The squeeze film force in a circular Newtonian squeeze film has been theoretically predicted by using the elliptical velocity profile assumption in the squeeze film by three different approximation methods. As examples, the numerical results for the sinusoidal squeeze motion, constant velocity squeezing state, and constant force squeezing state have been obtained and the results have been found to be in good agreement with those obtained using experimental test coefficients predicted by the spectral analysis techniques for Newtonian circular squeeze film geometry. The validity of applying the energy integral method (EIM) or the successive approximation method (SAM) has been justified and the effectiveness of EIM or SAM in predicting squeeze film force using the elliptical velocity profile assumption in the squeeze film for large-amplitude motion has been demonstrated.