A moving mass and a circuit do not look alike. Yet both can store energy, lose energy and respond to a push. Their equations reveal a reusable pattern that makes unfamiliar systems easier to model.
Start with a mass, spring and damper
Let x be the mass's displacement and v=dx/dt its velocity. An external force F must overcome inertia m d²x/dt², viscous damping b dx/dt and spring force kx: F=m d²x/dt²+b dx/dt+kx. Inertia resists changes in velocity; the spring stores energy when displaced; the damper dissipates energy as heat.
This is a model choice. Dry friction, slack, impacts and nonlinear springs are absent. The equation is most useful where those omissions are acceptable for the question at hand.
Put a series RLC circuit beside it
Let q be capacitor charge and i=dq/dt be current. Kirchhoff's voltage law gives V=L d²q/dt²+R dq/dt+q/C. The order of derivatives matches the mechanical equation exactly when we compare displacement x with charge q and velocity v with current i.
Under this force–voltage, or impedance, analogy the pairs have corresponding roles but different physical units. A numerical parameter is not copied from kilograms into henries; scaling and units must be defined for an actual analog model.
| Mechanical | Series RLC | Shared role |
|---|---|---|
| Force F [N] | Voltage V [V] | External effort |
| Displacement x [m] | Charge q [C] | Accumulated flow |
| Velocity v [m/s] | Current i [A] | Flow |
| Mass m [kg] | Inductance L [H] | Stores kinetic/magnetic energy |
| Damping b [N·s/m] | Resistance R [Ω] | Dissipates energy |
| Stiffness k [N/m] | Inverse capacitance 1/C [F⁻¹] | Stores spring/electric energy |
Check the analogy with energy
The moving mass stores ½mv², the spring stores ½kx², and the damper dissipates power bv². The inductor stores ½Li², the capacitor stores q²/(2C), and the resistor dissipates Ri². This second comparison explains why the matching terms are more than typographic coincidence.
There is also a force–current, or mobility, analogy with a different mapping. Pick one convention and keep it throughout a derivation. Neither convention says a spring is literally a capacitor or that their units are equal.
Do not mistake an analogy for the motor model
EIGENROOM's DC motor has an electrical current state coupled to a rotating mechanical speed state through torque and back EMF. That is a physical electromechanical connection, not merely the mass–RLC equation relabeled. The Lab does not simulate a spring or a standalone RLC circuit today.
Use the shared pattern to ask good modeling questions: what stores energy, what dissipates it, what is the input, and what state must be remembered? Then return to the real device's equations and measurements.
The force–voltage analogy matches equations and energy roles, not physical units or a real motor's wiring. State the convention before transferring an insight.
CHECK YOUR UNDERSTANDING: If x corresponds to q, which electrical quantity corresponds to mechanical velocity dx/dt?
Since i=dq/dt, mechanical velocity corresponds to current. This is a role in the force–voltage analogy, not equality of units.