  We present a theoretical description of energy transfer processes between two
noncontact quasi- twodimensional crystals separated by distance a, oscillating
with frequency omega0 and amplitude rho0 , and compare it with the case of two
quasi-twodimensional crystals in uniform parallel motion. We apply the theory
to calculate van der Waals energy and dissipated energy in two oscillating
slabs where each slab consists of a graphene monolayer deposited on SiO2
substrate. The graphene dielectric response is determined from first
principles, and SiO2 surface response is described using empirical local
dielectric function. We studied the modification of vdW attraction as function
of the driving frequency and graphene doping. We propose the idea of
controlling the sticking and unsticking of slabs by tuning the graphene dopings
EF i and driving frequency omega0 . We found simple rho02 dependence of vdW and
dissipated energy. As the Dirac plasmons are the dominant channels through
which the energy between slabs can be transferred, the dissipated power in
equally doped EF1 = EF2 = 0 graphenes shows strong omega0 = 2omegap peak. This
peak is substantially reduceed when graphenes are deposited on SiO2 substrate.
If only one graphene is pristine (EFi = 0) the 2omegap peak disappears. For
larger separations a the phononic losses also become important and the doping
causes shifts, appearance and disappearance of many peaks originating from
resonant coupling between hybridized electronic-phononic excitations in
graphene-substrate slabs.
