I think I have a new way to determine twist center using beam theories - I have a new publication Semi-analytical solution of VABS-based Timoshenko beam model for free vibration of composite structures which provides an analytic solution of beam displacement (which can be used to recover 3D displacement with VABS) solved using a root finding algorithm. Theoretically, I can place the origin of the cross-section at the centroid for convenience, and obtain effective cross-sectional properties which can be used to solve for displacement on the centroid line (a line formed by connecting all the centroids of each cross-section, in my paper, it is called the reference line which can be any line chosen by the analyst) under free-free vibration (as it removes the need to apply constraints on the beam). If the twist center is indeed the centroid, then the torsion mode should show no displacement (u2=u3=0) but only twist (theta1 being nonzero) on the centroid line for all t (time).
Because this method has not been attempted before, I would like to see if you have any feedback on this.
This depends on how you constrain the warping function. If you constrained it to vanish at your reference line, then you will always have the reference line displacement vanish.
Attached is my handwritten elasticity solution for the twist center. The conclusion is that twist center is dependent on the boundary conditions, and therefore twist center is not an intrinsic property of the beam.
Nice to know that you are determined to get to the bottom of a seemingly classic problem.
I am not going to dispute the derivation you have been through as you must have double checked.
However, taking a step back, it is well-known that torsion problem is boundary condition dependent. This was precisely the reason why Saint-Venant came to his famous Saint-Venant principle. The spirit of the principle is that although the solution varies from boundary conditions to boundary conditions, the variations are observed only in a neighbourhood of the boundaries. Sufficiently far away from the boundaries, the solution as obtained by Saint-Venant will be unique and independent of boundary conditions.
I suspect that you conclusion would reproduce the same, i.e. the twist centre might vary from cross-section to cross-section, but only in a limited neighbourhood of a boundary. Sufficiently away from it, the variation would die out eventually. Having said so, I would be more than happy if I can be proven wrong.