The structures determined by a tensor field of $(1,1)$ type of constant rank, with the property $f^3+f=0$, was studied by many authors. R. Miron and Gh. Atanasiu determined the set of all connections compatible with $f$-structures, the integrability of $f$-structures and studied the case of $(f,g)$-structures in [1]. In the present paper we shall consider $f$-structures on the total space $E$ of the vector bundle $\xi=(E,\pi M)$ and we shall find the nonlinear connections $N$ on $E$ so that the tensor field $f$ has a particular form. In this manner $f$-structures of type I and type II are defined and in these cases the compatible $d$-conncctions from general case [2] are determined.