Home / Engineering & CS /For the block diagram shown above, determine the following: 1) Transfer function \( Y(s) / R(s) \) 2) The resulting transfer function is a standard \( 2^{\text {nd }} \) order which has the following format. \[ H(s)=\frac{\omega_{n}^{2}}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}} \] Determine the value of \( \mathrm{K} \) that would result in a maximum overshoot of \( 10 \% \) from a step input.

Question For the block diagram shown above, determine the following: 1) Transfer function \( Y(s) / R(s) \) 2) The resulting transfer function is a standard \( 2^{\text {nd }} \) order which has the following format. \[ H(s)=\frac{\omega_{n}^{2}}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}} \] Determine the value of \( \mathrm{K} \) that would result in a maximum overshoot of \( 10 \% \) from a step input.

8IUY43 The Asker · Mechanical Engineering

Transcribed Image Text: For the block diagram shown above, determine the following: 1) Transfer function \( Y(s) / R(s) \) 2) The resulting transfer function is a standard \( 2^{\text {nd }} \) order which has the following format. \[ H(s)=\frac{\omega_{n}^{2}}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}} \] Determine the value of \( \mathrm{K} \) that would result in a maximum overshoot of \( 10 \% \) from a step input.
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Transcribed Image Text: For the block diagram shown above, determine the following: 1) Transfer function \( Y(s) / R(s) \) 2) The resulting transfer function is a standard \( 2^{\text {nd }} \) order which has the following format. \[ H(s)=\frac{\omega_{n}^{2}}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}} \] Determine the value of \( \mathrm{K} \) that would result in a maximum overshoot of \( 10 \% \) from a step input.
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【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/2Y(s) = G1(s) * G2(s) * R(s) - G2(s) * Y(s)Taking the Laplace transform of the above equation and rearranging, we get:Y(s) / R(s) = G1(s) * G2(s) / (1 + G2(s))Substituting G1(s) and G2(s) from the diagram, we get:Y(s) / R(s) = K / (s^3 + 3s^2 + 2s + K)To determine the value of K that would result in a maximum overshoot of 10% from a step, we need to first find the natural frequency and damping ratio of the transfer function. We can write the transfer function in the standard 2^(nd) order format as follows:H(s) = K / (s^3 + 3s^2 + 2s + K)H(s) = omega_n^2 / (s^2 + 2zetaomega_n*s + omega_n^2)where omega_n is the natural frequency and zeta is the damping ratio.Comparing the coefficients of the standard form and the transfer function, we get:omega_n^2 = K2zetaomega_n = 2solving for zeta, we get:zeta = 1 / (omega_n * sqrt(2))We want the system to have a maximum overshoot of 10% from a step. The maximum overshoot is given by:M_p = exp((-zeta*pi)/sqrt(1-zeta^2))From the above equation, we can solve for zeta to get the desired maximum overshoot:zeta = sqrt((ln(M_p))^2 / (pi^2 + ln(M_p)^2))Substituting M_p = 0.1, we get:zeta = 0.591Now, we can solve for K using the natural frequency and damping ratio:omega_n = sqrt(K) = sqrt(omega_n^2) = sqrt(zeta^2 + (2*pi/T)^2)Substituting T = 1, we get:omega_n = sqrt(zeta^2 + 4*pi^2)Substituting zeta = 0.591, we get:omega_n = 6.201 rad/sFinally, we can solve for K:K = omega_n^2 = 38.398Therefore, the transfer function is:Y(s) / R(s) = 38.398 / (s^3 + 3s^2 + 2s + 38.398Explanation:Please refer to solution in this step.Step2/2Y(s) = G1(s) * G2(s) * R(s) - G2(s) * Y(s)Taking the Laplace transform of the above equation and rearranging, we get:Y(s) / R(s) = G1(s) * G2(s) / (1 + G2(s))Substituting G1(s) and G2(s) from the diagram, we get:Y(s) / R(s) = K / (s^3 + 3s^2 + 2s + K)To determine the value of K that would result in a maximum overshoot of 10% from a step, we need to first find the natural frequency and damping ratio of the transfer function. We can write the transfer function in the standard 2^(nd) order format as follows:H(s) = K / (s^3 + 3s^2 + 2s + K)H(s) = omega_n^2 / (s^2 + 2zetaomega_n*s + omega_n^2)where omega_n is the natural frequency and zeta is the damping ratio.Comparing the coefficients of the standa ... See the full answer

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