Question

Problem 4.7. Please give some explanation.

houc nejen 8 Note that X is the coordinate of the centre of mass of the mode is the reduced mass μ of the system where 1/μ 1/m + 1/m2 Problem 4.7. The diagram shows an oscillatory force Fa cos wt acting on a mass M which is part of a simple harmonic system of stiffness k and is connected to a mass m by a spring of stiffness s. If all oscillations are along the x axis show that the condition for M to remain stationary is w2-s/m. (This is a simple version of small mass loading in engineering to quench undesirable oscillations.) system whilst the effective mass in the Y Fo cos ot

Answer 1

Given that the Force to cos omega t acting on a mass H equation of motion for M. M middot d^2 x_1/dt^2 = -k x_1 + F_0 cos omega t since M is stationary d^2 x_1/dt^2 = 0 x_1 = F_0/k cos omega t___(1) equation of motion for M M d^2 x_2/dt^2 = -5 k (x_1 - x_2) d^2 x_2/dt^2 = -S/M x_1 + s/M x_2 d^2 x_2/dt^2 -S/M x_2 = -S/M times F_0 /k cos omega t Now taking omega_0^2 = S/M \r\n Solving above differential equation d^2 x_1/dt^2 = omega_0^2 x_2 x_2 (t) = Ae^- omega_0 t + B e^t omega_0 t x_2 (t) = -F_0/k omega_0^2/D^2 - omega_0^2 cos omega t = -F_0/k omega^_0^2 cos omega t/- omega^2 - omega_0^2 x_2 (t) = F_0/2k cos omega t total x_2 (t) = Ae^-omega_0 t + Be^t omega_0 t + F_0/2k cos omega_0 t omega = omega_0 = squareroot S/M omega = squareroot S/M Proved