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During my last bike holiday I have happened to ride downwhill on roads which had transversal stripes to mark the approaching of a turn.

The stripes were made with road paint, and were slightly higher than the road plane, so that passing on them makes you feel the bump.

At the speed one can reach while on downhill the repeated "ta-ta-pum" is non comfortable, so I was induced to slow down.

However I was wondering: could it happen that, for some particular speed of the bike, the bump frequency resonates with the frame Eigen frequency leading to potential catastrophic frame failure?

A resonant frequency encountered on a road surface would be for the whole bike system, i.e. frame, wheels, rider etc. The rider is effectively attached to the frame via elements with spring, damping and active control properties (arms and legs). Bumps encountered at a specific frequency might buck the rider off.

Bicycle frames are pretty stiff so any undamped resonant frequencies they have would be relatively high. Certain not at 10 or so Hertz you may get from riding on regularly spaced road bumps.

You assume that all structures fail when excited at resonant frequencies. This is plainly not true, otherwise acoustic musical instruments would self destruct when played. There certainly are structures where resonances can contribute to damaging oscillations of course, but I don't think bicycle frames suffer from this.

There is a phenomenon called Speed wobbles https://en.wikipedia.org/wiki/Speed_wobble that can cause a loss of control of the bicycle. A speed wobble is "a quick (4–10 Hz) oscillation of primarily just the steerable wheel(s) of a vehicle. Initially, the rest of the vehicle remains mostly unaffected, until translated into a vehicle yaw oscillation of increasing amplitude producing loss of control". The frame will not brake, but due to the loss of control a catastrophic crash at high speed could occur.

The question is, whether those strips can induce these oscillations and I do not have an answer for that. There are probably enough perturbations of many frequencies present all the time and the main problem is to induce the instability, no to get the initial perturbation.

You can see a nice video of a bicyclist getting a speed wobble at

Resonance frequency is an unchangeable design feature. But it's rather unlikely that it will destroy the frame, at least one from a reputable builder. I would not make the same statement for a BSO which might suffer from weak welding and poor material.

What you experienced on this stretch of road however might not have been proper frame resonance but rather resonance of the system bike/rider that causes a lot of discomfort. The way to deal with it depends on the type of bicycle, sprung or unsprung. And if it is sprung, adjusting the setting of the dampers will help a lot. After all it is part of their job.

With a rigid frame and fork the resonance frequency is given by the design and the frame material. The only bit that can be changed here would be the tyre size within the limits given by the largest fitting size. And to some extent by the tyre pressure.

There's still the option of going much faster or much slower, as you would do on a corrugated road with a motor vehicle. If you have had that opportunity it means either flying over the ripples at speed or following them with no speed at all.

Also, take a relaxed position on your bike. Don't grab the bar too tightly and lift your bottom slightly to keep more pressure on the pedals with the cranks horizontal.

(Last, but not least the purpose of these stripes is to slow down traffic, unfortunately they tend to cause more discomfort to cyclists and interfere with a safe trajectory than to car drivers who might just be annoyed by the noise.)

Below are a couple of hypothetical graphs of the response of a bike to being acted on by a driving force at a particular frequency.

The unrealistic one on the left is the kind of graph that would lead to the pop-culture scenario where the bike breaks apart due to resonance. The resonance has to be at a pretty low frequency to match the frequency of the bumps in the road. There is very little damping (internal friction in the bike-rider system), and this shows up in two ways: (1) as you'd expect intuitively, the response is higher because energy isn't being dissipated rapidly into heat; (2) less intuitively, it makes the resonance peak very narrow.

The second graph is more like what I would expect for a system like a bike frame coupled to a rider's body. Because the frame is a complicated object, it has many resonant frequencies, not just one, and each one is fairly weak. I expect that the resonant frequencies would be rather high, probably 100-1000 Hz, and higher, because the frame is stiff and light. There is a lot of damping, due to coupling of the frame to the rider's soft squishy body, coupling to the tires, and possibly also due to internal friction in the frame. This damping makes the peaks not very tall, and also makes them wide, so they blend together.

If you want some experimental evidence about this, take your bike and bang on it with your fist in various spots and in various directions. (I guess you want to do this while sitting on it, since your body affects the system with its mass and damping.) If there are resonances in the audio region, you'll hear those frequencies as audible sound. If there are low-frequency resonances, you'll feel those as vibration. You can tell the amount of damping based on how long the ringing sustains. If it rings like a bell, you have low damping. If it's a quick thud or "dink" that immediately drops off, you have high damping.

Even if you assume the worst-case scenario of the left-hand graph, the peak is very narrow. (To make it tall enough to destroy the bike, we have to make damping low enough so that the width becomes very small.) This means that the hazard would occur only at a very, very specific, small range of frequencies. I think if most of us experienced this while riding, we would instinctively slow down a little. That would cause the driving frequency to drop and immediately take us too far from the resonant frequency to excite any significant response.

Designers clearly would not want to make a bike that would vibrate with low damping in response to a vertical force from a bump in the road. Such a bike would oscillate up and down for a long time, which would be annoying. Even in the case of a mountain bike that has shock absorbers, there is probably a ton of damping designed into the shocks in order to avoid just this kind of effect.