Question

Determine the natural frequencies and vibration modes of the two degree of freedom rectilinear system shown in the following figure.

k m, ww m2 DCL LEE LFF

please detail all the steps

ans:

Answer 1

Given question txi. ft2 2 10 Fox Pinding natural frequencies, we have to draw the FBD(Free Body diagram) of each mass. . was?.. . . Klarte) Xcarta) k ma mă with reference to FBD's, we lead to differential equations of motion as feillows mož, + (xp-Y)=0 → cm -k(Fr*2)=0 - Foy the normal motion mode of oscillation, each mass undergoes harmonic motion, passing through equilibrium position, for such motion we let = Arsinet ; * = A, Sint So, , - - Nywa siwt ; %ZAWslour Substituting these values in equations we get mith, w sio wt) + (A Siwt - Anstowt)=0 mal-Awsint-K(Asinel-Ansiot) 20. take sinwr as common and eliminate in both the equations we get -MA +KA-KAL=0 KAUKA=0. KA+ KA2 = 0. . ..-mg A9 Se, (K-mw2,-KA2=0 3 and CK-My %) 4q-KA 200 Cra

Writing these equations in matxise tein Kamw2 T -k Tai k-malu2 [A] -k A2 Pf the This equation is satisfied for any a, and determinant of the above equation is 2010 So, K-M7? -K K km, ZO Leraw ) Kami O . -K K Determinant of 2x2 is ad-bezo - So, CK-MACKM1)-(K2)=0 KP-Km.) – Km) X+97**2*xf 0 fen, som ) X2 (k my+km) =0 solving this by ax²-båtezo . – х. -b Ju-цас la = (km, +km2) + CkM +km.) 2-4707 QCM ) (kmi tkm2) - ( km tkm2). So, 29 > 2mine d=0 (kmi tkm2) + (km time) қn, +Yon, - 2 muna

11 = 6*=>713=0) and tp = Life" => wp = K[mu-me) V , Fey finding vibration niceles, consides equation Az(K-w2) – KA=0 KA = Ag (K-mowa) For w, (first natural frequency) - (Kumpig?) -> = Kamp (o)? fer log (secouel Natural frequency) I take myne Josef -(*==} ( kendime )) 19 vekankan) XI.-12 ->X2=