Characteristic Parameters of Underdamped Second Order Systems

Flashcards covering the characteristic parameters, mathematical relationships, and step-response definitions of underdamped second order systems.

Step response of underdamped second order systems

The graphical response curve $C(t)/K$ depicting key transient response specifications, including delay time ($t_d$), rise time ($t_r$), peak time ($t_p$), overshoot ($M_p$), and stabilization time ($t_s$).

Slope at origin

The initial derivative of the response curve at time zero, defined as $y'(0) = 0$.

Stabilization time ($t_s$)

The time required for the system response to stay within a specified error band, approximated by $t_s \thickapprox \frac{\pi}{\sigma} = \frac{\pi}{\zeta \omega_n}$, where $\sigma = \zeta \omega_n$.

Rise time ($t_r$, 10%-90%)

The time required for the response to rise from $10\%$ to $90\%$, given by $t_r \thickapprox \frac{\pi - \theta}{\omega_d}$, where $\theta = \cos^{-1}(\zeta)$.

Peak time ($t_p$)

The time required to reach the first peak of the overshoot, calculated as $t_p = \frac{\pi}{\omega_d}$, where $\omega_d = \sigma \tan(\theta)$.

Overshoot ($M_p$)

The maximum peak value of the response measured from unity, given by $M_p = e^{-\frac{\zeta \pi}{\sqrt{1-\zeta^2}}} = e^{-\frac{\pi}{\tan(\theta)}}$.

$\sigma$ (Sigma)

A parameter defined in relation to damping ratio $\zeta$ and natural frequency $\omega_n$ as $\sigma = \zeta \omega_n$.

$\theta$ (Theta)

An angle parameter defined in relation to the damping ratio $\zeta$ as $\theta = \cos^{-1}(\zeta)$.

$\omega_d$ (Damped natural frequency)

The oscillation frequency related to $\sigma$ and $\theta$ by $\omega_d = \sigma \tan(\theta)$.