An energy harvester is being designed to harvest energy from a MEMS system. The harvester consists of a micro-cantilever beam vibrating in a viscous liquid such that its damping ratio is 0.2 . The micro-cantilever beam is made of silicon \(\left(E=1.9 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\right.\) ) is \(30 \mu \mathrm{m}\) long, is rectangular in cross section, has a base width of \(2 \mu \mathrm{m}\), and a height of \(0.5 \mu \mathrm{m}\). The mass density of silicon is \(2.3 \mathrm{~g} / \mathrm{cm}^{3}\).

(a) What is the natural frequency of the energy harvester using a SDOF model? Use the equivalent mass of a cantilever beam at its end.

(b) What energy is harvested over one cycle of motion if the harvesting occurs at the natural frequency with a vibration amplitude of \(1 \mu \mathrm{m}\) ?

(c) What is the average power harvested over one cycle?

(d) What is the power harvested over one hour?