The rotation induced inhomogeneous problem is non-trivial and inevitable. In this paper a generic framework is developed to investigate the inhomogeneous condensate in a system of fermions under the presence of rotation. It is a self-consistent method basing on a set of relativistic BdG equations solved with typical iteration algorithm. Taking the chiral condensate for example we study rotational effects numerically and discover two inhomogeneous effects, the local rotational suppression effect and centrifugal effect. They may have significant impacts on the phase structure of various kinds of matter. Several systems in different physics branches have been discussed in the paper.Introduction.-Inhomogeneous phenomena emerging in rotating matter have attracted a lot of interest in different branches of physics. In cold atom physics, the rotation would not only induce significant modifications on phase structures[1] but also excite novel inhomogeneous states [2][3][4], such as the topological nontrivial vortex state [5]. In astrophysics, neutron star has kept people a long-last interest on its spinning effect for decades [6,7]. Besides the usual structure studies which related to the phase identification in each layer, pulsar, which rotating rapidly and persistently, shows more mysterious properties, such as the glitch behavior which may indicate the rotational speed distribution among different layers and coupling of them [8][9][10][11]. In the highenergy nuclear physics, motivated by the new record of vorticity on LHC, i.e. around 0.006GeV [12], more studies have been performed to explore both the vorticity distribution [13][14][15][16][17][18] and its impacts on the phase structure of the dense and hot strongly interacting Quantum Chromodynamics(QCD) matter [19][20][21][22][23][24][25][26][27]. According to simulations of transport models [16][17][18] the lifetime of the vorticity is relatively long and the spatial fluctuation is very large, i.e. in the range of 0∼0.2GeV in the early stage of the evolution of quark gluon plasma(QGP) fireball produced in relativistic heavy ion collision(HIC). Hence it is tempting to ask what kind of influence would the inhomogenousness of vorticity bring on the phase structure and the parton transport in QGP. Recently a series of novel chiral transport phenomena, such as chiral magnetic/vortical effects(CME/CVE) [28][29][30][31] and chiral magnetic/vortical wave(CMW/CVW) [32][33][34], are of great interest as potential determinate signals of chiral restoration in QGP. Therefore studying the inhomogeneous rotational effects self-consistently is necessary for characterizing these chiral transport phenomena quantitatively.