Question

Problem 2. Eigenvalue and Eigenvector Consider the mass-spring system in Fig. P13.5. The frequencies for the mass vibrations can be determined by solving for the eigenvalues and by applying Mi + kx = 0, which yields m 0 07/31 (2k -k -k X1 (0 0 m2 0 {2}+{-k 2k -kX{X2} = {0} LO 0 m3] 1 iz) 1-k -k 2kJ (x3) lo Applying the guess x = xoeiat as a solution, we get the fol- lowing matrix: 52k - m102 -k -k XL - 2k - m202 -k X02 eicot = L -K -k 2k - M302] | Xo3 nom mammz- FIGURE P13.4 Let m1 = m2 = m3 = 1 kg, and k = 2 N/m. In this case, 1 = 0 , find eigenvalue . Show the x1, x2, x3 as function of t with unsolved coefficients. (which would require initial conditions) (Optional) Use MATLAB 'eig' command to solve for the eigenvalues of thek - mo? matrix above. Then use these eigenvalues to solve for the frequencies (0)

Answer 1

O Cilven i hel m, m, n, lry errol t = 2 1m To rivol eigen value & and show to , Xn X as fumetton oht with unsolved Colbichert Solution FBD of figure Mix + k, = k1%-20) mx tk x + kg (20-%g) = 0 0 m₂ % + k (x2-x) = Kgl Xy - x) .. m, x + kalx-1) + Kg (-x) = 0 (11 kile mik Ky ng min + kg (X-X) + Ku T z = 0 0 Rearanging the above coluation we get m, si + (kotka) x - K₂ K₂ = 0 m x - kqx + (K ₂t Kg) x - kg X₂ = D. miz - Kollz + 1 kzt KyJX₂ = 0

equation may be written in matty All above form as m, oo7% 2 0 m₂ o g Looms (4) + ( Kitka -k, o 200 - kg Katte kg salos for implicity of expression let us say that m, = m₂ = m = m Kaakaakg =k Again equation may be written as moo7802 (2k kose mon + K 2K * 9 99 O -K 2 mg get Applying the guess x= xe hut as a solution, we SC = xiw. eint x = X 22 w² eint =-xou eint rm o of werd 2k -k o 750, et 2. o mol-1 w lutyti ke akk xor email Loom₂-x w? ¿ut / lokai Rearranging the above madoin F2K -- Mw2 -k 752017 . Co -K 2k-mur- k inje - o E o k 2k-m wypos

But value of mi= m = mg=18k=2 we get -2 4-w² -2 . Lo -2 4-w²x3 put x = w? u-box -2 on To -2 207 4- xe (4-x) x = 2X₂ 4-X= 20 it you put gen = 60.8 then x=4-280.8 - 2.6 Again (4-xx, .- 20 iwl e 20 (4% -2%, - dx, Deat=0 Ave Hence them Solve figure All Equation to collation Convent în then & Values find by FBD of madrix from and