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

Deformation of algebroid bracket of differential forms and Poisson manifold