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 single degree-of-freedom 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 first 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., 2DOFs TMD and 3DOFs 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 (1R2 T) DOFs is designed by embodying the geometric constraint patterns by flexible beams, and changeable elastic elements are incorporated. Experiments show that the first, second, and third bending modes of the cantilever beam are suppressed by 80.0%, 67.5%, and 61.2%, respectively, by the 3DOFs TMD for multiple modes suppression.