1. Frequency Response Analysis
Chapter 8

2. Outline Segway from earlier work Fundamental Theorem of Linear Systems
Graphical representations of TFs
Bode Diagrams (Section 8.2)
Bode Diagrams with Matlab (Section 8.3)
Phase and Gain margins; Resonant Peak Magnitude and Resonant Frequency; Correlation between step transient response and Frequency Response in standard 2nd order system (some of Section 8.9) See FE Reference
Ungraded HW. A problems: 15, 16 (def of bandwidth), 19 (need def of miniphase)
Graded HW. B problems: 1, 2, 4 (check with MatLab), 6, 26, 27, 29

3. Segway from earlier work
We have solved several forms of a problem in the past.
We will re-solve a form of this problem now.
The solution process reviews several issues that allow us to solve the central result that forms the foundation of frequency analysis.
Consider computing the acceleration that a spring-mass-damper system experiences when it is released from rest.

4. Hanging Spring Mass Damper
Prototype form
Prototype parameters and problem parameters:
Ink/chalk/lead/space saving names:

5. Transfer Functions Relating position to input
Relating acceleration to input
Acceleration due to a step input
Inverse Laplace transforming

8. Fundamental Theorem of Linear Systems
Summarizes the steady state response of a stable linear time invariant system to a sinusoidal input.
Sinusoidal inputs are common in nature, are easy to obtain in the lab.
Consider a stable system with transfer function G(s) and input

10. Compare input function to steady-state output.
Draw conclusions.
Magnitude and phase of G(s) are important.

11. Graphical Representation of TFs
Bode plots.
Nyquist plots.
Polar plot: parameter

12. Bode Diagrams (Section 8.2)

13. Bode Diagrams with Matlab (Section 8.3)

14. Phase and Gain margins
See FE manual.
Compare with 562 and 563.

15. Resonant Peak Magnitude and Resonant Frequency
Consider 2nd order prototype system.
Put TF in Magnitude/phase (I.e. polar) form.
Find frequency of maximum magnitude (resonant frequency).
Find corresponding Maximum magnitude (resonant peak magnitude).
A large resonant peak magnitude indicats the presence of a pair of dominant closed loop poles with small damping ratio. (BAD)
A smaller resonant peak magnitude indicates a well damped closed loop system. (GOOD).
Resonant frequency is real only for damping ratio less than .707.
No closed loop resonance for damping ratio greater than .707.

16. Correlation between step transient response and Frequency Response in standard 2nd order system
Consider unity gain negative feedback system with
Phase margin is
See pg. 570.
Cross over frequency is