CDS 140a Winter 2013 Homework 3

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R. Murray, D. MacMartin	Issued: 19 Jan 2011 (Tue)
ACM 101/AM 125b/CDS 140a, Winter 2013	Due: 26 Jan 2011 (Tue)

__MATHJAX__

Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading).

1. Perko, Section 2.2, problem 5:
2. Perko, Section 2.3, problem 1:
3. Perko, Section 2.5, problem 4:
4. Perko, Section 2.5, problem 5:
5. Choose one of the following systems and determine all of the equilibrium points for the system, indicating whether each is a sync, source, or saddle.
   (a) Moore-Greitzer model: The Moore-Greitzer equations model rotating stall and surge in gas turbine engines are given by <amsmath>

    \aligned 
      \frac{d\psi}{dt} &= \frac{1}{4 B^2 l_c}\left(\phi - \Phi_T(\psi) \right), \\
      \frac{d\phi}{dt} &= \frac{1}{l_c}\left( \Phi_c(\phi) - \psi + \frac{J}{8} 
        \frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\
      \frac{dJ}{dt} &= \frac{2}{\mu + m} \left(
         \frac{\partial \Phi_c}{\partial \phi} + \frac{J}{8}
            \frac{\partial^3 \Phi_c}{\partial \phi^3} \right) J,
    \endaligned
   

   </amsmath>

   where

   <amsmath>

    \aligned 
        B &= 0.2, & \Phi_T(\psi) &= \sqrt{\psi},\\
        l_c &= 6, & \Psi_c(\phi) &= 1 + 1.5 \phi - 0.5 \phi^3, \\
        \mu &= 1.256, &\qquad\qquad m &= 2.
    \endaligned
   

   </amsmath>

   This is a model for the dynamics of the compression system (first part of a jet engine) with $\psi$ representing the pressure rise across the compressor, $\phi$ representing the mass flow through the compressor and $J$ representing the amplitude squared of the first modal flow perturbation (corresponding to a rotating stall disturbance). (Hint: There is more than one equilibrium point and not all of them are stable.)

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Notes:

• The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.