For the system shown in figure below use Lagrange equation to write the equation of motion, then use the result mass and stiffness matrices in Rayleigh method to estimate the fundamental frequency of the system. Taken the displacement vector to be 2 unit for first displacement coordinate and increment by 1 unit for the rest displacement coordinate. Let a=b. 2K Massless Beam K K E Ww E

3. Consider the vibrating system described below (t) 4 -2 -2 1 [ói]x+ [ $ ? ]x= 11 * 0 Compute the mass - normalized stiffness matrix, the eigenvalues, the normalized eigenvectors, the matrix P, and show that PTP I and PTKP is the diagonal matrix of eigenvalues A.

For point a, suggest a different form but the same equation without removing u

The dependence of the tensions t (units: N) in a structure on the external forces f (units: N) follows the linear system Ct=f, where both t and f are 5 by 1 column vectors. The coefficient matrix C has been determined to be [ 11 14 20 -5 -19 ] -5 7 -3 5 -6 4 -5 13 -2 -10 0 0 0 12 -10 0 0 0 0 14 By how many N would the external force f2 need to be decreased (while all other external forces stay the same) in order to decrease the tension t3 by 94 N?

2- Derive the rule-of-mixtures expression for the composite extensional modulus E₁ assuming the existence of an interphase region. The starting point for the derivation would be the model shown below. For simplicity, assume the interphase, like the matrix, is isotropic with modulus E¹. With an interphase region there is a volume fraction associated with the interphase (i.e.,V;). For this situation: vf + vm + vi = 1 H |w²||wm|