Damping Ratio⚠ unverified

Understanding the Parameters

• Mechanical — $c$ is the viscous damping coefficient, $m$ the mass, $k$ the stiffness. Damping is measured relative to the critical value $c_c = 2\sqrt{km}$.

• Electrical — $R$ is the loss, $L$ and $C$ the energy‑storage elements; $\zeta$ compares $R$ to the characteristic impedance $\sqrt{L/C}$.

• $\zeta$ — inversely related to sharpness: $\zeta = \dfrac{1}{2Q}$ (see Q Factor).

Derivation (Approaching a Proof)

Both systems obey the same canonical second‑order equation. Writing it as

$$\ddot{x} + 2\zeta\omega_0\,\dot{x} + \omega_0^2\,x = 0,$$

the damping ratio is read off by matching coefficients.

• Mechanical ($m\ddot{x} + c\dot{x} + kx = 0$): divide by $m$ to get $\omega_0 = \sqrt{k/m}$ and $2\zeta\omega_0 = c/m$, so $\zeta = \dfrac{c}{2m\omega_0} = \dfrac{c}{2\sqrt{km}}$.

• Electrical series RLC ($L\ddot{q} + R\dot{q} + q/C = 0$): divide by $L$ to get $\omega_0 = 1/\sqrt{LC}$ and $2\zeta\omega_0 = R/L$, so $\zeta = \dfrac{R}{2L\omega_0} = \dfrac{R}{2}\sqrt{\dfrac{C}{L}}$.

The two expressions are the same $\zeta$ — a direct consequence of the mechanical–electrical analogy ($m\leftrightarrow L$, $c\leftrightarrow R$, $1/k\leftrightarrow C$). The critical value $\zeta=1$ marks the boundary between ringing and non‑oscillatory decay.

History

The damping ratio is a cornerstone of vibration theory and control (Routh, Rayleigh, and the classical second‑order system) and, through the electrical–mechanical analogy formalised in the early 20th century, of circuit and servo design alike. $\zeta$ (or its inverse $Q$) is the parameter that tunes overshoot, settling time, and resonance in any second‑order system.

Related Concepts: Q Factor, Quality Factor, Rlc Series Resonant, Resonance Frequency LC Circuit

Notes: Serves two registry calculators — vibration-analysis-damping-ratio (mechanical) and filters-damping-ratio (electrical); same dimensionless $\zeta$. Name collision noted in Known Issues.