Insects are virtually airborne oscillators. Many use the dynamic properties of their thorax $($a centralized exoskeleton structure that contains the flight muscles$)$ to lower the energy requirements of flight. It has long been hypothesized that insects flap at the resonant frequency of their thorax to make flight more efficient. To assess this hypothesis, Dr$.$ Jankauski conducted frequency response testing of a honey bee thorax ${?}^1.$

Consider the graphic below. A euthanized honeybee is clamped between to push rods. The top push rod is fixed, and the lower push rod is moved vertically via an electrodynamic vibration shaker. A signal generator is used to excite the vibration shaker across a wide frequency range that contains the known honeybee wingbeat frequency. The motion of the shaker $($and the lower push rod$)$ is measured via an accelerometer. A piezoelectric force sensor is used to measure the force applied at the thorax. With the thorax acceleration and force known, one can estimate the frequency response function relating input force to output acceleration, or $G(j)=\frac{A(j)}{F(j)}.$ The magnitude and phase of $G(j)$ are shown below. for problem $1.$ Please answer the following questions.

a$)$ Estimate the insect thorax damping ratio $$ assuming that terms of $O({}^2)$ are negligible.

b$)$ Consider the case where the acceleration has a constant magnitude between $200-600Hz.$ Based on the magnitude plot, what does this imply about the input force in the same frequency range?

c$)$ Consider the steady$-$state acceleration $a(t)=1cos(2500t)\frac{m}{s^2}.$ What is the expression for $f(t)$ required to achieve this acceleration? You can assume the units of the FRF with a linear $y-$scale are $\frac{\frac{m}{s^2}}{N}.$

Magnitude vs$.$ Frequency

Question $2$: Magnitude and phase of $G(j)$