  We optomechanically measure the vibrations of a nanomechanical system made of
a graphene membrane suspended on a silicon nitride nanoresonator. When probing
the thermal noise of the coupled nanomechanical device, we observe a
significant deviation from the normal mode expansion. It originates from the
heterogeneous character of mechanical dissipation over the spatial extension of
coupled eigenmodes, which violates one of the fundamental prerequisite for
employing this commonly used description of the nanoresonators' thermal noise.
We subsequently measure the local mechanical susceptibility and demonstrate
that the fluctuation-dissipation theorem still holds and permits a proper
evaluation of the thermal noise of the nanomechanical system. Since it
naturally becomes delicate to ensure a good spatial homogeneity at the
nanoscale, this approach is fundamental to correctly describe the thermal noise
of nanomechanical systems which ultimately impact their sensing capacity.
