The open loop transfer function of a unity feedback control system is given by \(G(s)=\frac{25}{s(s+5)}\). The natural frequency of oscillator is fixed. Arrange the damped frequency of oscillation for the following damping ratio in ascending order A. \(\xi = 0.5\) B. \(\xi = 0.1\) C. \(\xi = 0.3\) D. \(\xi = 0.25\) E. \(\xi = 0.4\) Choose the correct answer from the options given below :
\(A \lt E \lt C \lt D \lt B\)
Step 1 — find the closed-loop form. With unity feedback and \(G(s)=\dfrac{25}{s(s+5)}\):
\(T(s)=\dfrac{G}{1+G}=\dfrac{25}{s^{2}+5s+25}\)
Comparing with the standard second-order denominator \(s^{2}+2\xi\omega_n s+\omega_n^{2}\) gives
\(\omega_n=\sqrt{25}=5\ \text{rad/s}\)
and the question explicitly holds this natural frequency fixed for all five cases.
Step 2 — the quantity being ranked. The damped frequency of oscillation is the imaginary part of the closed-loop poles \(s=-\xi\omega_n \pm j\omega_n\sqrt{1-\xi^{2}}\):
\(\omega_d=\omega_n\sqrt{1-\xi^{2}}\)
With ωn constant, ωd decreases monotonically as ξ increases — heavier damping means slower ringing.
Step 3 — evaluate for each damping ratio (ωn = 5 rad/s):
| Item | ξ | \(\omega_d=5\sqrt{1-\xi^{2}}\) |
|---|---|---|
| A | 0.50 | 4.33 rad/s — smallest |
| E | 0.40 | 4.58 rad/s |
| C | 0.30 | 4.77 rad/s |
| D | 0.25 | 4.84 rad/s |
| B | 0.10 | 4.97 rad/s — largest |
Step 4 — read off the ascending order.
\(A \lt E \lt C \lt D \lt B\)
i.e. the ascending order of ωd is exactly the descending order of ξ.
Points worth noting. All five values satisfy ξ < 1, so every case is under-damped and genuinely oscillatory. Notice how weakly ωd depends on ξ for small ξ (at ξ = 0.1 it is still 99.5 % of ωn) — the square root flattens the dependence. What ξ does change strongly is the overshoot, \(M_p=e^{-\pi\xi/\sqrt{1-\xi^{2}}}\), and the decay rate ξωn.
Hence, the ascending order of damped frequency is \(A \lt E \lt C \lt D \lt B\).
Similar Questions
consider a unity feedback control system for an open loop transfer function \(G(s)=\frac{5}{s(s+1)}\). Arrange the time constant in ascending order at different value of damping ratio of
A. \(\xi=3\)
B. \(\xi=7\)
C. \(\xi=1\)
D. \(\xi=10\)
E. \(\xi=5\)
Choose the correct answer from the options given below :
The addition of a pole to the forward path transfer function of a closed loop system, generally has the effect of
Match the following lists :
List - I List - II a. Negative real and simple roots i. Sustained oscillatory b. Negative real and equal roots ii. Overdamped c. Complex conjugate roots iii. Critically damped d. Imaginary conjugate roots iv. Underdamped Correct codes are :
The step, ramp and parabolic test input signals can respectively be expressed as
Assertion (A) : In control systems, steady state response in the final requirement for calculating the efficiency of the system.
Reason (R) : The transient response is also critical for the determination of the steady state response.
If the characteristic equation of a closed loop system is S2 + 2S + 2 = 0, then the system is
Important Questions from Time Response Analysis
Match List I with List II:
List I
(Effeet of ξ)
List II
(Condition of System)
(A) 0 < ξ < 1 (I) Over damped (B) ξ > 1 (II) Undamped (C) ξ = 0 (III) Unstable (D) ξ = −1 (IV) Under damped Choose the correct answer from the options given below:
What is the value of ωn in the given transfer function?
\(G\left( s \right) = \frac{{36}}{{{s^2} + 4.2s + 36}}\)
Which of the following is correct for over-damped and under-damped system, respectively?
What will be the time response expression for a standard first order system having unit step function \(\frac{1}{s}\) as the input
A second order control system is NOT required to satisfy the following specification:
