We studied coherent phonon dynamics in Co-Mo-catalyst grown single-walled carbon nanotubes by using sub-5-fs visible pulses. Vibrational wave-packets corresponding to radial breathing mode for four chiral systems, (6,4), (6,5), (7,5) and (8,3), can be separately observed and achieved exact chirality assignments without ambiguity in coherent phonon spectra. Coherent phonon generation of radial breathing modes is in-depth studied by analyzing the probe energy dependent amplitude profiles, which indicates that the real and imaginary parts of the third-order susceptibility can both contribute to the modulation of the probed difference absorbance.