If the modal damping ratio for the lowest mode of Chapter Problem 8.13 is 0.03 , determine the modal damping ratio for the higher modes and determine the response of the system if the machine is displaced \(2 \mathrm{~mm}\) and released.

Data From Chapter Problem 8.13:

A \(400 \mathrm{~kg}\) machine is placed at the midspan of a 3-m-long, 200-kg simply supported beam. The beam is made of a material of elastic modulus \(200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and has a cross-sectional moment of inertia of \(1.4 \times 10^{-5} \mathrm{~m}^{4}\). Use a three degree-of freedom model to approximate the system's lowest natural frequency.