Low-frequency vibration suppression is challenging in practical engineering problems due to the harsh requirement for vibration reduction devices, which requires constant low stiffness over a wide amplitude range. A passive tuned mass damper (TMD) composed of a positive stiffness module (PSM) in parallel with a negative stiffness module (NSM) is proposed, which are implemented by serial double-parallelograms (DP) and parallel-DP, respectively. The PSM has a large deflection range of constant stiffness for a given beam length, while the NSM offers negative stiffness within a certain deflection range when applied with axial load above the critical threshold. Based on the closed-form modeling of the stiffness modules using the beam constraint model (BCM), the design and analysis of the PSM and NSM are carried out considering the nonlinearity under large deflections. Afterward, with the structure of TMD implemented, its stiffness characteristics and low-frequency tunability are experimentally validated. Finally, the application on a suspension bridge model shows that a maximum of 29.8-dB vibration reduction of low-frequency mode is attained within the frequency range of interest. The proposed TMD well attenuates the vibrations excited by sweep sinusoidal and harmonic excitations under prespecified threshold levels of acceleration.