Modeling and Simulation (Spring, Mass, Damper system)

[jtucker]

Homework Statement

1. The first part of the problem was to find a transfer function. Output is the displacement of the mass (mass of pinion is negligible)

2.The next thing to do was find the poles, which I believe means set the denominator=0 and solve for s.

3. The next thing to do was find the damping ratio (z) and natural frequency (Wn)

4. Here is where I am stuck:
Determine values of k and b such that the following are met:
m=0.1kg
r=0.01m
600msec<=rise time (tr)<=800msec
and %OS <= 10%

Homework Equations

The transfer function I came up with:

1. G(s)=(1/mr)/(s^2+b/m*s+k/m)

Poles I found:

2. Poles=(-b/m +- sqrt((b/m)^2-4(k/m)))/2

3. z and Wn I found:

z=b/(m*2*sqrt(k/m))
Wn =sqrt(k/m)

The Attempt at a Solution

4. To attempt to find values I used:

Tr~1.8/Wn


substituting into rise time equation (4) above I came up with
0.50625<=k<=0.9
I also concluded that z must be >= 0.6 for the overshoot condition.
and solving (2) for b I have
b>=0.12 *sqrt(k/m)

I have tried rearranging these equations every way I can think of to come up with a solution. I have also tried making graphs by hand and using Matlab (though I'm not great with Matlab). The thing I am running into is that there seems to be any number of solutions that will work, but I am assuming that I am wrong and that there should only be one valid solution.

Any help will be greatly appreciated, I've spent many hours trying to figure this out!

 

[donpacino]

first write out the equations for overshoot and rise time.

then plug in your equations for zeta and the natural frequency. you'll have two equations, two unknowns, which is a match made in heaven!

pick a PO and tr, and go to town.

 

[jtucker]

Thanks donpacino. I appreciate your response. It turns out that the values were in fact arbitrary as long as they fit within the given conditions. So I was solving the problem correctly. I should have trusted myself and just picked some values. Instead I was under the assumption that the professor wanted specific values so I wasted a bunch of time assuming I was wrong. In fact the professor acknowledged after I asked him about it that he should have given more restrictions or been clear that it was more of a design problem than a find a specific value problem.

 

[donpacino]

Awesome! just a note:

In the real world you are almost never given absolute requirements. you are given mins and maxes. the term 'good enough' is often used in real engineering. When designing a system with a max rise time and percent overshoot you can go below them as much as you want or can.

Just remember that in many cases, to get those "better" responses you sometimes need more expensive, larger, or more complex components. So in many cases, barely meeting the design criteria may be your best option! (keep in mind barely meeting the criteria means barely meeting criteria with tolerance and margin in mind.

 

[rezeew]

I understand your struggle to find a solution to this problem. It can be frustrating when there seem to be multiple solutions that could work. However, in this case, it is important to remember that the values of k and b must also satisfy the conditions of the rise time and overshoot. This means that the values you have calculated so far may not be the only solutions, but they must be within a certain range to meet the given conditions.

One approach you could take is to use a trial and error method, where you start with a certain value for k and b, and then check if it satisfies the conditions. If it does not, you can adjust the values until you find a combination that works. This method may take some time, but it can help you narrow down the range of values that could work.

Another approach is to use a numerical optimization method, such as gradient descent or genetic algorithms, to find the optimal values of k and b that satisfy the conditions. These methods can be more efficient and accurate, but they may require some coding or software implementation.

Overall, the key is to keep in mind the conditions that must be met and continue to refine your values until you find a suitable solution. It may also be helpful to consult with a colleague or seek assistance from a tutor or online resources for additional guidance. Good luck with your problem solving!