Question

1. Springs and a mass are attached to a rigid bar, as shown in Fig 1. The hinges are free to rotate. 0 denotes the rotational angle of the rod, and 0-0 when all springs are not stretched. The mass of the bar and the size of the mass are negligible. Neglect gravitational force, and assume 0 is very small. 1) Derive the equation of motion for this system with Lagrange's method. (20pt) 2) Find the natural frequency of the system. (Spt) ) Obtain the response 0(t) when a- m, b-2 m l-3 m, k1-4 N/m, k2- 1 N/m, -4 N.s/m, m -0.889 kg, 140)-0 rad, and θ(0)-1 rad/s. (20pt) ml

Answer 1

Given a very small disturbances to the system show below kinetic energy T = 1/2 theta^2 when i_0 = means inflate about potential energy V = 1/2 k, s^2 + 1/2 k_2 s_2 where s_1 = theta o s_2 = theta b according small angle a gumption therefore V = 1/2 , k, a^2 theta^2 + 1/2 F-2 b^2 theta^2 damping energy D = 1/2 s^2_2 where s_2 = theta h D = 1/2 c b^2 theta^2 lag ranging L = T - U L = 1/2 theta ^2 - 1/2 h, s^2 - 1/2 k_2 s_2 ^2 = 1/2 f theta^2 - 1/2 k, a^2 theta^2 - 1/2 k_2 b^2 theta^2 lagrangely equation (for free vibration ) rho/ rho t (rho l/rho q) - dL/dq + rho D/ rho q = 0 where q = generalixed cooidinate for thin problem q = theta rho/rho t (rho L/d theta