Question

Consider an undamped system where the vector-matrix form of the system model is: [F(t) [18 07ž Mx + Kx = 083, + [18000 -72007x -7200 8000X, E]-[] The input to the system is F(t) = 6300 sin (30t). Use modal decomposition to find the system's frequency response. Note that the frequency response is the particular solution, and also called the steady-state response.

Answer 1

The solution is provided in the images below. Revert back in case of any query.

We first solve it as a free vibration problem to find the natural frequencies and the mode shapes of the systems. After that, we apply the properties of orthonormal modes to get the mass orthonormal modes and the orthonormal modal matrix. This would convert this 2 DOF system into 2 sdof systems which can be easily solved.

Mitka (Fle) [: :1:11 18000 -7200 -7200 8000 ol dr Flt) = 6300 sin (30+).. to We need and mode shape for. first find find the natural frequenues the We know that, 1- m{w3] + kl: 0. natural frequences, will give us so 18000 18w? -7200 -7200 8000 -8w2 118000-18w2) (8000 - 862) -(72094 -0. This gives us, w, = 20 radis. W 40 rad's.

Substituting values in (-m ]w² + k) {x} = 0 we will get the nomal modes. For W = W, = 20 18000 -188400 -7200 LO 7200 10800 -7200 X -0 -7200 4800 x,? (X2J is Second normal mode, for w 40 radi. 18009. - 18x1600 8000 - 8x1600 -200 X -7200 2002 -10800-2200 -7200. -48oo. 0,- { ? :-1- modal matrix, U: [u, uz] 1.5 -1.5

We know that, Ulm ] Ui mu where mi is generalised mass m, u. Ilmu, For U, -> mi : PARDOD. 36 Similarly, UI [m] = M₂ my : 500.000.36 mass otthonormal mode (Un). is given leg, Uni - Uilmi. U f t m 6 -75313 L moob Uz Doool beto fans] -1.5

Ortho normal modal matrix Una lun. Une] --] Un 1 6 1.5 all HH For the queltion, we have 18 18000 -7200 X 6300 sin (30+) -7200 8000 x o let the solution be x: Euniqi: Una bulestatuting 8incia > mlung + Sok] Winq = {E}. multiplying (1) by un un [m] Un et Un' [k] Unq = unfo } But un mun > = 1 T And so we get, lila Twnlq = {e} where {2} Generalised 6) force

{a} !! 1.5 -1.5 6300 sin (30+)] 6 [ 는 {2} = 1050 sin (30+)] sin (30t). Equation (iii) con now be_expressed oss 4 + -92- + = 1050 sin 30t TOSO sin 327 _Wa q-2 These are single dot equations like mätka - Fasad sinut la) the_respoude of Given initial conditions are zero, la) is gusen ley, alt) oo E sinut (k-mw² Ist sinwt. 1-Y? so 4,4+) 92 It). 1050 sin 3ot. loso sinzot 6,2-(30) -soo 2.1 sinzot. lose 51307 1.5 sinot wX-13012 {Un, }q, + { Unz}q Una سري لم allt) : Suniqi alt): sin 30t -0.1 -0.9