# FREE VIBRATION ANALYSIS OF THIN CYLINDRICAL SHELLS SUBJECTED TO INTERNAL PRESSURE AND FINITE ELEMENT ANALYSIS

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2 On the other hand, if a shell vibrates without any external loading it is called free vibration (Ramachandran, 1993). The destructive effect of resonance with nearby oscillating or rotating equipment may be avoided if the fundamental frequencies of the structures are known beforehand. Previous investigations on the vibration of thin-walled cylinders seem to be limited mostly to the unpressurized case. The free vibrations of thin cylinders in the case of negligible bending stiffness have been considered by Rayleigh (Timoshenko, 1940). The solution is relatively simple inclusion of the bending stiffness of the walls of the shells make problem much more complicated. Rayleigh derived an expression for the frequencies of thin cylinders in which the motion of all cross sections was identical. This corresponds to the fundamental axial form for a free ended cylinder. The general equations of flexural vibration of the walls of cylinders were later investigated by Love but no frequency equations for any specific end conditions were given. The frequency equation for a cylinder with freely supported ends was first proposed by Flugge (Flugge, 1934). For freely supported ends, frequency equations based on strain energy relations by Timoshenko were experimentally verified with considerable accuracy (Arnold and Warburton, 1953). The natural frequency of thin walls may actually decrease as the number of circumferential nodes increases due to the proportion of strain energy. The vibration of pressurized cylinder has been discussed by Stern under the hypotheses that the skin vibrates normal to the static position and that the stresses in the vibration modes are effectively equal to the stresses generated by the internal pressure Later Serb has solved the problem for the lowest frequencies for nearly- inextensible vibration modes by Rayleigh method. Rayleigh showed that a great simplification in the analysis of shell vibrations can be achieved if tangential inertia forces can be neglected. It is then necessary to consider only one component of displacement-the transverse, or radial component. Reissner shows that the simplified equations as shallow shell theory that gives an accurate description of the transverse vibration of cylinders provided that the number of circumferential waves, n, is sufficiently large (Reissner, 1955). Reissner developed theoretical models and applied to thin walled circular cylinders subjected to internal pressure in which numbers of circumferential modes are relatively small and by considering different pressure values. The frequency spectra and vibration modes obtained from theoretical models are compared with numerical techniques (Reissner, 1955). 2. MATHEMATICAL MODELING OF THIN CYLINDRICAL SHELLS FOR DYNAMIC CHARACTERISTICS A set of simplifying assumptions that provide a reasonable description of the behavior of thin elastic shells is used to derive the equilibrium equations that are consistent with the assumed displacement field. The assumptions, known as LOVE s first approximations are (a) the thickness h is quite small when compared with other dimensions and also with its radii of curvature. (b) the elastic displacements are small in comparison with the thickness of the shell. (c) the stress normal to the middle surface is negligible. (d) The normal to the middle surface before deformation remains normal even after deformation. The curvilinear coordinate system shown in Fig.1. and Fig.2 represents a thin cylinder with freely supported ends. The x- axis is directed along the generator of the cylinder, y = aθ is measured clockwise in the circumferential direction, and the z-axis is directed inward along the positive normal to the middle surface of the shell. Let u, v, w be the components of the displacement of a point on the middle surface of the shell in the x, y and z directions respectively. The governing equations of vibration of shells are obtained starting from the Hamilton s principle. Fig.1. Coordinate system The basic assumptions of Love s first approximation are used. To account for the effect of internal pressure, the interaction of the membrane stresses and the change of curvature are included in the equations of equilibrium. By considering the Hamilton s principle the governing equations for the vibration of shell is determined. By applying the Donell s assumptions to the obtained governing equations and by considering the axisymmetric case, natural frequencies of thin cylindrical shells under free vibrations are determined. Fig.2. Stress resultants The Hamilton s principle states that among the set of all admissible configurations of a system, the actual motion makes the quantity stationary, provided the configuration is known at the limits t = t 1 and t = t 2. According to Hamilton s principle Volume: 05 Special Issue: 13 ICRAES-2016 Sep-2016, 41

3 (3.1) where T is kinetic energy, U is strain energy and V is work potential of all applied loads. The governing differential equations, the strain energy due to loads, kinetic energy and formulations of the general dynamic problem are derived on the basis of Hamilton s principle. The equation of motion is obtained by taking a differential element of shell with internal forces like membrane (N 1, N 2 and N 6 ) shearing forces (Q 1 and Q 2 ) and the moment resultants (M 1, M 2 and M 6 ). 3. EQUATION OF EQUILIBRIUM By applying the dynamic version of the Hamilton s principle and substituting the parameters of strain energy and the kinetic energy as given in equation (3.1) and integrating the displacement gradients by parts, the resulting equation is (3.2) The governing equations derived in orthogonal curvilinear coordinates for general shell element is reduced for circular cylindrical shell. The equations of motion are given as (3.3) - (3.4) 4. UNPRESSURIZED CYLINDRICAL SHELLS According to Donnell s assumptions, (a) Neglecting the effect of transverse shear force, while considering equilibrium of forces in circumferential direction (b) The effect of stretching displacement may be neglected while considering curvature displacement relations. Frequency parameter, (3.5) Membrane stress, (3.6) (3.7) Axial wavelength factor, (3.8) Shell parameter, (3.9) The following three equations are obtained (3.10) For a non trivial solution the determinant of the coefficients of U,V,W in above equations must vanish resulting to the following equation (3.11) The characteristic equation (3.11 ) thus obtained from model is a cubic equation and results in three roots for three dimensional shell. The three roots results in Eigen Values and Eigen Vectors in axial, circumferential and radial directions. Frequencies and mode shapes are generated for both pressurized and unpressurised conditions. 5. PRESSURIZED CYLINDRICAL SHELLS: In the case of pressurized thin cylindrical shells the frequency equation is similar to that of unpressurized cylindrical shells except that the membrane stresses are replaced with the pressure terms as (3.12) then the characteristic equation becomes Where reverting to the notations Here (3.13) When the internal pressure vanishes, the coefficients reduce to. The expressions are: Volume: 05 Special Issue: 13 ICRAES-2016 Sep-2016, 42

5 Table.5. µ=0.3, a/h = 170/7.89 = 21.54, λ=0.756, ΔP=200bar Axial frequency f 1, Radial frequency f 3, Table.6. µ=0.3, a/h = 370/7.89 = 46.89, λ=0.756, λ=0.756, ΔP=200bar Axial frequency f 1, Radial frequency f 3, Theoretical results: Unpressurized condition for different values of l/a ratio Table.7. µ=0.3, l/a = 706/170 = 4.15, λ=0.756, ΔP=0bar Axial frequency f 1, Radial frequency f 3, Table.8. µ=0.3, l/a = 906/170 =5.329, λ=0.589, ΔP=0bar Axial frequency f 1, Radial frequency f 3, Pressurized condition for different values of l/a ratio Table.9. µ=0.3, l/a = 706/170 = 4.15, λ=0.756, ΔP=100bar Axial frequency f 1, Radial frequency f 3, Table.10. µ=0.3, l/a = 906/170 =5.329, λ=0.589, λ=0.589, ΔP=100bar Axial frequency f 1, Radial frequency f 3, Table.11. µ=0.3, l/a = 706/170 = 4.15, λ=0.756, λ=0.756, ΔP=200bar Axial frequency f 1, Radial frequency f 3, Table.12. µ=0.3, l/a = 906/170 =5.329, λ=0.589, ΔP=200bar Axial frequency f 1, Radial frequency f 3, Volume: 05 Special Issue: 13 ICRAES-2016 Sep-2016, 44

6 Table13. Theoretical calculation of Eigen vectors at different Pressures Pressure ΔP, bar U/W V/W Numerical Results (ANSYS codes) Fig.6 Displacement in Y-direction at a pressure of P=5 bar Fig.3 Displacement in x-direction at a pressure of P=5 bar Fig.4 Displacement in x-direction at a pressure of P=5 bar Fig.7 Displacement in Z-direction at a pressure of P=5 bar Fig.5 Displacement in Y-direction at a pressure of P=5 bar Fig.8 Displacement in Z-direction at a pressure of P=5 bar Comparison of ANSYS results with the theoretical results, For Pressure P=5 bar and at nodal value n=5 Volume: 05 Special Issue: 13 ICRAES-2016 Sep-2016, 45

7 Table.14. Validation of Eigen vectors with numerical technique Theoretical results ANSYS results U/W V/W U/W V/W Fig.12 Frequency variation in axial direction for different pressure values, with varying number of lobes. Fig.9 Frequency variation in axial direction for varying pressure values and a/h ratios. Fig.10. Frequency variation in circumferential direction for varying pressure values and a/h ratios. Fig.13. Frequency variation in circumferential direction for different pressure values, with varying number of lobes. Fig.11. Frequency variation in radial direction for varying pressure values and a/h ratios. Fig.14. Frequency variation in radial direction for different pressure values, with varying number of lobes. Volume: 05 Special Issue: 13 ICRAES-2016 Sep-2016, 46

9 per unit length stress resultants in shell, force membrane stresses due to internal pressure membrane stresses due to internal pressure thickness parameter shell parameter frequency parameter axial wave length factor Poisson s ratio Volume: 05 Special Issue: 13 ICRAES-2016 Sep-2016, 48

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