I recently was attempting to shift up a hill and observed that
- The shifting levers on the handlebars displayed significant resistance to my pushing them.
- Despite several rotations of the crank, the bicycle had not shifted gears.
Googling around, I've seen people claim that pedaling under power increases chain tension and that this chain tension will affect how easily the chain moves between gears.
Intuitively, this does not make sense to me, as how easily the chain moves orthogonal to the direction of travel (i.e., the ease of shifting) would seem to be independent of the chain tensions parallel to the direction of travel. This seems to imply there's a causal relationship between the relative magnitudes of these forces.
Moreover, given that the chain itself is the same length regardless of the power transmitted, it's unclear to me how pedaling increases chain tension. Is it that one part of the chain becomes more tense, and the other becomes less tense? A free-body diagram type explanation would be very helpful.
How might one understand or model the shifting action and mechanism in a way that explains why chain tension affects shifting?