  This paper focuses in providing a Melnikov-like function that controls the
existence of periodic solutions bifurcating from period annuli present in some
families of the second-order discontinuous differential equation given by
$\ddot{x}+\alpha\; \text{sgn}(x)=\eta x+\varepsilon \;f(t,x,\dot{x})$. This
family has garnered extensive attention from various researchers, especially
when considering specific instances of $f(t,x,\dot{x})$. The interest in
studying this type of differential equation is due to its relevance in modeling
systems with abrupt state changes in both natural and engineering contexts.
