# ECE 590 -- Design Project #2

## Nonlinear Turbojet Engine -- Runge-Kutta Integration

ECE 590,
Fall 1996, Due 12/12/96

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The third-order nonlinear model for a turbojet engine developed by
Brennan and Leake consists of 2 algebraic and 3 differential equations,
which are given by equations 7-58 through 7-62 on page 212 of the text. The
input signal for the system is the fuel mass rate W_fd. The time history
for this input signal is:

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W_fd = 1.5 from t = 0 to 0.5 seconds;

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W_fd = 0.75 from t = 0.5 to 1.0 seconds;

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W_fd = 1.25 from t = 1.0 to 1.5 seconds;

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W_fd = 0.5 from t = 1.5 to 2.0 seconds.

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The purpose of the project is to study the stability and accuracy properties
of various explicit Runge-Kutta integration algorithms when simulating a
moderately stiff system. The algorithms to be investigated will be: the
2-stage, second-order accurate modified Euler method; several 2-stage
first-order accurate methods; and the 1-stage, first-order accurate (Euler)
method. For each of the 2-stage methods, the value of the Runge-Kutta
coefficient a_2 is to be 1/2.

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The initial condition for the state vector is x_0 = [0.5; 1.8; 0.55]. The
state response of the 2-stage, second-order accurate method with a timestep
of 0.001 seconds can be used as the basis for evaluating the accuracy of the
other methods and timesteps.

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All of the 2-stage methods to be considered in this project, with the
coefficient a_2 constrained to be 1/2, can be parametrized by the design
parameter g. When g = 2, the modified Euler method results. For g > 0 and
not equal to 2, the order of accuracy of the 2-stage method is 1.

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For the modified Euler method (g = 2), for several 2-stage, first-order
accurate methods (as a minimum, g = 1, 4, 6, 8), and for the 1-stage,
first-order accurate method (normal Euler), investigate the stability and
accuracy properties of the methods in simulating the given system as
functions of T and g.

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Report your results, discussing how the stability and accuracy of the methods
compare, providing simulation and stability region plots as necessary to
support your discussion and conclusions.

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