xc(t) is general solution of ¨x+ω2nx = 0 ...have already seen ? xp(t) is any solution of (1) Try xp(t) = Xsinωt ? X = F0/k1?(ω/ω n)2 = δst1?(ω/ω n)2 2 Undamped Forced Vibration Particular Solution xp(t) =Xsinωt= δst1?(ω/ω n)2 sinωt -6 -4 -2 0 2 4 6 0 0.5 1 1.5 2 2.5 3 ω<ωn ω>ωn 3 Damped Forced Vibration ¨x+2ζωn˙x+ω2nx = F0m sinωt (2) General Solution x(t) = xc(t)+xp(t) ? xc(t) is general solution of ¨x+2ζωn˙x+ω2nx = 0 ...have already seen ? xp(t) is any solution of (2) Try xp(t) =X1cosωt+X2sinωt or, xp(t) = Xsin(ωt?ψ) 4 Damped Forced Vibration ? xc(t) is general solution of ¨x+2ζωn˙x+ω2nx = 0 ...have already seen ? xp(t) is any solution of (2) Try xp(t) =X1cosωt+X2sinωt or, xp(t) = Xsin(ωt?ψ) ? ... X = F0/k{[1?(ω/ω n)2]2 +[2ζω/ωn]2}1/2 ψ = tan?1 bracketleftBigg 2ζω/ω n 1?(ω/ωn)2 bracketrightBigg 5 Damped Forced Vibration X = δst{[1?(ω/ω n)2]2 +[2ζω/ωn]2}1/2 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 6 Damped Forced Vibration ψ = tan?1 bracketleftBigg 2ζω/ω n 1?(ω/ωn)2 bracketrightBigg 0 1 2 3 7 Vibration Isolation How much of the applied force is transmitted to the wall? Transmissibility = |Transmitted force ||Applied Force| Applied Force Fext = F0sinωt Transmitted Force (spring + dashpot) Fw = kx+c˙x = kXsin(ωt?ψ)+cωXcos(ωt?ψ) = X radicalBig k2 +(cω)2 sin(ωt?σ) 8 Vibration Isolation Transmissibility = X radicalBig k2 +(cω)2 F0 = radicaltpradicalvertex radicalvertexradicalbt 1+(2ζω/ωn)2 (1?(ω/ωn)2)2 +(2ζω/ωn)2 0 1 2 0 1 2 3 Transmissibility For ω/ωn > √2 damping increases transmissi- bility !! For ω/ωn <√2 having a spring increases trans- missibility !! 9