I am interested into understanding the behaviors of weakly coupled ascillators. Several books I have looked into give some really good insights and I wanted to look into a specific idea I had a bit further.

Most harmonic oscillators setups that I have seen has masses attached by springs in a series. I was wondering how the equations would change (and their behaviors) if you contained a subset of those masses and springs in their own system like a box.

For example, look at this figure with two different setups:

enter image description here

In the top setup $m_1$ and $m_2$ function in their own contained system which is then also influenced by $m_3$. In the second system, $m_1$ and $m_2$ act as one subsystem while $m_3$ and $m_4$ act as another subsystem.

The setup I have for both are as follows but I am not sure if they are correct. Let $a^2$, $b^2$, $c^2$, and $d^2$ be measures of the coupling strength.

System 1:

\begin{align} \frac{d^2 x_1}{dt^2} + a^2 x_1 + b^2(x_1-x_2) + c^2(x_1+x_2-x_3)&=0 \\ \frac{d^2 x_2}{dt^2} + a^2 x_2 + b^2(x_2-x_1) + c^2 (x_2+x_1-x_3)&=0 \\ \frac{d^2 x_3}{dt^2} + a^2 x_3 + c^2(x_3-x_1-x_2)=0 \\ \end{align}

System 2:

\begin{align} \frac{d^2 x_1}{dt^2} + a^2 x_1 + b^2(x_1-x_2) + c^2(x_1+x_2-x_3-x_4))&=0 \\ \frac{d^2 x_2}{dt^2} + a^2 x_2 + b^2(x_2-x_1) + c^2(x_1+x_2-x_3-x_4)&=0 \\ \frac{d^2 x_3}{dt^2} + a^2 x_3 + d^2(x_3-x_4) + c^2(x_3+x_4-x_1-x_2)&=0 \\ \frac{d^2 x_4}{dt^2} + a^2 x_4 + d^2(x_4-x_3) + c^2(x_3+x_4-x_1-x_2)&=0 \\ \end{align}

I am not sure how how close I am to these solutions being correct. Also,, I am looking for a general way to set up hierarchical systems of this nature. If anyone has some insights I would be very appreciative.

  • 1
    $\begingroup$ The Bond-Graph technique gives good support to couple those kind of sub-systems. $\endgroup$
    – Cesareo
    Jul 3, 2022 at 13:01

1 Answer 1


A useful way to think about it is to generalize to a "network" of springs, where the Hookean spring equation $m\ddot{x} =- k x$ is generalized to a vector equation $M \ddot{\vec{x}}=-K{\vec{x}}$ where $M,K$ are matrices that define "generalized" mass and stiffness. Typically, the mass matrix $M$ is diagonal, but the stiffness can couple different masses. In your first example: $$\left[ \begin{array}{ccc} 1,0,0\\0,1,0\\0,0,1\\ \end{array} \right] \left[ \begin{array}{ccc} \ddot{x_1}\\\ddot{x_2}\\\ddot{x_3} \end{array} \right] = -\left[ \begin{array}{ccc} a^2+b^2+c^2&&c^2-b^2&&c^2\\ c^2-b^2&&a^2+b^2+c^2&&-c^2\\ -c^2&&-c^2&&a^2+c^2 \end{array} \right] \left[ \begin{array}{ccc} x_1\\x_2\\x_3 \end{array} \right]$$

This allows you to identify blocks of "almost" decoupled spring systems and decide if treating them as subsystems is helpful.


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