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Consider a system with transfer function G(s) = s+6/Ks2+s+6. Its damping ratio will be 0.5 when the values of k is:
- a)2/6
- b)3
- c)1/6
- d)6
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Consider a system with transfer function G(s) = s+6/Ks2+s+6. Its dampi...
Answer: c
Explanation: s+6/K[s2+s/K+6/K] Comparing with s2+2Gw+w2
w= √6/K
2Gw=1/K
2*0.5*√6/K =1/K
K=1/6.
Explanation: s+6/K[s2+s/K+6/K] Comparing with s2+2Gw+w2
w= √6/K
2Gw=1/K
2*0.5*√6/K =1/K
K=1/6.
Most Upvoted Answer
Consider a system with transfer function G(s) = s+6/Ks2+s+6. Its dampi...
Given transfer function: G(s) = s^6 / Ks^2 + s^6
Damping ratio is given by: ζ = 0.5
To find the value of K at ζ = 0.5, we need to use the following formula:
ζ = damping ratio = ζ_n / √(1 - ζ^2)
where ζ_n = natural damping ratio
For a second-order system, the natural damping ratio is given by:
ζ_n = 1 / (2ζ)
Substituting the given values, we get:
0.5 = (1 / (2ζ)) / √(1 - 0.5^2)
Simplifying the above equation, we get:
ζ = 1 / (2√3)
Now, substituting the value of ζ_n in the transfer function, we get:
ζ_n = √(K / 6)
ζ_n = 1 / (2ζ)
K / 6 = (1 / (2ζ))^2
Substituting the value of ζ, we get:
K / 6 = (1 / (2 * 1 / (2√3)))^2
K / 6 = 1 / (1/3)
K / 6 = 3
K = 6 * 3
K = 18
Therefore, the value of K at ζ = 0.5 is 1/6 or option (c).
Damping ratio is given by: ζ = 0.5
To find the value of K at ζ = 0.5, we need to use the following formula:
ζ = damping ratio = ζ_n / √(1 - ζ^2)
where ζ_n = natural damping ratio
For a second-order system, the natural damping ratio is given by:
ζ_n = 1 / (2ζ)
Substituting the given values, we get:
0.5 = (1 / (2ζ)) / √(1 - 0.5^2)
Simplifying the above equation, we get:
ζ = 1 / (2√3)
Now, substituting the value of ζ_n in the transfer function, we get:
ζ_n = √(K / 6)
ζ_n = 1 / (2ζ)
K / 6 = (1 / (2ζ))^2
Substituting the value of ζ, we get:
K / 6 = (1 / (2 * 1 / (2√3)))^2
K / 6 = 1 / (1/3)
K / 6 = 3
K = 6 * 3
K = 18
Therefore, the value of K at ζ = 0.5 is 1/6 or option (c).
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