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A while ago I witnessed a machine that consisted of a mass hanging from a weak spring. One could pull it down a little bit and twist the spring too (torsional pendulum style) and it would oscillate between 2 modes—springing up and down and acting like a torsional pendulum (energy goes back and forth). At a certain initial energy, each mode has a certain energy that doesn't change with time. How does this work (i.e. which equations govern it)? My feeling is that the spring constant k depends somewhat on the torsion of the spring, but I'm not sure.

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  • $\begingroup$ Energy of a spring obeying Hooke's law is $E=\frac{1}{2}kx^2$, where $x$ is spring's displacement. For system many springs which are not transfering energy between each other, energy is $E=\frac{1}{2}(k_1 x_1^2+...+k_n x_n^2)$. To obtain simplest model of interaction between springs, add some polynomial (possibly of order higher than $2$) in variables $x$ to the formula for energy. For example term of the form $\frac{1}{2}gx_1^2 x_2$ will lead to transfer of energy between first and second spring. $\endgroup$ – Blazej Aug 24 '16 at 23:40
  • $\begingroup$ I did do some research, but all I could find was coupled spring systems with arbitrary masses and spring constants. I understand this leads to a coupled eigenvalue DE system, but is this isomorphic to the mechanism I proposed (coupled torsion pendulum and spring)? The many-spring system you proposed does actually have eigenmodes where each state has a constant energy that does not get transferred (ideally). $\endgroup$ – Marcus Aurelius Aug 24 '16 at 23:42
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Energy of a spring obeying Hooke's law is $E=\frac{1}{2}kx^2$, where $x$ is spring's displacement. For system of many springs which are not transfering energy between each other, energy is $E=\frac{1}{2}(k_1 x^2_1+...+k_n x^2_n)$. To obtain simplest model of interaction between springs, add some polynomial (possibly of order higher than $2$) in variables $x$ to the formula for energy. If mixed terms such as $gx_1x_2$ or $gx_1x_2^2$ are introduced, they will cause transfer of energy between different springs in the system. Let us analyze what different terms can do.

Firs consider term of zero order - addition of constant to the energy formula. This changes nothing because only differences of energy matter. Therefore this is immaterial.

If you add linear term such as $gx$ then you will merely shift equilibrium position of the system. To see this just note that $\frac{1}{2}kx^2+gx = \frac{1}{2}k(x+\frac{g}{k})^2 - \frac{g^2}{2k}$, so this system has the same dynamics as previous one but with $x$ variable replaced by $x+\frac{g}{k}$. So equilibrium position is shifted to $- \frac{g}{k}$.

Quadratic terms such as $gx_1x_2$ will introduce genuinely new action and can be used to model coupling between different springs. If you add such term eigenmodes of the system will change and both eigenmodes will involve oscillations of both springs. Therefore if you initially disturb only one of the springs you will observe that second will start moving. This is because the initial configuration (only one spring moving) is necessarily a mixture of both eigenmodes.

If you want to model more interesting phenomena, such as dependece of one spring's $k$ on other spring's discplacement you need higher order terms. For example term of the form $\frac{1}{2}gx_1x_2^2$ with $g$ small can be interpreted by saying that second spring constant depends on displacement of first spring and in fact it is equal to $k_2+gx_1$. Systems with cubic and higher order interactions are nonlinear and therefore notion of eigenmodes loses meaning. If at first approximation you neglect higher order terms and then reintroduce them with perturbative techniques you will see that effect of nonlinearities is to transfer the energy between different modes of the linearized system.

Remark: If you don't understand some of the maths in this post (say, eigenmodes) then you need to study physics of linear systems such as harmonic oscillators. This is explained in many introductory classical mechanics books. Most likely also in basic books about differential equations.

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