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Ultrasonic-assisted magnetic abrasive flow machining (UAMAFM) process shows enhanced finishing performance compared to conventional abrasive flow machining (AFM). In this present research paper, mathematical models for MR and Ra have been developed for the UAMAFM process by considering both steady-state and transient phenomena. The external ultrasonic and magnetic field assistance enhanced the velocity and length of contact of active abrasives, calculated from the kinematic analysis. The resultant finishing forces have also been evaluated by considering these external aids. The steady-state material removal per finishing cycle remains constant and depends on the velocity of motion, length of contact, resulting forces, number of active abrasives, and work material hardness. The transient material removal per finishing cycle was calculated in terms of the volume of irregularities present over the work surface, i.e., initial surface roughness. The mathematical model for surface roughness was developed in terms amount of material removed (MR), and initial (Ra0) and critical surface roughness (Racr)?. The predicted values of material removed and surface roughness from developed mathematical models agreed with experimental results with a deviation of 7.80% and 2.44%, respectively. [ABSTRACT FROM AUTHOR]