ME375 Dynamic System Modeling and Control

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MESB374 System Modeling and AnalysisFrequency
Response
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Frequency Response

• Forced Responses to Sinusoidal Inputs
• Transient and Steady-State Response
• Frequency Response
• Steady-State Response to sinusoidal inputs at
  various input frequencies
• Bode Plots
• A convenient graphic display of frequency
  response at all input frequencies

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Forced Response to Sinusoidal Inputs

• Ex Lets find the forced response of a stable
  first order system
• to a sinusoidal input
• Forced response in s-domain
• PFE
• Use ILT to obtain forced response in time-domain

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Forced Response to Sinusoidal Inputs
Input is sin(2t)
Output
Input
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Forced Response to Sinusoidal Inputs

• In-class Ex Given the same system as in the
  previous example, find the forced response to
  u(t) sin(10 t).

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Forced Response to Sinusoidal Inputs
Input is sin(10t)
Output
Input
Transient
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Forced Response to Sinusoidal Inputs

• Ex Lets revisit the same example where
• and the input is a general sinusoidal input
  sin(w t).
• Use the residue formula to find Ais

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Forced Response to Sinusoidal Inputs

• Ex Forced response in time-domain
• The steady state sinusoidal response in
  time-domain

Stable LTI System
Sinusoidal input
Sinusoidal output
Phase shift Changed Magnitude
What happens to frequency?
No Change!
where
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Forced Response to Sinusoidal Inputs
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Frequency Response

• Frequency response is used to study the steady
  state output ySS(t) of a stable LTI
  system to sinusoidal inputs at different
  frequencies.
• In general, given a stable system
• If the input is a sinusoidal signal with
  frequency w , i.e.
• then the steady state output ySS(t) is also a
  sinusoidal signal with the same frequency as the
  input signal but with different magnitude and
  phase
• where G(jw) is the complex number obtained by
  substitute jw for s in G(s) , i.e.

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Frequency Response
LTI System G(s)
Input u(t) U(s)
Output y(t) Y(s)
ySS
2p/w
t
?
?
?

• A different perspective of the role of the
  transfer function

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Frequency Response
G
Sinusoidal Input u(t)
Steady-state Sinusoidal Output y(t)
G
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In Class Exercise

• Ex 1st Order System
• The motion of a piston in a cylinder can be
  modeled by a 1st order system with force as input
  and piston velocity as output
• The EOM is
• (1) Let M 0.1 kg and B 0.5 N/(m/s), find the
  transfer function of the system

• (2) Calculate the steady state output of the
  system when the input is

f(t)
0)0
2
(-63.4349))
0.8944
v
(-75.9638))
0.4851
(-80.5377))
0.3288
(-82.8750))
0.2481
(-84.2894))
0.1990
0.1661
(-85.2364))
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In Class Exercise
(3) Plot the frequency response plot
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Example - Vibration Absorber (I)

• Let M1 10 kg, K1 1000 N/m, B1 4
  N/(m/s).Find the steady state response of the
  system for f(t) (a) sin(8.5t) (b) sin(10t)
  (c) sin(11.7t).

• Without vibration absorber
• EOM
• TF (from f(t) to z1)

(-6.9852))
0.0036
0.025
(-90.000))
(-172.7699))
0.0027
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Example - Vibration Absorber (I)
Poles
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Example - Vibration Absorber (II)

• With vibration absorber
• EOM
• TF (from f(t) to z1)

• Let M1 10 kg, K1 1000 N/m, B1 4 N/(m/s),
  M2 1 kg, K2 100 N/m, and B2 0.1 N/(m/s).
  Find the steady state response of the system for
  f(t) (a) sin(8.5t) (b) sin(10t) (c)
  sin(11.7t).

0.023
(-66.5))
0.001
(-90))
0.016
(-88.5))
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Example - Vibration Absorber (II)
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Example - Vibration Absorber (II)

• Characterizing Transient Response
• The characteristic equation
• Q1 Can you make a good guess of the
  duration of the transient period?

Q2 Can you explain the observed steady-state
sinusoidal responses?
If the frequency of input is near the imaginary
part of one of poles, resonance will possibly
happen. If the frequency of input is near the
imaginary part of one of zeros, the effect of
input will probably be absorbed.
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Example - Vibration Absorbers

• Frequency Response Plot
• Absorber tuned at 10 rad/sec

• Frequency Response Plot
• No absorber added

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Example - Vibration Absorbers

• Bode Plot
• No absorber added

• Bode Plot
• Absorber tuned at 10 rad/sec

A better way to graphically display Frequency
Response !
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Example -- SDOF Suspension

• Simplified Schematic (neglecting tire model)

• Suspension System
• How does the traveling speed influence the
  magnitude of joggling?

Car body
z
K
B
Suspension
v
Wheel
Ap
xp
Reference
L
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Example -- SDOF Suspension
Find
Given
How does the traveling speed influence the
magnitude of joggling
All parameters are normalized