nrp wrote:
I was in a 172M club that had a bad first order shake appear after an
engine overhaul. It turned out the prop had to be indexed to 45
degrees @ TDC to get rid of it. The orientation requirements are in
the Cessna manual, but there is no explanation.

As an ME with some background in dynamics, this is the only reason I
can see for this:

All 4 cylinder horizontally opposed engines with offset (in contrast to
directly opposite) cylinders will have a second harmonic yaw motion
about their CG due to the short connecting rods. Without a double
crank speed counterbalance shaft system, there is no way to eliminate
this. This causes a yaw vibration or swinging of the engine at 80 Hz
at 2400 rpm for example. I think it is also the reason the vibration
environment on the front of a Lycoming engine is so severe.

I think this is called a "rocking couple".

It's an issue with aftermarket Harley engines designed to use offset
cylinders so as not to need female (fork) rods. When Colombo laid out
the "short block" Ferrari V12 he wanted to have the bores on centers
and fork/blade rods but Enzo said since Packard did not to so with the
Twin Six it was unnecessary.

A little bit off topic, as always. Certified a/c are stuck with LyCon
but homebuilders who use 'em are idiots. They belong with the OXX-6,
Pobjoy, Menasco, Hirth, Siemens-Halske, Guiberson and Caminez on a
museum wall.