(Solved): The dynamics of a machine is modeled by the mass-spring-damper system shown, which has mass \( 40 ...

The dynamics of a machine is modeled by the mass-spring-damper system shown, which has mass \( 40 \mathrm{~kg} \) and stiffness \( 20,000 \mathrm{~N} / \mathrm{m} \). Due to other operating machines nearby, the foundation is vibrating with amplitude \( 10 \mathrm{~mm} \) and frequency \( 3 \mathrm{~Hz} \). It is desirable to limit the vibration (displacement) amplitude of the machine to be equal to or less than 15 \( \mathrm{mm} \). (a) Find the minimum damping ratio to achieve that. (b) Determine the associated damping coefficient \( c \).