Question: An instrument is being shipped in a container with foam packaging that has an effective vertical stiffness and provides an effective damping factor. Model this package system as a mass (instrument) on a spring (k = stiffness of foam packaging) as in Fig. P5.2, with an added viscous dashpot (ζ = damping factor due to foam packaging).
Gravity g acts downward. (a) Derive the equation of motion for the mass, with displacement u(t) measured positive downward from the position of the mass when the spring and dashpot first come in contact with the ground. Show your free-body diagram.
(b) Determine an expression for the vertical position of the mass as a function of time during the time that the spring remains in contact with the ground.
(c) If the instrument weighs 40 lb, k = 100 lb/in, and ζ = 0.05. and if the container and its contents hit the ground with a vertical speed V = 150 in/sec, determine the maximum total force exerted on the instrument by the packaging, that is, the maximum value of the combined force of the "spring" and the "dashpot."
