Non-uniqueness in cylindrical shells optimization

Abstract

Cylindrical shells are widely used in many structural designs, such as offshore structures, liquid storage tanks, submarine hulls, and airplane hulls. During the optimization process of cylindrical shells, one is faced with a set of very unique challenges. Unlike that of simpler structures such as beams or plates, the modal spectrum of cylindrical shell exhibits very unique characteristics. Mode crossing, uniqueness modal spectrum, and redundancy of modal constraints are just a few of the unique attributes faced during the optimization process. In cylindrical shells, the lowest natural frequency is not necessarily associated with the lowest wave index. In fact, the natural frequencies do not fall in ascending order of the wave index either. Solution of the vibration problem of cylindrical shells also indicates repeated natural frequencies. These modes are referred to as double peak frequencies. Mode shapes associated with each one of the natural frequencies are usually a combination of (i) radial (flexural), (ii) longitudinal (axial), and (iii) circumferential (torsional) modes. When a uniform shell is segmented longitudinally along its axis, the thickness optimization of the segment thicknesses will yield in a segmented shell with varying thickness segments. In this paper, the non-uniqueness in optimum design of cylindrical shells for vibration requirements is presented, and its implications are discussed. Related issues such as the new mode sequence, mode crossing, repeated natural frequencies and stationary modes are also discussed. The numerical results were compared with those obtained analytically. The analytical expressions for the equations of motion for segmented circular cylindrical shells are derived using Donnell–Mushtari. The eigenvalue problem of the analytical solution is then solved using a MATLAB program script to predict the natural frequencies and strain energy distribution for the various mode shapes in the modal spectrum.