Question

Problem 2) For a 2 DOF system the equations of motion are given as: [mi 0 0 m2 (X2 mig L -m29 L -m29 L m29 L Where m1 =m2 =m g=gravity and L =length a) Determine the frequencies and mode shapes. b) Verify that the natural modes are orthogonal. c) Determine the response fX:(0) Note: x1(t) = xo , x2(t) = 0 and xi(t) = xo , iz(t) = 0 d) If the system is excited by a harmonic force Fi(t) =F1 sinot, determine maximum amplitudes X1 and X2

Answer 1

solution: 6.8 “ทาง 2 - mag 2 {:} mag mag 쁜 2 and Filt) = F sin ut. 1a) Let r; = x; sin(wt +6.) -x; we sin(wt+&.) = -x, w² -mw²+mg/ -mg/l -ng/L -ww + mg/l 7 - B X À lal=0 22 2 mw wg L =0 yos 望 (w² - 29 272) w?=0 and 22 w = 1 1 1 1 29 2 frequencies. For W=0 11 X, N - ngh ondot mg/ 3-1, 96) From giver matrix, we we can write, + + -0 시 M29 M, Ltmg 22 mil mag +1]

(-929/L) x + (M2 + M29/L) 2 (5+1) Az L+9 9 When me = ma x=1 The modes are on orthogonal. 1e) Given, x, (t) = 20 Nolt) = o 1 alt=0 Sylt) = Xosinot sinut txo 19 Given, Fi (t) = F, sin wt Maximum amplitude I FON W 20 x = 0 X2 0 For wa 9 - 29 L x = Fi VILD 22 w2 X = 25, =Xq 29 2 Lo Stay Home - Stay Safe.