Homework 6, due 4:00pm, March 6, 2008.

[goodwine]

Reading
For the homework only, all of Chapter 8, sections 9.2-9.5 and 9.7. For the class coverage through break, we will go through 9.8.

Exercises
8.15, 8.17, 8.18, 9.1, 9.2, 9.4, 9.5 and 9.6.

[mnguye10]

For 8.17 part 1, figure 8.40 doesn't have the distance from the mass to the pivot point labeled, are we allowed to create an arbitrary constant for that distance or is there a way to write the equation of motion in terms of given constants and variables only?

[sagarwal]

also for 8.17, is it fair to assume that m is a point mass?

[mnguye10]

For Problem 9.4, Figure 9.21 and the text under it suggests that:

2. If the real part of the pole is decreased and the imaginary part is held constant, then the damping ratio is increased and the frequency of the response will be constant.

(Page 276)

I don't see how that makes sense. Shouldn't the frequency be increasing?

- If the damping ratio is decreasing and the frequency is held constant, then according to the real axis coordinate of negative (zeta times omega-n), shouldn't the pole be moving right, which increases the real component of the pole instead of decreasing it?

- If the magnitude of the position vector of the pole on the real-imaginary axes is the frequency and there is a pole that's free to move along a fixed horizontal line at a constant on the imaginary axis, shouldn't the magnitude of that position vector increase as the pole is moved leftward along that line? (Thus increasing frequency, which suggests it can't be constant?)

[mnguye10]

Oh, wait, I read that wrong. I was thinking decreased damping for the first point, but the second point still suggests that the frequency wouldn't be constant?

[Nick S]

On problem 9.5, what do the locations of the zeros tell you? I can't find anything in the book that really explains what these are for...

[goodwine]

For 8.17 part 1, figure 8.40 doesn't have the distance from the mass to the pivot point labeled, are we allowed to create an arbitrary constant for that distance or is there a way to write the equation of motion in terms of given constants and variables only?

It should be l. If you called it something different, that's o.k. There isn't any way to get the equation of motion without that distance.

[goodwine]

also for 8.17, is it fair to assume that m is a point mass?

Yes.

[goodwine]

For Problem 9.4, Figure 9.21 and the text under it suggests that:

2. If the real part of the pole is decreased and the imaginary part is held constant, then the damping ratio is increased and the frequency of the response will be constant.

(Page 276)

I don't see how that makes sense. Shouldn't the frequency be increasing?

The natural frequency increases, but the damped natural frequency is constant. The latter is what you will see when you plot the response since it is the frequency inside the sine and cosine functions in the response.

- If the damping ratio is decreasing and the frequency is held constant, then according to the real axis coordinate of negative (zeta times omega-n), shouldn't the pole be moving right, which increases the real component of the pole instead of decreasing it?

What you said is correct... but I'm not sure what the question is.

- If the magnitude of the position vector of the pole on the real-imaginary axes is the frequency and there is a pole that's free to move along a fixed horizontal line at a constant on the imaginary axis, shouldn't the magnitude of that position vector increase as the pole is moved leftward along that line? (Thus increasing frequency, which suggests it can't be constant?)

This is the same as my first answer. The natural frequency increases, but the damped natural frequency remains constant if the imaginary component remains constant.

[goodwine]

On problem 9.5, what do the locations of the zeros tell you? I can't find anything in the book that really explains what these are for...

The effect of a zero on the second order step response is discussed in the book in the section on the second order response.

[goodwine]

For problem 8.15 part 2, I am having a horrible time trying to find how to write the equation that relates v_in to e_sp. I cannot find any KVL type stuff for the voltage drop across a capacitor or an inductor that doesn't involve an integral. I am completely lost on this one and any help would be great.

Take the Laplace transform of those equations. The beauty of the approach is that it eliminates (in a sense) derivatives and integrals.