A Mechanical Formulation for Resistive Network Analysis: The Spring-Conductance Matrix Method

We introduce a formal and systematic analogy between static mechanical spring networks and electrical resistor networks, establishing a one‑to‑one mathematical correspondence where spring displacements map to electric potentials, spring constants to conductances, and forces to currents. This equivalence transforms the static equilibrium equations of point-mass-spring systems into the nodal equations of resistor networks, enabling the construction of a conductance matrix that is identical in form to a mechanical stiffness matrix and coincides with the weighted graph Laplacian. We demonstrate the versatility of this framework through a series of examples from elementary series and parallel combinations to non-planar networks such as the 3D resistor cube and the Petersen graph. We show that the method provides both an intuitive mechanical interpretation of circuit concepts and a systematic, matrix-based computational algorithm for calculating equivalent resistance. The approach naturally extends to AC networks containing resistors, capacitors, and inductors, offering a unified treatment of linear networks in both DC and AC regimes. We discuss the pedagogical value of the analogy for teaching circuit theory and network analysis, as well as its connections to graph theory and computational implementation. Limitations of the method, including its restriction to linear elements and the numerical considerations of matrix inversion, are briefly discussed.