I know that time dilation effects everything that moves in relation to everything else that's around the thing that moving.
No it doesn't.
Time dilation is the name for the phenomena that two observers will disagree about the time period between two events when they are in motion relative to each other. The two observers do not affect each other or the events being timed. It is similar to how buildings are smaller the further away from them you are... your distance to the building has no effect on the building itself.
Does that also include vibrating and spinning thing...
Yes.
...and if it does how so?
The same way as it works for any moving thing.
The trick with relativity is to be painfully specific: generalities seldom work.
1. Two observers in motion wrt each other will observe different periods for vibration and rotation for vibrating and rotating objects - that is a time dilation effect.
Two observers in different positions on a vibrating or rotating object will have their clocks disagree a lot.
2. If you have two Einstein synchronized light-clocks, stationary wrt each other, and one is vibrating or rotating, then ... I suppose I could set up a vibration or rotation that affects the tick of one wrt the other. I'm not sure how much of this would be a relativity effect though... indirectly sure. To see what I mean - sketch out two, otherwise identical, light clocks and see different ways you can define "vibration" and "rotation" for each clock.
You can have a go looking, if you like, at the related effect of length contraction as it applies when a non-rotating observer measures the circumference of a rotating disk (when standing at the center of the disk).
But you want to be careful that time dilation is a specific outcome of special relativity and the Lorentz transformation. The best way to cope with more general situations is to use the transformation itself instead ... everything that involves a change of perspective involves this transformation.
