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

 

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.

 

 

Do you think this solution is right?

 

Do you think this solution is right?

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?

 

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.

 

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...

 

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?

 

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?

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...

 

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...

I'm asking about it at a more fundamental level.

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?

 

I'm asking about it at a more fundamental level.

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?

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...

 

I was asking specifically about an impulsive force in mechanics.

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?

 

W

I was asking specifically about an impulsive force in mechanics.

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?

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...

 

What do you know about the mass after the impulse is applied?

I think this is the answer we're gonna find after solving the problem mathematically...

 

I think this is the answer we're gonna find after solving the problem mathematically...

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?

 

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...

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.

 

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?

 

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.

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?

 

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?

I think the second photo I've posted can answer your question...I just solved it mathematically...

 

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?