On the statistics of natural frequency splitting for rings with random mass imperfections

This paper considers the statistical distribution of natural frequency splits for an initially perfect ring with different types of random mass imperfection. The analysis used to derive analytical expressions for the natural frequency splits is based on a Rayleigh-Ritz approach, in which it is assumed that the mode shapes of the imperfect rings are identical to those of a perfect ring. The types of random mass imperfection investigated are: (i) random harmonic variations in the mass per unit length around the circumference of the ring; (ii) the attachment of random point masses at random locations on the ring; and (iii) the attachment of random point masses at uniformly spaced positions on the ring. For case (i) it is found that the frequency splits always have a Rayleigh distribution. For case (ii) an expression for the statistical distribution is deduced which tends to a Rayleigh distribution as the number of attached masses increases. For case (iii) it is found that the frequency split distribution is dependent upon the mode considered and the number of attached masses, and that in some situations the frequency splits have a “half Gaussian” (i.e., non-Rayleigh) distribution.