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The natural frequency of an undamped second-order system is 40 rad/s. If the system is
damped with a damping ratio 0.3, the damped natural frequency in rad/s is ________.
damped with a damping ratio 0.3, the damped natural frequency in rad/s is ________.
Correct answer is '38.15'. Can you explain this answer?
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The natural frequency of an undamped second-order system is 40 rad/s. ...
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The natural frequency of an undamped second-order system is 40 rad/s. ...
Undamped Second-Order System
An undamped second-order system is a system that does not experience any damping. It is characterized by a natural frequency (ωn) which determines the rate at which the system oscillates. In this case, the natural frequency is given as 40 rad/s.
Damped Second-Order System
A damped second-order system is a system that experiences damping, which causes the system to lose energy and slows down its response. The damping ratio (ζ) determines the amount of damping in the system. The damping ratio is given as 0.3 in this case.
Damped Natural Frequency
The damped natural frequency (ωd) is the frequency at which the damped system oscillates. It can be calculated using the formula:
ωd = ωn * sqrt(1 - ζ^2)
where ωn is the natural frequency and ζ is the damping ratio.
Substituting the given values into the formula:
ωd = 40 rad/s * sqrt(1 - 0.3^2)
= 40 rad/s * sqrt(1 - 0.09)
= 40 rad/s * sqrt(0.91)
≈ 38.15 rad/s
Hence, the damped natural frequency is approximately 38.15 rad/s.
An undamped second-order system is a system that does not experience any damping. It is characterized by a natural frequency (ωn) which determines the rate at which the system oscillates. In this case, the natural frequency is given as 40 rad/s.
Damped Second-Order System
A damped second-order system is a system that experiences damping, which causes the system to lose energy and slows down its response. The damping ratio (ζ) determines the amount of damping in the system. The damping ratio is given as 0.3 in this case.
Damped Natural Frequency
The damped natural frequency (ωd) is the frequency at which the damped system oscillates. It can be calculated using the formula:
ωd = ωn * sqrt(1 - ζ^2)
where ωn is the natural frequency and ζ is the damping ratio.
Substituting the given values into the formula:
ωd = 40 rad/s * sqrt(1 - 0.3^2)
= 40 rad/s * sqrt(1 - 0.09)
= 40 rad/s * sqrt(0.91)
≈ 38.15 rad/s
Hence, the damped natural frequency is approximately 38.15 rad/s.
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The natural frequency of an undamped second-order system is 40 rad/s. If the system isdamped with a damping ratio 0.3, the damped natural frequency in rad/s is ________.Correct answer is '38.15'. Can you explain this answer?
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