  We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the
tangent bundle $T^*M$ to a Poisson manifold $(M,\pi)$ starting from an
algebroid bracket of differential forms. We use these brackets to generate
Poisson structures on the tangent bundle $TM$. Next, in the case when $M$ is
equipped with a bi-Hamiltonian structure $(M,\pi_1, \pi_2)$ we show how to
construct another family of Poisson structures. Moreover we present how to find
Casimir functions for those structures and we discuss some particular examples.
