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MECH431_topic5_Frequency

# MECH431_topic5_Frequency - K G(s Frequency-response...

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K G(s) + - MECH 431 Frequency response Slide 1 Frequency-response analysis Chapter 8

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K G(s) + - MECH 431 Frequency response Slide 2 Introduction Frequency response refers to the steady state response of a system to a sinusoidal input. In frequency-response methods, we vary the frequency of the input signal over a certain range and study the resulting response. The information that is obtained from the frequency response is different from the information we get from the root locus. Both methods complement each other.
K G(s) + - MECH 431 Frequency response Slide 3 Introduction The Nyquist stability criterion enables us to investigate both the absolute and relative stabilities of linear closed-loop systems from a knowledge of their open-loop frequency response characteristics. Frequency-response tests are in general simple and can be made accurately by use of readily available sinusoidal signal generators and accurate measurement equipment.

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K G(s) + - MECH 431 Frequency response Slide 4 Introduction Often the transfer functions of complicated components can be determined experimentally by frequency- response tests. In designing a closed-loop system, we adjust the frequency-response characteristic of the open-loop transfer function by varying several design criteria in order to obtain appropriate transient-response characteristics for the system.
K G(s) + - MECH 431 Frequency response Slide 5 Obtaining steady-state outputs G(s) It will be shown next that after waiting until steady- state, the frequency response can be calculated by replacing s in the closed loop transfer function by jw . The transfer function becomes: Phase shift between input sine and output sine Amplitude ratio

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K G(s) + - MECH 431 Frequency response Slide 6 Obtaining steady-state outputs In the frequency response test, the input frequency w is varied until the entire frequency range of interest is covered. (1) L ( X sin wt )
K G(s) + - MECH 431 Frequency response Slide 7 Obtaining steady-state outputs The inverse Laplace transform of (1) yields: For the system to be stable, - s 1 , - s 2 ,…,- s n have negative real parts and they all drop out of the equation as t .

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K G(s) + - MECH 431 Frequency response Slide 8 Obtaining steady-state outputs The constants can be found from: and: a = G ( s ) wX s 2 + w 2 s + jw ( ) s = jw = XG jw ( ) 2 j = a = G ( s ) wX s 2 + w 2 s jw ( ) s = jw = XG jw ( ) 2 j = magnitude angle
K G(s) + - MECH 431 Frequency response Slide 9 Obtaining steady-state outputs And we can re-express y ss (t) as: and using Euler’s formula: Different amplitude Same frequency Different phase Recall:

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K G(s) + - MECH 431 Frequency response Slide 10 Obtaining steady-state outputs So a linear time-invariant system subjected to a sinusoidal input will, at steady state , have a sinusoidal output of the same frequency as the input but with different amplitude and phase .
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MECH431_topic5_Frequency - K G(s Frequency-response...

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