We study the large-time behaviour of the nonlinear oscillator \[ \hskip-20mm m\,x'' + f(x') + k\,x=0\,, \] where m, k>0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case  $f(x')= A\,|x'|^{\alpha-1}x'$  with α real, A>0. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.