When I was going to my school with my ID card hanging around my neck, it started doing oscillations like a pendulum. I was moving forward and it was oscillating left to right and right to left. What forces are at play here?
[Physics] Why does the ID card oscillate sideways when walking
coupled-oscillatorseveryday-lifenewtonian-mechanicsoscillators
Related Solutions
This is actually an interesting problem in classical mechanics, dating back to Huygens. We'll work with the three variables you define in the question, namely $(x,\theta_1, \theta_2)$. Also we'll set your $m = l = g = 1$ for simplicity.
Kinetic Energy
The position vector of the first pendulum bob is
$$\mathbb{r}_1 = (x+\sin\theta_1, \cos\theta_1)$$
whence we deduce its kinetic energy to be
$$2T_1 = \dot{x}^2 + 2\dot{x}\dot{\theta_1}\cos\theta_1 + \dot{\theta_1}^2$$
We can similarly find the kinetic energy of the second bob.
Finally we must take into account the kinetic energy of the support with mass $M$.
$$2T_3 = M\dot{x}^2$$
It's interesting to keep $M\neq 0$ since different values of $M$ give different behaviour.
Potential Energy
We assume gravity acts on the bobs as usual, producing potential energy terms of form $\cos\theta$. We also assume an elastic potential $kx^2$ pulling the table back to equilibrium. Overall we have
$$V = kx^2 - \cos\theta_1-\cos\theta_2$$
Equations of Motion
One can easily write down the Lagrangian $L=T-V$ and from it deduce the equations of motion
$$\ddot{\theta}+\ddot{x}\cos\theta+\sin\theta-\dot{x}\dot{\theta}\sin{\theta} = 0$$
$$(M+2)\ddot{x}+\ddot{\theta_1}\cos\theta_1+\ddot{\theta_2}\cos\theta_2 - \dot{\theta_1}^2\sin\theta_1-\dot{\theta_2}^2\sin{\theta_2} + kx = 0$$
Interestingly when I put these into Mathematica, there was no synchronization! It turns out the missing ingredient is damping.
Damping
Intuitively the phase difference between the pendulums must drift in a periodic way in the absence of any dissipative effects. Indeed that's what you see when numerically solving the above equations with Mathematica.
Recall that damping is usually modelled as an additive term proportional to the velocity. Adding in such terms for $\theta_1$, $\theta_2$ and $x$ now does produce the desired synchronization behaviour. For my initial conditions and choice of constants we get antiphase locking.
Summary of the Physics
Momentum transfer through a connecting medium coupled with dissipative effects leads to synchronization.
Better Models
To fully model the video mentioned you'd need a forcing term from the escapement mechanism of the metronomes. You can read about such an approach here. See also this Wolfram demonstration and the papers it references.
Towards Chaos
Evidently this setup is nonlinear and so generically displays chaotic behaviour. The study of such systems is particularly important in chemistry and biology. Here is a good introduction.
If you want to play around with this behaviour yourself, here's my rudimentary Mathematica code. Try playing with the constants and initial conditions.
sol = NDSolve[{30 x''[t] + y''[t] Cos[y[t]] + z''[t] Cos[z[t]] -
y'[t]^2 Sin[y[t]] - z'[t]^2 Sin[z[t]] + 30 x[t] + 2 x'[t] == 0,
y''[t] + x''[t] Cos[y[t]] + Sin[y[t]] - x'[t] y'[t] Sin[y[t]] +
0.02 y'[t] == 0,
z''[t] + x''[t] Cos[z[t]] + Sin[z[t]] - x'[t] z'[t] Sin[z[t]] +
0.02 z'[t] == 0, x[0] == 0, x'[0] == 0, y[0] == Pi/10,
y'[0] == 0, z[0.5] == 1, z'[2] == 0}, {x, y, z}, {t, 0, 1000}]
Plot[{Evaluate[y[t] /. sol], Evaluate[z[t] /. sol]}, {t, 0, 250},
PlotRange -> All]
Related questions
Observer stationary
A vehicle passing a stationary vehicle can produce a complex pressure wave
From MEASUREMENT OF THE AERODYNAMIC PRESSURES PRODUCED BY PASSING TRAINS
In this you can see that the stationary vehicle is first pushed away and then sucked back towards the passing vehicle. Lastly the opposite sequence occurs at the tail end of the passing vehicle.
Observer moving
When the observer's vehicle is moving relatively slowly, I don't have an explanation for why any objectively measured forces would be much reduced.
However a slowly moving vehicle is producing a lot more shaking and vibration than a stationary one and this may mask a human observer's perception of additional motions. Our perception may be affected in a non-linear way by a base level of shaking induced by motion. Also, as a driver, I notice bumping and shaking much much less than I do as a passenger, this is perhaps due to my concentration being focussed elsewhere and my anticipation of motion induced by control inputs. If so the former might be a factor in the passing vehicle observation when the observer is actively driving.
Maybe some experimentation with phone accelerometers would produce some useful data?
Best Answer
As humans we oscillate left and right when we walk because we have two legs. You can get a resonance when the length of the cord is such that your pace matches the period of the swing.
(Like pushing a child on a swing a little higher each time they approach you.)
Whilst walking we also oscillate up and down - this can also contribute to driving the resonance.