For certain values of $k$ it is known that $\mbox{Diff}(S^k)$ is not homotopy equivalent to $O(k+1)$. So there are sphere bundles that do not arise from vector bundles.


Since I've never (knowingly) come across such a sphere bundle I'm interested in seeing some enlightening examples of sphere bundles which do not come from vector bundles.


Thank you for any contribution.



Examples of sphere bundles
As far as I know the only explicitly-described such bundles are in Hatcher's paper:


* Hatcher. Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ, Stanford Calif 1976), Part 1, pp. 321.


I think in Igusa's Higher Franz Reidemeister Torsion book there might also be these examples, although I don't have the book at home with me so I can't check. But that seems likely as Igusa has also developed these examples. 


