Energy exchange between a bell and a hammer

Imagine a bell that has only 3 ideal resonant frequencies (no damping) and I am hitting it with a point like ideal hammer, does this mean that all of the energy of the hammer is distributed among those 3 resonances?

I think not, i.e. the energy stored in the resonant modes are less than the overall energy transferred from the hammer to the bell. The hammer is like a Dirac delta function with a flat spectrum, i.e. it has a component on every frequency. So on those frequencies where the bell is non resonant, it is maximally damping, that means the energy is converted to heat.

So the bell will ring forever on its 3 resonances with a total energy less than what was exchanged with the hammer.

Am I wrong?

256bits
Gold Member
Surprised no one has responded to your post because I think it is a great physics and mathematical intellectual challange, especially modelling the hammer hit as the Dirac delta function.

You do say thought that there is no damping in the bell at its resonant frequencies, but then go on to say that at other frequencies there is maximal damping, which I am sure sure both statements could both be true, but a physics major could give most likely give it a thumbs up or down either way.

I would not think that all of the energy of the hammer is transferred to the bell, as this is an elastic collision.

Consider an ideal ball bearing being dropped from a height h onto an ideal anvil - ideal meaning no deformation of either object. If the collision is completely elastic, the ball bearing should rebound back up to its original height h, and the ball would not have imparted any energy to the anvil, in which case no sound should be heard either.

Once deformation is allowed and then depending upon the rigidity of ball and anvil, the ball would rebound to a height somewhat less than h, and energy would have been exchanged between the two objects both of which would both experience ringing.

Now how that relates to the Dirac delta function, is beyond me at the moment as I have not delved into solving the mathematics as of yet, but may if time permits.

I would wonder if modelling the bell and hammer as two spring- mass systems exchanging energy would be a path to follow.

Hopefully a knowledgeable individual chimes in with mathematical insight.

A bell, like any other object, has infinite resonance frequencies. Restricting the analysis to three modes may lead to wrong conclusions.
Each of these infinite frequencies has its own vibration pattern (nodes and anti-nodes).
When you hit with the hammer the bell's shape changes a bit. The new shape can be expanded in a Fourier series over the allowed modes. The amplitude of each mode depends on where you hit the bell, that´s why the perceived sound changes.
In a lossless situation the modes oscillate forever, an unrealistic situation for an actual bell.

I mean really ideal. we can also do it more abstract, imagine there is a system whose frequency response consists of 3 Dirac deltas, say at ω1, ω2 and ω3.

Now I am exciting it with a Dirac delta in time domain with a certain energy E. By definition, this corresponds to a flat spectrum that contains the same amount of energy. As far as I remember, this is guaranteed by the Bessel inequality/equality and the completeness of the vector space and so on.

From the flat spectrum, the system chooses only those three, and damping all other to mathematical zero which I understand is maximum heat generation (or is it reflection?) on every frequency other than the resonance frequency, and forever oscillating on resonance frequencies.

So the overall stored energy must be less than that of the impulse.

Am I wrong?

I mean really ideal. we can also do it more abstract, imagine there is a system whose frequency response consists of 3 Dirac deltas, say at ω1, ω2 and ω3.

Now I am exciting it with a Dirac delta in time domain with a certain energy E. By definition, this corresponds to a flat spectrum that contains the same amount of energy. As far as I remember, this is guaranteed by the Bessel inequality/equality and the completeness of the vector space and so on.

From the flat spectrum, the system chooses only those three, and damping all other to mathematical zero which I understand is maximum heat generation (or is it reflection?) on every frequency other than the resonance frequency, and forever oscillating on resonance frequencies.

So the overall stored energy must be less than that of the impulse.

Am I wrong?

I would like to think that you're right, unless there's something I'm unaware of (which there are a lot of things). This is the way we were taught signals and systems, to imagine a hammer strike as an impulse, which is another term for the delta function.

If the bell's impulse response is defined by a delta function, or multiple ones, then its output will match its impulse response when an impulse is applied at the input (time domain delta function), which is flat with infinite bandwidth in the frequency domain. The delta function's infinite gain will perfectly store the energy without losses in those frequencies. You might be better off talking about this in a purely mathematical model, rather than an "ideal" bell to eliminate any confusion, and then apply that model to your "ideal" version of a bell. I think Gordianus had a really good answer and it is realistic.

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You might be better off talking about this in a purely mathematical model, rather than an "ideal" bell to eliminate any confusion, and then apply that model to your "ideal" version of a bell.

so considering the pure mathematical model, is it ok to say that the portion of the energy that does not fall on one of the ideal eigenfrequencies ω1, ω2 or ω3 is turned to heat?

I am confused whether this portion of energy turns into heat, or it is not absorbed at all, i.e. it is reflected back.

Let me ask something simple: Have switched from a three-dimensional object (the bell), that has infinite resonance modes, to a zero-dimension system with three resonances?
Needless to say they're vastly different. Their impulse responses are quite different.

so considering the pure mathematical model, is it ok to say that the portion of the energy that does not fall on one of the ideal eigenfrequencies ω1, ω2 or ω3 is turned to heat?

I am confused whether this portion of energy turns into heat, or it is not absorbed at all, i.e. it is reflected back.

That is not something a transfer function can tell you directly. You must do further work to characterize the system before you can say where the attenuated energy goes.

sophiecentaur
Gold Member
2020 Award
Imagine a bell that has only 3 ideal resonant frequencies (no damping) and I am hitting it with a point like ideal hammer, does this mean that all of the energy of the hammer is distributed among those 3 resonances?

I think not, i.e. the energy stored in the resonant modes are less than the overall energy transferred from the hammer to the bell. The hammer is like a Dirac delta function with a flat spectrum, i.e. it has a component on every frequency. So on those frequencies where the bell is non resonant, it is maximally damping, that means the energy is converted to heat.

So the bell will ring forever on its 3 resonances with a total energy less than what was exchanged with the hammer.

Am I wrong?

I think you are looking for a flaw in conventional thinking. I don't think there is one - it's your statement that "The hammer is like a Dirac delta function" that needs examination.
The hammer just has Kinetic Energy (unchanging) whilst it's in motion.
When it hits the bell, the bell will distort and be set in motion according to its natural modes of vibration. If you could look at the variation of Force with Time, you would see, not an Impulse but fast variations of force (at the ringing frequencies) contained in an envelope of time corresponding to the contact time of the hammer Plus a non-periodic impulse. There will be Energy transferred to the final net motion of Hammer and Bell and energy into the various modes of vibration and these will carry on after the impulse has finished.
There is no frequency shifting of energy - as your post seems to suggest. Where the mechanical impedance of the bell is appropriate (low, at resonance), energy will be transferred and where the impedance is high (off resonance) no vibrational. energy will be transferred. I guess the secret of a 'good bell / hammer combination' will be one where there is as little as possible net KE ater the contact.

Relating this to the simplest system I can think of, a simple pendulum can be excited with all of the KE of a mass striking it if the two masses are equal and the collision is elastic. In this case, only one mode of vibration would take on all the energy of the striker.

thanks for the replies so far. Now things are becoming more clear. I also like the notion of mechanical impedance which is similar to the electrical case. An ideal elastic collision would resemble a perfect impedance matching, such that all of the power of the impulse is transferred into the system.

On those frequencies where the mechanical impedance of the system is low, oscillations will damp very soon with a damping coefficient α. Where the impedance is high, then they will oscillate for a very long time (ideal case being forever for $Z_{bell}(\omega_1)=∞$, where ω1 is one of the ideal resonance frequencies).

From the above discussion one may conclude the following for the conservation of energy in the system in the case of perfect power transfer:

$\mathcal{E}_{input}=\mathcal{E}_{stored}+{heat}$

thanks for the replies so far. Now things are becoming more clear. I also like the notion of mechanical impedance which is similar to the electrical case. An ideal elastic collision would resemble a perfect impedance matching, such that all of the power of the impulse is transferred into the system.

On those frequencies where the mechanical impedance of the system is low, oscillations will damp very soon with a damping coefficient α. Where the impedance is high, then they will oscillate for a very long time (ideal case being forever for $Z_{bell}(\omega_1)=∞$, where ω1 is one of the ideal resonance frequencies).

From the above discussion one may conclude the following for the conservation of energy in the system in the case of perfect power transfer:

$\mathcal{E}_{input}=\mathcal{E}_{stored}+{heat}$

Why would $Z_{bell}(\omega_1)=∞$?

Also, I think energy can go other places than just heat. It can go into reshaping areas of the bell for example, which would be a reconfiguration of the atom positions. Also, the bell would only be able to resonate indefinitely if it were in a vacuum I think. The medium it transmits sound on, like air, would dampen the bell's oscillations.