The acoustic vibration modes of spherical composite nanoparticles formed by a core and an arbitrary number of shells are computed using a semianalytic approach based on elastic theory. The modes observed in time-resolved pump-probe experiments are identified in the case of metal-dielectric-based spherical composite nanoparticles, and results are illustrated in the case of Ag@SiO2 core-shell nanoparticles. The presence of a light dielectric shell is shown to only weakly shift the frequency of the dominant mode observed in time-resolved experiments. Moreover, a large impact of the mechanical contact between the different materials forming the particle on the vibrational mode frequency and damping is predicted, offering the possibility of experimentally addressing this parameter.