A classification of the possible free wave motions in a three-layered composite thick cylinder is presented. The governing equations of motion for free wave motion for isotropic and orthotropic elastic media are given and a combined solution is developed for a thick cylinder made of three different materials, such as orthotropic-isotropic-orthotropic layers. The use of Bessel and special Frobenius series is required to obtain a correct, closed form solution for the propagating waves. Numerical results are given in the form of dispersion curves and these are discussed. A correct approach to the calculation of the cut-off frequencies is presented. It is shown that the wave speeds of decoupled longitudinal-shear motion in orthotropic shells is dependent upon the ratio of the different shear moduli, Ci66/Ci55. The influence of a thin, soft rubbery material at the centre of a sandwich configuration is thoroughly analyzed. The effect of the isotropic core properties (its stiffness and thickness) on the dynamics of the wave motion is investigated.
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Classification of wave motions in thick orthotropic cylinders
Medium Kind of waves Functions used References Isotropic Axisymmetric Bessel functions [3, 4] and Jn , Yn , In and Kn asymmetric Transverse Axisymmetric Bessel functions [5] isotropy, Jn and Yn hexagonal symmetry, C11 = C33 Asymmetric Bessel functions [7, 8] Jn , Yn , In and Kn General Axisymmetric Frobenius functions, [6] orthotropy, real indices C11 $ C22 Asymmetric Frobenius functions, [9] C22 $ C33 real and complex indices
structural parts of robotic machines. The existing literature on the three-dimensional equations of elastodynamics permits a classification to be made of wave motions in isotropic and anisotropic circular cylinders. This is presented in Table 1. All the analyses mentioned in this table deal with three-dimensional wave propagation in thick cylinders of infinite length. Without exception, they allow steady state harmonic waves to travel along the x-axis of the isotropic or orthotropic cylinder.
The following points are noted about the classification scheme of Table 1. The solutions for isotropic media and transverse isotropy, with hexagonal elastic symmetry were published in references [3][4][5][6][7][8]. It was shown that exact solutions can be obtained in terms of three displacement potentials, each satisfying a partial differential equation of second order. The stress-free boundary conditions for hollow cylinders result in a characteristic (6 × 6) determinant for asymmetric motions, and in a (4 × 4) determinantal equation for axisymmetric motions. These analyses also reveal that the cylinder having material which is transversely isotropic and hexagonally symmetric (i.e., when E f = E r , or C 11 = C 33which means that the plane of symmetry is in fact the cross-section of the cylinder) is a specially simple case. It is only in this case of orthotropy that the basic equations of motion can be decoupled. This was first recognized by Mirsky [7,8], by introducing displacement potentials. Mirsky [6] was evidently aware of the difficulties involved in the analysis, especially when considering more complicated orientation of the elastic symmetry plane. In reference [6] he used an entirely different approach by introducing Frobenius power series into the analysis. In this way he was able to deal with general orthotropic materials. This method is the only one which copes with a more complicated transverse isotropy, where the plane of elastic symmetry is the x-F plane. However, Mirsky's treatment was restricted to axisymmetric wave motion only. A method has been presented recently by the authors [9] for dealing with asymmetric wave motion in generally orthotropic thick cylinders and can be regarded as a benchmark solution.
The aim of this paper is to show the kind of problems which arise when a more complicated structure is analyzed, namely a layered cylinder, which consists of alternate orthotropic-isotropic-orthotropic layers. This combination of different media requires the combining of different kinds of solutions (Bessel functions with Frobenius power series). The analysis then becomes very complicated when closed form solutions are sought.