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19.4: B.4- Analogies Between an m-c-k Mechanical System and an LRC Electrical B-8 Circuit

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    7750
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    By comparing ODEs Equation 19.2.6 and Equation 19.3.15 and the many other mechanical and electromagnetic equations developed in this appendix, we can identify some clearly analogous variable quantities and constants between a forced \(m\)-\(c\)-\(k\) mechanical system and a powered \(LRC\) electrical circuit (Halliday and Resnick, 1960, Section 38-2). The most obvious analogies are listed in the following table.

    Forced \(m\)-\(c\)-\(k\) mechanical system Powered \(LRC\) electrical circuit
    mass \(m\) inductance \(L\)
    viscous damping constant \(c\) resistance \(R\)
    stiffness constant \(k\) inverse capacitance \(1/C\)
    position of the mass \(x(t)\) charge on the capacitor \(q(t)\)
    velocity of the mass \(v_{x}(t)=\dot{x}(t)\) current through the circuit \(i(t)=\dot{q}(t)\)
    excitation force \(f_{x}(t)\) input voltage \(e_{i}(t)\)
    kinetic energy \(E_{K}=\frac{1}{2} m v_{x}^{2}\) magnetic energy \(E_{M}=\frac{1}{2} L i^{2}\)
    strain energy \(E_{S}=\frac{1}{2} k x^{2}\) electrical energy \(E_{E}=\frac{1}{2}(1 / C) q^{2}\)
    power dissipation \(P_{c}=-c v_{x}^{2}\) power dissipation \(P_{R}=-R i^{2}\)
    power input \(P_{f}=f_{x}(t) v_{x}\) power input \(P_{e}=e_{i}(t) i\)

    The mathematical similarity of an \(m\)-\(c\)-\(k\) system to an \(LRC\) circuit is one of the simpler analogies between physical systems and circuits, and there are many other analogous electrical circuits of greater complexity (Fifer, 1961, Vol. III, Chapter 19; MacNeal, 1962). In fact, during the 1940s and 1950s, methods of direct analog computation were developed extensively for analysis of many physical “prototype” systems with use of electrical circuits. In the introductory chapter of a 1962 book entitled Electric Circuit Analogies for Elastic Structures, Richard H. MacNeal wrote (pages 6-7): “The … direct analog computer is … a device in which each electrical component (or group of components) is equivalent to a physical component of the prototype system.” MacNeal stated further: “This technique has been frequently used for the solution of problems concerning mechanics and elasticity, the conduction of heat, fluid flow, and electromagnetic waves.” However, during and after the mid-1960s, direct analog computation was completely eclipsed by the rapidly advancing capabilities of digital computers and software written for digital processing. It is an ironic twist of history that MacNeal himself became an innovative pioneer in the development and implementation of the finiteelement method for high-precision digital computation of structural behavior.


    This page titled 19.4: B.4- Analogies Between an m-c-k Mechanical System and an LRC Electrical B-8 Circuit is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.