Industrial processes often require the damping of vibrations. Consider a workbench that uses four large ideal springs in place of traditional legs, with each spring supporting one corner of the workbench and exhibiting a force constant of k = 47500 N/m. This workbench is used when creating products that involve highly unstable chemicals. According to specifications, the acceleration of the manufacturing equipment must be held to less than 21.6 m/s² for the manufacturing process to proceed safely. The amplitudes of vibrations are expected to remain under 0.592 mm throughout. Given that the workbench and maufacturing equipment have a combined mass of m= 5.95 kg, find the maximum acceleration amax expected of the equipment and workbench. Ignore the mass of the springs. An engineer decides to damp the oscillations by applying one shock absorber to each spring. These shock absorbers apply a velocity-dependent drag force F = -bu that opposes the motion of the bench. For proper damping, the engineer decides that the motion of the bench should drop to half-amplitude within 0.479 s. In other words, within 0.479 s, the amplitude of motion should drop by a factor of one half. When purchasing shock absorbers, what damping coefficient b should the engineer specify to the industrial equipment supplier? Assume that the oscillations are underdamped.

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Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.