A \(500 \mathrm{~kg}\) reciprocating machine is mounted on a foundation of equivalent stiffness \(5 \times 10^{6} \mathrm{~N} / \mathrm{m}\). When operating at \(800 \mathrm{rpm}\), the machine produces an unbalanced harmonic force of magnitude \(50,000 \mathrm{~N}\). Two cantilever beams with end masses are added to the machine to act as absorbers. The beams are made of steel \(\left(E=210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\right)\) and have a moment of inertia of \(4 \times 10^{-6} \mathrm{~m}^{4}\). A \(10 \mathrm{~kg}\) mass is attached to each beam. The absorbers are adjustable in that the location of the mass on the absorber can be varied.

(a) How far away from the support should the masses be located when the machine is operating at \(800 \mathrm{rpm}\) ? What is the amplitude of the absorber mass?

(b) If the machine operates at \(1000 \mathrm{rpm}\) and produces a harmonic force of amplitude 100,000 N, where should the absorber masses be placed and what is their vibration amplitude?