Analogies between spring–mass–dashpot systems, electrical systems, and fluidic systems

The direct stiffness method of the finite element method turns a system of springs into a system of linear equations, imposes boundary conditions by removing some of the variables to make the system nonsingular, and then solves the system.

If we restrict our attention to systems of Hookean springs in one dimension connected at nodes, we have essentially two relationships:

• The net force from the springs on any node, except for those held fixed by boundary conditions, is zero.
• The force through a spring is its rigidity times the difference in displacement between its endpoints.

This immediately suggests comparison to electric circuits. For example:

• The net current from the resistors into any node is zero, except for those connected to current sources.
• The current through a resistor is its conductance times the difference in voltage between its endpoints.

And the resulting series, parallel, and bridge laws are the same for springs and for resistors.

This suggests extending the analogy {force = current, rigidity = conductance, displacement = voltage, springs = resistors} to other components. In this analogy, a dashpot, whose force is its “dashpoticity” times the rate of change in the displacement between its endpoints, or equivalently the difference in velocity between its endpoints, is analogous to a capacitor, whose current is its capacitance times the rate of change in voltage between its endpoints. This extends the analogy to the following:

| Mechanical         | Electrical  |             |                   |
|--------------------+-------------+-------------+-------------------|
| force              | current     | F = kd      | I = GV            |
| stiffness/rigidity | conductance | k = F/d     | G = I/V           |
| spring             | resistor    |             |                   |
| displacement       | voltage     | d = F/k     | V = I/G           |
| dashpot            | capacitor   | F = x dd/dt | I = C dV/dt       |
| dashpot-stiffness  | capacitance |             | C = Q/V = ∫I dt/V |

For reasons of conservation of energy, an inductor has no passive mechanical equivalent in this analogy; it would need to have a displacement proportional to the rate of change of force.

But that’s not the only possible analogy, since other linear circuit elements (inductors and capacitors) combine in series and parallel in the same way. For example, making capacitors instead of resistors analogous to springs:

• The net charge at any node is zero, except for those connected to current sources.
• The charge across a capacitor is its capacitance times the difference in voltage between its endpoints.

That is:

| Mechanical             | Electrical              |     |     |
|------------------------+-------------------------+-----+-----|
| spring                 | capacitor               | ??? | ??? |
| compliance? stiffness? | capacitance? elastance? |     |     |
| velocity               |                         |     |     |
| force                  |                         |     |     |

In this analogy (force = charge, displacement = voltage, rigidity = capacitance) if we differentiate with respect to time, we find that velocity is rate of change of voltage and FUCK I HAVE NO FUCKING CLUE. It seems like inductors should be masses, what? Since they’re 180° away from springs? On a mass, the derivative of velocity is force divided by mass. On an inductor, the derivative of current is voltage divided by inductance. So does that mean {velocity = current, force = voltage, inductance = mass}?

Presumably you can make the analogy equally valid in exactly the opposite direction, with springs representing inductors, masses representing capacitors, and dashpots representing resistors. In this analogy, I think charge represents displacement, velocity represents current, and force represents voltage: the rate of change of displacement of a mass is its velocity, and the rate of change of velocity of a mass is the force divided by the mass, thus {force = voltage, velocity = current, capacitance = mass}. This is exactly contradictory to what I came up with in the previous paragraph, so clearly I am smoking crack.

This analogy has:

| force        | voltage     |               |               |        |
| velocity     | current     | v = dd/dt     | I = dQ/dt     |        |
| displacement | charge      |               |               |        |
| spring       | inductor    |               |               |        |
| dashpot      | resistor    |               |               |        |
| mass         | capacitor   |               |               |        |
| stiffness    | inductance? | k = F/(dv/dt) | L = V/(dI/dt) | !?!?!? |
|              |             |               |               |        |

Fuck, I have no idea.