Consider the microgravity isolation suspension system shown

Consider the microgravity isolation suspension system shown below to be used for isolating sensitive experiments in space. The system consists of an outer frame of mass M = 10 kg and an inner experiment box of mass m = 4 kg. The inner box can slide vertically inside the frame without friction and its motion is isolated by 4 springs of stiffness k = 500 N/m. The outer frame is excited by a vertical disturbance force f(t). a) Write the equations of vertical motion of the outer frame of mass M and the inner experiment box of mass m. b) Find the transfer function from the disturbance force f(t) to the vertical displacement x(t) of the inner box. c) Consider a sinusoid disturbance input force f(t) = 4 sin(5t) (in N). Determine the steady-state amplitude of the inner box displacement |x|.

Solution

GIVEN:- M = 10kg, m = 4kg, Stiffness k = 500N/m

PART (a):-

When a disturbing force f(t) is applied vertically upwards on the outer frame, the inner box slides downwards by a displacement x(t). To counter this force f(t) and equal but opposite reaction is produced in the inner box = R(t) causing an acceleration a. This reaction is balanced by the spring force f(s) of 4 springs acting on the inner box.

Summing the forces in the y-direction for inner box, we get-->

R(t) = -4 . f(s)

So, m.x(t)\" = -4.k.x(t) which gives m.x(t)\"+ 4.k.x(t) = 0 (Equation of motion for inner box)

Also since f(t) = R(t), So, M.x(t)\" = -4.k.x(t) which gives

M.x(t)\" + 4.k.x(t) = 0 (Equation of motion for outer frame)

PART (b):-

Referring standard books we find the transfer function for the inner box, given by x(t) = A. sin(n.t + ) where A & are constants and n is the natural frequency.

PART (c):-

Input force f(t) = 4 sin(5t). TO FIND Steady state amplitude of inner box displacement IxI = ?

So, m.x(t)\" = -m.A.n^2.sin(nt) is comparable with m.x(t)\" = 4 sin(5t)

That is -m.A.n^2 = 4, which gives A = -4/ (4x5^2)   

So Amplitude A = - 0.04 (ANSWER)