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In the following problem, we ask the reader to some of the details left out of the above discussion, to analyse the closed-loop system for the stability properties, and to conduct a numerical simulation of the nonlinear system.
Work out the details leading toEq. (1).
![Check Mark](/static/check-mark.png)
The expression for
Answer to Problem 1P
Solution:
The expression for the angle between the fly ball connecting arm and the vertical shaft is:
Explanation of Solution
Given information:
The model for the fly ball governor follows from taking into account all of the forces acting on the fly ball and applying Newton’s law,
Explanation:
Let
By using the Newton’s second law of motion,
There are two forces acting on the point mass, one is gravity, which is in downward direction, so it is
The magnitude of the sum is easily found from the figure as
Also, for upward direction, the gravitational force is
Thus, the total magnitude of the sum is
But Newton’s second law of motion tells that the net force is the mass times the acceleration.
Let
Thus,
Therefore, the expression for the angle between the fly ball connecting arm and the vertical shaft is
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Chapter 5 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
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