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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