Question

Consider an undamped system where the vector-matrix form of the system model is: [18 072 Mä +Kx = 08 |, [18000 - 7200 x -7200 8000 1-10 The system is initially at rest in equilibrium when it is put into motion with an initial velocity (0) = 120 while x,0) = x, 0) = 0 and «,(0) = 0. Use modal decomposition to find the system's free response.

Answer 1

Mä + ka = 18 i servo -72007 :]X] 0 8 -7200 svoll X2 ali: co 988u of me suming that it is possible to have Harmonic motion and me at the same brego w and the same phase angle & take the solution X = X_ Cos (wt +0) X = X, Com Wt ++) ar [where X and X₂ are constants] -18 w+ 18 ovo -7200 X -7000 -8w+ BODO X2 lo for Non-trivical solution of X, and to the determinant of the coeff. of X, and I must be zero.

- 186² +1 8vo. -7200 det co -7200 -8W2dopo F18w"+18000)(-8w++ sobo) -(7800)? 20 144234-888000 wa +(244*100) -(5184x10%) = 0 14424- 288000 w?+ 92160000 20 04-gotow + 640000 = 0 (62;?-2000 W + 640 090 =0 sol. of above eq. are W, = 20 centi; W, = 40 (rad) © (-18w?+ Bood) x +47800 X2 ) = 0 (-8w? +8000) x2 - 7900X1 X2 - 18 w²+ 8cro r = XL 7800

(0) re x" -18W + 8000 7 op = 0.011 XI 2 -18x W + 8000 ولا tre --0,89 Xem 700 2 to 2 abomal Mocks of vibration corresponding to us and we (1) x 1 (1) 1 XL ix (2) х" Olli Ky 12) Х (2) (7/2 X - let / 12:27 Walt) = 244) + egye) = xos" Cop (w37 +0)+ xy C8 (wszt +0,1 ) Hylt) = Pada) (+ mai mult = texas un Car(wit+02) + x 1 Ceplegt +o) (+) = 0.11 Xi !" Cop (w.t +0.) - 9.89 x^2) Cor count +0)

in above equation equation there = 120 , there are your unknown constante xl, x2). I and I these can be determined from the initial conditions: X (0) = %10) = 0 _co) Sg (0) 0 these initial conditions teads to- 2,(0) = 0 = x + CORO Kg (0) = 0.11 X!" CORO -8.89 xf2"Cosa *_(O) = 120 = - 20 xalas sind -20 yali) sind. - Ho xf2) Bino 3 (0) = 0 = -24a x Bing + 15.6 x n ring solving abone equation XI Cox of, = 0 Corby = 0 .X2 Cor de =0 6 (2) 12

XW sind 9.89x180)+ 0 201-9,89–8:110) -5.78 you someone O'lll x 120 -0 Follo H0 (-9,89-01111) [xt" Coso, ) ? cesty Aiman Rimoy?] 5178 222) Xiz Oillo sino = -5.78 --4 Soma = -0.llo = -1 0110 5.78 37) sa = Feloo än = 5.78 Cop laot + ) + O'll.COM (407 + 31 ) k's - 0.64 Com (dot + ) 70-318 Cox (406 Floo