Resonant Frequency MCQ Quiz - Objective Question with Answer for Resonant Frequency - Download Free PDF

The transfer function of the standard 2nd order system is defined as

\(T.F = \frac{{ω _n^2}}{{{s^2} + 2ξ {ω _n}s + ω _n^2}}\)

The resonant frequency ω0 is given by

\({ω _0} = {ω _n}\sqrt {1 - 2{ξ ^2}} \)

Where ωn = undamped natural frequency

ξ = Damping ratio

Resonant peak M0 is given by:

\({M_0} = \frac{1}{{2ξ \sqrt {1 - {ξ ^2}} }},\;0 \le ξ \le \frac{1}{{\sqrt 2 }}\)

Resonant frequency (ωr) is the frequency at which the magnitude characteristic of frequency response curve has its peak value.

The above equation is meaningful only for 1 – 2ζ2 ≥ 0, i.e.

\(zeta \le \frac{1}{\sqrt{2}}~or~\zeta \le 0.707\)

Note: If ζ > 0.707, then ωr = 0

Resonant frequency, \(\rm \omega _r={\omega _n}\sqrt {1 - 2{\xi ^2}}\)

Damping Ratio \(\rm \xi =\frac{1}{{2Q}}\)

Resonant Peak  \(\rm M_r = \frac{1}{{2\xi }}\sqrt {1 - {\xi ^2}}\)

Bandwidth \(\rm BW={\omega _n}\sqrt {1 - 2{\xi ^2} + \sqrt {2 - 4{\xi ^2} + 4{\xi ^4}} } \)

The transfer function of the standard 2nd order system is defined as

\(T.F = \frac{{ω _n^2}}{{{s^2} + 2ξ {ω _n}s + ω _n^2}}\)

The resonant frequency ω0 is given by

\({ω _0} = {ω _n}\sqrt {1 - 2{ξ ^2}} \)

Where ωn = undamped natural frequency

ξ = Damping ratio

Resonant peak M0 is given by:

\({M_0} = \frac{1}{{2ξ \sqrt {1 - {ξ ^2}} }},\;0 \le ξ \le \frac{1}{{\sqrt 2 }}\)

Resonant frequency (ωr) is the frequency at which the magnitude characteristic of frequency response curve has its peak value.

The above equation is meaningful only for 1 – 2ζ2 ≥ 0, i.e.

\(zeta \le \frac{1}{\sqrt{2}}~or~\zeta \le 0.707\)

Note: If ζ > 0.707, then ωr = 0

The transfer function of the standard 2nd order system is defined as

\(T.F = \frac{{ω _n^2}}{{{s^2} + 2ξ {ω _n}s + ω _n^2}}\)

The resonant frequency ω0 is given by

\({ω _0} = {ω _n}\sqrt {1 - 2{ξ ^2}} \)

Where ωn = undamped natural frequency

ξ = Damping ratio

Resonant peak M0 is given by:

\({M_0} = \frac{1}{{2ξ \sqrt {1 - {ξ ^2}} }},\;0 \le ξ \le \frac{1}{{\sqrt 2 }}\)

Resonant frequency (ωr) is the frequency at which the magnitude characteristic of frequency response curve has its peak value.

The above equation is meaningful only for 1 – 2ζ2 ≥ 0, i.e.

\(zeta \le \frac{1}{\sqrt{2}}~or~\zeta \le 0.707\)

Note: If ζ > 0.707, then ωr = 0

The characteristic equation for the given open-loop transfer function is defined as:

1 + G(s)H(s) = 0

∴ For the given open-loop transfer function, we will get the characteristic equation as:

\({s^2} + \frac{1}{T}s + \frac{K}{T} = 0\)     ---(1)

The standard form of the characteristic equation of the second-order system will be:

\({s^2} + 2\xi {\omega _n}s + \omega _n^2 = 0\)

Comparing this with Equation (1), we get:

\({\omega _n} = \sqrt {\frac{K}{T}}\)    ---(2)

Also \( \;2\xi {\omega _n} = \frac{1}{T}\)

Using Equation (2), we get:

\(\xi = \frac{1}{{2\sqrt {KT} }}\)      ---(3)

From the given problem, we have the maximum overshoot as:

M_p = 20%

Since the maximum overshoot is defined as:

\({M_p} = \frac{{ - \xi \pi }}{{e\sqrt {1 - {\xi ^2}} }} = \frac{{20}}{{100}}\)

Putting on the respective values:

\(\frac{{\xi \pi }}{{\sqrt {1 - {\xi ^2}} }} = 1.6\)

ξ = 0.45

Now, the resonant frequency is given by:

\({\omega _r} = {\omega _n}\sqrt {1 - 2{\xi ^2}} = 6\)

Therefore, the natural frequency will be:

\({\omega _n} = \frac{6}{{\sqrt {1 - 2{{\left( {0.45} \right)}^2}} }}\)

= 7.82 rad/sec

Substituting the values of ξ and ω_n in Equations (1) and (2), we get:

\({\left( {7.82} \right)^2} = \frac{K}{T}\;and\)

\(0.45 = \frac{1}{{2\sqrt {KT} }}\)

Solving the above equations, we obtain the values of K and T as:

K = 8.67 and T = 0.14 

Resonant frequency, \(\rm \omega _r={\omega _n}\sqrt {1 - 2{\xi ^2}}\)

Damping Ratio \(\rm \xi =\frac{1}{{2Q}}\)

Resonant Peak  \(\rm M_r = \frac{1}{{2\xi }}\sqrt {1 - {\xi ^2}}\)

Bandwidth \(\rm BW={\omega _n}\sqrt {1 - 2{\xi ^2} + \sqrt {2 - 4{\xi ^2} + 4{\xi ^4}} } \)

Resonant frequency, \(\rm \omega _r={\omega _n}\sqrt {1 - 2{\xi ^2}}\)

Damping Ratio \(\rm \xi =\frac{1}{{2Q}}\)

Resonant Peak  \(\rm M_r = \frac{1}{{2\xi }}\sqrt {1 - {\xi ^2}}\)

Bandwidth \(\rm BW={\omega _n}\sqrt {1 - 2{\xi ^2} + \sqrt {2 - 4{\xi ^2} + 4{\xi ^4}} } \)

Range ξ is \(0<ξ<1/√2\)

\({\left. {{\phi _r}} \right|_{\xi = 0}} = - {\tan ^{ - 1}}\left[ {\frac{{\sqrt {1 - 2{\xi ^2}} }}{\xi }} \right] = - 90^\circ \)

\({\left. {{\phi _r}} \right|_{\xi = \frac{1}{{\sqrt 2 }}}} = - {\tan ^{ - 1}}\left[ {\frac{{\sqrt {1 - 2{\xi ^2}} }}{\xi }} \right] = 0^\circ \)

Range ξ is \(0<ξ<1/√2\)

\({\left. {{\phi _r}} \right|_{\xi = 0}} = - {\tan ^{ - 1}}\left[ {\frac{{\sqrt {1 - 2{\xi ^2}} }}{\xi }} \right] = - 90^\circ \)

\({\left. {{\phi _r}} \right|_{\xi = \frac{1}{{\sqrt 2 }}}} = - {\tan ^{ - 1}}\left[ {\frac{{\sqrt {1 - 2{\xi ^2}} }}{\xi }} \right] = 0^\circ \)

\(\phi = {\tan ^{ - 1}}\left[ {\frac{{\sqrt {1 - 2{\xi ^2}} }}{\xi }} \right]\)

\(=64.12°\)

since control systems are Low pass filters that give more negative phase. So Ø = -64.12 degrees

Note :

Cos-1ξ  cannot be used here because this is resonance condition

The transfer function of the standard 2nd order system is defined as

\(T.F = \frac{{ω _n^2}}{{{s^2} + 2ξ {ω _n}s + ω _n^2}}\)

The resonant frequency ω0 is given by

\({ω _0} = {ω _n}\sqrt {1 - 2{ξ ^2}} \)

Where ωn = undamped natural frequency

ξ = Damping ratio

Resonant peak M0 is given by:

\({M_0} = \frac{1}{{2ξ \sqrt {1 - {ξ ^2}} }},\;0 \le ξ \le \frac{1}{{\sqrt 2 }}\)

Resonant frequency (ωr) is the frequency at which the magnitude characteristic of frequency response curve has its peak value.

The above equation is meaningful only for 1 – 2ζ2 ≥ 0, i.e.

\(zeta \le \frac{1}{\sqrt{2}}~or~\zeta \le 0.707\)

Note: If ζ > 0.707, then ωr = 0

Concept:

DC gain: DC Gain of a system is the gain at the steady-state which is at t tending to infinity i.e., s tending to zero.

Resonant peak (M_r): The maximum value of the magnitude of closed-loop transfer function in the frequency response is called the resonant peak.

\({M_r} = \frac{{DC\;gain}}{{2\zeta \sqrt {1 - {\zeta ^2}} }}\)

Resonant frequency (ω_r): The frequency at which the resonant peak occurs is termed as the resonant frequency.

\({\omega _r} = {\omega _n}\sqrt {1 - 2{\zeta ^2}} \)

Calculation:

DC gain calculation:

\(T\left( 0 \right) = \frac{{10K}}{K} = 10\)

Resonant peak = 20 (as shown in the figure)

Damping factor calculation:

\(20 = \frac{{10}}{{2\zeta \sqrt {1 - {\zeta ^2}} }}\)

\(4\zeta \sqrt {1 - {\zeta ^2}} = 1\)

\(16{\zeta ^2}\left( {1 - {\zeta ^2}} \right) = 1\)

Let ζ² = x

16x (1-x) = 1

16x² – 16x + 1 = 0

\(x = \frac{{16 \pm \sqrt {{{16}^2} - 4 \times 16\;} }}{{2 \times 16}}\)

X = 0.933 and x = 0.067

∴ ζ = 0.965 and ζ = 0.259

If resonant peak > 1 then \(\zeta \le \frac{1}{{\sqrt 2 }} = 0.707\)

∴ ζ = 0.259

Gain and natural frequency calculation:

The standard characteristic equation of a second order system is:

s² + 2ζ ω_n s + ω_n² = 0

Comparing the characteristic equation of the given T.F:

s² + 2ζ ω_n s + ω_n² = s² + 2s + K

ω_n = √ K

2 ζ ω_n = 2

2 ζ √ K = 2

K = \(\frac{1}{{{\zeta ^2}}} = \frac{1}{{0.067}} = 15.02\)

ω_n = √ 15.02 = 3.8 rad/s