Question

how to solve this problem?(not use energy methods)

1. Determine the two natural frequencies and mode shapes of the two-mass systems shown mi m 2k m22m 2k

Answer 1

Kx_1 - 2K(x_1 - x_2) = mx_1 (1) 2K(x_1 - x_2) - 2K x_2 = 2 mx_2 (2) For normal mode of oscillation each mass undergoes harmonic motion of same frequency passing through equilibrium position simultaneously. For such motion we can let x_1 = A middot sin wt x_2 = A_2 sin wt from (1) -K A_1 sin wt - 2K (A_1 sin wt - A_2 sin wt) = m(-A_1 w^2 sin wt) -KA_1 - 2K (A_1 - A_2) = -m A_1 w^2 -3KA_1 + 2A_2 k = -mA_1 w^2 (3k - mw^2) A_1 - 2A_2 k = 0 (3) from (2) 2k(x_1 - x_2) - 2kx_2 = 2 m x_2 2K(A_1 sin wt - A_2 sin wt) - 2k A_2 sin wt = 2m (-a_2 w^2 sin wt) 2K A_1 - 2K A_2 - 2KA_2 = -2 m A_2 w^2 KA_1 - 2KA_2 = -m A_2 w^2 KA_1 + (mw^2 - 2k)