There is an interaction between nanocrystals forming clusters, different from isolated nanocrystals. For the study of the interaction between nanocrystals, a nanocluster containing 3080 nanocrystals is constructed with a diffusion limited aggregation method. The cluster vibrational modes are calculated for the simulated nanocluster with the harmonic approximation and the size of the cluster is sufficiently large to generate a smooth cluster vibrational modes spectrum at low frequencies. The nanocrystal mass and the force constant between nanocrystals are necessary to obtain the actual values of the cluster modes. An internal vibrational mode resonant with a cluster mode is damped due to the coupling between the internal vibrational modes of nanocrystals and the cluster modes. The electronic-relaxation process emitting a damping nanocrystal mode exhibits a nonexponential or an exponential decay rate depending on the ratio of the damping rate, of the internal vibrational mode to the strength of the electron--phonon coupling. This work suggests the damping of internal vibrational modes as a possible mechanism responsible for the suppression of the electronic energy relaxation in Y$_2$O$_3$ nanocrystals observed in optical measurement.