How close should frequencies be to show 'beating'?

[Master1022]

Homework Statement

If we have an applied force of $F(t) = A \sin(\omega t)$. If $\omega_n = 1$, then what value of $\omega$ may show beating?

Relevant Equations

$f = \abs{f_2 - f_1}$

This is for a standard mass-spring-damper system which is being modeled in MATLAB.

I have done some research on the internet and it just says that the frequencies should be close to show the beating phenomenon. Is there a general rule of thumb that I should follow to know how close the frequencies should be?

I have just been plugging in random values, but I am unable to really see an ouput that looks like any of the examples on the internet, nor am I able to justify why I have chosen said values.

Would appreciate it if someone could explain this idea of the frequencies being 'close' (and perhaps quantify that) or point me in the right direction, then I would appreciate that very much.

Thanks in advance.

 

[hutchphd]

What is the system? It is not clear to me what you are doing here.

 

[Master1022]

What is the system? It is not clear to me what you are doing here.

It is a standard mass-spring-damper system (with \zeta = 0) with an applied force F(t) = F_{0} \cos(\omega t).

That was just the context, but the question was in two parts:
1) How close (quantitatively) do frequencies need to be to show the beat phenomenon?
2) How to think about the question: "If the resonant (angular) frequency is \omega_n = 1, what value of \omega do you think might show beating?"

 

[hutchphd]

The question seems to me unanswerable. Like "how high is up?"or "how blue is the sky?". The amplitude of the sum of two oscillator frequencies gives the difference frequency.
Perhaps I misunderstand...

 

[Master1022]

The question seems to me unanswerable. Like "how high is up?"or "how blue is the sky?". The amplitude of the sum of two oscillator frequencies gives the difference frequency.
Perhaps I misunderstand...

Thanks for responding. So how do we know whether the summation will show a beating pattern? Is there a requirement on the difference of the frequencies (not all summed up pairs of frequencies will show beating)?

 

[hutchphd]

Carefully define "a beating pattern" and you will probably answer your own question.

 

[Master1022]

Carefully define "a beating pattern" and you will probably answer your own question.

Well the definition in the referenced text (S.S Rao Mechanical Vibrations) says: "When two harmonic motions, with frequencies close to one another, are added, the resulting motion exhibits a phenomenon known as beats... (then it goes on to show the mathematical equations of the wave)".

However, what does 'close' mean? Is there a general rule of thumb for this? Am I missing something from the definition that is holding me back?

Thanks in advance.

 

[RPinPA]

I think we're missing some information here.

I have just been plugging in random values,

Into what?

I'm suspecting there's an equation here that you haven't shown us, something more than just the trig identity that shows the sum of two frequencies can be represented as the product of sinusoids at the sum and difference frequencies.

 

[Master1022]

I think we're missing some information here.
Into what?

I'm suspecting there's an equation here that you haven't shown us, something more than just the trig identity that shows the sum of two frequencies can be represented as the product of sinusoids at the sum and difference frequencies.

sorry, I mispoke - the 'plugging in' of values was for something separate. I was plugging in values for the frequency of the forced vibrations, \omega into my MATLAB mass-spring-damper with forced vibrations model.

Even if I use the trig identity you mention above, is there a way for me to theoretically determine whether a given pair of frequencies will cause a beat pattern?

Thanks.

 

[RPinPA]

The trig identity which demonstrates the beat phenomenon works for any pair of frequencies. I think my description of it was not accurate but it's easy enough to derive from the sum and difference identities.

$$\cos(a +b) = \cos a \cos b - \sin a \sin b \\
\cos(a -b) = \cos a \cos b + \sin a \sin b \\
\Rightarrow 2\cos a \cos b = \cos(a - b) + \cos(a + b)$$

Define $c = a + b$ and $d = a - b$ so $a = (c + d)/2$ and $b = (c - d)/2$
$$ \cos c + \cos d = 2 \cos \left ( \frac {c + d}{2} \right ) \cos \left ( \frac {c - d}{2} \right )$$

Pretty sure I did that right. There is no restriction on $c$ and $d$. You'll get a "beat" pattern for any two frequencies.

But you said that just plugging in frequencies into an equation like this is not what you're doing. I gather these frequencies are related or restricted in some way you haven't told us. What is $\omega$? What is $\omega_n$? What equations govern this "mass-spring-damper system" that you are plugging frequencies into? In those equations, how do these two frequencies appear?

The only other thing I can think of is perceptual. If the "beats" are below our audible range, say around 20 Hz, we'll hear the amplitude modulation. Above that, it just registers as a low-frequency sound.

 

[FactChecker]

The beat frequency will be there at the frequency given by the equation. The two frequencies will always mix together that way. The pattern of how the frequencies add and subtract as time goes on is guaranteed.

 

[haruspex]

The trig identity which demonstrates the beat phenomenon works for any pair of frequencies. I think my description of it was not accurate but it's easy enough to derive from the sum and difference identities.

$$\cos(a +b) = \cos a \cos b - \sin a \sin b \\
\cos(a -b) = \cos a \cos b + \sin a \sin b \\
\Rightarrow 2\cos a \cos b = \cos(a - b) + \cos(a + b)$$

Define $c = a + b$ and $d = a - b$ so $a = (c + d)/2$ and $b = (c - d)/2$
$$ \cos c + \cos d = 2 \cos \left ( \frac {c + d}{2} \right ) \cos \left ( \frac {c - d}{2} \right )$$

Pretty sure I did that right. There is no restriction on $c$ and $d$. You'll get a "beat" pattern for any two frequencies.

But you said that just plugging in frequencies into an equation like this is not what you're doing. I gather these frequencies are related or restricted in some way you haven't told us. What is $\omega$? What is $\omega_n$? What equations govern this "mass-spring-damper system" that you are plugging frequencies into? In those equations, how do these two frequencies appear?

The only other thing I can think of is perceptual. If the "beats" are below our audible range, say around 20 Hz, we'll hear the amplitude modulation. Above that, it just registers as a low-frequency sound.

The modulus sign is the key. It means that beats are sensed at frequency |c-d|.
For that to work perceptually, this needs to be rather less than the perceived high frequency, |c+d|/2.
If we say there need to be two complete periods of the high frequency per beat and choose c>d, this gives c+d > 4(c-d).
Plot some examples either side of that constraint and see if it bears out my guess.

 

[willem2]

It is a standard mass-spring-damper system (with \zeta = 0) with an applied force F(t) = F_{0} \cos(\omega t).

That was just the context, but the question was in two parts:
1) How close (quantitatively) do frequencies need to be to show the beat phenomenon?
2) How to think about the question: "If the resonant (angular) frequency is \omega_n = 1, what value of \omega do you think might show beating?"

You won't get any beating with this setup. The system will vibrate with frequency \omega, with a largeramplitude if \omega is closer to \omega_n = 1,

 

[TSny]

The question does seem ambiguous.

You can have some fun here:
https://academo.org/demos/wave-interference-beat-frequency/
You can play around with when you can no longer decipher hearing beats and compare to when you can no longer decipher beats visually in the superposition of the waves (bottom graph). Of course, there is no exact point at which these occur and different people are not likely to agree.