The Electrical-Mechanical Analogy — Question 5

Question 5

Mechanical damping maps to a resistor with the same damping ratio

A mechanical oscillator has damping ratio $\zeta=b/(2\sqrt{mk})$. An RLC analogue uses the force-voltage correspondence. Let L=.250 H and C=.00400 F.

1. Derive its damping ratio and critical resistance.

2. Find R for damping ratio .300.

Question 5 - Solution

Mechanical damping maps to a resistor with the same damping ratio

(a) Method and calculation

$$\ddot q+(R/L)\dot q+q/(LC)=0\Rightarrow\zeta_e=\frac R2\sqrt{C/L}.$$$$\boxed{R_{\mathrm{crit}}=2\sqrt{L/C}=15.8114\,\Omega}.$$

(b) Interpretation and further calculation

$$R=2\zeta_e\sqrt{L/C}.$$$$\boxed{R=.300R_{\mathrm{crit}}=4.74342\,\Omega}.$$

Physical check

The resistor plays the role of viscous drag; critical damping is a dimensionless comparison, not a fixed resistance.