Vibration of spinning ring-stiffened thin cylindrical shells

Abstract

The vibration characteristics (frequencies and mode shapes) of a spinning thin cylindrical shell with multiring stiffeners are investigated. The theory is first derived for the spinning shell and spinning rings, in which the concept of harmonic line load is introduced. The general frequency equation for a spinning ring-stiffened shell is derived. Numerical results for no spinning are compared with those of previous investigations, and precise agreement is obtained. For the spinning case, the results, obtained from the present approach, show that the ring stiffeners stiffen only the modes of higher circumferential wave number n. The effects of rotation on the reinforced shell and the phenomena of traveling mode shapes are also illustrated in detail in the paper. PINNING shell elements have been widely used in me- chanical, aeronautical, and marine engineering. The spin- ning of structures generates the Coriolis effects, and in conse- quence, the frequencies bifurcate and the modes become time dependent.1'2 Ring elements are frequently used to stiffen the deflective response of shell. The vibration behavior of the shell-rings structures is, hence, of practical importance. Many researches have been devoted to the dynamic analysis of sta- tionary cylindrical shell with ring stiffeners. Nevertheless, the analysis of spinning cylindrical shell-rings structures has not been seen in the literature. Because of Coriolis effects, the vibration behavior of the spinning shell-rings structure may differ significantly from the stationary one. The purpose of this research is to develop an analysis technique for joined spinning shell-rings structures and to investigate the effects of spinning on the structures' characteristics. The major difficulty arising from the combination of shell and ring elements is the absence of exact shape functions. Wah and Hu3 sketched the natural modes of the cylindrical shell reinforced with uniform, evenly spaced ring stiffeners. Garnet and Levy4 investigated the same problem but treated the inter- action between ring and shell as displacement- dependent forces. Al-Najafi and War burton5 employed the finite ring element to a similar problem, where partial experimental re- sults were shown to verify their numerical results. Forsberg6 cut a nonspinning shell-ring into shell and ring segments. Between segments, the compatibility and equilibrium condi- tions were enforced, and the concept of transfer matrix was applied to yield an overall system matrix. His derivation was, in a sense, exact; hence, his results provide a basis for validat- ing our approach for the nonspinning case. Beskos and Gates7 solved for the free and forced responses of ring-stiffened shells with the aid of dynamic stiffness influence coefficients. In fact, they employed the transfer matrix method in the paper. The method, which was introduced by Bishop and Johnson,8 was also developed to study the free vibration of the ring-stiffened shells by Wilken and Soedel9'10 and of the rein- forced plates by Kelkel. 11 The validity of the method in evaluating the natural frequencies for such com- bined structures has been verified in their papers. Note that all of the papers just mentioned dealt with only nonspinning shells. In this paper, the approach is adopted for the analysis of a spinning ring-stiffened cylindrical shell. Unlike the traditional receptances, the authors introduce the concept of line receptance in the paper so that the diffi- culty arising from the traveling modes is overcome. The fre- quency equation of the spinning shell with an arbitrary num- ber of ring stiffeners (equal or unequal spacing) is then derived. Numerical results specialized for the stationary case (0 = 0) are first compared with those available in other investi- gations. Frequencies and mode shapes for up to four ring stiffeners are demonstrated. The effects of stiffeners and spin- ning speed on the frequencies of the reinforced shell are partic- ularly discussed. Mathematic Mode