  A new Thermodynamic Uncertainty Relation (TUR) is derived for systems
described by linearly coupled Langevin equations in the presence of non-linear
frictional forces. In our scheme, the main variable represents the velocity of
a particle, while the other coupled variables describe memory effects which may
arise from strongly correlated degrees of freedom with several time-scales and,
in general, are associated with thermal baths at different temperatures. The
new TUR gives a lower bound for the mean-squared displacement of the position
of the particle, including its asymptotic diffusion coefficient. This bound, in
several examples worked out here, appears to be a good analytical estimate of
the real diffusion coefficient. The new TUR can be also applied in the absence
of any external force (with or without thermal equilibrium between the baths),
a case which usually goes beyond the scope of original TURs. We show
applications to non-linear frictional models with memory, such as the Coulomb
and the Prantdtl-Tomlinson models, usually representative of friction at the
nano-scale and within atomic-force microscopy experiments.
