  Several mechanical systems are modeled by the static momentum balance for the
displacement $u$ coupled with a rate-independent flow rule for some internal
variable $z$. We consider a class of abstract systems of ODEs which have the
same structure, albeit in a finite-dimensional setting, and regularize both the
static equation and the rate-independent flow rule by adding viscous
dissipation terms with coefficients $\varepsilon^\alpha$ and $\varepsilon$,
where $0<\varepsilon \ll 1$ and $\alpha>0$ is a fixed parameter. Therefore for
$\alpha \neq 1$ $u$ and $z$ have different relaxation rates.
  We address the vanishing-viscosity analysis as $\varepsilon \downarrow 0$ of
the viscous system. We prove that, up to a subsequence, (reparameterized)
viscous solutions converge to a parameterized curve yielding a Balanced
Viscosity solution to the original rate-independent system, and providing an
accurate description of the system behavior at jumps. We also give a
reformulation of the notion of Balanced Viscosity solution in terms of a system
of subdifferential inclusions, showing that the viscosity in $u$ and the one in
$z$ are involved in the jump dynamics in different ways, according to whether
$\alpha>1$, $\alpha=1$, and $\alpha \in (0,1)$.
