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- 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

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

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