Exercises ^[1]

1. Indicate if each of the following systems is a dynamic system: A spacecraft coasting through deep space (gravitational effects are not significant). A logic circuit consisting of ideal boolean AND, OR, and NOT gates. A filter circuit consisting of resistors and capacitors. An automotive suspension system consisting of the frame, wheels, springs, and shock absorbers.

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   Yes. Possibly, if the circuit includes feedback (either directly or indirectly) from a gate's output to its input, such as in a flip-flop circuit. When this is true, the circuit output following an input signal transition depends on the input signals as well as the previous state of the circuit. Yes. Yes.

2. Write the following differential equation as a system of first-order equations: x'' = ax' + bx' + cx + d

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   x' 3 = ax 3 + bx 2 + cx 1 + d x' 2 = x 3 x' 1 = x 2 where x = x 1.

3. Show how Equation 2.14 (page 32) is changed if wind resistance is included. The angular acceleration due to wind resistance is modeled as a constant C multiplied by the square of the bob velocity, acting in the direction opposite to the velocity.

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4. Using the results of Exercise 3, add the effect of a steady wind to the model. The wind is modeled as moving with constant velocity v _w from left...