Question

3. A two-story building is represented in Fig. 3 by a lamped mass system in which mi=1/2m2 and ki = ½ k2. Use Lagrange's equations to derive the differential equations governing the motion of the building and find its normal modes (characteristic frequencies and mode shapes).-r ,m/ Fig. 3 4. In Problem 3 determine the equation of motion of each mass by the normal mode summation method in two cases: (a) if a force is applied to mi to deflect it by unity and the system is releases from this position and (b) if a force is applied to m2 to deflect it by unity and the system is releases from this position

Please answer question number 4, the answer to question 3 is unnecessary.

Answer 1

Let a 2 story building is represented by a mass system m_1 = 1/2m_2 k_1 = 1/2 k_2 Let the Lagrange's equations to drive motion of the building Free body diagram: (P.E) = V = 1/2 k_1(x_2 - x_1)^2 + 1/2 k_2x_2^2 X.E = 1/2 m_1x_1^2 + 1/2 m_2 x_2^2 the generalized coordinates are x_1, x_2 L = T - V = 1/2 m_1 x_1^2 + 1/2 m_2x_2^2 - 1/2 k_1 (x_2 - x_1)62 - 1/2 k_2 x_2^2 Therefore partialdifferential L/ partialdifferential x_1 = -1/2 k_1 2(x_2 - x_1)(-1) = 1/2 K_1(x_2 - x_1) partialdifferential L/ partialdifferential x_2 = -1/2 2k_1(x_2 - x_1) - 1/2 k_2 2x_2 = K_1(x_1 - x_2) = k_2x_2 partialdifferential L/ partialdifferential x_1 = m_1x_1, partialdifferential L/ partialdifferential x_2 = m_2 x_2 The differential equation along the x_1 d/dt(partialdifferential L/ partialdifferential x_1) - partialdifferential L/ partialdifferential x_1 = 0 \r\n m_1 x_1 + k_1(x_1 - x_2) = 0