Question

The model of the mechanical system in Figure A.4 (no friction, no damping) is governed by Smiy" = -kıyı + kz(y2 – yı) Im2y = -kz(y2- yı) – k3y2 Let mı = m2 = 10 kg, kų = k3 = 20 kg/secz, k2 40 kg/sec. Find the solution satisfying the initial conditions yı(0) = y2(0) = 0, yí(0) = 1 m/sec, yż(0) = -1 m/sec. 0 y - 92 k2 R3 | wvim.

Answer 1

m. y + (kytkely - Kaya may + (kg + kg) - kad = 0 = mi Kitka ako kit ka ) Ji + تو مع หา -ks Katka 0 kg se² Lim, = M = 10 ki = kg = 20 (60X2); K = 40 likes [initial conditions , 10) = 4; 10) = 0 8, 10) = 1 (Mbe) d'(0) = -1 (26) 2 let solution | response of the system y = Yi Cos lut+o) where $- Shares angle E-m, w + (k, + kg) -K2 _mw + (kytky) se 'Y, Cos (wtto) Y Collat +d) ka

for the Nonitrivial solution of the system- -Low't w +(60) - 40 det -40 -low +60 (-low?+60 12-462 -0 Loow" + 3600 - 12000² - 1600 =0 we + 36-16 12 w² w"-12 W² + 20 20 =0 2 18+ 144 - 00 1,2 2 2 W] =2 i = 10 frequencies. wi 0, Ws = 1.414 god) ; We = 3:16 and are the Natural in Now we find the modes of the system diff, Natural freg. lie az and use system at

3 x : 1 xili (-10 3 + 6 ) Y - Go Y, co 103 460 مراسم = Go J

at w/ -2 heya (-10x2) +60 y!!! 4,609 1 4o at w/= 10 (2) y (2 2 aug (tox 10)+60 yo -1 - Now Nosomal Moodles of vibration corresponding to to we and we cole -

ya ( I Y!!! yo You 4, 121 Y 2 = 14 (2) yours Lyceu 1(2) -L response of the mystem total Malt) = level y en a yolud . Coplayt +0 ) + 49C(63,476, y(t) = you Con (1-9148 +Q2)+ y su Cost ( 3164+42) 7(4) = yles Cos(1-914440) -Yel Coo(3:16+02 ) In above equations four unknown constants you, yol 24 de They can be determined by using initial condition (t) there are equations there A

IC's: yo) = 4260) (0) b) == ; 40 = 1 yu ca Cos d. + y2) Cor de y al cosa - Vic al cos de co 6 -Leady uz sino - Y22.00,. 8inde = 1 -1.494 y sind - 3.16 Y and Bing at -1.414 years since + 3.16 yild sing a fom eq. y used @ and D - Y CON BO from eg. © and @ you. Simo zo ; y un singu =-0-316 ce cas in ons j 3 +(x"'Coxo, = 0 ya) = (0+0.3162)2 = 0.316 ya co

(1) (2) y!) 20 0 316 CO 0 34 2 2 ang - 1 , sind = R 20 - Ox Cost + 0) + 0.16 Cog ( 3·16 + + II ) 20 = 0.36 Cop(3-6 +) 3 395 pse - 0316 Cos (36t +