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The damping ratio of a system having the characteristic equation s2 + 2s + 8 = 0 is
- a)0.300 (underdamped)
- b)0.353 (underdamped)
- c)1.24 (overdamped)
- d)1.02 (overdamped)
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The damping ratio of a system having the characteristic equation s2 + ...
Given, s2 + 2s + 8 = 0
Here, ωn = √8 rad/s = 2√2 rad/sec
and
or,
Since,
therefore
system is underdamped.
Here, ωn = √8 rad/s = 2√2 rad/sec
and
or,
Since,
thereforesystem is underdamped.
Most Upvoted Answer
The damping ratio of a system having the characteristic equation s2 + ...
Characteristic Equation:
The characteristic equation of a system is given by:
s^2 + 2ζω_ns + ω_n^2 = 0
where,
s = Laplace variable
ζ = damping ratio
ω_n = natural frequency
Given Characteristic Equation:
s^2 + 2s + 8 = 0
Comparing this equation with the characteristic equation, we can see that:
ζ = 1
ω_n^2 = 8
Damping Ratio:
The damping ratio (ζ) is a measure of how fast the oscillations in a system die out. It determines the behavior of the system response.
Damping Ratio and System Response:
The value of the damping ratio determines the type of system response. There are three main cases:
1. ζ < 1="" (underdamped):="" the="" system="" response="" oscillates="" before="" settling="" down="" to="" the="" steady-state.="" the="" response="" is="" characterized="" by="" decaying="" />
2. ζ = 1 (critically damped): The system response reaches the steady-state as quickly as possible without oscillating.
3. ζ > 1 (overdamped): The system response slowly approaches the steady-state without any oscillations.
Solution:
In the given characteristic equation, ζ = 1.
Therefore, the system is critically damped.
The correct answer is option 'B' (0.353), which corresponds to an underdamped system. This is an incorrect answer.
The given options do not match the characteristics of the system. The correct answer should be 'C' (1.24), which corresponds to an overdamped system.
The characteristic equation of a system is given by:
s^2 + 2ζω_ns + ω_n^2 = 0
where,
s = Laplace variable
ζ = damping ratio
ω_n = natural frequency
Given Characteristic Equation:
s^2 + 2s + 8 = 0
Comparing this equation with the characteristic equation, we can see that:
ζ = 1
ω_n^2 = 8
Damping Ratio:
The damping ratio (ζ) is a measure of how fast the oscillations in a system die out. It determines the behavior of the system response.
Damping Ratio and System Response:
The value of the damping ratio determines the type of system response. There are three main cases:
1. ζ < 1="" (underdamped):="" the="" system="" response="" oscillates="" before="" settling="" down="" to="" the="" steady-state.="" the="" response="" is="" characterized="" by="" decaying="" />
2. ζ = 1 (critically damped): The system response reaches the steady-state as quickly as possible without oscillating.
3. ζ > 1 (overdamped): The system response slowly approaches the steady-state without any oscillations.
Solution:
In the given characteristic equation, ζ = 1.
Therefore, the system is critically damped.
The correct answer is option 'B' (0.353), which corresponds to an underdamped system. This is an incorrect answer.
The given options do not match the characteristics of the system. The correct answer should be 'C' (1.24), which corresponds to an overdamped system.
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The damping ratio of a system having the characteristic equation s2 + 2s + 8 = 0 isa)0.300 (underdamped)b)0.353 (underdamped)c)1.24 (overdamped)d)1.02 (overdamped)Correct answer is option 'B'. Can you explain this answer?
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