Question #42298

A mechanical spring undergoing cyclic deformation exhibits elastic hysteresis, dissipating energy $\Delta E = \alpha x_0^2$ per cycle, where $x_0$ is the displacement amplitude. In a direct electromechanical analogy, this spring is modeled as an inductor with a lossy material core in a series $RLC$ circuit. The displacement $x$ maps to the charge $q$ on the capacitor, and the velocity $v$ maps to the current $i$. The hysteresis loss is represented by an equivalent series resistance $R$. If $\alpha_e$ is the electrical analog of the constant $\alpha$, what is the value of $R$ in terms of $\alpha_e$, the inductance $L$, and the capacitance $C$? Assume the damping is small.