The multi-degrees-of-freedom (MDOF) tuned mass damper (TMD) has proven its ability to suppress multiple modes of interest, and it possesses less mounting space than multiple SDOF TMDs of equal damping mass. However, it is challenging to implement the exact design of MDOF TMDs having expected vibration modes. The conceptual design of MDOF TMD containing visualized DOFs is firstly presented by the graphical approach, and the visualization of the quantitative relationship between the freedoms and constraints of TMD is attained. Then dynamics modeling is analytically formulated by incorporating experimental data, and optimization of MDOF TMD considering background modes is performed. Two scenarios of MDOF TMD (i.e., two-DOF TMD and three-DOF TMD) are simulated. Vibration suppression of single dominant mode and multiple modes are achieved, corresponding to the case when the primary structure is subjected to wide and narrow band harmonic excitations, respectively. Afterward, a TMD with one rotational and two translational (1R2T) DOFs is designed by embodying the geometric constraint patterns by flexible beams, and changeable elastic elements are incorporated. Experiments show that the 1st, 2nd, and 3rd bending modes of the cantilever beam are suppressed by 80.0%, 67.5%, and 61.2% respectively by the three-DOF TMD for multiple modes suppression.