Frequency Response

Chap8 - Freq 1

Agenda

• Preliminaries

• First order systems

– Frequency response

– Low-pass filter

• Second order systems

– Classical solutions

– Frequency response

• Higher order system

Chap8 - Freq 2

Frequency response

• Steady-state behavior of systems to harmonic excitations over a range of input frequencies

• Determination of important behavioral characteristics of dynamic systems by subjecting them to harmonic inputs and observing the response– Experimentally,

– Analytically, or

– Numerically.

Chap8 - Freq 3

Preliminaries - ODE

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Chap8 - Freq 4

Preliminaries - Laplace transform

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Chap8 - Freq 5

First Order Systems - Classical Solution

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Chap8 - Freq 6

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Chap8 - Freq 7

Frequency Response

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Chap8 - Freq 8

Frequency Response

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Chap8 - Freq 9

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Low-Pass Filter

Typical system response

Chap8 - Freq 10

Second-order Systems

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Chap8 - Freq 11

Second-Order Systems

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Chap8 - Freq 12

Second-Order Systems

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Chap8 - Freq 13

Second-Order System• The transfer function of a

2nd-order system:

• The frequency response of this

system can be modeled as:

• When :

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• The critical frequencies

are = 2 (zero), 10 (pole),

and 50 (pole).

• MATLAB (exact resp.):

w = logspace(-1,3,300);

s = j*w;

H = 1000*(s+2)./(s+10)./(s+50);

magdB = 20*log10(abs(H));

phase = angle(H)*180/pi;

• MATLAB (Bode):

num = [1000 2000];

den = conv([1 1o], [1 50]);

bode(num, den);

• Bode plots are useful as an

analytic tool.

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Examples 1

Find the time constant for the following RC circuit

that has parameters R=8200 Ω and C=2.2μf. Sketch the amplitude frequency response and determine the corner frequency. Approximately how long does it take the transient part of the solution to this system to die out?

Chap8 - Freq 16

Solution

Chap8 - Freq 17

Example 2

A harmonic signal of amplitude 1 and frequency 70 Hz is the input to a linear first-order system whose time constant is 0.5 second. What is the amplitude of the output? What is the phase of the output with respect to the input? Does the output lag behind the input or lead it?

Chap8 - Freq 18

Solution

Chap8 - Freq 19

Pair-Share: Example 3

Answer the same questions as in the previous problem, except that now the system is a second-order linear system with a natural frequency of 21 rad/sec and a damping ratio of 0.25

Chap8 - Freq 20

Solution

Chap8 - Freq 21

Pair-Share - Example 4It is proposed to attach an additional spring-mass-damper system to a primary spring-mass-damper system as shown in figure below. Find the steady-state amplitude of the displacement response of the primary mass, and plot it with respect to the input frequency for the cases with an without the attached system. Such attached systems can be used to absorb unwanted vibrations. Comment on the effectiveness of reducing vibration for this system. The parameters of the problem are:

Chap8 - Freq 22

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Solution

Chap8 - Freq 23

Solution (cont.)

Chap8 - Freq 24

Solution (cont.)

Chap8 - Freq 25

Example 5

Determine the transfer function relating the output voltage to the input voltage for the following RLC circuit:

Develop the corresponding Bode plot using data from Example 4.3. What is the natural frequency of the system? Redesign the circuit so as to increase its natural frequency by 30 percent, but keep the damping ratio at 0.707.

2e0e

Chap8 - Freq 26

Solution (cont.)

Chap8 - Freq 27

Solution

Chap8 - Freq 28

Solution (cont.)

Chap8 - Freq 29

Example 6

Find the roots of the characteristic equation of the following systems. What are the time constants and/or natural frequencies? What are the damping ratios? Comment on the stability of each. Solve by and check by digital computation.

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)(153282.

Chap8 - Freq 30

Solution

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Chap8 - Freq 31

Solution (cont.)

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)(4542. tfzzzzd

Chap8 - Freq 32

Example 7 Problem 8.12 a,c,e,g

Find the transfer function for each of the systems below, relating the output x or z to the input u or f.

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Chap8 - Freq 33

SolutionTake Laplace transform of each equation and solve for the response variable divided by the input variable. For d, first put into classical form.

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Chap8 - Freq 34

Example 8

Find the set of state-space equations for each of the below systems:

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tuxxc

tuxxb

tuxxxa

Chap8 - Freq 35

Solution

Chap8 - Freq 36

Solution (cont.)

Chap8 - Freq 37

Solution (Cont)

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Chap8 - Freq 38

Solution (Cont)

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Chap8 - Freq 39

Example 9Determine the transfer function coefficients for the circuit :

If:

Find the eigen values, and check the stability of the system. If the system is stable, determine its Bode plot.

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Chap8 - Freq 40

Solution

Chap8 - Freq 41

Solution (cont.)

Chap8 - Freq 42

Solution (cont.)

The Bode plot is

Chap8 - Freq 43