Underdamped System MCQ Quiz - Objective Question with Answer for Underdamped System - Download Free PDF

Explanation:

In system modeling using a transfer function, the roots of the characteristic equation are the Poles of the transfer function.

• The characteristic equation of a system is typically obtained by setting the denominator polynomial of the closed-loop transfer function to zero.
• The poles of a transfer function are the values of 's' that make the denominator of the transfer function equal to zero (assuming no common factors with the numerator).
• The locations of the poles in the s-plane are critical for determining the stability and transient response of a system. If any pole lies in the right half of the s-plane, the system is unstable.

The correct answer is Undamped
Concept:

The general expression of the transfer function of the standard second-order system is:

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}\)

Where,

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation:

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

Roots of the characteristic equation are: 

\(- \zeta {\omega _n} + j{\omega _n}\sqrt {1 - {\zeta ^2}} = - \alpha \pm j{\omega _d}\)

α is the damping factor

The nature of the system is described by its ‘ζ’ value

calculation:
Given:
characteristics equation  is s2 + 2 = 0
\({s^2} + 2\zeta {\omega _n} + \omega _n^2 = 0\)
Comparing it with CE.
since , 2ζ ωn = 0,
ζ=0
so nature of the system is Undamped.

The general expression of the transfer function of the standard second-order system is:

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}\)

Where,

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation:

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

Roots of the characteristic equation are: 

\(- \zeta {\omega _n} + j{\omega _n}\sqrt {1 - {\zeta ^2}} = - \alpha \pm j{\omega _d}\)

α is the damping factor

The nature of the system is described by its ‘ζ’ value

• Negative-feedback amplifier subtracts a fraction of its output from its input so that negative feedback opposes the original signal.
• The applied negative feedback improves its performance namely gain stability, linearity, frequency response, step response and reduces sensitivity to parameter variations due to manufacturing or external.
• So, a two-stage amplifier with negative feedback is always stable.
• A two-stage amplifier with negative feedback has an overshoot when damping factor K is Less than unity.
   

​Explanation:

Underdamped system has the damping factor K is less than 1 and the peak overshoot occurs.

Explanation:

The general expression of the transfer function of the standard second-order system is:

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}\)

Where,

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation:

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

Roots of the characteristic equation are: 

\(- \zeta {\omega _n} + j{\omega _n}\sqrt {1 - {\zeta ^2}} = - \alpha \pm j{\omega _d}\)

α is the damping factor

The nature of the system is described by its ‘ζ’ value

Important Points

• Negative-feedback amplifier subtracts a fraction of its output from its input so that negative feedback opposes the original signal.
• The applied negative feedback improves its performance namely gain stability, linearity, frequency response, step response and reduces sensitivity to parameter variations due to manufacturing or external.
• So, a two-stage amplifier with negative feedback is always stable.
• A two-stage amplifier with negative feedback has an overshoot when damping factor K is Less than unity.
   

​Explanation:

Underdamped system has the damping factor K is less than 1 and the peak overshoot occurs.

Concept:

The standard second-order system is given by \(\frac{{\omega _n^2}}{{{s^2} + 2\xi {\omega _n}s + \omega _n^2}}\)

Where ξ is the damping ratio.

If ξ = 1, then the system is critically damped.

If ξ < 1, then the system is underdamped.

If ξ > 1, then the system is order damped.

Calculation:

1. \(H\left( s \right) = \frac{1}{{{s^2} + 4s + 4}}\)

By comparing with a standard second-order transfer function,

ωn2 = 4 ⇒ ωn = 2

\(2\xi {{\rm{\omega }}_n} = 4 \Rightarrow \xi = \frac{4}{{2\times2 }} = 1\)

So, it is a critically damped system.

2. \(H\left( s \right) = \frac{1}{{{s^2} + 5s + 4}}\)

ωn2 = 4 ⇒ ωn = 2

2 ξ ωn = 5 ⇒ ξ > 1

So, it is over damped system.

3. \(H\left( s \right) = \frac{1}{{{s^2} + 4.5s + 4}}\)

ωn2 = 4 ⇒ ωn = 2

\(2\xi {{\rm{\omega }}_n} = 4.5 \Rightarrow \xi > 1\)

So, it is overdamped system.

4. \(H\left( s \right) = \frac{1}{{{s^2} + 3s + 4}}\)

ωn2 = 4 ⇒ ωn = 2

2 ξ ωn = 3 ⇒ ξ < 1

So, it is under damped system.

Concept:

The pole location for a general underdamped second-order system is given as:

The peak time is given by:

\({t_p} = \frac{\pi }{{{\omega _n}\sqrt {1 - {\xi ^2}} }} = \frac{\omega }{{{\omega _d}}}\)

\({t_p} \propto \frac{1}{{{\omega _d}}} \propto \frac{1}{{imaginary\;part\;of\;pole}}\)

And the setting time is given by:

\({t_s} = \frac{4}{{\xi {\omega _n}}} = \frac{4}{{{\sigma _d}}}\)

i.e. \({t_s} \propto \frac{1}{{{\sigma _d}}} \propto \frac{1}{{Real\;part\;of\;pole}}\) 

ω_d = Damped frequency of oscillation

σ_d = exponential damping  

Analysis:

From the given step response of the system, peak times are related as:

\({t_{{p_1}}} > {t_{{p_2}}} > {t_{{p_3}}}\)

∴ \({\omega _{{d_1}}} < {\omega _{{d_2}}} < {\omega _{{d_3}}}\)

Also, given that the envelope is the same for all three systems, i.e. exponential damping (σ_d) or real parts of poles will be constant.

∴ The poles of the system will be:

\({s_1} = - {\sigma _d} \pm j{\omega _{{d_1}}}\)

\({s_2} = - {\sigma _d} \pm j{\omega _{{d_2}}}\)

\({s_3} = - {\sigma _d} \pm j{\omega _{{d_3}}}\)

Where \({\omega _{{d_1}}} < {\omega _{{d_2}}} < {\omega _{{d_3}}}\) 

∴ Plot in option (4) is correct. 

Explanation:

The general expression of the transfer function of the standard second-order system is:

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}\)

Where,

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation:

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

Roots of the characteristic equation are: 

\(- \zeta {\omega _n} + j{\omega _n}\sqrt {1 - {\zeta ^2}} = - \alpha \pm j{\omega _d}\)

α is the damping factor

The nature of the system is described by its ‘ζ’ value

Important Points

Concept:

If the open-loop transfer function G(s) is connected in positive feedback with a feedback gain of H(s), then the transfer function of the closed-loop system is given as:

 \(T(s)=\frac{{G\left( s \right)}}{{1 - G\left( s \right)H\left( s \right)}}\)

If the open-loop transfer function G(s) is connected in negative feedback with a feedback gain of H(s), then the transfer function of the closed-loop system is given as:

 \(T(s)=\frac{{G\left( s \right)}}{{1 + G\left( s \right)H\left( s \right)}}\)

The second-order standard characteristic equation is given as:

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

Where ζ is the damping ratio

ωn is the natural frequency

• For ζ = 0, system is undamped
• For 0 < ζ < 1, system is underdamped
• For ζ = 1, the system is critically damped
• For ζ > 1, the system is overdamped

Calculation:

The closed-loop transfer function,

 \(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{3}{{s\left( {s + 4} \right) + 3\left( {s + 12} \right)}} = \frac{3}{{{s^2} + 7s + 36}}\)

On comparing the above equation with a standard transfer function, we have,

\(\omega _n^2 = 36 \Rightarrow {\omega _n} = 6\;rad/sec\)

2ξω_n = 7

\(\xi = \frac{7}{{2{\omega _n}}} = \frac{7}{{12}} < 1\)

Comparing the given equation with standard 2^nd order response

\(c\left( t \right) = 1 - \frac{{{e^{ - {\rm{\zeta \;}}{{\rm{\omega }}_{nt}}}}}}{{\sqrt {1 - {\rm{\zeta }}{{\rm{\;}}^2}} }}\sin \left( {{\omega _{d}t} + {{\cos }^{ - 1}}{\rm{\zeta \;}}} \right)\)

\(y\left( t \right) = 1 - \frac{2}{{\sqrt 3 }}{e^{ - {\rm{ \;}}{{\rm{}}_{t}}}}\cos \left( {\sqrt 3 t - \frac{\pi }{3}} \right)\)

\(\sqrt {1 - {\rm{\zeta }}{{\rm{\;}}^2}} = \frac{{\sqrt 3 }}{2}\)

Squaring both sides

\(1 - {\rm{\zeta }}{{\rm{\;}}^2} = \frac{3}{4}\)

\(\frac{1}{4} = {\rm{\zeta }}{{\rm{\;}}^2}\)

\({\rm{\zeta \;}} = \frac{1}{2}\)

\({\rm{\zeta \;}}{{\rm{\omega }}_n} = + 1\)

\(\left( {\frac{1}{2}} \right){\omega _n} = 1\)

The general expression of the transfer function of the standard second-order system is:

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}\)

Where,

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation:

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

Roots of the characteristic equation are: 

\(- \zeta {\omega _n} + j{\omega _n}\sqrt {1 - {\zeta ^2}} = - \alpha \pm j{\omega _d}\)

α is the damping factor

The nature of the system is described by its ‘ζ’ value

• Negative-feedback amplifier subtracts a fraction of its output from its input so that negative feedback opposes the original signal.
• The applied negative feedback improves its performance namely gain stability, linearity, frequency response, step response and reduces sensitivity to parameter variations due to manufacturing or external.
• So, a two-stage amplifier with negative feedback is always stable.
• A two-stage amplifier with negative feedback has an overshoot when damping factor K is Less than unity.
   

​Explanation:

Underdamped system has the damping factor K is less than 1 and the peak overshoot occurs.

Concept:

The standard second-order system is given by \(\frac{{\omega _n^2}}{{{s^2} + 2\xi {\omega _n}s + \omega _n^2}}\)

Where ξ is the damping ratio.

If ξ = 1, then the system is critically damped.

If ξ < 1, then the system is underdamped.

If ξ > 1, then the system is order damped.

Calculation:

1. \(H\left( s \right) = \frac{1}{{{s^2} + 4s + 4}}\)

By comparing with a standard second-order transfer function,

ωn2 = 4 ⇒ ωn = 2

\(2\xi {{\rm{\omega }}_n} = 4 \Rightarrow \xi = \frac{4}{{2\times2 }} = 1\)

So, it is a critically damped system.

2. \(H\left( s \right) = \frac{1}{{{s^2} + 5s + 4}}\)

ωn2 = 4 ⇒ ωn = 2

2 ξ ωn = 5 ⇒ ξ > 1

So, it is over damped system.

3. \(H\left( s \right) = \frac{1}{{{s^2} + 4.5s + 4}}\)

ωn2 = 4 ⇒ ωn = 2

\(2\xi {{\rm{\omega }}_n} = 4.5 \Rightarrow \xi > 1\)

So, it is overdamped system.

4. \(H\left( s \right) = \frac{1}{{{s^2} + 3s + 4}}\)

ωn2 = 4 ⇒ ωn = 2

2 ξ ωn = 3 ⇒ ξ < 1

So, it is under damped system.

The correct answer is Undamped
Concept:

The general expression of the transfer function of the standard second-order system is:

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}\)

Where,

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation:

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

Roots of the characteristic equation are: 

\(- \zeta {\omega _n} + j{\omega _n}\sqrt {1 - {\zeta ^2}} = - \alpha \pm j{\omega _d}\)

α is the damping factor

The nature of the system is described by its ‘ζ’ value

calculation:
Given:
characteristics equation  is s2 + 2 = 0
\({s^2} + 2\zeta {\omega _n} + \omega _n^2 = 0\)
Comparing it with CE.
since , 2ζ ωn = 0,
ζ=0
so nature of the system is Undamped.