What Is Natural Frequency & How Does It Affect Objects?

[mutineer123]

It may be a bit late in the day to point out than, when discussing the motion of a free oscillator, it is,in fact, normal to discuss what happens when it is displaced by a certain amount and then released. Not, 'struck', in order to get it going. The analysis is much easier because the initial energy is only in the form of Potential at t=0 and it is easy to define and calculate. Hitting with a hammer is more complicated than necessary! Any basic analysis of 'driven' oscillators will use smooth (single frequency) driving waveforms - not regular 'hitting'.

@mutineer:
Do you have a specific AS level text / course book or is your book a general A level source'?
AS level can be a big jump from the very abbreviated Physics experience that GCSE gives students. Reading around is very useful but you should not worry when something outside your specification content is a bit confusing. Your A2 course next year will explain the basis of the Simple Harmonic Oscillator (ideal) and also the (Not Simple) Harmonic Oscillator, in the form of a pendulum. Sufficient unto the day is the evil thereof. . . .

I just made this comment on another thread but - keep responding if you find the thread you initiate is wandering off the direction you wanted. We often ***** amongst ourselves and need to be brought to heel.

[omg, I just discovered that the word for a female dog is not permitted on this forum - well ***** me!]

I am not sure what you mean by a 'specific' A level book. Our school has given us a Cambridge University Physics course book. And that's what we are using.

 

[PeterO]

Thanks, this has helped me a lot, but I have a few doubts.

"Now if we set a swing going and stood back and timed it, we may find the period is 2.5 seconds - that is its natural Period [so its natural frequency is 0.4].
[at this point I plan to ignore the fact that the period changes slightly if the amplitude gets large - let's pretend it doesn't.]"

The period for a complete oscillation will change right? because like you said the swing, looses energy. So the period will increase, and 'not' remain at 2.5 seconds. So how do we determine the natural period, if the period keeps changing? or do we just take the period we get from the first push?

"You may reach a point where the little push you are giving merely replaces the energy lost during a single swing, so the amplitude of the swing settles on some value. You have really hit resonance then."

I did not quite get this part. Can you please explain it tom again, how the swing 'settles' on some value?

I don't think it changes in the way you are thinking.

Simple harmonic Motion has a constant period.
If you set the swing going such that the amplitude is, say, 15^o , then as the oscillations decay to a mere 5^o [which will take quite some time] the Period will not perceptively alter.

Only if you get up to more exciting [for the rider] amplitudes like 35^o to 45^o will you get a measurable change in Period, and even then it is only slight.

It sounds like you may be confusing the speed of the swing, as it passes through the mean position, with the Period [a not uncommon link people make in error]

When the the swing has a very small amplitude, it takes the full 2.5 seconds to get from one extreme position, through the mean to the other extreme, then back through the mean to where it started. That might only be a total distance of 20 cm. If the swing travels 20cm in 2.5 seconds, it cannot be traveling very fast at any time.

With a larger amplitude, the full oscillation might be a total of 8m [starting 2m to one side - swinging to 2m the other ide, then swing back to 2m this siad again.
If the swing is going to travel 8m in 2.5 seconds AND it is stationary at the extreme positions, it is not surprising that it passes through the mean position at quite a high speed.
At such amplitudes, the Period will have changed - but only by a small amount -perhaps 0.1 seconds, so the 8m has to be covered in approx 2.6 seconds. It still has to get up to high speed to cover the whole trip.

When we go to a playground to give a friend/child a ride on the swing - we just know to give a little push every time the swing has come back, and is about to move away from us - we don't use a watch or physics - we use common sense.

NOTE: As the swing loses energy, it actually completes the oscillation in a shorter amount of time - so one might say it speeds up! But That really refers to the fact it does more oscillations in a given time, rather than traveling through the air at higher speed

 

[sophiecentaur]

@mutineer
You keep asking the same question "What is the the natural frequency?"
The answer is just the frequency that the oscillator will have when it has been released from a non equilibrium position and then left to itself. Forget about hitting it and seeing what happens. For a true Simple Harmonic Oscillator like a mass on an ideal spring there is just one natural frequency of oscillation and consequently the motion will be sinusoidal, whatever the amplitude. The definition of a simple harmonic oscillator is that the restoring force is always towards the equilibrium position and is directly proportional to the displacement - the simplest rule you could apply and it yields the simplest form of motion.
Now you can't get much more simple than that, can you?

If you want to talk about resonance at the most basic level, you have to think what will happen if another SINUSOIDALLY varying force is applied to this simple system. Pushing a swing every so often may possibly be a very familiar concept but is is a snare and a delusion because it is all too complicated, involving impulses (collisions) and a lot of arm waving in the explanation. Applying a sinusoidal force means that the force is always there - constantly pulling or pushing the oscillator. It surely can't be too taxing on the imagination to consider dangling a mass on a spring from your hand and then moving your hand up and down slightly. Get a long rubber band out of a drawer and hang a mug on it if you want an idea of what I mean. This is much nearer to the ideal SHM situation so, imho, is a better model to work with (If you really want to ignore the maths of the situation)

Start with the mass at rest.
If you move your hand up and down at just the right frequency, the oscillator is continually 'chasing' this movement because the force from your hand (combined with the weight)) is always in a direction to accelerate the oscillator. You are steadily adding energy to the system and the amplitude will gradually build up. The maximum amplitude of oscillation will be when friction losses during each cycle balance the energy you are adding. If there's no energy loss (not a practical situation, ever) then there's no limit to the amplitude it will reach eventually (until the spring coils close completely). If the frequency of your movement is just above the natural frequency, you start off accelerating the mass but, even during the first cycle, you are pulling back 'early' so you are forcing the oscillator to go back before it would naturally do. So its frequency will be higher than natural. This also limits the amount of energy it can 'take' from your hand movement. It will NOT ever move at its natural frequency but at the impressed frequency and its amplitude will never build up to the same maximum. If you excite it at a lower frequency, your force changes later than the natural oscillation and, again, it will be out of step at some stage in each cycle, taking some of the energy out again. The frequency of oscillation will, again, be at the driving frequency and NOT at the natural frequency.

Until you can appreciate this very simple model, other models are likely just to confuse you. When a swing is hit every so often, it spends some of its time in natural movement and then gets disturbed - a nightmare to analyse, in fact. Don't go there!

SHM is a typical example of a Physical process that seems to require acres of written explanation but can be taken care of with two or three lines of Maths. No surprise then that all serious Physicists are prepared to get their hands dirty asap with the dreaded M word!