Free vibrations of freely supported oval cylinders

Abstract

A study was made of the free vibration frequencies and mode shapes for freely supported oval cylindrical shells. Cross section curvatures were expressed in terms of a single eccentricity parameter that allowed a wide range of doubly symmetric ovals to be studied. Kinematic equations employing both the Love and the Donnell assumptions from thin shell theory were used in this study and results of the two formulations were compared. Little difference was observed between the results obtained from the two theories for a wide range of shell configurations. Comparisons were also made between the results obtained from this study and those from two previous approximate analyses. It was found that one of the approximate analyses (a Rayleigh-Ritz technique) was quite accurate for all ranges of eccentricities studied. The other approximate analysis (a perturbation technique) was found to be reliable for ovals with eccentricities in the range ( -0.5 < € < 0.5). A study was also made to determine the effects of eccentricity of oval cross sections. The frequencies and mode shapes were found to vary significantly with increasing eccentricities. Irregularities in the frequency vs wave-number curves and a localized "cupping" in the region near the minimum frequency were observed. In-plane inertias were retained yielding the expected three frequencies for each combination of longitudinal and circumferential wave numbers. However, unlike the unstiffened circular cylinder, more than one set of three natural frequencies and associated mode shapes were found for some combinations of longitudinal and circumferential wave numbers. However, although the wave numbers (i.e., number of crossings) were the same in these cases, the wave shapes were obviously different. Nomenclature 2a,2b-minor and major axes of shell, respectively Amn,B m n,C mn = displacement constants e x ,e s ,e x s = axial, circumferential, and shear strain components, respectively E -Young's modulus h = shell thickness k -number of terms in circumferential direction for convergence L =1, for Love's equations; = 0, for Donnell's equations L S ,L X = circumferential and longitudinal length of the shell, respectively m,n = indices on displacement summation n -number of full circumferential waves r = radius of curvature 7*0 = mean radius of curvature, L s /2w= frequency matrix = (1 + M )/2 (3 = L S /L X 7 = (1 -M)/2 $/ = Kronecker delta € = eccentricity parameter (|e| < 1) i)= nondimensional x coordinate, x/L x \ -circular frequency of a noncircular cylindrical shell /x = Poisson's ratio £ = nondimensional s coordinate, -s/L 8

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“…The nonuniform shell curvature associated with a noncircular cross section is represented by using trigonometric series for the coordinates of an oval cross-section shell reference surface (1958). The aspect ratio, or out-of-roundness, of the cross section is represented in the analysis using an eccentricity parameter introduced by [Romano and Kempner 1962] and later used by [Culberson and Boyd 1971;Chen and Kempner 1976]. This parameter is defined in the subsequent section, and the aspect ratio, related to the eccentricity parameter, represents the ratio of the minor axis to the major axis.…”

A new vibration approach of an elastic oval cylindrical shell with varying circumferential thickness

“…The vibration problems of cylindrical shells with an oval profile are rather limited, which is due to the difficulty introduced in governing equations of motion. Some researchers conducted their studies to such problems with constant or variable thickness, but they have used a simple form of curvature radius -one eccentricity parameter (such as Chen and Kempner, 1978;Culberson and Boyd, 1971;Ganapathi et al, 2003;Koumousis and Armenakas, 1983;Kumar and Singh, 1991;Soldatos and Tzivanidis, 1982). A treatise on the use of the transfer matrix approach for mechanical science problems is presented by Tesar and Fillo (1988).…”

Simplified equations and solutions for the free vibration of an orthotropic oval cylindrical shell with variable thickness

“…The vibration problems of the cylindrical shells of the oval profile are rather limited and are due to the difficulty introduced in governing equations of motion. Some researchers conducted their studies using different methods and theories to such problems with constant or variable thickness such as . A treatise on the use of the transfer matrix approach for mechanical science problems was presented by Tesar and Fillo .…”