Angular momentum of a solid cylinder rotating around axis

1. The problem statement, all variables and given/known data

I’ll provide picture for clearer understanding. The solid cylinder of mass ##m## and radius ##r## revolves about its geometric axis at an angular rate ##p## rad/s. Simultaneously, the bracket and shaft revolve about x-axis at the rate ##\omega## rad/s. Determine the angular momentum about O.

3. The attempt at a solution

Due to symmetry it follows that ##I_{xy}=I_{xz}=I_{yz}=0## so the angular momentum reduces to ##\boldsymbol{L_{O}}=I_{xx}\omega \boldsymbol{\hat{i}}+I_{yy}p\boldsymbol{\hat{j}}##.

Further thoughts: If the ##mh^2## is removed, does not that calculate the rotation of the cylinder IF the geometrical axis coincide with the ##y##-axis ??

I have the calculated ##I_{xx}## correctly but ##I_{yy}## is incorrect.

##I_{yy}=\overline{I}_{yy}+md^2=\frac{1}{2}mr^2+mh^2## but there shound NOT be an ##mh^2## ….and I want to understand why

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