This study explores the dynamics of a strongly overdamped particle situated in a cosine potential when subjected to harmonic excitation. The primary focus is on the existence and stability of the particle’s steady-state solutions. Using the harmonic balance method up to the first order and incorporating a constant bias term, our analytical approach reveals interesting dynamics influenced by the particle’s mass. For a massless particle, steady-state solutions oscillate symmetrically around the potential well’s minima or maxima. However, considering the particle’s mass introduces a significant modification: new equilibrium positions emerge for the oscillation center, linking the extremities of the potential. This leads to a continuous biasing of the oscillation center solely by modulating the amplitude of the harmonic force. Our findings have potential implications for the precise control of particle positioning in various fields, including microscale and nanoscale applications.