  We consider generalized gradient systems with rate-independent and
rate-dependent dissipation potentials. We provide a general framework for
performing a vanishing-viscosity limit leading to the notion of parametrized
and true Balanced-Viscosity solutions that include a precise description of the
jump behavior developing in this limit. Distinguishing an elastic variable $u$
having a viscous damping with relaxation time $\varepsilon^\alpha$ and an
internal variable $z$ with relaxation time $\varepsilon$ we obtain different
limits for the three cases $\alpha \in (0,1)$, $\alpha=1$ and $\alpha>1$. An
application to a delamination problem shows that the theory is general enough
to treat nontrivial models in continuum mechanics.
