Electrical to Mechanical 1 with Ungrounded Capacitors

The diagram shown below is the same as the previous example for Electrical to Mechanical 1, but two of the branches have been switched (since the branches are in parallel, this is an equivalent circuit).  The currents are now chosen such that two loop currents flow through the inductor.

We can write the node equations.

We can rewrite the equations using analogous quantities.

This set of equations presents a problem because the acceleration of the mass is no relative to a fixed reference, but is relative.  We can fix this by defining a new set of positions.

Now we can rewrite the equations of motion using these new variables.

After a little thought, and the realization that:

, and

the analogous system can be built.  The dimension z3 is the extension of the spring.  The position left side of the spring and friction element (B2) is only known relative to z2, the position of the mass.

Note that this is the same system (with different coordinates) that was created originally when the circuit parameters were defined such that only one current passed through the inductor.

References