  We study spin dynamics of a spin-Peierls chain with nearest-neighbor and
next-nearest-neighbor Heisenberg spin exchange interactions together with a
gapped and dispersionless phonon. The dynamical spin correlation function and
phonon excitation spectrum are calculated at zero temperature by using
dynamical density-matrix renormalization-group method. We find a new spin
excitation assisted by non-softening phonon. The excitation is located above
phonon in energy and shows a dispersive feature with strong intensity near the
momentum $\pi$. The phonon excitation spectrum is also influenced by the
spin-phonon interaction. We discuss the possibility of observing the
spin-phonon coupled features in inorganic spin-Peierls compound CuGeO$_3$.
