Preliminary orthotropic elastic model for the study of natural frequencies and mode shapes of a 3D printed Onyx thin circular cylindrical shell

The linear vibrations of a 3D printed Onyx thin circular cylindrical shell are considered. A model based on Sanders-Koiter shell theory and orthotropic elastic constitutive equations is adopted to obtain elastic strain and kinetic energy. The deformation of the middle surface of the shell is described in terms of longitudinal, circumferential and radial displacements, which are expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction of the shell. Free-free boundary conditions are considered. The Rayleigh-Ritz method is applied to calculate approximate natural frequencies and mode shapes. An isotropic elastic model is first adopted to obtain initial reference values for natural frequencies and mode shapes of the 3D printed shell. An experimental modal analysis is then performed to verify the accuracy of the initial isotropic elastic model and to find exact values for natural frequencies and mode shapes of the 3D printed shell. A more effective orthotropic elastic model is finally applied assuming different values of Young’s modulus along the longitudinal and circumferential directions of the shell. A parametric analysis is carried out by assuming a constant circumferential Young’s modulus and varying the longitudinal Young’s modulus. The goal is to minimise the difference between analytical and experimental results, in order to identify the actual orthotropy degree of the 3D printed shell.