In this paper an almost-Poisson structure is constructed for the non-self-adjoint dynamical systems，which can be decomposed into a sum of a Poisson bracket and the other almost-Poisson bracket. The necessary and sufficient condition for the decomposition of the almost-Poisson bracket to be two Poisson ones is obtained. As an application，the almost-Poisson structrue for generalized Chaplygins systems is discussed in the framework of the decomposition theory. It is proved that the almost-Poisson bracket for the systems can be decomposed into the sum of a canonical Poisson bracket and others two noncanonical Poisson brackets in some special case，which is useful to integrate the equations of motion.