# Homework Help: How does resonance work?

1. Aug 1, 2006

### Delzac

Just a qns,

how does resonance work? i know about the natural frequency stuff, but why and how does it occur? why in the 1st place do every object has a natural frequency and different natural freq.

And by the way, if we know the natural frequency of a man, couldn't we potentially can kill a person?

Thx

2. Aug 1, 2006

### HallsofIvy

"Resonance" is a lot like pushing a child on a swing. If you exert the force while the child is still coming toward you, all you will do is slow the swing. If you apply the force just as the child is starting away from you you will add the total force to the swing, doing work and adding energy. If you apply the force at "random" times, sometimes it adds energy, sometimes takes it away. It's repeatedly applying the force at just the right time ("natural frequency") that causes resonance. For a swing, the period of the swing (so its "natural frequency") is determined by the length of the lines supporting it. If you struck a steel rod, the kind of situation I think you are referring to, the frequency with which it vibrates is determined by the material. Things made from a single, crystaline, material will have specific "natural frequencies".

A building, or a bridge, will be built of or supported by iron girders, for example that have a specific "natural frequency".

A person, and in fact most things, are made of many different materials. Each material has its own "natural frequency"- that's why you don't get resonance with such things- there are so many "natural frequencies" involved, they conflict with each other.

3. Aug 1, 2006

### Delzac

But how does it actually works in a mircoscopic level? Different lengths of a prong have different natural frequency, but there are made of the same material isn't it? so y not same natural frequency?

4. Aug 1, 2006

### Andrew Mason

As Halls said, objects often have a natural vibration period. All you need for such a natural vibration to occur is a restoring force that is roughly proportional to displacement: eg. pendulum. When you displace the tines of a fork, the metal stretches and exerts an opposing force that tries to restore the original position. The restoring force is proportional to displacement: F = -kx.

$$F = ma = m\frac{d^2x}{dt^2} = -kx$$ has a general solution:

$$x = Acos\omega_0t$$ where A is the maximum amplitude at time t = 0 and $\omega_0 = \sqrt{k/m}$. This is simple harmonic motion with natural period of vibration $\omega$.

But striking a tuning fork or causing something to vibrate by giving it an initial push is not resonance. Resonance occurs in an oscillator when a periodic force is applied and the period of the force is the same as the natural period of vibration (which, as explained above, depends on the restoring force and mass of the vibrating parts). Look up: http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html#c1"

AM

Last edited by a moderator: Apr 22, 2017
5. Aug 1, 2006

### andrevdh

For resonance to occur in an object it is necessary that the incident vibration create a standing wave in the object. A standing wave have regions of maximum and minimum oscillations occuring at fixed positions in the object. The object is most likely to break at the regions of maximum oscillation if the amplitude are of sufficient magnitude.

Standing waves are created in an object by multiple reflections of the disturbance, bouncing off the inner walls of the object. These reflecting disturbances interfere with each other. The positions of maximum and minimum interference of these reflections need to occur at fixed positions in the object. This is highly unlikely if the object has a non-symmetrical structure, since the reflections will reflect at odd angles off the walls and rarely meet at regular intervals and in the same regions.

Another criteria is that the frequency of the incident vibration need to be such that the standing wave will form - which is determined by the physical dimensions and speed of propagation of the vibration inside of the object.

So, sorry to disappoint you, but you will need to find a different way to get rid of your physics teacher.

Last edited: Aug 1, 2006
6. Aug 1, 2006

### Delzac

but then again how can the atoms in dif length of a prong react differently to dif. freq. of incoming waves?

7. Aug 1, 2006

### Andrew Mason

You have to study simple harmonic motion. Your question is not very clear. The atoms do not react individually. The tine acts as a mechanical unit.

AM

8. Aug 2, 2006

### andrevdh

The prongs vibrate transversely to their length. Long prongs will bend more readily and therefore tend to vibrate at a lower frequency when you hit it. Shorter prongs will form standing waves at higher frequencies due to their stiffness. You can demonstrate it to yourself with a ruler over the edge over a table. Push the end of the ruler down and release it. Try it with a shorter length. I think one can see the standing wave in a vibrating tuning fork if you hold it in front of a computer monitor.

Last edited: Aug 2, 2006
9. Aug 2, 2006

### desA

I like 'andrevdh' 's explanation. I hadn't hought about the standing wave aspect. Thanks, that's got me thinking about my current research...

I'm going to add something into the mix here. 'Resonance' occurs at singularities in the system function ie. when the denominator tends towards zero. This can be seen in 2nd order systems, in Laplace domain, for instance.

Such phenomena show up in some strange places & some complex pde's.

Question: Is there a relationship between 'resonance' & chaos?

10. Aug 2, 2006

### Delzac

Pardon me for my seemingly lack of understand of the topic. :P but can someone explain to me what is the meaning of the words in bold?

BTW i have finish my Simple Harmonic Motion lecture, but they nv mentions these terms.

11. Aug 2, 2006

### desA

Oops, perhaps I loaded you with too much jargon.

When the denominator (lower term of fraction) of a function tends to zero, then the function becomes undefined & can zoom off to either +- infinity. This is often termed a singularity condition.

2nd order system eg. mass-spring-damper with a derivative of order 2 eg. the acceleration term.

Is there a relationship between 'resonance' & 'chaos'? Thinking out loud in terms of certain critical points in a pde which displays chaotic behaviour. Would this pde also display resonance?

12. Aug 2, 2006

### Delzac

Erm what function specifically? is it the formular X= X_0 sin wt ?
can u give an example thx

13. Aug 2, 2006

### J77

1. What does this has to do with resonance? Do you mean that at the resonant frequency, the amplitude of the vibration tends to infinity?

2. ???

3. Dynamical systems can display resonance - again due to some external, periodic forcing. However, resonance, afaik, has nothing to do with transitions to chaotic dynamics.

14. Aug 2, 2006

### desA

1. Take a look at the Laplace transform work on system instability - transfer functions etc. There are a few ways that the transfer function for a forced system can run into trouble. One way is when the terms balance each other, then the transfer function becomes undefined. Another is that certain terms eg. damping factor, become less than a certain critical value which results in a sqrt(-ve).

If the amplitude is to go to infinity, then either the numerator travels to infinity, the denominator to zero, or a combination of both.

Control folks design to steer away from such conditions in case some odd forcing function finds its way into the system - say via a disturbance, & then causes the system to resonate. They use tricks to cotrol these out of the system. A first-order plant, with appropriate forcing function can, display 2nd-order or higher behaviour.