What Is Natural Frequency & How Does It Affect Objects?

[mutineer123]

I am having trouble understanding resonance. But before that I need to understand natural frequency of an object. I saw similar questions posted in the forum, but I don't understand the answers fully. So what exactly is a natural frequency?
One answer which I have read says
"Nearly all objects, when hit or struck or plucked or strummed or somehow disturbed, will vibrate. If you drop a meter stick or pencil on the floor, it will begin to vibrate. If you pluck a guitar string, it will begin to vibrate. If you blow over the top of a pop bottle, the air inside will vibrate. When each of these objects vibrate, they tend to vibrate at a particular frequency or a set of frequencies. The frequency or frequencies at which an object tends to vibrate with when hit, struck, plucked, strummed or somehow disturbed is known as the natural frequency of the object."
So is the natural frequency always present in an object? Like for instance a tuning fork, does it have a natural frequency when 'not' struck? Also if i hit it, i will have a frequency, but then if i hit it harder, there will be more ossicliations per unit cm, so the frequency will get higher, won't it? So the natural frequency, if there is one, changes?

 

[Bobbywhy]

Mutineer123, Welcome to Physics Forums!

The natural frequency of a tuning fork remains whatever it is, regardless if it is struck or not. If you strike it it will vibrate at its natural frequency. You already have all this knowledge down pat!

Only one misconception...if you hit the tuning fork harder, it just vibrates at a greater AMPLITUDE, but the frequency remains fixed.

 

[Studiot]

Hello mutineer and welcome to Physics Forums.

Before any great explanation do you understand what plain old 'frequency' is?

 

[tiny-tim]

welcome to pf!

hi mutineer123! welcome to pf!

So is the natural frequency always present in an object?

yes

Also if i hit it, i will have a frequency, but then if i hit it harder, there will be more ossicliations per unit cm, so the frequency will get higher, won't it? So the natural frequency, if there is one, changes?

no

(you may like to have a look at http://www.phys.unsw.edu.au/jw/basics.html … no particular reason … it's just so cool that i like to refer people to it! )

 

[mutineer123]

Mutineer123, Welcome to Physics Forums!

The natural frequency of a tuning fork remains whatever it is, regardless if it is struck or not. If you strike it it will vibrate at its natural frequency. You already have all this knowledge down pat!

Only one misconception...if you hit the tuning fork harder, it just vibrates at a greater AMPLITUDE, but the frequency remains fixed.

Yes that's true, but I saw an animated video in youtube some time back, where if the tuning fork is hit harder then the frequency increases. This makes sense, because if the fork is hit harder, it vibrates faster(backward and forward), so vibrating the surrounding air molecules faster, thus increasing oscillation, increases the frequency.

 

[mutineer123]

Hello mutineer and welcome to Physics Forums.

Before any great explanation do you understand what plain old 'frequency' is?

Thank you for welcoming me, and yes I know what frequency is.

 

[Studiot]

Hello again mutineer.

OK so there are three ways you can make something vibrate or oscillate.

Firstly with a single blow or impulse.

For instance ringing a bell (once).

Secondly by repeated impulses.

For instance sitting on a swing and working it back and fore, higher and higher.

Thirdly by continually forcing it to vibrate.

For instance playing a wind instrument by continual blowing.
Or for instance waving the end of a rope up and down.

All except the last example are examples of resonance exciting the natural frequency of a mechanical system.

The last example is the only way to cause a mechanical system to vibrate at some other frequency or a whole number multiple of its natural frequency.

It takes energy to vibrate, this is supplied by the exciting or driving agent, not by the system itself.
Thus in order to vibrate there must be energy transfer to the system itself from the agent.

Resonance at the natural frequency or some whole number multiple of it is by far and away the most efficient transfer mechanism.

Note I said 'or some multiple' since a system can resonate at higher frequencies that the fundamental or natural frequency if driven hard enough.
So yes if you blow your instrument hard enough you will get frequency doubling etc.

That is enough for this post, if you have followed this so far we can do more.

 

[sophiecentaur]

A few layers involved here.
The natural frequency of a basic 'mass on a spring' oscillator is independent of amplitude and is how the system will oscillate freely, (i.e. undriven) however it's been excited.
A system with a non-linear relationship between force extension will have a natural frequency which depends upon amplitude.
A more complex system (strings etc.) Will oscillate naturally at a number of overtone frequencies as well as a fundamental.
Introduce some damping (air friction) and loading due to additional air mass and the natural frequency will be reduced.
It all depends at what level you're thinking.

 

[Bobbywhy]

When an object is in free oscillation, it vibrates at its natural frequency. For example, if you strike a tuning fork, it will vibrate for some time after you struck it, or if you hit a pendulum, it will always oscillate at the same frequency no matter how hard you hit it. All oscillating objects have a natural frequency, at which they will vibrate at once they have been moved from the equilibrium position.

The tuning fork is a useful instrument for investigating sound because it vibrates at only one frequency, in contrast to most musical instruments that produce several different frequencies simultaneously. A struck tuning fork vibrates at a natural frequency that depends upon the fork's manufacture—the dimensions and the material from which it is made. The natural frequency is characteristic of the object’s shape, size, & composition.

The displacement at a certain point in time is the distance of the object away from the centre point. The displacement is 0 at the centre, at its maximum at one end (usually on the right when right is taken as positive), and at its greatest negative value on the opposite end (usually left but, again, only when right is taken as positive). Displacement is given the symbol s or x.

The amplitude is the greatest displacement of an oscillating object. It is measured from the center point to one of the maximum points of displacement. The amplitude can increase or decrease with time. Amplitude is represented by the symbol A
Period is the time taken for a single oscillation. the frequency is the number of oscillations per second.

A mass resonates, when the driving frequency of oscillations is equal to the natural frequency of the object. This means that work is done to keep drive the oscillations.
If the driving frequency is less than the natural frequency, the amplitude decreases to a much smaller value.

 

[sophiecentaur]

You need to make it clear that the oscillatios of a 'driven' oscillator will be at the driving frequency. The amplitude will depend on how near the driving frequency is to the natural frequency and to the Q (quality) factor of the oscillator.

 

[mutineer123]

Hello again mutineer.

OK so there are three ways you can make something vibrate or oscillate.

Firstly with a single blow or impulse.

For instance ringing a bell (once).

Secondly by repeated impulses.

For instance sitting on a swing and working it back and fore, higher and higher.

Thirdly by continually forcing it to vibrate.

For instance playing a wind instrument by continual blowing.
Or for instance waving the end of a rope up and down.

All except the last example are examples of resonance exciting the natural frequency of a mechanical system.

The last example is the only way to cause a mechanical system to vibrate at some other frequency or a whole number multiple of its natural frequency.

It takes energy to vibrate, this is supplied by the exciting or driving agent, not by the system itself.
Thus in order to vibrate there must be energy transfer to the system itself from the agent.

Resonance at the natural frequency or some whole number multiple of it is by far and away the most efficient transfer mechanism.

Note I said 'or some multiple' since a system can resonate at higher frequencies that the fundamental or natural frequency if driven hard enough.
So yes if you blow your instrument hard enough you will get frequency doubling etc.

That is enough for this post, if you have followed this so far we can do more.

Well I did follow you, but as i posted in my question, I am a bit shaky about resonance.(I don't understand how the frequency starts to increase, and keep on increasing when the frequency matches the natural frequency of the object)

 

[mutineer123]

When an object is in free oscillation, it vibrates at its natural frequency. For example, if you strike a tuning fork, it will vibrate for some time after you struck it, or if you hit a pendulum, it will always oscillate at the same frequency no matter how hard you hit it. All oscillating objects have a natural frequency, at which they will vibrate at once they have been moved from the equilibrium position.

The tuning fork is a useful instrument for investigating sound because it vibrates at only one frequency, in contrast to most musical instruments that produce several different frequencies simultaneously. A struck tuning fork vibrates at a natural frequency that depends upon the fork's manufacture—the dimensions and the material from which it is made. The natural frequency is characteristic of the object’s shape, size, & composition.

The displacement at a certain point in time is the distance of the object away from the centre point. The displacement is 0 at the centre, at its maximum at one end (usually on the right when right is taken as positive), and at its greatest negative value on the opposite end (usually left but, again, only when right is taken as positive). Displacement is given the symbol s or x.

The amplitude is the greatest displacement of an oscillating object. It is measured from the center point to one of the maximum points of displacement. The amplitude can increase or decrease with time. Amplitude is represented by the symbol A
Period is the time taken for a single oscillation. the frequency is the number of oscillations per second.

A mass resonates, when the driving frequency of oscillations is equal to the natural frequency of the object. This means that work is done to keep drive the oscillations.
If the driving frequency is less than the natural frequency, the amplitude decreases to a much smaller value.

See, Bobby,From your explanation I get why amplitude increases when you hit something harder, but I still don't get why the frequency does 'not' increase! I mean if I hit the tuning fork harder right, the two metal ends, will move back and forth faster, creating more oscillations! So frequency should increase with force.

 

[Delta Kilo]

I mean if I hit the tuning fork harder right, the two metal ends, will move back and forth faster, creating more oscillations!

The ends will move faster but they will also travel longer distance each time. The two factors cancel each other out. Try it with a simple pendulum (a ball on a string will do). As long as the angle is not too big the period is independent from the amplitude.

 

[Studiot]

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.

 

[sophiecentaur]

See, Bobby,From your explanation I get why amplitude increases when you hit something harder, but I still don't get why the frequency does 'not' increase! I mean if I hit the tuning fork harder right, the two metal ends, will move back and forth faster, creating more oscillations! So frequency should increase with force.

When 'driven' with a signal with a higher or lower frequency than the natural freq. the oscillator will oscillate at the driving frequency. The amplitude will depend on the degree of 'coupling' which will relate to where or how the force is applied. When uncoupled, the oscillator will revert to its natural frequency and the amplitude will decay.

 

[vin300]

Let's assume the body vibrates as a simple harmonic oscillator.
The velocity at any instant t is v-kxt/m, where x is displacement from mean, v is the velocity at mean.The velocity becomes zero at t=mv/kx at the extremity.

Now, equating potential energy at the extremity to kinetic energy at the mean,
0.5kx^2=0.5mv^2, we get v/x= sqrt(k/m), which is constant.
Hence, the time period after which velocity becomes zero i.e., mv/kx remains constant irrespective of the magnitude of vibration.

 

[sophiecentaur]

There is a fantastic and unassailable argument that the frequency of an ideal 'mass and spring' oscillator is independent of the amplitude of the oscillation. It is because, when you solve the differential equation which you get by writing down the force and acceleration on the mass, the result is a frequency which is independent of amplitude.
Any other, verbal / arm waving, argument 'may' sort-of-justify the fact but, without the Maths, it tends just to be 'chat'. If you aren't prepared either to learn the Maths or accept what it tells you then there is no prospect of a good understanding. Modern Science hangs, entirely, on the associated Maths and you can't ignore it, I'm afraid.

 

[Bobbywhy]

“The main reason for using the (tuning) fork shape is that, unlike many other types of resonators, it produces a very pure tone, with most of the vibrational energy at the fundamental frequency, and little at the overtones (harmonics). The reason for this is that the frequency of the first overtone is about 52/22 = 25/4 = 6¼ times the fundamental (about 2½ octaves above it).[2] By comparison, the first overtone of a vibrating string or metal bar is only one octave above the fundamental. So when the fork is struck, little of the energy goes into the overtone modes; they also die out correspondingly faster, leaving the fundamental. It is easier to tune other instruments with this pure tone.”
http://en.wikipedia.org/wiki/Tuning_fork

“A physical system can have as many resonant frequencies as it has degrees of freedom; each degree of freedom can vibrate as a harmonic oscillator. Systems with one degree of freedom, such as a mass on a spring, pendulums, balance wheels, and LC tuned circuits have one resonant frequency. Systems with two degrees of freedom, such as coupled pendulums and resonant transformers can have two resonant frequencies. As the number of coupled harmonic oscillators grows, the time it takes to transfer energy from one to the next becomes significant. The vibrations in them begin to travel through the coupled harmonic oscillators in waves, from one oscillator to the next.
Extended objects that experience resonance due to vibrations inside them are called resonators, such as organ pipes, vibrating strings, quartz crystals, microwave cavities, and laser rods. Since these can be viewed as being made of millions of coupled moving parts (such as atoms), they can have millions of resonant frequencies.”
http://en.wikipedia.org/wiki/Resonance

“An acoustically resonant object usually has more than one resonance frequency, especially at harmonics of the strongest resonance. It will easily vibrate at those frequencies, and vibrate less strongly at other frequencies. It will "pick out" its resonance frequency from a complex excitation, such as an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies other than its resonance.
Acoustic resonance is an important consideration for instrument builders, as most acoustic instruments use resonators, such as the strings and body of a violin, the length of tube in a flute, and the shape of a drum membrane.” http://en.wikipedia.org/wiki/Acoustic_resonance

 

[sophiecentaur]

The reason for choosing a the 'U' shape for a tuning form is that it is inherently balanced and is not affected by how it is held or mounted. The oscillations of the tips of the prongs are, basically, apart and together and pretty much independent of the position of the mid point. Any common, side-to side motion soon dies away after it's been struck. Of course, there is a small degree of coupling of energy from the oscillations of the prongs to the pointed handle, because that is how the fork is connected to the sounding board / table / etc (very little sound goes directly from the prongs into the air).

If you want a predictable frequency from an oscillating system, you connect it so that it is only lightly coupled to the outside world, which is why a Xylophone / Glockenspiel bar rests where it does on the knife edge supports and why a Quartz crystal is mounted similarly in its little metal envelope. A clock pendulum or watch hair spring oscillate freely with only a minimum of energy exchange from driving spring or to the clock movement. We're talking 'High Q' systems.

@bobbywhy
Please, when you are discussing multiple modes of oscillation, make a point of not confusing / equating Harmonics (integer multiples of the fundamental frequency) with Overtones, which are not necessarily harmonically related. The distinction can be Very Relevant in some applications. You clearly know this but I don't want other people to get it wrong.

 

[Studiot]

Bobbywhy, don't you think you are obscuring the difference between oscillation and resonance with your tuning fork?

For oscillation you require a minimum of one oscillating system.

For resonance you require a minimum of two.

 

[sophiecentaur]

Bobbywhy, don't you think you are obscuring the difference between oscillation and resonance with your tuning fork?

For oscillation you require a minimum of one oscillating system.

For resonance you require a minimum of two.

Yes- it's the difference between Free and Driven oscillations.

 

[Studiot]

Yes- it's the difference between Free and Driven oscillations.

There is a third way and that is taken by a tuning fork.

 

[sophiecentaur]

?
How is it other than multiple modes?

 

[Studiot]

?
How is it other than multiple modes?

I already spelt it out in simple terms in post#7.

Resonance occurs when the input from the driving system has the same frequency as the natural or resonant frequency of the driven system.

A single blow or impulse does not constitute an oscillating or periodic system nor does it posses a frequency.

A tuning fork is is usually used in this mode.

Therefore no resonance is involved.

The single impulse, which may take many different forms, excites the tuning fork to oscillate at its resonant frequency. Hence the confusion.

 

[Bobbywhy]

Sophiecentaur, Thank you for your observation. You are exactly correct: It is indeed important to stress the difference between “overtones” and “harmonics”.

“The term harmonic has a precise meaning - that of an integer (whole number) multiple of the fundamental frequency of a vibrating object. The term overtone is used to refer to any resonant frequency above the fundamental frequency - an overtone may or may not be a harmonic. Many of the instruments of the orchestra, those utilizing strings or air columns, produce the fundamental frequency and harmonics. Their overtones can be said to be harmonic. Other sound sources such as the membranes or other percussive sources may have resonant frequencies which are not whole number multiples of their fundamental frequencies. They are said to have non-harmonic overtones.”
http://hyperphysics.phy-astr.gsu.edu/hbase/music/otone.html

 

[Bobbywhy]

@Studiot, Thank you for your comment. Excuse me if my post obscured the difference between oscillation and resonance…clearly two distinctly different phenomena. You are exactly correct to point out that there is a difference between oscillation and resonance in tuning forks.

“Resonance occurs in any oscillatory system when the frequency of the driving force is near the natural frequency of the system. At the resonant frequency of a linear system the amplitude of oscillation is at its maximum value. Therefore, in a linear system of oscillation there is a direct correspondence between the amplitude of oscillation and the driving frequency, and thus a periodic input always results in a periodic output. However, a non-linear system may oscillate at various amplitudes for one particular driving frequency because the elements of the driving force may not vary linearly with the space parameters. Therefore, there is no unique maximum resonant amplitude in a non-linear oscillatory system and a periodic input may not result in a periodic output. In such a non-linear oscillatory system one may observe chaotic behavior in the output of the system.”
http://www.phy.davidson.edu/stuhome/derekk/resonance/pages/main.htm

An excellent discussion of this subject occurred right here on PF in May, 2005. Please see “Resonance and natural oscillations”
https://www.physicsforums.com/showthread.php?t=77161

 

[B.E.M]

I remember it seeming strange to me that so many objects turn out to oscillate at a constant frequency. I think there are two facts contributing to this.

(1) One specific example, a simple oscillator, has this property. A simple oscillator is defined by a simple linear relationship: f=-k(x-x0).

(2) Pretty much every oscillator will approximate a simple oscillator if amplitude is not too excessive.
This is just because any function of force f=F(x) that is a smooth curve will approximate a straight line for a small enough interval, and for oscillation, we must be centered around a portion of the curve that includes both positive and negative values for the force. So we get a straight line passing through (f=0,x=x0) This is a simple oscillator, f= -k(x-x0)

 

[Studiot]

A simple oscillator is defined by a simple linear relationship: f=-k(x-x0).

This relation is not enough (a necessary but not sufficient condition) and is therefore is not a definition.

Consider the system at x = x₀.

Then f = 0 and a perfectly satisfactory solution is that the system is not moving (ie not oscillating).

 

[B.E.M]

This relation is not enough (a necessary but not sufficient condition) and is therefore is not a definition.

Consider the system at x = x₀.

Then f = 0 and a perfectly satisfactory solution is that the system is not moving (ie not oscillating).

or is it oscillating with exactly the natural frequency and an amplitude of zero?

To which the only sensible response is.. "whatever..".

 

[sophiecentaur]

That sort of observation can be applied to the analysis of most systems. You have merely stated the situation in the limit as maximum displacement approaches zero. Continuity is there.

 

[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.