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A close up of what Lou's efforts. The now machined area is the very spot that the steady rest jaws touched the very irregular (lumpy) rough round stock while the current rough center-hole was drilled. Why is the current hole not of good quality? Very simply, the bumpy shaft was bouncing back and forth among three basically non-forgiving jaws of the steady rest while the center drill was shoved against the vibrating end of a rotating shaft: the resulting hole is more triangular than truly round.
In this shot, the end of the piece is being supported by a lathe center, held by the tailstock, jabbed rather rudely into the crappy center hole previously explained. As bad as this sounds, the short region that is being machined will be close to perfectly round, so that the shaft shall turn without shaking and bouncing when the Steady rest is returned to this location. When the shaft is supported by the steady rest the second time, the roughly conical center hole can be cleaned up and made true for a good fit against the ball-bearing center that will support the shaft through the rest of the machining process.
Photo by Bill Liebman.