Abstract The dynamic behaviors of a two-degrees-of-freedom rigid body with vibrating support are studied in this paper. Both analytical and computational results are employed to obtain the characteristics of the system. By using the Lyapunov direct method the conditions of stability of the relative equilibrium position can be determined. The incremental harmonic balance method (IHB) is used to find the stable and unstable periodic solutions for the strongly non-linear system. By applying various numerical results such as phase plane, Poincaré map, time history and power spectrum analysis, a variety of periodic solutions and the phenomena of the chaotic motion is presented. The effects of the changes of parameters in the system can be found in the bifurcation diagrams. Further, the chaotic behavior is verified by using Lyapunov exponents. The modified interpolated cell mapping method (MICM) is used to study the basins of attraction of periodic attractors and the fractal structure. Besides, additions of a constant torque, a periodic torque, addition of dither signals, delayed feedback control, adaptive control, and bang–bang control are used to control the chaos phenomena effectively.