Confusion About Oscillating Mass

[person123]

I'm sure there's an obvious answer to this, but this problem has been confusing me for some time.

Imagine there were a massive object attached to the end of a crankshaft. A force is applied to accelerate the crankshaft, causing the mass to oscillate. Assuming there is no friction, what would happen to the system when the force is no longer applied? It shouldn't be possible for it to slow down if there's no energy loss, but wouldn't it require a force to continue to accelerate the mass back and forth?

 

[zwierz]

picture is needed

 

[person123]

picture is needed

I'm not exactly sure what you're looking for, but this is the basic idea.

 

[zwierz]

the system will oscillate This is a Hamiltonian system with one degree of freedom

 

[person123]

And it will continue to oscillate at the same rate even when the torque is no longer applied?

 

[zwierz]

What does the "rate" mean? This will be nonlinear oscillation. Is gravity applied?

 

[person123]

I mean the frequency of the oscillation of the mass. I don't understand what you're asking.

 

[zwierz]

The motion of the system will be periodic

 

[Ibix]

One thing to think about that may not have occurred to you - whatever your axle is attached to will also wobble so that the centre of mass does not move. This is analogous to the Earth-Moon system, which revolves around a point (the barycentre) slightly offset ftom the centre of the Earth.

Does that help?

 

[person123]

Is gravity applied?

I should have mentioned that. It would make sense not to include gravity just for the purpose of simplifying things, I suppose.

 

[zwierz]

whatever your axle is attached to will also wobble so that the centre of mass does not move

that is wrong

I should have mentioned that. It would make sense not to include gravity just for the purpose of simplifying things, I suppose.

ok anyway the motion will be periodic

 

[Ibix]

that is wrong

How is momentum conserved if the centre of mass moves?

 

[zwierz]

ow is momentum conserved if the centre of mass moves?

who does say that momentum conserved?

 

[person123]

who does say that momentum conserved?

How can momentum not be conserved? I assume that even if the mechanism were fixed on the earth, it would cause the Earth to oscillate very slightly in response (as ibix described).

 

[zwierz]

How can momentum not be conserved. I assume that even if the mechanism were fixed on the earth, it would cause the Earth to oscillate very slightly in response (as ibix described).

Oh if you include the Earth in the system then I withdraw my objections :)

 

[person123]

One thing to think about that may not have occurred to you - whatever your axle is attached to will also wobble so that the centre of mass does not move. This is analogous to the Earth-Moon system, which revolves around a point (the barycentre) slightly offset ftom the centre of the Earth.

Does that help?

I think that makes sense, but I still have one more question which relates to the initial purpose of this thread.
I was trying to calculate the torque required to oscillate the mass at a constant frequency. So I found the velocity of the mass, and I then found the derivative, which gave me the acceleration. This allowed me to find the torque at the point in which the velocity of the mass was zero and the acceleration was at its maximum. (It gave me τ=Mω^2l^2, in which l is the length of the short rod). However, it doesn't seem to make sense to calculate this when there isn't any torque required to maintain that constant oscillation. I can't seem to reconcile this confusion with the explanation.

 

[A.T.]

I think that makes sense, but I still have one more question which relates to the initial purpose of this thread.
I was trying to calculate the torque required to oscillate the mass at a constant frequency. So I found the velocity of the mass, and I then found the derivative, which gave me the acceleration. This allowed me to find the torque at the point in which the velocity of the mass was zero and the acceleration was at its maximum. (It gave me τ=Mω^2l^2, in which l is the length of the short rod). However, it doesn't seem to make sense to calculate this when there isn't any torque required to maintain that constant oscillation. I can't seem to reconcile this confusion with explanation.

Are you assuming that all parts, except the block are masses? In that case the total energy is zero when the block is at rest, so the oscillation won't be maintained without an applied torque.

 

[person123]

Are you assuming that all parts, except the block are masses? In that case the total energy is zero when the block is at rest, so the oscillation won't be maintained without an applied torque.

Yes, I'm assuming all other parts are without mass. Isn't that contradictory to ibix's argument, or is this just a misunderstanding on my part?

However, let's say the mass of the other parts were just extremely small. Would the frequency of oscillation decrease or remain constant over time?

Also, how can there be no energy in the system if it was initially in motion and no energy was lost?

 

[A.T.]

Also, how can there be no energy in the system if it was initially in motion and no energy was lost?

Your assumptions are obviously contradictory then.

However, let's say the mass of the other parts were just extremely small. Would the frequency of oscillation decrease or remain constant over time?

Without energy dissipation, constant.

 

[person123]

Without energy dissipation constant.

But can't you approximate what would occur in the system by considering the other parts without mass if the mass of those parts are in reality very small?

 

[A.T.]

But can't you approximate what would occur in the system by considering the other parts without mass if the mass of those parts are in reality very small?

You have already answered this question, by pointing out the contradiction it leads to. How are massless rigid parts supposed to store energy, in order to conserve it, while the block stops at each turnaround?

 

[Ibix]

The problem is that if nothing except the mass has mass then any forces applied cause infinite acceleration. You can certainly consider your crankshaft to be massless, but it has to be attached to something with mass in order for it to have been able to produce a torque in the first place without itself instantly spinning up to infinite speed.

 

[Nidum]

I haven't quite locked on to what the problem is but let's try a few things :

The configuration is like that of a normal I/c engine . Crankshaft , connecting rod and notionally a piston . The piston has a large mass but the other components are massless . Conventional arrangements of bearings etc. but with zero friction .

If the crankshaft is initially rotated at constant rpm by an external power source then the piston will move back and forth in a straight line . Motion will be something like SHM but with some higher order motion components added .

When the external power source is disconnected the piston will continue to move until it reaches an extremity of it's travel where it will stop .

To make the motion continue beyond that stop point there needs to be a heavy flywheel fitted to the crankshaft .

 

[person123]

I asked what would happen to the frequency of oscillation if the mass of the rods were extremely small.

Without energy dissipation, constant.

To make the motion continue beyond that stop point there needs to be a heavy flywheel fitted to the crankshaft .

According to A.T., ignoring friction, wouldn't any mass cause the oscillation to be maintained at a constant frequency?

 

[Nidum]

I don't see any contradiction between what AT has said and what I have said .

When the external drive is disconnected the piston will just continue in motion until it can go no further . There is then no torque on the crankshaft to make it rotate and no force on the piston to start it moving again .

To make the motion keep going there needs to be enough residual rotational KE in the crankshaft to carry it past dead centre and then generate enough torque to produce a force in the connecting rod sufficient to start the piston moving again .

Zero mass crankshaft components can't store any KE at all .

In a real situation the crankshaft components themselves could be made sufficiently massive to store enough KE but more commonly the crankshaft components are kept relatively light and a massive flywheel is fitted instead .

 

[person123]

All right. So here's my understanding of how it would work.

If the mass of the flywheel were extremely large and the oscillating mass extremely small, the angular velocity of the flywheel would be almost constant. As the former decreases and the latter increases, the angular velocity would fluctuate to a greater extent.

If the system were connected to the earth, it would cause the Earth to oscillate (extremely slightly).

If the mass of the rods and flywheel were nonexistent, when the velocity of the oscillating mass reaches zero it would remain still because there is no stored kinetic energy.

If there were no friction in the system and the connecting rods had any mass at all, it would oscillate perpetually.

Is this the right idea?

 

[A.T.]

If the system were connected to the earth, energy would be conserved because the oscillating mass would pull on the earth, causing it to accelerate.

No, it doesn't work like that. In the rest frame of the total center of mass (Earth and block), there is a time point when both masses are at rest, so total KE is zero. This is obviously inconsistent with the other assumptions (etc. total rigidity) which preclude energy dissipation.

 

[person123]

No, it doesn't work like that. In the rest frame of the total center of mass (Earth and block), there is a time point when both masses are at rest, so total KE is zero. This is obviously inconsistent with the other assumptions (etc. total rigidity) which preclude energy dissipation.

OK. I think I get that—I'll edit my previous post. I'm assuming this inconsistency is because energy conservation would require completely elastic collisions which wouldn't happen if the parts were perfectly rigid.

 

[Nidum]

If there were no friction in the system and the connecting rods had any mass at all, it would oscillate perpetually.

Is this the right idea?

Not quite . There is a minimum value of energy storage capacity needed in the flywheel / crank / con rod assembly to ensure continuous motion .

Please note that trying to explain exactly how a theoretical system that can't actually exist works is difficult and often misleading . You would be better off from this point on in this discussion if you drew up a realistic system which we could then analyse more definitively . When you see how the energy in a real system transfers between components you should be able to understand all of this problem much more easily .

Anyway if you want to continue we need to develop a proper mathematical model of this type of system .

 

[person123]

All right. I think the problems I'm used to dealing with are simple enough that I never have to deal with the inconsistencies of these theoretical cases (I'm only in high school taking regents physics).

Should I provide specific values for the mass of the rods (and possibly the friction) or should I just assume they're low but not nonexistent?

If I assume that they are low, would my equation τ=mω^2l^2 (which didn't take into account the mass of the rods) still be a good approximation.

 

[Nidum]

No need for numbers . Algebra variables are what we will use . We can keep the actual maths quite simple .

Getting a bit late now here in UK so I'm signing off pro tem .

Nos dda .

 

[jbriggs444]

No need for numbers . Algebra variables are what we will use . We can keep the actual maths quite simple .

Seems simple enough. Finite mass on the piston, rigidly mounted crank-case/engine block. Frictionless bearings and frictionless piston rings. Watch what happens as the mass of the crankshaft and connecting rod decrease toward zero. There is always an increase in crankshaft angular velocity near top dead center and bottom dead center. As the mass of the crankshaft and connecting rod decrease toward zero, the peak angular velocity diverges toward infinity.

There is no convergence toward a physically reasonable limiting behavior. Something will break first.

 

[person123]

Seems simple enough. Finite mass on the piston, rigidly mounted crank-case/engine block. Frictionless bearings and frictionless piston rings. Watch what happens as the mass of the crankshaft and connecting rod decrease toward zero. There is always an increase in crankshaft angular velocity near top dead center and bottom dead center. As the mass of the crankshaft and connecting rod decrease toward zero, the peak angular velocity diverges toward infinity.

There is no convergence toward a physically reasonable limiting behavior. Something will break first.

All right. My train of thought before making the thread was this:
I know that when the mass is momentarily still, the acceleration is at its maximum. I can find that acceleration, and then find what the angular acceleration would be at that point. This would allow me to find the maximum torque. It did give me an equation (τ=ω^2l^2m) which passes dimensional analysis and seems reasonable, but I never considered the mass of the rod. Does this calculation make any sense to do, or should I go back to the drawing board completely?

 

[person123]

Also what is exactly meant by:

There is no convergence toward a physically reasonable limiting behavior. Something will break first.

What is meant by something breaking? Is this meant in a physical sense?

 

[jbriggs444]

Also what is exactly meant by: What is meant by something breaking? Is this meant in a physical sense?

How are you going to manage infinite acceleration with a material with a finite strength to mass ratio?