The second composite is of somewhat more complex spline object (altered with an envelope) with what appears to me to be set up identically to the first. The orientation of the second object is 90 degrees different from the first, hence the different faces in the flat mapping component. The top of the object took the rivet map fine but the bottom, shown patched into the screenshot, appears black. Since most of this was done by trial and error I'm not sure how I got to these two points, but I did try to replicate what I did on the first working on the second. I'll post screenshots of the cylinder, which seems like it should be much easier to solve if I have a few minutes later.

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