Lecture d33. Forced vibration Fo sin wt Spring force Fs==k k>0 Dashpot Fd=-ca Forcing Feat= Fo sin wt Newton's Second Law(mi=2F) ma+ ca f ka Fo sin wt Wn=vk/m, S=c/(2mwn) Equation of motion F 十2n+nc 0 sin wt
Lecture D33 : Forced Vibration Spring Force Fs = −kx, k > 0 Dashpot Fd = −cx˙, c > 0 Forcing Fext = F0 sin ωt Newton’s Second Law (mx¨ = P F) mx¨ + cx˙ + kx = F0 sin ωt ωn = q k/m, ζ = c/(2mωn), Equation of motion x¨ + 2ζωnx˙ + ω 2 nx = F0 m sin ωt 1
Undamped forced vibration sin wt General solution c(t)=rc(t)+Ep(t) (t) is genera/ solution of ctni=0 have already seen p(t) is any solution of (1) Tryp(t)= X sin wt→ Fo/k (u/un)21-(u/un)2
Undamped Forced Vibration x¨ + ω 2 nx = F0 m sin ωt (1) General Solution x(t) = xc(t) + xp(t) • xc(t) is general solution of x¨ + ω 2 nx = 0 . . . have already seen • xp(t) is any solution of (1) Try xp(t) = X sin ωt ⇒ X = F0/k 1 − (ω/ωn) 2 = δst 1 − (ω/ωn) 2 2
Undamped forced vibration Particular solution t=X sin wt st sin wt (w/wn) W/G Cosine
Undamped Forced Vibration Particular Solution xp(t) = X sin ωt = δst 1 − (ω/ωn) 2 sin ωt -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 cWna+wnic sin wt (2) General solution c(t)=ac(t)+Ep(t) (t)is general solution of a+2CWn C+Wn=o have already seen Ep(t)is any solution of(2) Ep(t)=X1 CoS wt+ X2 sin wt or mp(t=X sin(wt-p)
Damped Forced Vibration x¨ + 2ζωnx˙ + ω 2 nx = F0 m sin ωt (2) General Solution x(t) = xc(t) + xp(t) • xc(t) is general solution of x¨ + 2ζωnx˙ + ω 2 nx = 0 . . . have already seen • xp(t) is any solution of (2) Try xp(t) = X1 cos ωt + X2 sin ωt or, xp(t) = X sin(ωt − ψ) 4
Damped forced vibration cct is general solution of +2Swn +wna=o have already seen (t) is any solution of(2) p(t=X1 cos wt+ X2 sin wt or Ep(t)= X sin(at-) Fo/k {[1-(u/u)2]2+[2cu/m]2 tan 2w/w
Damped Forced Vibration • xc(t) is general solution of x¨ + 2ζωnx˙ + ω 2 nx = 0 . . . have already seen • xp(t) is any solution of (2) Try xp(t) = X1 cos ωt + X2 sin ωt or, xp(t) = X sin(ωt − ψ) ⇒ . . . X = F0/k {[1 − (ω/ωn) 2] 2 + [2ζω/ωn] 2} 1/2 ψ = tan−1 " 2ζω/ωn 1 − (ω/ωn) 2 # 5