Question

1: For the single degree of freedom system driven by a harmonic base motion we discussed in the class. The governing equation is given by mž + ci + kx = cy + ky Where y(t) = Y sin wt and w is the driving (excitation) frequency. Given the initial conditions are x(0) = x, and (0) = v.. Combine the homogeneous and particular solutions and satisfy the initial conditions to obtain the complete solution for x(t). Do so for all damping cases (< < 1,3 = 1,5 > 1).

NOTE: this is base excitation not force vibration.

Answer 1

Solution:

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Given mä+Cåtkv = fo sinut +4)* ët Cons) é + x = sinut Ź +2 5 won x + 12x = sinut wher &= 6/2m)? C wn2 g/km con = naterratural frequency Solution of this clifferential equation include CF and PI Damping factor or Damping xatto x = CF +PI Finding .CF jë + 2 5 wne + coñx=0

+ B e 927 ef 4,792 Root of this equation is say 41,2 -2&wn t J (2 & wn]2 -4 0² 4,2 2 4,2=(-&s=ſ&2-1) un = (A +Bf) eat (18x1 =%2=1] over Damped System &=1 Damped System E<I under damped System If é>1= critically 독기 CFS A exitt t Betzt CE = AetestJ53-1 Jwn Ble! 14-8-12 )wnt) CF = ATBA) eat &=1 Lizd=-Wn = (A+B+) e whit

E <I E essent [; ze-Escont 21,2=-sunt irr- &² Wo =-& wnt i wd CF = A pat +Betzt = A et En +iwd)t +Bel-son-iwala se&cond [Aetlost +Bededit] -&ont [ Acconcosttekinwordt) +B(coowdf-Leinwd t) ] =é sont [(A+B) COB Cd +(A+B) sin ud t] x 320 pcoswd+ + Xcos sinwat] [x sin(Wdt +) CF exe-&sont gion (Wit+0) for sin wt Doperator D2 + Q & wn)D +wn? fo sin wt [Wn2-w2) - (25Wn)D] 662-62) + (2&wn)D [(Wm2-w2)-(2&wn)p] fo [wn?-62) 8190t -125W Wn) con wit] (com2-62) ?+ (25 WWn)? fo [Rconoswt - Ramd coowt] (on²-62) 2+Q&weon)? Rcord=wn2.62 Rsmo = 2 & won PI = D2=-w2 m Х D²=-w2 2 where

So PI = For R sin (wt-o) R2 do sin (wt-o) con²/m form R sm for sin (wt-o) Ilona-w2) +(263ww)? folk sin (wt-Ⓡ) - 2)2 +254) So, x(+) for s<1 X(+) = e-Sant [x smlodt-02] + folk sin (wt-o] V1-100) 12 +(260) for '&=1 X (A) = (A +8+) e folk + sin (WF-) J[4-4.)+72+2 for Es> (-&+1521)wnt x(t)= Ảe "+Bel-8-1&2) Wat -wat (28_1012 wn + folk sin(wt-o) | [1- ) 27² +(22 .Esw12 wn