We propose a scheme that exploits the effect of normal-mode splitting in a three-mode optomechanical system (OMS) to manipulate phonons in a single-excitation level. The three normal modes of the OMS are formed by coupling two mechanical modes to a cavity mode, which is driven by a strong red-detuned pump field via radiation pressure. It is shown that the Landau–Zener dynamics of the three-normal-mode configuration is versatile for quantum information engineering. Explicitly, we show that it is feasible to transfer a single phonon between two mechanical modes with distinct frequencies, and an entangled W state among the three modes can be prepared. The effect of damping is analyzed by numerical simulation, which confirms the validity of our proposal.