Time Response in Overdamped Case Calculator

Calculate Time Response







Formula

The formula to calculate the Time Response in Overdamped Case is:

\[ C_t = 1 - \left( \frac{e^{-\left( \zeta_{\text{over}} - \sqrt{\zeta_{\text{over}}^2 - 1} \right) \left( \omega_n T \right)}}{2 \sqrt{\zeta_{\text{over}}^2 - 1} \left( \zeta_{\text{over}} - \sqrt{\zeta_{\text{over}}^2 - 1} \right)} \right) \]

Where:

Definition

Time Response for Second Order System is defined as the response of a second-order system towards any applied input.

Example Calculation

Let's assume the following values:

Using the formula:

\[ C_t = 1 - \left( \frac{e^{-\left( 1.12 - \sqrt{1.12^2 - 1} \right) \left( 23 \cdot 0.15 \right)}}{2 \sqrt{1.12^2 - 1} \left( 1.12 - \sqrt{1.12^2 - 1} \right)} \right) \]

Evaluating:

\[ C_t = 0.807466086195714 \]

The Time Response is 0.807466086195714.

Conversion Chart

Overdamping Ratio Natural Frequency (Hz) Time Period (s) Time Response
1.1 23 0.15 0.814233809677
1.11 23 0.15 0.810886982647
1.12 23 0.15 0.807466086196
1.13 23 0.15 0.803997860260
1.14 23 0.15 0.800501279431
1.15 23 0.15 0.796990083321
1.16 23 0.15 0.793474386082
1.17 23 0.15 0.789961732784
1.18 23 0.15 0.786457811676
1.19 23 0.15 0.782966945783