The reported anomalies of the proton mean kinetic energy, Ke(H), in nanoconfined water, as measured by deep inelastic neutron scattering (DINS), constitute a longstanding problem related to proton dynamics in hydrogen-bonded systems. A considerable number of theoretical attempts to explain these anomalies have failed. The mean vibrational density of states (VDOS) of protons in water nanoconfined inside single wall carbon nanotubes (SWCNTs) is calculated as a function of temperature and SWCNT diameter, D-CNT, by classical molecular dynamics (MD) simulation using the TIP4P-2005f water model. The calculated VDOS are utilized for deducing the mean kinetic energy of the water protons, Ke(H), by treating each phonon state as a harmonic oscillator. The calculation depicts a strong confinement effect as reflected in the drop of the value of Ke(H) at 5 K for D-CNT < 12 angstrom, while absent for larger diameters. The results also reveal very significant blue and red shifts of the stretching and bending modes, respectively, compared to those in bulk ice, in agreement with experiment.