Question

m К . Х3

make the transformation of the system to the original coordinate

Answer 1

3) mK ht I mm * - on inson som monit egn of motion of the first mass: = - K + KLX-4) - K KK - - (2-4) = -2KG + KA2 + 2K - KR2=0.... -K(1-34) K (N-12) of motion of second man: = -K (₂-4) + K (1₃-X2) = + K24 - 2 KN2 + KA₂ ... 9 - in to – Kh + 2 K N 2 - Krz egn of motion for thired mens. -K (R3-R2) EGE-- -- K ( - 12) - KX3 - Klla - 2 K 23 => m -K mg +26*3 = 0... ct From eg" , & D we can write, X2 + 10 co 3 2K - O - K oln To 2K - Knol -K 2X] [x o Lo o me LOGO let, 1 the normel mode, & with normal frequency a. then a t = -2 then

o wint 2K -mwnt2K -K K -mw. M2+2K ng then if we want to find normal which are non-trivial (ie 4 + 8 -Wm+26 - 0 - mu+2K k = 0 o k mw +2K| Now, look at egh . My = - Kx; m- 10 oso 300 Note that, eqr ip same as finding the eigen values, & of the K metrix. Moreover finding the normal modez is same as finding the eigen vectorn of ke metrix. So, if we are in the eigen basip of K, then R 12) = a 127 basis eigenvector. DPPPPPPPPP eigenbarin

so, eqh reducep to. Mlay = -K 127 = -127 where 127= [ ] → eigen barip on normel modes. 3 mox -- ** ; maintenan; max-ax The nign up due to convension, you can alway? redefre k by absorbing the inve sign] here a = wao, fram , 12K-2 -k - K - 24-7 = 0 x So, -K 2K-2 a= 2K (2+82) K, (2-82) K the k metrix in diagonal form is. 2K o o I Ricare = 0 (2+2) K o = sfest 100 (2-13]K! w similarity tramformetion matrix of co-ordinate trasformation. matrix made by eigen vectore corresponding to the eigenvalues.

V12 12 V / - V ve p S o - 13/2 12 12/2 1/2) vuz in diagonal basis, 12=2K) = 19 = (2+J£)K), - 17 = (2-12) k) d - - in original co-ordinate, a=2K) - A = 27. s 12=2K) P ^ - (152 o 1/2 1/2 -5212 1/2 1/2 12/ 2 1/2 /11 0 o | A=(24) x) = | 9 = (2+2)")- 19 = (2-2)x) = $ 12 = (18) m)- (12)