  A linear Weingarten surface in Euclidean space ${\bf R}^3$ is a surface whose
mean curvature $H$ and Gaussian curvature $K$ satisfy a relation of the form
$aH+bK=c$, where $a,b,c\in {\bf R}$. Such a surface is said to be hyperbolic
when $a^2+4bc<0$. In this paper we classify all rotational linear Weingarten
surfaces of hyperbolic type. As a consequence, we obtain a family of complete
hyperbolic linear Weingarten surfaces in ${\bf R}^3$ that consists into
periodic surfaces with self-intersections.
