Bill Goodwine's Courses

All problems are from the 10th edition of the course text.

1. E2.22
2. AP2.2
3. E3.15
4. P3.3
5. P3.10
6. DP3.1
7. DP3.3
8. MP3.5

On DP3.3, the equations are non-linear with respect to x1 due to the angle the cable makes as it is pulled away from x1=0. Are we supposed to solve the equations numerically and guess values of Kd until it meets the given criteria or is there something I'm missing that will allow us to solve these non-linear equations such that we can find Kd analyitically? Thanks.

bmertz wrote:On DP3.3, the equations are non-linear with respect to x1 due to the angle the cable makes as it is pulled away from x1=0. Are we supposed to solve the equations numerically and guess values of Kd until it meets the given criteria or is there something I'm missing that will allow us to solve these non-linear equations such that we can find Kd analyitically? Thanks.

I haven't worked it out fully myself, but I'm pretty sure what happens is that the nonlinearity due to the angle that the force is applied to the airplane from the cables is canceled by a corresponding nonlinearity in the amount that the spring is stretched. I think if you fully work out m \ddot x = forces, while some nonlinearities show up in the forces, they eventually cancel.

I can't get the nonlinearities to cancle out. There are two that are causing me problems

1: the hypotinuse length sqrt(h^2 + x[1]^2)

2: x[1] x[2] / hypotinuse from the angle

should we attempt to numerically solve these nonlinear equations or
should we try to solve this by linearizing the hypotinuse length? A problem with linearizing the hypotinuse is that it is not a good approximation for all x[1] values from 0 - 30. Any recommendations on dealing with this?

jmengers wrote:I can't get the nonlinearities to cancle out. There are two that are causing me problems

1: the hypotinuse length sqrt(h^2 + x[1]^2)

2: x[1] x[2] / hypotinuse from the angle

should we attempt to numerically solve these nonlinear equations or
should we try to solve this by linearizing the hypotinuse length? A problem with linearizing the hypotinuse is that it is not a good approximation for all x[1] values from 0 - 30. Any recommendations on dealing with this?

Linearizing really isn't an option since, if I recall correctly, the angle of the cable goes from 0 to 45 degrees. If necessary, you may resort to numerics.

Dr. Goodwine,

I think this is the correct state-variable representation of the RLC circuit we did in class:


Sorry it took so long; I forgot my controls notebook this morning and had to rework it.