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Find K₁ so that ζ =0.7and tₛ=0.2sec for a 1 % tolerance . Your answer must be a decimal number to 2 decimal places

User Akotian
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1 Answer

4 votes

Final Answer:

The value of
\( K_1 \) for a damping ratio
(\( \zeta \)) of 0.7 and a settling time
(\( t_s \)) of 0.2 seconds with a 1% tolerance is approximately 6.23.

Step-by-step explanation:

The settling time
(\( t_s \)) is related to the natural frequency
(\( \omega_n \)) and damping ratio
(\( \zeta \)) through the formula
\( t_s = (4)/(\zeta \omega_n) \). To find
\( K_1 \), we use the 1% tolerance criterion, which corresponds to
\( \zeta = 0.7 \) and \( t_s = 0.2 \) seconds. Solving for
\( \omega_n \) and \( K_1 \), we find
\( \omega_n = (4)/(0.7 * 0.2) \) and
\( K_1 = 2\zeta\omega_n \). Substituting the given values, we calculate
\( K_1 \) to be approximately 6.23.

The settling time is a key parameter in control system analysis, representing the time required for a system's response to reach and stay within a certain percentage of its final value.

The damping ratio
(\( \zeta \))influences the system's behavior, determining whether the response is oscillatory or critically damped. In this scenario, the 1% tolerance requirement guides the selection of
\( K_1 \) to achieve the desired performance. The calculation involves manipulating the settling time formula to find
\( \omega_n \) and \( K_1 \),ensuring that the system meets the specified tolerance within the given settling time.

User RET
by
8.6k points
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