I understand the Transmission Error is the displacement error of the gear shafts, but if measured how would the signal over time show and why? if its a sinusoid would that mean that only a few times per cycle the displacement occurs or is it always present?
Transmission error would usually be at the tooth mesh frequency. i.e. if you have N teeth on the gear, then you would see N cycles of oscillation over one revolution. It is probably not a pure sinusoid, depending on your type of gear, but it is periodic. If you just google "dynamic transmission error", you can find many papers with plots of time histories. e.g. https://core.ac.uk/download/pdf/337497.pdf