Passive-Element Analog Circuits for Three-Dimensional Elasticity

This paper considers the derivation of passive-element electrical analogs for three-dimensional elasticity in rectangular, cylindrical, and spherical polar coordinates. The analogs represent, in finite-difference format, the elastic behavior of differential volumes of material defined by the partial differential equations of linearized elasticity theory. Basically, the circuits are derived by equating the elastic strain energy to the power dissipation in the analog circuit. The analogs discussed here are force-current, displacement-voltage circuits. As compared to the more widely known operational amplifier or “active” electrical analog, these analogs simulate the appropriate equations of the elastic system by “passive” electrical components; i.e., by some combination of resistors, inductors, capacitors, and transformers. Strictly speaking, the circuits shown in this discussion are for static elasticity problems. However, as referenced in the text, these analogs can be converted readily into dynamics circuits by visual inspection of the static analogs.