What Is Natural Frequency & How Does It Affect Objects?

[Studiot]

But most systems do not have an entire branch of physics/maths called statics or equilibrium attached.

To offer something as a definition you must show that it is valid for all possible cases.

Nor does it address my original observation that the initial start of oscillation comes from another system to the oscillator. Energy is initially transferred from another system to initiate oscillation.

So your equation refers to a system already in oscillation only.

go well

 

[sophiecentaur]

Would you also apply that argument to an electronic circuit - giving us two types of circuit: one with the power switched on and one with it switched off? OK, we have AC and DC conditions in electricity and we sometimes draw a distinction, for convenience, but is there a difference?
I think you are trying, here, to draw a distinction where one does not exist.

 

[mutineer123]

Hello again mutineer, don't rush things.

I'm glad you understand frequency and my last post. We need to build up to the answer to your question in stages. It is essential to understand each stage before tackling the next.
It is not helped by others here saying that the only frequency that an object will resonate at is its natural frequency.
Any musician will tell you this is not true and I will cover this as well. However there are very special conditions for this so let's deal with the simple resonance first.

As I said resonance is about energy exchange from one system to another. So let's look at a system slow enough to watch in real time. Let's take a pendulum made of a heavy weight hanging on a light rod. I say a rod to avoid complications due to a string flopping about.

Let us set the pendulum swinging to and fro with a single tap.

Now consider what happens when we apply a second, third, fourth in fact a whole succession of taps as shown in the sketches.

If the second tap comes at B then the blow is less effective than the first as it is now acting against the direction of the swing and may stop or even reverse the motion. Less energy is transferred as a result to the motion of the bob.

Point C is the worst case for this as the bob is moving with maximum velocity against the hammer.

When the bob has reached the top of its swing at D it is momentarily stationary just before it reverses direction. This is the most effective point to hit the bob a second time, since all the effect of the hammer is received by the bob and adds to what the bob is already doing.

I'm sure you can see that the same thing happens with each successive blow. If you always strike at point D then you will always get reinforcement of the pendulum's swing.

To strike always at D we need to strike at a regular time interval, known as the period.
Hopefully you know that the frequency is the reciprocal of the period?
So that if our striking rate or frequency matches the natural frequency of the swing we get maximum total energy transfer.
At any other rate the strikning can sometime add and sometimes reduce the energy transfer.

I'm sure you have noted that I missed A so far. That is because whilst the second blow adds to the motion of the bob, it also accelerates the bob. So the bob will meet the hammer sooner on the third/fourth etc blows and situation B or C will occur sooner cancelling out any temporary input.

I have avoided the use of the term momentum in this description because I am not sure if you understand it?

So resonance is all about the timing of a series of small energy pulses, matching a system's ability to absorb them and add them to its oscillation.

If this helps we can proceed to what happens with wind instruments to get multiples of the fundamental or natural frequency.

Firstly I thank you with all my heart for taking time from your life for explaining this to me, and I am sorry for the late reply, my net was down. I had one doubt.

"So that if our striking rate or frequency matches the natural frequency of the swing we get maximum total energy transfer."

From these posts(as well as from other's) I comprehend that the natural frequency refers to the frequency at which the pendulum oscillates when normally tapped. But you have probably noted my doubt about this in other replies. Because if I tap the pendulum harder, the (natural)frequency changes. So how do I know the precise amount of force, that gives me the natural frequency?

 

[PeterO]

Firstly I thank you with all my heart for taking time from your life for explaining this to me, and I am sorry for the late reply, my net was down. I had one doubt.

"So that if our striking rate or frequency matches the natural frequency of the swing we get maximum total energy transfer."

From these posts(as well as from other's) I comprehend that the natural frequency refers to the frequency at which the pendulum oscillates when normally tapped. But you have probably noted my doubt about this in other replies. Because if I tap the pendulum harder, the (natural)frequency changes. So how do I know the precise amount of force, that gives me the natural frequency?

A pendulum is the odd ball of most oscillating devices mentioned in this thread, since the "restoring force" - the force driving the bob back to its original position, is a component of the weight of the bob, which is only approximately proportional to the displacement from the mean position. It is a really good approximation when the amplitude is small.

A pendulum has many natural frequencies, depending on amplitude.

When oscillating at small amplitude, you would most probably not be able to measure the difference in "natural frequency" between amplitudes of 1 degree and 2 degrees, but would measure a difference with the natural frequency for an amplitude of 45 degrees.

Lets suppose that the natural frequency at 20 degrees was 0.6, and the natural frequency at 45 degrees was 0.55. If the pendulum was swinging with an amplitude of 45 degrees, it will begin to decay due to air resistance and losses in the "string" and eventually be swinging at 20 degrees. Wait long enough and it will stop. All you need to do is replace the little bit of energy it loses during the swing(s) it made before you decided to replenish.

Even a guitar string is affected by amplitude. If you extremely displace the string when plucking, you increase the tension in the string, so the "Natural frequency" is slightly higher. In that case, the amplitude also decays quite quickly, and the sound drops quickly back to the note you expected. During the normal operation of the guitar as an instrument, such extreme distortions while plucking are avoided - though many guitarists deliberately distort the string with the "no-plucking hand" to achieve a musical effect [glissandi].

 

[sophiecentaur]

These threads all seem to introduce more confusion than enlightenment, I'm afraid.
A simple pendulum is not, in fact a simple harmonic oscillator because the restoring force is not actually proportional to the displacement (so it's hardly the best example for getting things sorted out with). The frequency does alter with amplitude (once the angle gets more than about +/- 10°). Not a good model to start with.

Moreover, we tend to excite a pendulum / playground swing with impulses, rather than a smooth, sinusoidally varying force, which again clouds the issue. If, for instance, you push a pendulum about once every ten swings, it will swing between pushes with a frequency of oscillation which is near its natural frequency (it will keep swinging between pushes). It will have been excited by the tenth harmonic of the excitation force fundamental. If you, however, excite it with a sinusoidally varying force at one tenth of its natural frequency, it will move at this low frequency - but with a low amplitude. Why not stick to the 'ideal' excitation force (with a spectrum of just one frequency) first and get to understand that situation?

This is why there is a lot to be said for considering a simple mass-on-spring, excited by a sinusoidal displacement of the mount point. This sort of resonator will have a very sharp resonance at its natural frequency. When there is some friction or energy loss (say your mass is under water, it will resonate off-frequency at a significant amplitude and, at the resonant frequency, the amplitude maximum will be less. The final amplitude at resonance, as the oscillator energy builds up, is reached when the amount of energy lost each cycle is the same as the energy supplied each cycle. When you try to drive it off frequency, there is less energy absorbed each cycle so the amplitude (running total) will be less - hence the well known bell-shaped resonance ('response') curve.

Having cleared all that up, it is then reasonable to move on to more complex situations involving multiple natural modes of resonance, non-linear force laws and non-sinusoidal driving waveforms. Take yer pick. BUT sort out the basic bit first!

[Edited to make second para make more sense, by rearranging the sentences]

 

[Michael C]

From these posts(as well as from other's) I comprehend that the natural frequency refers to the frequency at which the pendulum oscillates when normally tapped. But you have probably noted my doubt about this in other replies. Because if I tap the pendulum harder, the (natural)frequency changes. So how do I know the precise amount of force, that gives me the natural frequency?

For most objects that give a musical note when hit or strummed (tuning forks, piano strings, xylophone bars...), any possible change in frequency is so tiny that it is not noticeable. I have a tuning fork that is calibrated to vibrate at 440 Hz. It does this no matter how hard I strike it. My piano is tuned to the frequency of this tuning fork: if I play an A on my piano, it always vibrates at 440 Hz, no matter how loud or soft I play it.

If the frequency of vibration of a piano string depended upon the force with which it was struck, it would be impossible to tune a piano!

 

[sophiecentaur]

Because if I tap the pendulum harder, the (natural)frequency changes. So how do I know the precise amount of force, that gives me the natural frequency?

A simple pendulum is NOT a simple harmonic oscillator and its natural frequency does depend on swing amplitude. However, I wonder whether you actually measured the effect you describe or is it just what you think will happen. The discrepancy is not large for a simple pendulum over a limited range of amplitudes.

@MichaelC. Not just tuning but Playing a piano would be a nightmare! Doinnnnggg!

Read my post (two posts up from here).

 

[Studiot]

This is surely at variance with current teaching and syllabuses at mutineer's level.

Surely most of this nitpicking obscures basic understanding for someone meeting this subject formally for the first time.

The important concept to get over is that the characteristic of wave motion is that it will only accept energy at certain frequencies.
Mutineer is asking very reasonably for a simple explanation as to why.

It is a very important question that greatly affects many branches of Physics.

I am trying to build up to a non mathematical answer to this, but anyone else is welcome to offer one.

 

[sophiecentaur]

@Studiot

I'm not sure what you are saying is at variation with current secondary School teaching but the original question(s) take us way out of that range of knowledge from the start and there's no way of answering some of the points by arm waving. I know some people can't or won't get into the Maths of this sort of topic but, the fact is, you just have to accept some things on trust if you exclude the Maths.
There is a basic incompatibility between a very simple model of 'kids on swings' and the idea of forced vs. free oscillation. People have made assertions which are not correct about the nature of oscillations and they have tried to extend ideas from a simple model to regions where they are totally misleading.
This is why I posted my 'back to basics' post, above. I have pointed out the danger of leaping in with both feet, half way through the topic and my worries have been well justified.
I would go so far as to say that the non-mathematical approach can't take you much further than pushing a swing descriptions. Oscillating systems and resonance are, as you say, an integral part of our understanding of the physical world. It is pretty important to get ideas about them as rock solid as possible before moving on.


Incidentally, this thread is about oscillations and not actually about Waves, which involve yet more complexity.

 

[Studiot]

I'm not sure what you are saying is at variation with current secondary School teaching

I am saying that the pendulum is presented as one of the first examples of SHM in current secondary texts and syllabuses. You seem to be suggesting otherwise.

Thank you for the correction this thread is about oscillation, not waves. I should ahve said that.

 

[sophiecentaur]

I am saying that the pendulum is presented as one of the first examples of SHM in current secondary texts and syllabuses. You seem to be suggesting otherwise.

Thank you for the correction this thread is about oscillation, not waves. I should ahve said that.

I just checked out the Pendulum in my "current secondary text" (AQA Physics A2) and it is, as it was in 1962, discussed including an approximation of sin(theta) to theta for small angles. It is not discussed in any detail, if at all, at lower levels. I think my "suggestion" is fairly well justified? Do you teach Physics?

 

[Studiot]

Hello mutineer, sorry for the digression but we now come to your question about amplitude and frequency.

Vin300 gave you a perfectly good mathematical answer (post#16) and Sophie Centaur has been urging you to do some maths.

So how is your maths?
You need to help us too you know.

Meanwhile to look at the physics I will stick with mechanical oscillations. To obtain an answer we need to look inside a single cycle of oscillation and understand what is happening.

The initial impulse applies a force which accelerates the mass.
But the initial force does not of itself result in oscillation and further it is only of short duration.
So the mass is sent on its way by an original kick.
The original kick is external to the system.
The harder the kick the faster and further the mass travels.
For it to change its motion from the original kick it has received it must be subject to another force by Newton's law.
This force is supplied by the configuration of the system itself.

The simplest force that can be acting would be a constant force.
This would not lead to oscillation as the force must reduce to zero somewhere for the cycle to pass through zero.
Further the force must be capable of reversing itself since ther are positive and negative halves to the cycle.

BEM introduced the next simplest force in post#27.
This one fits the bill a treat.
This force is proportional to the displacement in such a way that it becomes zero at sero displacement.
It is capable of acting in reverse for displacements in the other direction.
It is usually called the 'restoring force'.

We find that working out the equations this restoring force is

a property of the system alone

It is independent of the initial kick.

The initial kick determines the amplitude but not the frequency.

As a matter of interest the equations tells us that the natural frequency of oscillation is

The square root of the ratio of the stiffness to the mass in the system, which if you like is the ratio of elastic to inertial forces.

 

[Studiot]

Extract from the current A level syllabus clearly states that the simple pendulum is to be studied as an example of SHM.

My emboldening.

13.1 Oscillations and Waves
13.1.1 Simple harmonic motion:graphical and analytical
treatments
Characteristic features of simple harmonic motion
Exchange of potential and kinetic energy in oscillatory motion
Understanding and use of the following equations
a = −(2πf ) 2x
x = Acos2πft
v = ±2πf A2 − x 2
Graphical representations linking displacement, velocity, acceleration ,
time and energy
Velocity as gradient of displacement/time graph
Simple pendulum and mass-spring as examples and use of the
equations
g
T = 2π l
k
T = 2π m
Candidates should have experience of the use of datalogging
techniques in analysing mechanical and oscillatory systems

The following australian animation is thanks to dlgoff

http://www.animations.physics.unsw.edu.au/mechanics/chapter4_simpleharmonicmotion.html

Note that it starts with a pendulum!

 

[sophiecentaur]

And how would you propose to do an "Analytical" treatment of the Pendulum without introducing mglSin(θ) as the restoring force and then making the explicit approximation?
These days, people are so fussy and litigious that the 'Qualification Awarding Bodies' publish a 'Specification' for their qualifications and, certainly in the Sciences, they all make available a Course Book, which contains details of what they expect to be taught and it a very good indication of all the students are required to know. The AQA course book, of course, Analyses the pendulum in the classical way; there is no other, in fact. Have you ever taught it in a simpler way?

I know that the Specifications are all available on the internet but I also know that they are not sufficient, on their own, for getting a good grade. A specific Text book is also required. Do you have access to an actual approved A level text? There was a brief time when some A level Specs. included some very 'noddy' items but they seem to be back on track these days.

PS, it's no good watching an Australian treatment of a pendulum - they all go upside down in the antipodes. But if you listen carefully, you'll hear that he says the pendulum is HArmonic motion but that an example of Simple Harmonic Motion is a mass on a spring. He doesn't actually commit himself to saying a pendulum is SHM.

 

[mutineer123]

Extract from the current A level syllabus clearly states that the simple pendulum is to be studied as an example of SHM.

My emboldening.



The following australian animation is thanks to dlgoff

http://www.animations.physics.unsw.edu.au/mechanics/chapter4_simpleharmonicmotion.html

Note that it starts with a pendulum!

To everyone taking the time, to answer my questions, I appreciate it, but so many answers( and each with different examples) is unfortunately not helping me come to terms with resonance/natural frequency.

"This is surely at variance with current teaching and syllabuses at mutineer's level.

Surely most of this nitpicking obscures basic understanding for someone meeting this subject formally for the first time."

I couldn't have explained this better myself, and thank you for bringing up the level. I am in AS level right now, so my syllabus is:

http://www.cie.org.uk/qualifications/academic/uppersec/alevel/subject?assdef_id=758

It is the " 2012 Syllabus - Revised" one. See page 27 and 28. That is it!

"So how is your maths?
You need to help us too you know."
I am sorry if I am late in replying, but as i said my nets been fluctuating lately, and I will try my best to help you guys. If you have seen the syllabus link above, you would find that I know precious little about all the mathematical terms posted by most of you (−(2πf ) 2x, x = Acos2πft, v = ±2πf A2 − x 2...) I think I will learn about these terms in A2, not now.

I would just like a simple AS level explanation in physics terms, how is resonance created,and what exactly is a natural frequency. I would not mind trying out math if it is essential, but as I said, I have not yet been introduced to all that Acos2nft type sums.

@ studiot, If you don't mind, can we just stick to the pendulum example you had given(in #14)? I feel as if this is going too fast for my comprehension. And like you said, I don't want to rush things.

 

[PeterO]

@ studiot, If you don't mind, can we just stick to the pendulum example you had given(in #14)? I feel as if this is going too fast for my comprehension. And like you said, I don't want to rush things.

OK here goes: using the swing in the park as your pendulum, with a person sitting on the swing.

Suppose you decide to push forward at what ever is there, once every second.
The first push will push the swing away.
The second push may be a "fresh air job" as the swing will actually be happily still moving away from you [swings in my park have a period of 2-3 seconds].
Your 3rd push might contact the swing, but there is actually a chance it will merely stop a swing that is coming towards you.

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.]

Armed with that knowledge, we can instead set out to push the swing once every 2.5 seconds - just a little push, nothing too violent.

The first push will set the swing moving with an amplitude perhaps only a few cm.
The second push will happen at just the right time to help the swing on its way, and increase the amplitude slightly.
The third push does the same.
Pretty soon, with hardly any effort on your part, the swing will be sailing away with an amplitude of a couple of metres - especially if you have been sensible enough to step back slightly after each push so that you don't get struck be the returning spring.

While the swing moves forward then back each time, it will lose some energy through friction primarily - the faster it travels, the greater the losses.

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.

NOTE: with the above swing, it would have been possible to push every 5 seconds instead - it would still excite the swing, you would just be "topping up" the energy after every 2 oscillations.

Similary, if you found another, shorter, swing with a period of only 1.25 seconds [frequency 0.8], you applying a push every 2.5 seconds would work on it.

Indeed your excitation if "one push every 2.5 seconds" would work on a swing with frequency 0.4, 0.8, 1.2, 1.6, 2.0, ... and you start to move into the area of harmonics.

 

[Studiot]

Hello mutineer,

Thank you for the link to your syllabus.

I see that all of what you are asking and we are trying to help with is on next year's A2 syllabus.

This year you should be studying the basics of wave motion.

So I'm not sure why you are worrying about the topic of oscillations. Hopefully by next year you will also have covered some more maths which will also help.

I also note you have or will cover Newton's laws in AS ie this year.

About the pendulum,

In post#42 I called it 'mechanical oscillations' but that includes a simple pendulum. PeterO has given a slightly different view of the same oscillation (as have others). How did you get on with these?

 

[sophiecentaur]

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!]

 

[mutineer123]

OK here goes: using the swing in the park as your pendulum, with a person sitting on the swing.

Suppose you decide to push forward at what ever is there, once every second.
The first push will push the swing away.
The second push may be a "fresh air job" as the swing will actually be happily still moving away from you [swings in my park have a period of 2-3 seconds].
Your 3rd push might contact the swing, but there is actually a chance it will merely stop a swing that is coming towards you.

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.]

Armed with that knowledge, we can instead set out to push the swing once every 2.5 seconds - just a little push, nothing too violent.

The first push will set the swing moving with an amplitude perhaps only a few cm.
The second push will happen at just the right time to help the swing on its way, and increase the amplitude slightly.
The third push does the same.
Pretty soon, with hardly any effort on your part, the swing will be sailing away with an amplitude of a couple of metres - especially if you have been sensible enough to step back slightly after each push so that you don't get struck be the returning spring.

While the swing moves forward then back each time, it will lose some energy through friction primarily - the faster it travels, the greater the losses.

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.

NOTE: with the above swing, it would have been possible to push every 5 seconds instead - it would still excite the swing, you would just be "topping up" the energy after every 2 oscillations.

Similary, if you found another, shorter, swing with a period of only 1.25 seconds [frequency 0.8], you applying a push every 2.5 seconds would work on it.

Indeed your excitation if "one push every 2.5 seconds" would work on a swing with frequency 0.4, 0.8, 1.2, 1.6, 2.0, ... and you start to move into the area of harmonics.

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?

 

[mutineer123]

Hello mutineer,

Thank you for the link to your syllabus.

I see that all of what you are asking and we are trying to help with is on next year's A2 syllabus.

This year you should be studying the basics of wave motion.

So I'm not sure why you are worrying about the topic of oscillations. Hopefully by next year you will also have covered some more maths which will also help.

I also note you have or will cover Newton's laws in AS ie this year.

About the pendulum,

In post#42 I called it 'mechanical oscillations' but that includes a simple pendulum. PeterO has given a slightly different view of the same oscillation (as have others). How did you get on with these?

Yes i guess you are right, but i still need to understand resonance, but for that I need to get 'natural frequency' which I sadly don't. When posting this question about natural frequency I did not know, so much math was involved. I just wanted a simple understanding of natural frequency.

 

[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!