Real life example of the composition of two SHMs with same angular frequency

[Hamiltonian]

Under the topic of simple harmonic motion comes the composition of two SHM's with the same angular frequency, different phase constants, and amplitudes in the same directions and in perpendicular directions.

composition of SHM's in same direction:
say a particle undergoes two SHM's described by the following equations:
$$x_1 = A_1sin(\omega t)$$
$$x_2 = A_2sin(\omega t + \phi)$$
then the resulting SHM is described by the equation:
$$x = x_1 + x_2 = Asin(\omega t + \theta)$$
we can find the values of $A$(amplitude of resulting SHM) and $\theta$(phase constant of resulting SHM) quite easily.
I wanted to know if there were any real-life(by "real-life" I simply mean something with springs and blocks and wedges and what not) examples where these equations could be applied instead of just a hypothetical particle undergoing 2 hypothetical SHM's. the first thing that came to my mind was something like this

(assume the two springs have a different value of spring constant(k) say $k1$ and $k2$)
then the equations of motion are:
$$x_1 = A_1 sin(\sqrt{\frac{k_1}{m}} t)$$
$$x_2 = A_2 sin(\sqrt{\frac{k_2}{m}}t)$$
but I am unsure as two how to (or if I even can) proceed to prove that this the composition of SHM's in the same direction. from the above equations of composition of SHM's the resulting amplitude of the motion should be $A_1+ A_2$

I didn't have any problem with the composition of SHM's in the perpendicular direction.composition of two SHM's

 

[hutchphd]

What you have drawn is just an Harmonic Oscillator with spring constant $k_1+k_2$. Please reconsider your question.

 

[Hamiltonian]

What you have drawn is just an Harmonic Oscillator with spring constant $k_1+k_2$. Please reconsider your question.

I thought maybe it could be considered to be a composition of two separate SHM's but I realize now that's not the case.
so I am still looking for some kind of example of a composition of two SHM's in the same direction(if that's even possible in real life)

 

[hutchphd]

Sure there are. For small oscillations a double pendulum is one that comes to mind. Two masses connected wall-spring-mass-spring-mass are what you had in mind I think. Such arrangements show up (more or less) in auto suspensions for instance.

 

[Hamiltonian]

Sure there are. For small oscillations a double pendulum is one that comes to mind. Two masses connected wall-spring-mass-spring-mass are what you had in mind I think. Such arrangements show up (more or less) in auto suspensions for instance.

I don't see how there is a composition of two SHM's in the same direction in a double pendulum(I assume you mean one pendulum attached to another(its a chaotic system I don't think that it perform SHM for small angles, so it can't be a composition of two SHM's)) now I am pretty confused is wall-spring-mass-spring-wall a composition of SHM's in the same direction or not I tried to prove it in my original post.

 

[hutchphd]

For small oscillations.
Perhaps you need to define "composition " more precisely. I don't know what you are looking for.

 

[Hamiltonian]

For small oscillations.
Perhaps you need to define "composition " more precisely. I don't know what you are looking for.

I am looking for real-life situations where two separate SHM's can be composed into a single SHM.
maybe this composition of two SHM's will help(this is only the math of composing two SHM's with the same angular frequency but I think maybe It will give an idea on as to what I am asking).

 

[hutchphd]

Thank you for the reference. I was assuming this was freely moving particles...perhaps you were also.. think of driven motion instead. This would describe a particle of air as two different colinear sound waves impinged or that of a charged particle subject to two similarly polarized electromagnetic waves. I would say a cork on the water with water waves but the wave motions are actually elliptical...not quite linear. Incidentally I think the text you point to is awful for not motivating this subject better. Hope that helps!

 

[tech99]

I am looking for real-life situations where two separate SHM's can be composed into a single SHM.
maybe this composition of two SHM's will help(this is only the math of composing two SHM's with the same angular frequency but I think maybe It will give an idea on as to what I am asking).

The difficulty with mechanical systems for this is that the two frequencies must be locked together, so basically we have a sine wave plus a harmonic. To do this requires something with a non-linear input/output characteristic. Two resonators alone do not do it (at least in a pefect world). It is easy in the electronic world; for example, to create the second harmonic it is only necessary to pass a sine wave through something with square law characteristic, a diode. The spectrum of the sine wave and its harmonic can then be seen on a spectrum analyser and the wave can be seen on an oscilloscope. It is also very easy to draw the graph showing the addition of two such waves.

 

[Merlin3189]

If the two SHMs have the same frequency, then you just get a constant amplitude at the same frequency with a different phase. And it's not very interesting. You can't tell it's not just a single SHM.

If you want different frequencies, whether harmonically related or not, you could have two driven oscillations such as two turntables, one mounted upon the other.

If the frequencies are not harmonically related, you'll get a complicated pattern, but not (I think) chaotic. If they are harmonically related, you get Lissajous patterns. (A projection of which patterns, onto a straight line, will give your sum of two SHMs.)

IMO the chaotic behaviour of the double pendulum arises because the two interact so that neither is SHM any more. You shouldn't get that problem with driven systems.

If you don't like the simple turntable model, try a column of liquid with two oscillating pumps feeding the base with SHM flows. Then look at the depth in the column.

EDIT: (Actually, I'd quite like to see that with several inputs to approximate triangle or square waves. I think seeing the water move like that would be a much more convincing demo of Fourier series. But You'd probably get cavitation if you tried to get too close to a square wave.)

 

[jrmichler]

Summary:: looking for practical examples of composition of SHM's in the same direction.

Under the topic of simple harmonic motion comes the composition of two SHM's with the same angular frequency

Try a search with search terms tuned mass damper. Lots of good examples in the first page of hits.

Tuned mass dampers have a spring mass system attached to a larger spring mass system. Both spring mass systems are tuned to the same frequency. The resulting system has two masses and two springs, and has two separate natural frequencies. One of those natural frequencies is higher than the original natural frequency, the other is lower. The difference between the two natural frequencies is proportional to the ratio of the tuned mass to the mass of the original system. The springs and masses are connected as follows, where the primary system is K1 and M1, and the tuned mass is K2 and M2:

Since the resulting system has two natural frequencies, the phase relation between the two masses is variable. Search beat frequency to learn more. Example of beat frequency from the Wikipedia article:

This is exactly the response of a system with an undamped tuned mass damper.

 

[Hamiltonian]

Try a search with search terms tuned mass damper. Lots of good examples in the first page of hits.

Tuned mass dampers have a spring mass system attached to a larger spring mass system. Both spring mass systems are tuned to the same frequency. The resulting system has two masses and two springs, and has two separate natural frequencies. One of those natural frequencies is higher than the original natural frequency, the other is lower. The difference between the two natural frequencies is proportional to the ratio of the tuned mass to the mass of the original system. The springs and masses are connected as follows, where the primary system is K1 and M1, and the tuned mass is K2 and M2:
View attachment 275940

In this scenario(wall-spring-block-spring-block) I think there is no straightforward composition of two SHM's and to solve for the amplitude and phase of the oscillations produced is quite complicated.

I think what conclusion I have come to is that there are no mechanical systems where two SHM's can directly be composed into a single one(unless that's possible in a tuned mass damper but the equations of motion of the two blocks are quite complicated and I am unable to see how we could compose two SHM's in this case maybe if someone could set up the equations of motion(I mean for $x(t)$ ))
I have not yet studied oscillations in electrical circuits yet but from what @tech99 said I think the concept of composition of two SHM's in the same direction will be used in oscillations in an electrical circuit.

The reason why I asked if there were any mechanical systems that could be considered to be a composition of two different SHM's was because the process of composing two SHM's is stated under the mechanics part of my textbook and I thought maybe it could be applied to some kind of mechanical systems. All the problems based on this section never mention any real system, they simply just mention the equations of two SHM's($x(t)$) and ask questions related to composing them into a single one.

if someone could help compose the equations of motion of a spring-mass damper by using the method mentioned here composition of two SHM's then that would be amazing!

 

[tech99]

An example of a mechanical machine which adds a fundamantal and harmonics (or any other wave) is the Kelvin Tide Calculator. It can add the various waves by passing a string over a sequence of pulleys, each of which goes up and down in accordance with one of the waves. https://www.mt-oceanography.info/science+society/lectures/illustrations/lecture32/tidepredicter.html

 

[RajanGidwani]

Fairly easy.
You can imagine yourself in a swing in a boat.
You get two SHMs , one from swing and another from boat.
If both are in same direction, you get much higher amplitude. Whereas if both are out of phase, you merely get a movement. All Imagination. Sorry about that

 

[Hamiltonian]

Fairly easy.
You can imagine yourself in a swing in a boat.
You get two SHMs , one from swing and another from boat.
If both are in same direction, you get much higher amplitude. Whereas if both are out of phase, you merely get a movement. All Imagination. Sorry about that

at first glance I thought this made sense, the boat will bob up and down and will result in vertical SHM and the swing for small angles will result in horizontal SHM and hence we can compose the two SHM's into one, but won't the swing also act on the boat and hence disrupting its simple vertical motion thereby making the problem much more complicated.
Hence I think I will still stand by my earlier conclusion.

I think what conclusion I have come to is that there are no mechanical systems where two SHM's can directly be composed into a single one

 

[berkeman]

the process of composing two SHM's is stated under the mechanics part of my textbook and I thought maybe it could be applied to some kind of mechanical systems. All the problems based on this section never mention any real system, they simply just mention the equations of two SHM's() and ask questions related to composing them into a single one.

My initial thought was multipath EM reception, like with WiFi, Bluetooth, etc. You get a direct path from the transmitter to your receiving antenna, as well as one or more reflected path signals. When those multiple signals are received at your antenna, they have (basically) the same frequency but different phases (and maybe polarizations). They can interfere constructively or destructively, depending on the phase relationships.

A mechanical analogy is water waves (gravity waves) that arrive via a direct path from the source and via a reflected path from some obstacle. Is that an example of what you were looking for?

(sorry if this was suggested earlier in the thread -- I did not see it mentioned when I skimmed the thread)

 

[Hamiltonian]

A mechanical analogy is water waves (gravity waves) that arrive via a direct path from the source and via a reflected path from some obstacle. Is that an example of what you were looking for?

I posted the above question when I had just studied SHM and had learned about Lissajous curves, they looked visually appealing and they were formed by the superposition of multiple SHM's. I wondered if there could be a system which would result in a trajectory that resulted in that of a Lissajous curve. But I don't think there is any physical setup that could lead to such a beautiful trajectory. (Although in my original post I only mentioned 1D motion I didn't bring up Lissajous curves(those would require two SHM's orthogonal to each other superposed))

I hadn't studied about waves back then, now I know many examples (sound waves, matter waves, EM waves) in which we superpose them and they interfere constructively or destructively. I think that stuff with superposing SHM's was just to get us prepared for superposing waves, as when two harmonic waves(maybe those on a string) interfere then we are practically superposing two SHM's for each particle of the string. Hence I think now I know plenty examples where we superpose SHM's.

​

 

[ergospherical]

The general solution to a small oscillations problem, for instance, is a sum of normal modes $\boldsymbol{q}(t) = \displaystyle{\sum_{i}} c_i \boldsymbol{a}_i \cos{(\omega_i t - \alpha_i)}$, but this generally isn't even periodic (unless the ratios of all the $\omega_i$ are rational).

 

[Hamiltonian]

The general solution to a small oscillations problem, for instance, is a sum of normal modes $\boldsymbol{q}(t) = \displaystyle{\sum_{i}} c_i \boldsymbol{a}_i \cos{(\omega_i t - \alpha_i)}$, but this generally isn't even periodic (unless the ratios of all the $\omega_i$ are rational).

I don't know what normal modes are yet
also form what I understand in most cases $q(t)$ won't be periodic but I mentioned that I am only talking about cases where all the oscillations have the same $\omega$.

edit: sorry I think you mentioned that, I think I am a bit confused as to what you are trying to say, that such a physical system(where composition of various SHM's leads to yet another SHM?) is very unlikely to exist?

 

[ergospherical]

No worries, I'm not sure what the question is exactly.

You mentioned you'd looked at Lissajous figures, for instance; two independent oscillators will give a closed algebraic curve in $x_1, x_2$ space if $\omega_1/\omega_2$ is a rational number. If $\omega_1/\omega_2$ is instead irrational, the Lissajous figure will fill the rectangle densely. A sum of normal modes is generally not periodic!

 

[hutchphd]

I built one in junior high similar to the one here:

 

[berkeman]

Lissajous

I built one in junior high similar to the one here:

TIL that I've been mispronouncing that word ever since I learned it in university (I read about them when I was experimenting with an old oscilloscope that I'd purchased).