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Consider the characteristic equation of a control system given by s3 + (K + 0.5)s2 + 4Ks + 50 = 0. Find the value of the frequency if the system has sustained oscillations for a given K.
- a)ω = 50 rad/sec
- b)ω = 25 rad/sec
- c)ω = 3.63 rad/sec
- d)ω = 4.63 rad/sec
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Consider the characteristic equation of a control system given by s3 +...
To find the value of the frequency for sustained oscillations, we need to find the roots of the characteristic equation. The characteristic equation is given by:
s^3 + (0.5K)s^2 + 4Ks + 50 = 0
To find the roots, we can use the Routh-Hurwitz stability criterion. The Routh-Hurwitz table for this characteristic equation is as follows:
1 4K
0.5K 50
(-0.5K*50)/(4K) = -12.5K
From the Routh-Hurwitz table, we can see that for sustained oscillations to occur, we must have at least one sign change in the first column of the table. In this case, since the first column has two positive values (1 and 0.5K), there are no sign changes.
Therefore, the system does not have sustained oscillations for any value of K.
s^3 + (0.5K)s^2 + 4Ks + 50 = 0
To find the roots, we can use the Routh-Hurwitz stability criterion. The Routh-Hurwitz table for this characteristic equation is as follows:
1 4K
0.5K 50
(-0.5K*50)/(4K) = -12.5K
From the Routh-Hurwitz table, we can see that for sustained oscillations to occur, we must have at least one sign change in the first column of the table. In this case, since the first column has two positive values (1 and 0.5K), there are no sign changes.
Therefore, the system does not have sustained oscillations for any value of K.
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Community Answer
Consider the characteristic equation of a control system given by s3 +...
Characteristic Equation
s3 + (K + 0.5)s2 + 4Ks + 50 = 0
For cubic equation
Inner product = Outer product
(K+0.5)4K = 50
4K2 +2K-50 =0
K= 3.29
For Frequency of oscillation :
Auxillary Equation
3.79 S2 + 50 =0
S= 3.63 rad/sec
s3 + (K + 0.5)s2 + 4Ks + 50 = 0
For cubic equation
Inner product = Outer product
(K+0.5)4K = 50
4K2 +2K-50 =0
K= 3.29
For Frequency of oscillation :
Auxillary Equation
3.79 S2 + 50 =0
S= 3.63 rad/sec
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Consider the characteristic equation of a control system given by s3 + (K + 0.5)s2 + 4Ks + 50 = 0. Find the value of the frequency if the system has sustained oscillations for a given K.a)ω = 50 rad/secb)ω = 25 rad/secc)ω = 3.63 rad/secd)ω = 4.63 rad/secCorrect answer is option 'C'. Can you explain this answer?
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