# Why choosing for prime numbers eliminates vibration?

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.

• Can you give a citation for this claim? – ACuriousMind Jun 4 at 16:47
• There are 6 spoke car alloys, see performancealloys.com/alloy-wheels/alloy-results/6-spoke-wheels. Then there are aircraft with 4 bladed props and even up to 8 blades, see aerospaceweb.org/question/propulsion/q0039.shtml – user207455 Jun 4 at 16:52
• I bellieve that resonances will not happen with prime numbers involved in a closed boundary, and so where resonances are a nuisance as with unwanted vibrations of wheels , it might help.(sines and cosines have to close the loop) – anna v Jun 4 at 17:55
• for vibrations to happen a sine/cosine solution must exist in the 2π circle of a wheel, which should have the same value at the same φ angle, going around the wheel. If the boundary conditions of the spokes are a prime number The frequencies where this can happen will be much fewer, because at the nodes on the spokes there cannot be closure .this is just a mathematical intuition. If I had an example I would answer :). I may be wrong – anna v Jun 5 at 3:53
• What you claim about turbine blades is wrong. What is really done there is ensuring that the number of blades and stators are coprime, which ensures that the rotors do not line up at the same time with the stators. It appears to me that there is no real reason for spokes to be prime, five or more might me a matter of load distribution. If you look online you can see that there are much more cases of wheels with 10 spokes than those with 5, and by no means is there an exclusive number for this, the choice might be more due to aesthetics and possibly marketing. – S V Jun 12 at 19:46

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.

In trying to answer this question I came across a lot of interesting phenomena related to primes. This is not a very detailed answer but will hopefully I can share the intuition and feel of the concepts involved.

The phenomena we are dealing with is resonance.

In any machine, there are several parts. Each part has some resonant frequency(a natural frequency). Now a noticeable vibration occurs when the magnitude of this oscillation increases. How will that happen? If you have two parts whose resonant frequencies are the same or multiples of each other, when every one part vibrates it sort of hits the other and makes the other part vibrate with a larger amplitude. Therefore slowly the amplitude of oscillation in both parts will increase due to a feedback mechanism and eventually such a large vibration can damage the machine.

The same goes for a wheel. Actually you cannot just look at spokes. There will also be some mechanism underneath which is holding on to the wheel with certain number of extensions. For example for the axle of the car to have good grip on the wheels it must have some elements extending from the axle to lock with the wheel. Now let the number of extensions be n1 and number of spokes be n2. If n1 and n2 are multiples of each other, due to the feedback mechanism discussed, large vibrations could be caused. Hence in general n1 and n2 need to be co-prime and in most cases the numbers 3,5,7 are used. As far as I understand it, co-prime numbers reduce the vibrations because they cancel each other out.