Keywords: 
• Earth rotation equation /
• nutation /
• polar motion

Abstract: In Earth rotation theory, three reference frames are included, i.e.the spatial(inertial) frame, the terrestrial frame and the nutation frame.The nutation frame defines a celestial ephemeris pole.The motion of the celestial ephemeris pole with respect to the spatial frame is called nutation (including precession), and that with respect to the terrestrial frame is called polar motion.The rotation of the Earth can normally be classified to two categories.One is the forced rotation due to the torque of the luni-solar attraction and other kinds of forces.And the other is the free rotation which is the proper mode solution of the force free Earth rotation equations.Take the example of rigid Earth rotation problem.If one studies the Earth rotation using the Eulerien equations without introducing the nutation frame, one can find the components of forced rotation as well as the component of free rotation like Eulerien wobble, but one can not find the Tilt-Over-Mode.Smith (1977, 1980) presented a mode theory for the free rotation of the Earth.In his theory, a nutation frame is introduced, which is assumed rotating uniformly with respect to the spatial frame, and only the components of free rotation are obtained, including the Eulerien wobble and the Tilt-Over-Mode.In this paper the theory of Smith (1977, 1980) is generalized to include both forced and free rotations by introducing an arbitrarily rotating nutation frame.The Eulerien equations are transformed to include as unknowns the angular velocity of the nutation frame with respect to the spatial frame, which represents the nutation, and the angles defining the orientation of the Earth with respect to the nutation frame, which represents the polar motion.Together with the definition of the nutation frame (as the definition of the nutation frame is arbitrary at some extent), one can solve simultaneously for forced and free nutation and polar motion.As demonstrative examples, studies of nutation and polar motion are made by assuming the nutation axis to be the Earth's figure axis, rotation axis and angular momentum axis respectively.And the case of the celestial ephemeris pole is also studied.Particularly, the Tilt-Over-Mode which appears only in mode theory of Earth rotation can also be found in the theory of this paper as well.In summary, this paper gives a new uniformed formulation for the problems of forced and free Earth rotation by introducing an arbitrarily rotating nutation frame.Although no new result is obtained for the phenomenon of Earth rotation, the formulation of the Earth rotation problem of this paper may be advantageous for educational purpose.