Among the different techniques proposed for damage detection using changes in measured modal parameters, the damage index method was found to be the most effective in an experimental investigation on comparative evaluation of these techniques [1]. The damage index algorithm is based on the observation that the change in modal strain energy of one or more modes was a sensitive indicator of damage. An expression for the index was first developed for linear elastic beam structures [2] and subsequently for plate-type structures [3]. The only modal parameter that is required for using this technique is the mode shape. The method requires only that the mode shapes be normalized consistently, but does not require mass normalized modes. This makes it possible to use this method if ambient excitation tests were used for modal parameter identification.
For beams and plates, only one degree of freedom is required for expressing the damage index. This is the displacement component normal to the beam's neutral plane or the plate's surface. This fact is quite significant because the test measurements are dependent on which degrees of freedom need to be instrumented and recorded. Having to measure only one component greatly simplifies the modal testing aspect. In the case of cylindrical shells, the general expression for the strain energy is a function of all three components of the displacement of the mid-surface. These are the longitudinal, circumferential and radial displacement components. For the most general situation, these three components would have to be measured in tests making the testing effort rather expensive.
However, for thin shells with a length much greater than the radius, certain approximations could be made. Previous work by others on shell theories has justified assuming the hoop strain and shear strain at mid-surface to be zero. This assumption results in relating some of the derivatives of one displacement component to that of another. Furthermore, the assumed form for the mode shapes, previously developed by other investigators, enables expressing all the three displacement components in terms of a single function of the axial co-ordinate. Thus, it is possible to derive an expression for the damage index that requires the measurement of only the radial component of shell vibrations.
2. ๏ค๏ฅ๏ฒ๏ฉ๏ถ๏ก๏ด๏ฉ๏ฏ๏ฎ ๏ฏ๏ฆ ๏ค๏ก๏ญ๏ก๏ง๏ฅ ๏ฉ๏ฎ๏ค๏ฅ๏ธ ๏ฆ๏ฏ๏ฒ ๏ฃ๏น๏ฌ๏ฉ๏ฎ๏ค๏ฒ๏ฉ๏ฃ๏ก๏ฌ ๏ณ๏จ๏ฅ๏ฌ๏ฌ
Consider the free vibrations of a cylindrical shell of mean radius R, uniform thickness h, and length l. It is assumed the shell is thin, i.e., h/R is small. The elastic modulus and the Poisson ratio are denoted E and n, respectively. In the cylindrical co-ordinate system, the displacement components are given by (u, v, w), in the (x, u, r) directions, respectively.