Larger IGH cog question
 

I'm looking to lower gear ratios on my Nexus Inter3 by switching to a larger rear cog.

Currently running 42/16 with the rear axle about halfway in the drop outs. Can I go up to 19T without needing a longer chain?

Would also consider smaller chain ring - crank is 104 BCD.

.
...not enough information. Whether you can get by with the same chain depends on the length of your rear dropouts, and where the axle sits right now with respect to any further adjustment. It's certainly possible. Anyway, new chains are pretty cheap.

Treat yourself if you have to. :)

Forgot to mention, bike has disc brakes - not sure if enough adjustment available on the calipers.
I will be ordering parts as LBS only has 16T cog in stock and short 1/8" chains for BMX.

Simple math.

The hub moves forward roughly 1/8" per added tooth, likewise it moves back by the same amount with smaller sprockets.

Here's the reasoning.

Each tooth adds 1/2" to the circumference of the sprocket. The chain wraps half way so you need 1/4" more chain. That's divided between the upper and lower loops, so the hub has to move 1/8" to make up the difference.

So, measure the amount of usable room and make a decision. If you have the dough, you can reduce the chainring and increase the rear by the same # of teeth, and the hub will barely move. (you still get a bit of movement because the upper and lower loops aren't parallel, which screws up my simple calculation slightly)

You could just get a half-link. My bike has about the same GIs.
But anyway. 42/19T is a really poor choice at about 44. 60, 82 GI. 1st will wizout at 8 or 9 mph and 2nd at 16 or 17. An 18T would be far better with the gear change at 10 mph, so you would be using 2nd most of the time. I go at 13 mph into the wind a lot.
I am too often in less efficient 1st gear myself, up to 14 mph. My cranks are also too short on this bike, which makes a tougher upshift. My SA 5w started at 46.5 GI also, not bad at all.

Considering the Nexus Inter3 is 0.733:1, 1:1, 1.364:1 with 26" wheel - my current GI are 50, 68, 93; making 3rd too high for anything other than downhill or a strong tailwind. My goal is to get 3rd close to 76 GI (same as my single speed) to use for most riding. Second will be for short uphill stretches, bucking a strong head wind and most winter riding conditions; 1st for the 6 to 8 blocks of 9% grade hill I climb everyday on my commute to work and winter riding in loose snow with extreme wind chills.
I've also made several test rides with my 26" hardtail (heavier bike with suspension fork) using only the 42T chainring and moving through the 14/34 freewheel and am confident the 19T cog will be the best good compromise for my purpose.

Thanks FBinNY.
I hadn't really thought about the "circle math" and my 9th grade students just finished that unit -LOL.
Most of my calculations were on the ratios and GI I'm trying to acheive.

BTW- I didn't mention it earlier because I believe you;re still well within the limits, but anyone reading this needs to be aware that IGH hubs may have low gear ratio limits.

As you lower the primary gearing you increase your leverage, and therefore the maximum input torque. These are just guideline numbers because obviously rider strength varies. But heavier or stronger riders, and those riding steep hills need to be mindful of these limits, and even apply them more conservatively based on your conditions.

Good reminder - rule of thumb I remember from the 60's (Sturmey Archer days) was don't go lower than 2:1 drive ratio.