Pair of interacting systems, driven coupled harmonic oscillators

[ssqq]

TL;DR Summary

I am working with a pair of systems, each of which is a system of damped, driven, coupled harmonic oscillators, and I am trying to figure out what parameters—if any—could result in each system resonating with a different frequency.

Question:

I am working with a pair of systems, each of which is a system of damped, driven, coupled harmonic oscillators, and I am trying to figure out what parameters—if any—could result in each system resonating with a different frequency.

I’m wondering if anyone here has any intuitions regarding whether this would be possible. Specifically, for a single system, I am starting with the equations for the classic wall-spring1-mass1-spring2-mass2-spring3-wall system, with damping, so the first-order equations would be:

$$ \dot{x}_{1} = v_1 $$
$$ \dot{v}_{1} = -\beta v_1 – k_1 x_1 – k_2 (x_1-x_2) $$
$$ \dot{x}_{2} = v_2 $$
$$ \dot{v}_{2} = -\beta v_2 – k_1 x_2 – k_2 (x_2-x_1) $$

Where $x_1,x_2$ are the positions of the masses relative to their equilibria

$v_1,v_2$ are the velocities of the masses

$k_1$ is the stiffness of spring1 and spring3

$k_2$ is the stiffness of spring2

$\beta$ is the damping coefficient

mass = 1 for all masses

From what I understand, this system has two modes of oscillation, with different frequencies. Now say that I add a driving force that has two or more frequency components, e.g.:

$D(t) = sin(wd_1 2\pi t) + sin(wd_2 2\pi t)$, where wd1,wd2 are the frequencies

So the equations become:
$$ \dot{x}_{1} = v_1 $$
$$ \dot{v}_{1} = -\beta v_1 – k_1 x_1 – k_2 (x_1-x_2) + D(t) $$
$$ \dot{x}_{2} = v_2 $$
$$ \dot{v}_{2} = -\beta v_2 – k_1 x_2 – k_2 (x_2-x_1) + D(t) $$

As far as I understand, if one of the frequencies of the driving force is near to the frequency of a resonant mode of the system, that mode will have high amplitude in the oscillation of the system (although this would depend on the initial conditions).

Now say that I have two identical copies of the above system: Xa and Xb. What I want to do is couple them in a way such that, one of the frequency components of the driving force D(t) causes Xa to oscillate in one of its harmonic modes, and the other frequency component of D(t) causes Xb to oscillate in the other harmonic mode. The equation I have for this introduces coupling terms:

The force exerted by the masses of Xa on Xb:
$$ C_{ab} = x_{a1} + x_{a2} $$

The force exerted by the masses of Xa on Xb:
$$ C_{ba} = x_{b1} + x_{b2} $$

So the whole system is now:
$$ \dot{x}_{a1} = v_{a1} $$
$$ \dot{v}_{a1} = -\beta v_{a1} – k_1 x_{a1} – k_2 (x_{a1}-x_{a2}) + D(t) + C_{ba} $$
$$ \dot{x}_{a2} = va_2 $$
$$ \dot{v}_{a2} = -\beta v_{a2} – k_1 x_{a2} – k_2 (x_{a2}-x_{a1}) + D(t) + C_{ba} $$
$$ \dot{x}_{b1} = vb_1 $$
$$ \dot{v}_{b1} = -\beta v_{b1} – k_1 x_{b1} – k_2 (x_{b1}-x_{b2}) + D(t) + C_{ab} $$
$$ \dot{x}_{b2} = vb_2 $$
$$ \dot{v}_{b2} = -\beta v_{b2} – k_1 x_{b2} – k_2 (x_{b2}-x_{b1}) + D(t) + C_{ab} $$

My questions are:

1. Is this sensible and is the effect I am looking for possible—one system resonates at one of the harmonic modes because of one frequency component of the drive, and the other resonates at the other mode because of the other frequency component of the drive, and the cause of this is the coupling between the systems Xa and Xb?

2. How would I go about figuring out what parameters achieve the desired effect? (Hopefully not analytically). Do I need to worry about constraining the initial conditions in some way?

3. Are there any examples of this sort of thing online that I could look at? Does anyone have any other insights into this?

Thank you

 

[anuttarasammyak]

Why does $k_3$ not appear in your equation ?

 

[ssqq]

Why does $k_3$ not appear in your equation ?

Because $k_1$ is the stiffness of spring1 and spring3, i.e. $k_3=k_1$--the springs attached to the wall have the same stiffness

 

[BvU]

TL;DR Summary: I am working with a pair of systems, each of which is a system of damped, driven, coupled harmonic oscillators, and I am trying to figure out what parameters—if any—could result in each system resonating with a different frequency.

Now say that I have two identical copies of the above system: Xa and Xb. What I want to do is couple them in a way such that, one of the frequency components of the driving force D(t) causes Xa to oscillate in one of its harmonic modes, and the other frequency component of D(t) causes Xb to oscillate in the other harmonic mode. The equation I have for this introduces coupling terms:

I have a hard time trying to understand how you want to do this: can you provide a sketch, or something ?

In particular: you drive both systems with two frequencies; how would you achieve that one resonates at one frequency and vice versa.

From what I understand, this system has two modes of oscillation, with different frequencies.

Yes, one with $x_1=x_2$ and one with $x_1=-x_2$.

The force exerted by the masses of Xa on Xb:
Cab=xa1+xa2

Which is zero for the resonance mode with $x_1=-x_2$. Probably not what you intend.

As far as I understand, if one of the frequencies of the driving force is near to the frequency of a resonant mode of the system, that mode will have high amplitude in the oscillation of the system (although this would depend on the initial conditions).

With damping present, you can concentrate on the inhomogeneous solution.

$\$

 

[ssqq]

Thank you so much for responding.

Here is a diagram to illustrate what I mean. The two systems are $X_a$ and $X_b$. I changed the spring indices to avoid confusion. I also realized that it didn't make sense to drive both masses in a given system with the same driving force, because that would prevent them from obtaining the anti-phase mode.

I think you're right about the coupling force. If $X_a$ were in the anti-phase mode, then the coupling force $C_{ab}$ (the force exerted by $X_a$ on the component masses of $X_b$) would be zero, so that is no good.

Basically, what I want is an interaction between $X_a$ and $X_b$ that causes the two systems to resonate with different frequency components of $D(t)$. If you have any insights into a simple way to accomplish this, I would appreciate it.

 

[Baluncore]

If you have any insights into a simple way to accomplish this, I would appreciate it.

I think you must consider the flow and storage of energy in the system.

The frequency spectrum of that energy is important, since a tuned resonator may not pass sufficient energy, to drive a different frequency resonator.

A non-linear component, (such as a loose connection), may generate harmonics of a fundamental, but not an independent frequency. Each resonator waveform must therefore be phase-locked to another, or to the driving waveform.

 

[BvU]

Here is a diagram to illustrate what I mean.

Can't say the depiction of $D$ and $C$ is very helpful ...
I suppose $D$ can be realized by moving the wall on one end. But $C$ ?

Basically, what I want is an interaction between Xa and Xb that causes the two systems to resonate with different frequency components of D(t). If you have any insights into a simple way to accomplish this, I would appreciate it.

Other than doing something close to cheating, I don't think you can do this. The $x_1=-x_2$ mode of one system wouldn't do anything to excite the $x_1=x_2$ mode of the other. And vice versa.

And it would be cheating if some controller (a signal processor) filters out the signal with $f_1$ and converts it to a driving force with frequency $f_2$

Do you have some application in mind or is this just a thought experiment, albeit somewhat masochistic ?

$\$

 

[ssqq]

This is supposed to be a phenomenological model of a complex many-body system, not an actual physical system. So D(t) can be whatever I want it to be, and I don't need to care if the coupling force could be implemented in an actual mechanical or electrical system. I am implementing this numerically on a computer.

Regarding the cheating question: the system should evolve toward a stationary state in which each system has a different frequency oscillation. Either system needs to be able to attain either frequency, depending on initial conditions or random noise. The controller seems like cheating to me, and I don't know if it could generalize well to cases of more than two of these systems.