Flexible thin-walled elliptical bearing (FTEB) is a new type of bearing, and its elliptical inner and outer rings make its vibration signal exhibit very unique characteristics, which are completely different from ordinary bearing. The experiment results of vibration detection show that the vibration signal of healthy FTEB presents elliptical periodic shocks, while ordinary bearing has no such shocks. The frequency composition of elliptical periodic shocks is determined. Elliptical periodic shocks are great interference to the fault periodic shocks, masking the waveform of fault periodic shocks and the fault characteristic frequency, thus concealing the fact of bearing failure. A new signal separation algorithm, singular value decomposition of adding additional signal (SVDAAS), is proposed to separate elliptical periodic shocks. By adding specific frequencies to the original signal, singular value distribution of the original signal can be changed so that the singular values corresponding to the specific frequencies can be identified and the desired frequency components can be separated by inverse singular value decomposition. SVDAAS can accurately extract single or multiple frequency components distributed in any frequency band. The shock signal separation example of the damaged FTEB shows that SVDAAS accurately separates the elliptical periodic shocks from the complex mixed periodic shocks, while the weak fault periodic shocks are unaffected, so that the fault status of FTEB is revealed. Compared with the popular frequency separation methods, SVDAAS obtains more accurate fault periodic shocks and stronger fault characteristic frequency.