1. A machine is supported by a linear spring mounting with linear viscous damping. Due to the weight of the machine, the static deflection is found to be 2 mm. Obtain a differential equation from a lumped parameter model, showing clearly the free body diagrams, damping factor, damping ratio, natural frequency. a) Calculate the natural frequency, in HZ, of undamped vibrations. (5 marks) b) When the machine is working an effective unbalanced rotating mass, e induces a vertical harmonic alternating force at a frequency equal to the speed of the driving shaft. If the steady-state amplitude of forced vibration, is the same when the rotational speed of 1027 rev/min, use the non-dimensional graph provided in fig. Q1 to estimate the damping ratio, 5 and the natural frequency of the damped vibrations. What would be the ratio between amplitudes on the same side after (9 marks) 3 cycles with no excitation. c) If the steady state amplitude at the above two speeds is 0.9 mm and the effective unbalanced rotating mass is one twentieth of the total mass of the machine, estimate the eccentricity, e of the rotating mass and the phase lag at the higher speed. (8 marks)