$(35$ pts $)$ A seismometer is a transducer used to detect earthquakes. It uses the motion

of the ground, $V_g(t),$ as an input to the system and uses seismic mass is suspended

from the case, which has mass $m_c.$ A magnet is attached to the bottom of the seismic

mass to yield a total moving mass of $m.$ Motion of the ground causes relative motion

between the magnet and the coil. The coil is assumed to be attached to the case and

thus to have the velocity $V_g(t).$ The velocity between the magnet and the coil that

compresses the parallel spring and dashpot is therefore $V_m-V_g.$ The relative motion

between the magnet and the coil behaves as a gyrator with modulus tilde$(r),$ which gives

rise to a voltage and current in the electrical circuit, which are labeled as $e_{gy}$ and

$i_{gy},$ respectively. The circuit shown in the figure is a model of a long cable. Assume

gravity acts downward as indicated in the figure and that the positive $y-$direction

points upward $($not shown$).$ Draw a bond graph of this transducer. Label flows at all

$1-$junctions, efforts at the $0-$junctions, and both effort and flow on both bonds attached

to all $2-$ports. You must label the electrical output of the transducer: $e_{\text{o}\text{u}\text{t}}.$ You can

construct a bond for the output by assuming that a resistor with very large resistance

value, $R_{\text{o}\text{u}\text{t}},$ is connected at the open circles shown in the schematic below. Simplify

the bond graph as much as possible by removing flow$-$through junctions and reducing

"diamond" parallel constructions to $"$T$"$ constructions. Provide half$-$arrows on all

bonds such that flows and efforts sum to zero at all $0-$ and $1-$junctions, respectively.