# Having a hard time grasping the notion of a natural frequency(and also resonance)

1. Feb 3, 2012

### 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 dont understand the answers fully. So what exactly is a natural frequency?
"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, wont it? So the natural frequency, if there is one, changes?

2. Feb 3, 2012

### 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.

3. Feb 4, 2012

### Studiot

Hello mutineer and welcome to Physics Forums.

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

4. Feb 4, 2012

### tiny-tim

welcome to pf!

hi mutineer123! welcome to pf!
yes
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! )

5. Feb 4, 2012

### mutineer123

Yes thats 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.

6. Feb 4, 2012

### mutineer123

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

7. Feb 4, 2012

### 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.

8. Feb 4, 2012

### 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.

9. Feb 4, 2012

### 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.

Last edited: Feb 4, 2012
10. Feb 4, 2012

### 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.

11. Feb 4, 2012

### mutineer123

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)

12. Feb 4, 2012

### mutineer123

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.

13. Feb 4, 2012

### Delta Kilo

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.

14. Feb 5, 2012

### 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.

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15. Feb 5, 2012

### sophiecentaur

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.

16. Feb 5, 2012

### 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.

17. Feb 5, 2012

### 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.

18. Feb 5, 2012

### 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

19. Feb 5, 2012

### 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.

Last edited: Feb 5, 2012
20. Feb 5, 2012

### 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.