# Linear Systems Control

Discussion in 'Homework Help' started by arman19940326, Mar 4, 2015.

1. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
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In terms of force,How can I define "Stopping the system" in the problem below?

2. ### WBahn Moderator

Mar 31, 2012
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I assume u(t) is an arbitrary force function and not the unit step function.

You might ask what the effect of applying an impulse function is, in general.

Then ask what would happen if, when the system passes through the rest position (but with some velocity), you could apply a force that removed the momentum from the system at that moment.

arman19940326 likes this.

Jul 31, 2014
43
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4. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
1
Do you think this solution is right?

5. ### WBahn Moderator

Mar 31, 2012
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No. Does it make since that in order to make and keep the object stopped that you would have to apply a sinusoidally varying force on it for the rest of time?

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6. ### MrAl Distinguished Member

Jun 17, 2014
2,554
515
Hi,

Looks to me that at the very least the exponential damping multiplier is missing from the calculations ie the Inverse Laplace Transform will probably produce this (ie e^-at) given at least a little friction. That would take any sine part to zero over time. If there is no friction that would be unreal, but i guess you could do it theoretically, and that would theoretically be an oscillator producing a sine wave (maybe a cosine wave) without any counter force yet. But usually something that rolls is considered to have at least a little friction, which will make the sine (or cosine) diminish over time. The friction will be small relative to the mass and spring so the solution without any counter force yet will be a damped sinusoid, not just a diminishing monotonic wave.

It dawned on me that in post #3 it looks like the solution was for a unit step forcing function rather than an impulse. Maybe not exactly, but close.

Last edited: Mar 5, 2015
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7. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
1
Consider that the given problem is just theoretically true,I mean we have no friction and the applied force is a unit impulse in the problem, that in reality there are no such things...

8. ### WBahn Moderator

Mar 31, 2012
18,085
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Again, consider what it means to apply an impulsive force to a system. An impulse does what? What is different about a system before and after an impulse is applied?

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9. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
1
I think after an imaginary Impulse force applied to the above system,the change in the position of M will be sinusoidal as I derived it in my solution...

10. ### WBahn Moderator

Mar 31, 2012
18,085
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I have an object sitting on a table. I apply an impulsive force to it. What is the consequence of doing so?

Hint: What are the units on an impulsive force? Those are units of what?

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11. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
1
I didn't get your point thoroughly, you want the meaning of an impulsive force? or in general, an impulse function? that's here...
http://en.wikipedia.org/wiki/Dirac_delta_function

When you apply an impulse to a system,it will change the initial conditions of the system...for example before applying the unit impulse to the system given in the problem the initial condition for x is x=0,but after the unit impulse is applied to the system the initial condition changes to a sinusoidal change in x...

12. ### WBahn Moderator

Mar 31, 2012
18,085
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You have an object (say of mass M) sitting on a (frictionless) table. You apply an impulsive force (say I) to it. What do you know about the mass after the impulse is applied?

This is foundational. How can you work a problem involving impulsive forces applied to a system if you don't understand what an impulsive force is and what it does to a system when applied to it?

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13. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
1
W
Well I'm an electrical engineering student and I learned about impulsive function in circuits theory and know about its effects on a circuit not in mechanical systems...I learned, in circuits the impulse can change the initial conditions ...the given problem is related to one of my courses (Linear Control Systems) and I think in this course one purpose is find a relation between mechanical systems and circuits...I think we have to look at the problem in a mathematical way...the impulsive force have to be seen as a mathematical function for the force and then we apply newton's second law to solve the problem...

14. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
1
I think this is the answer we're gonna find after solving the problem mathematically...

15. ### WBahn Moderator

Mar 31, 2012
18,085
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Okay, so let's do it mathematically.

Again, let's go with a simple system so that we can learn about the fundamental relationships involved. Use the block (of mass M initially at rest on a frictionless surface) and then apply an impulse of magnitude I to it. Do the math and see what you get. In particular, after the impulse is applied what is the block's velocity, momentum, and energy. Do you see any simple relationship between any of them and the magnitude of the impulse?

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16. ### MrAl Distinguished Member

Jun 17, 2014
2,554
515
Hello again,

Yes you are right, the response with no counter force will be an unending sinusoidal because there is no friction (presumably). With some friction the sinusoid eventually dies out, but with no friction there is nothing to take energy away from the system. But to be more exact, it's not a sine, it may be a cosine, so you should do your math over again. If the original forcing function was a unit step then it would be a sine, but it's an impulse not a step.

Also, when you apply an impulse at t=0 you know now that you get a response that is continuously sinusoidal (more generally speaking and with no friction). So if you had a second network exactly the same but instead you apply a negative impulse at t=0, what would you get for that second network?

Simplified version:
You have a mass in deep space that is not moving relative to your position. You apply a large force for a very short time and it starts moving away. Your friend is some distance away and sees it coming toward him. What force does he apply to stop the mass?

The electrical circuit analog of this system is also interesting to look at, we can do that later.

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Jul 31, 2014
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18. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
1
If so,I've made a mistake in my math...Can you tell me which part of the solution in post #3 is not right?

19. ### arman19940326 Thread Starter Member

Jul 31, 2014
43
1
I think the second photo I've posted can answer your question...I just solved it mathematically...

20. ### WBahn Moderator

Mar 31, 2012
18,085
4,917
It would make it easier to see the significance if you had used a scaled impulse, but perhaps you can still see it.

You have x(t). Now see what v(t), p(t), and E(t) are. Which of those has a very simple relationship to the impulse that was applied.

Also, you need to start tracking your units properly. As it stands, you are claiming that the position of the of object, x(t), is at a location measured in 1/mass. Does that make any sense?

Last edited: Mar 6, 2015