During the mechanical loading and unloading process, Tb--Dy--Fe giant magnetostrictive materials can dissipate a mass of elastic energy due to the irreversible movements of non--180° domain walls, which is of interest to be applied in passive damping control systems. The magnetomechanical damping capacity of Tb--Dy--Fe compound is strongly sensitive to the stress magnitude as well as the external magnetic fields. As a new member of the Tb--Dy--Fe family, quaternary Tb_0.36Dy_0.64(Fe_0.85Co_0.15)₂compound has been developed as a good candidate in wide operating \temperature range applications. In order to realize the application of Tb_0.36Dy_0.64(Fe_0.85Co_0.15)₂ compound in passive damping control system, it is important to systemically investigate its damping capacity under coupled magnetomechanical loadings. In the present work, <110> oriented Tb_0.36Dy_0.64(Fe_0.85Co_0.15)₂ crystal was prepared with a growth velocity of 480 mm/h by zone melting directional solidification method. The damping capacity was studied by quasi--static stress--strain measurements under a series of constant magnetic fields up to 0.325 T. Stress ranges from 0 to -10, -30 and -50 MPa were used at room temperature. The results show that maximum damping capacity (Δ W/W) is obtained at zero field. Under certain stress amplitude σ_m, Δ W/W decreases with the increase of magnetic field. A critical magnetic field exists in the damping capacity--magnetic field (Δ W/W--H) curves, and seems independent on the stress magnitude. Under coupled magnetic--stress loadings, the magnetostriction--magnetization curves were measured to analyze the switching process of domains and movements of domain walls, by which an explanation on the variation of damping capacity was given.