Because of the deficiency of fixed kernel in bilinear time-frequency distribution (TFD), i.e. for each mapping, the resulting time-frequency representation is satisfactory only for a limited class of signals, a new adaptive kernel function named the radial parabola kernel (RPK), is proposed. The RPK can adopt the optimizing method to filter cross-terms adaptively according to the signal distribution, obtain good time-frequency resolution, and offer improved TFD for a large class of signals. Compared with traditional fixed -kernel functions, such as Wigner-Ville distribution, Choi-Willams distribution and Cone-kernel distribution, the superiority of the RPK function is obvious. At last, the RPK function is applied to the analysis of vibration signals of bearing, and the result proves the RPK function an effective method in analyzing signals.