System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 9, Problem 9.34P
To determine

The expression for the steady state response System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  1

The steady state response for the given function is given by

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  2

Given:

Fourier series approximation and is given by

Fourier series approximation = System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  3

The model equation is System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  4

Concept Used:

We first derive an expression for transfer function of the given system and then calculate System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  5 magnitude of the given transfer function, phase angle, bandwidth, and magnitude of phase angles of respective frequencies and finally calculate the expression for steady state response.

Calculation:

Fourier series approximation System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  6

The model equation is System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  7

Applying Laplace transform for the above equation we get,

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  8

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  9

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  10

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  11

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  12

Therefore, the transfer function of the given model is given by

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  13 (1)

Comparing the above equation with standard transfer function System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  14 we get

Time constant System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  15 0.1s

The bandwidth is given by

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  16

Substituting for System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  17 0.1s we get

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  18 10 rad/s

Considering the transfer equation in equation (1), the magnitude of transfer function is given by

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  19

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  20 (2)

The phase angle for the given model is given by

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  21 (3)

The bandwidth System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  22 for the given system should lie between 0 and 10 System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  23 as System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  24 10 rad/s

So, we consider only System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  25 4 and 8

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  26

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  27

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  28

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  29

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  30

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  31

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  32

The phase angle for the given model is given by

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  33

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  34

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  35

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  36

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  37

The steady state response for the given Fourier series,

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  38

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  39

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  40

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  41

Conclusion:

Therefore, the steady stat response for the given function is given by

System Dynamics, Chapter 9, Problem 9.34P , additional homework tip  42

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