For a second order system given by the following equation, damping coefficient \(\zeta\) is \(\frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+4 y=4 x\)

Concept:

A standard second-order system is represented as: \(\frac{d^2y}{dt^2} + 2\zeta\omega_n \frac{dy}{dt} + \omega_n^2 y = \omega_n^2 x\)

Where,

• \(\omega_n\) = natural frequency
• \(\zeta\) = damping coefficient

Given Equation:

\(\frac{d^2y}{dt^2} + 2\frac{dy}{dt} + 4y = 4x\)

Compare with the standard form:

\[ \omega_n^2 = 4 \Rightarrow \omega_n = 2 \]

\[ 2\zeta \omega_n = 2 \Rightarrow \zeta = \frac{2}{2 \cdot 2} = 0.5 \]

Hence, the correct answer is option 1