Question

1. A Three Degree of Freedom discretized lumped parameter system is shown in the figure. (a). Derive the equations of motion for the system using Newton's Second Law of Motion or Energy Methods. (b). Transform the ordinary differential equations obtained into the matrix form. (C). Estimate the fundamental frequency of vibration of the system, assuming the mode shape and the following system parameters: ka=k, k2= 2k, k3 = 3k, m1 = m, m2 = 2m, 113 = 3m (d). Identify which method you are using to estimate the fundamental frequency and whether the estimated fundamental frequency is higher or lower than the exact value of fundamental frequency? mi L m L.

Please provide references to the Model/Equations used from the textbook.

Answer 1

ki - Som :- civon. KIFK, 112=2k, K3=3k m,=m, m₂=2m, m3=3m for Three degree of freedom system: a. The The ean of motion can be written as m, si = -lix,- 62(1, -12) m₂ %₂ = 12 (11-12) - LezlN₂ -13) m3 Üz = 163 (M₂-23) K2 mz 11112 K3 m3 3 Roanangry the above equation, we get mini + (kitkan, - 6212=0 M₂ M2 - Uzu, t (103 + 12) 42 - lezn3=0 mziz - K3y2 + legnz=0

72 m2 O O m3 nz 0 b. Now, Transform ordinary differential equation in a matrix form - kitles) - 0 1 -ki leztle2 -kz - K3 43 the above oqeations can be written in A matrix form as - (-:] [] + [1] [m] = T mass matrix stiffness matrix [m] + [<J[n] =0 -ü where, [c]= m' [w] = Oynamic matrix [m] adj (m) for harmonic oscillation frequoney & II(-) -- [m] =0 (from us] say, w²=t, (c] [n] - [m] =0 GI - c] [n] =0 I = identity matrix The solution of equation may be obtained by, 111-C1=0

Now, Als -2k o SK -3K -3K 3k 12=0 2 m ng n3 J use, [c] = [m] '[K] o 3 - 2o Ek [6m² 6m3 0 3m² o 5-3 0 O 222 0-33 [c] = k18 3 -2 6 5 -3 0-33 o 3o 2 1 18 -12 le m O 6 IS 6 0 O - izle m -3le m -66 ס m Expend along Columm -1. 1 : 11-12 11-18k A-isk 1-ble we get, (H-184) (x²641 - 15k) + gole 1) +34-1 -1241 +720 =0 43 - 21 kx + golet - 18 6 12 + 21x18 621 - 1620 162 36631 • 183643 +36 427 39K 1+1 ( 74 + 20PRK - youtube 18x36 l - 1620 62 = 0 m +364_m 2 m ha m2 +2164² wzo P3 42164² m2 m3

So, after finding 7=? wo get froquoney, 4= w? w = n =?