# [Physics] Why choosing for prime numbers eliminates vibration

everyday-lifemathematicsnewtonian-mechanicsresonancevibrations

I have read that the spokes of a car wheel are usually five because, besides other substantial reasons, five being a prime number helps to reduce vibrations.

The same also happens with the numbers of turbine blades and the way a microwave grill is spaced. Prime numbers are always preferred.

Prime numbers are generally used to reduce the magnitude of resonances. These occur in a non-linear multi-frequency system when two of the frequencies $$\omega_1:\omega_2$$ match at a ratio $$p:q$$, where $$p,q$$ are comprime integers.

For simplicity, you can think of a minimal example of such a system as two (non-linear) oscillators that are coupled with a dimensionless strength $$\epsilon$$. Say that oscillator 2 is not oscillating, then the driving from oscillator 1 will generally cause it to oscillate with a kinetic energy proportional to $$\epsilon$$. However, at resonance, the response of oscillator $$2$$ scales as $$\sqrt{\epsilon}$$. Funny things can happen in dissipative systems, where you expect any vibration to be damped, but it turns out that sometimes a sustained resonance occurs that keeps the secondary oscillation "locked" in place for prolonged amounts of time that scale as $$\propto t_{\rm diss}/\sqrt{\epsilon}$$, where $$t_{\rm diss}$$ is the dissipative time scale (but generally the system stays in resonance for a $$\propto t_{\rm diss} \sqrt{\epsilon}$$ time).

On the other hand, this response is also exponentially suppressed by a factor $$\exp\left[-\alpha(|p|+|q|)\right]$$ with $$\alpha$$ some positive number, at least for reasonably smooth couplings between the oscillations. In other words, when the $$p,q$$ in the resonance are large numbers, the resonance is of "high order", and its magnitude will be much smaller and much less bothering. As a rule of thumb, you have to care about resonances with $$|p|+|q|$$ up to 5 or so.

Now consider the example of the wheel with 5 spokes. The contact of the wheel with the road will bring a driving with the rotation frequency $$\Omega$$ into the system. However, the next leading harmonic of the driving will have a frequency $$5\Omega$$ because of the spokes. Now if there are oscillators in the system with proper frequencies $$\omega$$ such that $$\omega/\Omega = p/q$$, then the secondary resonance $$\omega/(5\Omega) = p/(5q)$$ is a much higher-order resonance ($$|p|+|5q|\geq6$$) unless $$p$$ is a multiple of five. But if $$p$$ is a multiple of five, the primary resonance has $$|p|+|q|\geq 6$$ and should already be reasonably weak. So pushing the next harmonic to 5 times the main frequency seems to be a reasonable to choice to somewhat reduce resonant response, and these kinds of rules will apply for any prime.

On the other hand, this is not a big reduction in the resonant response, the only way to muddle resonances out is really to make sure the oscillations in the system are non-linear (their frequency spectrum is non-degenerate, the oscillators are highly anharmonic), they are not likely to match the driving frequencies or each other in low-order ratios ($$1:1$$, $$1:2$$), and that sufficient damping is present.

Consider also the fact that moving a lot of the power of the driving of the system through the wheel to the next harmonic $$5\Omega$$ means essentially making the wheel less round. But there is a lot of reasons why you want to have your wheel round, so I do not believe the power in the next harmonic will really be large.

So, I believe there must be a number of other reasons to choose the number of spokes, and 5 is really a compromise between a number of factors such as manufacturing and robustness as mentioned in some of the other answers here.