A stationary shaft is modeled as a second-order system:
mx + kx = h(t) (5)
where m and k are the mass and stiffness of the shaft, respectively. Also, h(t) is the external excitation driving the shaft. Answer the following questions.
(a) If m is 1 kg and the natural frequency of the shaft is 10 rad/s, what is the stiffness k of the shaft?
(b) An engineer mounts the shaft onto a set of bearings to form a spindle. The vibration of the spindle is then modeled through:
mi + ci + (k + k)x = h(t) (6)
where ci and ki are the damping and stiffness of the bearings. If the bearings are ball bearings, ki = 800 N/m and ci = 3 Ns/m. Calculate the natural frequency wn and viscous damping factor of the spindle. Is the system overdamped or underdamped?
(c) If the bearings are fluid bearings, ki = 20 N/m and ci = 33 Ns/m. Calculate the natural frequency wn and viscous damping factor of the spindle. Is the system overdamped or underdamped?
(d) If the spindle is subjected to an impulsive load with zero initial conditions, which bearing should you use in order to minimize the vibration x(t)? Please explain why.