I think this is a very basic question, but relativity is always not intuitive to me. When we say that time goes slow when moving (viewed by another inertial coordinate system), we just assume that there is a clock to measure time.
However, everyday clocks are mostly mechanical (like quartz clock). I know that quartz clock works using piezo-electric effect, so it's a tuning fork that vibrates when you apply electricity. So it's a mechanical effect, and the accurately tuned frequency depends on many things, among them specifically length of the fork. (Tuning fork is a two cantilevers basically)
So what seems confusing to me is that the length of the fork should be affected by relativity itself, (length contraction) and it would depend on the direction the clock moving. Is this effect real? If so, is it same as time dilation I've learned? But it seems to me that cannot be true, because time dilation is independent of moving direction but this seems not. I know that SR was tested many times, and I've not heard that they used special clocks (like atomic) in the experiments.