This paper is concerned with the unsteady flow of a disk performing non-torsional oscillation in its own plane and a Newtonian fluid at infinity while they are initially rotating with the same angular velocity about non-coaxial axes. For a more general study, it is considered that the disk executes non-torsional oscillation along any desired direction in its own plane. An exact solution obtained for the velocity field is compared with a periodic solution presented in order to find the time when the periodic flow starts. A very good agreement is found between the two solutions in the periodic state. It is an interesting result that the x- and y-components of the force per unit area exerted by the fluid on the disk vary in almost opposite direction when the non-torsional oscillation takes place along the eccentricity direction. Further, the change in the y-component of the translational velocity becomes noticeable in this case.