# Ringing a bell - How much energy is lost to sound?

• Chaos' lil bro Order

#### Chaos' lil bro Order

Hey,

I was wondering about a scenario, perhaps you can help me.
Imagine a bell hung from an arbitrary pole in which the friction of the bell-pole junction is known. Assume this bell and pole are in a clean room of known atomospheric density and pressure. Orthogonal to the plane of the pole is an actuator that thrust outwards a rod which strikes the bell and rings it. You also know the force of the rod that strikes the bell and you know the bell's mass. My question is, upon using the actuating rod to strike and ring the bell, can physicists measure how much of the energy goes into producing the ringing sound and how much goes into swinging the bell?

I know this scenario is very loose, but my intention is to understand just how much energy is lost to sound propogation vs. bell deflection. If someone can provide a rough estimate of the order of magnitude that would be nice.

You know, something like, 0.0001% of the energy is radiated as sound while 99.9999% of the energy goes into deflecting the bell.

Thanks.

It would actually be quite easy to determine experimentally. All you would really need is a few good (or one very good) measurement of the sound intensity at a certain point that is a known distance from the equilibrium position of the bell.

Intensity is measured as Watts per meter squared - so if the sound has an intensity of, say, 1 milliwatt per meter squared (roughly 80 dB) at a distance 3 meters from the bell, the overall power is this intensity times the area of the sphere - or about .113 watts.

$$$\begin{array}{l} Power = I \cdot 4\pi r^2 \\ Power = 10^{ - 3} \cdot 36\pi = 0.113W \\ \end{array}$$$

Keep in mind that this is the power for that instant in time. Since the bell would slowly come to a silence, you would ultimate have to write your equation for power as a function of time, then integrate. For example, if the sound intensity decreases linearly for 10s:

$$$\begin{array}{l} Intensity(t) = - .0001t + .001 \\ Power(t) = \left( { - .0001t + .001} \right)\left( {36\pi } \right) \\ \sum {Energy} = \left( {36\pi } \right)\int_0^{10} {\left( { - .0001t + .001} \right)} \\ \end{array}$$$

Alternatively, you could just measure the overall movement of the bell afterward, and assume that nearly everything else is lost to sound. I think you will find that the energy dispersed as sound is much higher than you imagine-keep in mind there are only two ways in which energy is eventually 'used' in this scenario - friction and sound.

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Very nice explanation King, much obliged.

No problem. I'll try to format the equations for you, if you'd like to see the actual formulas to do the calculations.

It will depend on just where you strike the bell, what you use to strike the bell and how hard you strike the bell.

IIRC there is a art to getting the clapper just the right length, if you do not hit the sweet spot on the bell you do not get as much ringing. I could even imagine finding a spot where you get NO ringing.

Without knowing the bells geometry and the properties of the striker I do not think it would be possible to answer your question.

Just a side note. Seems to me that bells that are struck from the outside are usually rigidly mounted (no swing). The bells that swing have an internal clapper and it is the swinging bell which strikes the clapper.

Integral

You raise valid points.

I think king's answer was sufficient given my unelaborate scenario. We don't need to nitpick it any further, but as you point out, one easily could further refine it.

Actually your scenario is the same as many more practical situations in dynamics. There are two really parts to the question:

1 If you apply a force to a structure which has several modes of vibration, how much energy goes into each different mode?

If you know the frequencies and deformed shape for each mode, there is a general (and mathematically simple) answer to that. In other situations it's a very practical problem - for example how to attach something that vibrates (e.g. an engine) to an object to minimize the amount of vibration transmitted.

2. The energy in a particular mode of vibration is partly converted to motion of the surrounding air (i.e. sound), partly into heat in the bell. Finding how much goes into sound and how much into heat is not so simple as the first part of the question!

As Integral said, it is entirely situationally dependent, but it can be measured from the other direction from what KingNothing said very easily: the KE of the striker is easy to determine and the energy in swinging the bell is just the potential energy change of the centroid, so the difference is the energy of vibration*.

Finding the energy in the sound directly is tough because it is released over time and you have to integrate it, as KingNothing said.

*Caveat - I'm not sure how much, if any, of the energy is quickly turned into heat at the impact point or in the bell. I suspect it is small compared to the other energies involved.

The standard way to measure what is going on would be a modal impact test (see a Dynamics 3 university course for details). Basically, you would attach an accelerometer to the bell, then hit it with a hammer that has a load cell in the head, and measure both the force and displacemnt against time.

With some standard signal processing techniques (FFT, deconvolution, etc) you can find the frequencies of all the vibration modes of the bell (including swinging on its pivot), the amount of response in each vibration mode for a given impact, and the amount of damping in each mode (i.e how long vibrations in each mode will take to die away)

Since the collision between the striker and the bell is unlikely to be elastic, a calculation using conservation of energy is not a good plan.

You can also do the modal analysis by taking the microphone data recorded earlier and running it through an FFT in a spectrum analyzer. We determine blade modes that way frequently.

Wonderful discussion guys, thanks for the input. So does anyone want to venture a ballpark guess on how much energy is radiated as sound? 1%? 0.1%? 0.01%? even less? I realize it is somewhat foolhardy to guess about such loose examples, but I'm really only looking for an equally loose answer.

Thanks.