Fourier Series In Circumferential Direction Research Articles

A semi-analytic model for vibro-acoustic analysis of functionally graded shells of revolution

A semi-analytic model with wider applications is developed for vibro-acoustics of submerged functionally graded material (FGM) shells of revolution, and independent or coupled FGM shells are specific cases. For structural model, the shell is divided to many narrow segments treated as conical shells. Materials vary continuously in thickness direction according to the four-parameter power-law distributions about volume fractions of two constituents. Based on the first-order shear deformation theory (FSDT), differential equations are solved via expanding displacements as power series and Fourier series in meridional and circumferential directions, respectively. Continuity and boundary conditions are assembled to vibration governing equation. For acoustic model, Helmholtz integral equation is simplified to one line integral after expanding velocity and acoustic pressure as Fourier series in circumferential direction, and the pressure is further expressed as displacements of the shell. Then, structural and acoustic models are coupled as adding the acoustic pressure to continuity conditions. Through comparing results of developed model with the ones of literature and coupled finite element–boundary element method for some FGM shells, rapid convergence and high accuracy are demonstrated. Furthermore, discrepancies between vibro-acoustic responses of FGM, homogeneous and layered shells are slight, and effects of heavy fluid become more obvious as the volume fraction of the material with large stiffness decreases.

A solution method for free vibration analysis of coupled laminated composite elliptical-cylindrical-elliptical shell with elastic boundary conditions

In this paper, an efficient, convenient and explicit method based on the Haar wavelet discretization approach for analyzing the free vibration of the coupled laminated composite elliptical-cylindrical-elliptical shells (ECESs) with elastic boundary conditions is presented. Two elliptical double curved shells are coupled on both end of cylindrical shell. Based on the first-order shear deformation theory the equations of motion for ECES are derived by means of Hamilton's principle. The separation of variables is first performed; i.e. displacement components and rotations of any point of the ECES are expanded to the Haar wavelet series in the meridian direction and Fourier series in circumferential direction. The constants appearing from the integrating process are determined by boundary conditions, and thus the partial differential equations are transformed into algebraic equations. By solving the characteristic equation, the natural frequencies and mode shapes of coupled laminated composite ECES are obtained. The present results have been compared with those of the published literature. The comparison results show that this method has high accuracy, high reliability and also a higher convergence rate in attaining the frequencies of the coupled laminated composite ECESs. Then, the effects of the main parameters such as material properties, geometrical parameters, and various boundary conditions, on the vibrational behavior of the coupled ECESs, are investigated. Finally, new free vibration analysis results of the coupled laminated composited ECES, which can be used as benchmark data for researchers in this field, are reported through the parameter study.

An analytical method for free vibrations of functionally graded cylindrical shells with arbitrary intermediate ring supports

An analytical method is presented to study free vibrations of functionally graded material (FGM) cylindrical shells with arbitrary intermediate ring supports. Material properties continuously vary in thickness direction in accordance with the four-parameter power-law distributions in terms of volume fractions of constituents. To establish the final governing equation, the shell is firstly divided into several shell segments according to locations of ring supports. Based on the first-order shear deformation theory (FSDT), differential equations about displacements are solved by expanding displacements as exponential functions in axial direction and Fourier series in circumferential direction. Correspondingly, five displacements and five forces at any cross section of segments are further expressed as 10 unknown coefficients for every circumferential mode number. The artificial spring technique is adopted to restrain displacements at boundaries and ring supports, and all segments are assembled to the overall shell through continuity conditions. To test the validity of the developed model, natural frequencies and mode shapes of several homogenous and FGM cylindrical shells with/without ring supports are compared with appropriate ones in the literature or calculated by finite element method (FEM), which demonstrates high accuracy and wide application of the proposed method. Furthermore, effects of ring supports and material parameters are analyzed. The results reveal that restraining radial and/or circumferential displacements at ring supports significantly affects natural frequencies, but mode shapes are mainly affected by radial displacements. In addition, natural frequencies of the FGM shell obviously vary with the power-law exponent, whereas the corresponding mode shapes almost keep unchanged.

A semi-analytic method for vibro-acoustic analysis of coupled propeller-shaft-hull systems under propeller excitations

A semi-analytic dynamic model is developed for vibro-acoustic analysis of submerged coupled propeller-shaft-hull systems under propeller forces. The propeller, shaft and hull are idealized as a lumped mass, multi-span beam and ring-stiffened shell of revolution, respectively. Stern, middle and thrust bearings are simulated as spring-damping systems. The hull is firstly decomposed into many segments, which are treated as conical shells and analyzed through combining Flügge shell theory and power series method. The shaft is divided into three Timoshenko beams according to the bearings. By expanding acoustic pressure and velocity as Fourier series in circumferential direction, surface Helmholtz integral equation is reduced to line integral, and acoustic pressure is further expressed as displacements of segments after meshing the line to some 3-node isoparameter elements. Boundary conditions and continuity conditions modified by external acoustic pressure are orderly assembled to the dynamic model. As considering the hull as a ring-stiffened conical-cylindrical-spherical shell, vibro-acoustic results of present method are compared with ones of coupled finite element-boundary element method, and the accuracy and efficiency of developed dynamic model are demonstrated. Furthermore, parameter analysis reveals that breathing and beam modes are predominant for vibro-acoustic responses, and sound power is mainly dominated by the cylindrical compartment.

Application of Short and Long (Enhanced Vlasov's) Solutions for Cylindrical Shell on Example of Concentrated Radial Force

Abstract Analytical approaches for cylindrical shell are mostly based on expansion of all variables in Fourier series in circumferential direction. This leads to eighth-order differential equation with respect to axial coordinate. Here it is approximately treated as a sum of two fourth-order biquadratic equations. First one assumes that all variables change more quickly in circumferential direction than in axial one (long solution), while the second (short) one is based on opposite assumption. The accuracy and applicability of this approach were demonstrated (Orynyak, I., and Oryniak, A., 2018, “Efficient Solution for Cylindrical Shell Based on Short and Long (Enhanced Vlasov's) Solutions on Example of Concentrated Radial Force,” ASME Paper No. PVP2018-85032) on example of action of one or two concentrated radial forces and compared with finite element method results. This paper is an improvement of our previous work (Orynyak, I., and Oryniak, A., 2018, “Efficient Solution for Cylindrical Shell Based on Short and Long (Enhanced Vlasov's) Solutions on Example of Concentrated Radial Force,” ASME Paper No. PVP2018-85032). Two amendments are made. The first is insignificant one and use slightly modified expressions for bending strains, while the second one relates to the short solution. Here we do not consider any more that circumferential displacement is negligible as compared with radial one. Eventually this improves the accuracy of results, as compared with previous work. For example, for cylinder with radius, R, to wall thickness, h, ratio equal to 20, the maximal inaccuracy for radial displacement in point of force application decreases from 5% to 3%. For thinner cylinder with R/h = 100, this inaccuracy decreases from 2.5% to 1.25%. These inaccuracies are related to larger terms in Fourier expansion, the significance of which decrease when length or area of outer loading becomes greater. The last conclusion is demonstrated for the case of distributed concentrated force acting along short segment on axial line.

A unified semi-analytic method for vibro-acoustic analysis of submerged shells of revolution

A unified semi-analytic method is developed to predict vibro-acoustic responses of submerged shells of revolution. The shell is firstly decomposed to several narrow shell segments to be treated as conical shells. By employing Flügge shell theory and power series, displacements and forces at any cross-section of conical shells are expressed as eight unknown coefficients. Then, expanding surface pressure and velocity as Fourier series in circumferential direction, the surface Helmholtz integral equation is reduced to the line integral along the generator line, and the pressure is further represented as displacements of all segments through meshing the generator line to some 3-node isoparametric elements and utilizing relationships of velocities and displacements. Last, continuity conditions between adjacent segments, which include modifications introduced by acoustic pressure, and elastic boundary conditions are assembled to the finial vibro-acoustic governing equation of submerged shells of revolution. Through comparing vibro-acoustic results of present method with ones in appropriate references or calculated by FEM/BEM for independent conical, cylindrical and spherical shells and combined spherical-cylindrical-spherical and conical-cylindrical-spherical shells, rapid convergence, wide application and high accuracy of the semi-analytic model are demonstrated. Meanwhile, contributions of different circumferential modes and different compartments of the combined conical-cylindrical-spherical shell to vibro-acoustic responses are investigated.

Nonlinear Mode Interaction for Thin Circular Cylindrical Anisotropic Shells

The Sanders shell equations for imperfect anisotropic circular cylindrical shells are used for the development of a research tool to study the nonlinear interaction of circumferential waves under different loading conditions. Therefore, a truncated Fourier series in circumferential direction is used as a possible solution of the nonlinear equilibrium equations in combination with the anisotropic constitutive relations. The envelope of the Fourier series coefficients in the axial direction is computed using a numerical integration technique satisfying the rigorous boundary conditions at the lower and upper shell edges. The case of an isotropic shell under axial compression is used to compare the results obtained by the current theory and the STAGS-A program.

Semi-Analytic Method for a Tri-Material Transversely Isotropic Full-Space in Frequency Domain

A linear elastic transversely isotropic full-space containing three different regions, which are an upper half-space, a lower half-space and a layer in between is considered in such a way, each region contains different material. The axes of symmetry of different regions are assumed to be normal to the interface of the regions and thus parallel. An arbitrary load in frequency domain is applied on an arbitrary patch located at the interface of the upper half-space and its underneath layer. Integral formulations are presented for the determination of the displacements and stresses in all regions. By means of the complete displacement potentials introduced by Eskandari-Ghadi (2005), Fourier series in circumferential direction and Hankel transform in radial direction, the displacements and stresses are determined in Fourier-Hankel space. Then, the displacements and stresses at any point are given in the line integral form. The solution is numerically evaluated for: (i) a half-space under an arbitrary surface load, (ii) a half-space under an arbitrary buried force, (iii) a half-space fixed at the top and under an arbitrary buried force, (iv) a half-space containing a layer bonded to the top of a half-space under an arbitrary force applied at the interface of two regions, (v) a full-space under an arbitrary load in it, (vi) a bi-material full-space under an arbitrary force applied at the interface of two half-space, and (vii) a layer of finite thickness fixed at the bottom and under an arbitrary surface load. To confirm the accuracy of the numerical evaluation of the integrals involved, some of the results are compared with existing solutions.

On refined analysis of bifurcation buckling for the axially compressed circular cylinder

We present extensive numerical results of bifurcation buckling analysis of the axially compressed circular cylinder. The analysis is based on the modified displacement version of the non-linear theory of thin elastic shells developed by Opoka and Pietraszkiewicz [Opoka, S., Pietraszkiewicz, W., 2009. On modified displacement version of the non-linear theory of thin shells. International Journal of Solids and Structures, 46, 3103–3110.]. To solve the buckling problem we apply the separation of variables and expansion of all fields into Fourier series in circumferential direction, with subsequent accurate calculations of eigenvalues of determinants of corresponding 8 × 8 complicated matrices. The numerical analysis of the buckling load is performed for the cylinders with length-to-diameter ratio in the range (0.05, 60), with eight sets of incremental work-conjugate boundary conditions analogous to those used in the literature and partly summarized in the book by Yamaki [Yamaki, N., 1984. Elastic Stability of Circular Cylindrical Shells. Elsevier, Amsterdam], and additionally with six sets of boundary conditions not discussed in the literature yet. The results allow us to formulate several important conclusions, such as: (a) omission in the non-linear BVP small terms of the order of error introduced by the error of constitutive equations leads to overestimated buckling loads for long cylinders with clamped boundaries; (b) for some relaxed boundary conditions the buckling load decreases for short cylinders with decrease of the cylinder length; (c) the results for additional six sets of boundary conditions reveal existence of several new cases, in which by relaxing geometric boundary conditions the buckling load falls down to about one half of the classical value in a wide range of the cylinder length-to-diameter ratios.

Elastic Response of a Thin Conical Shell to an Oblique Impact

Strain waves in a truncated conical shell generated by oblique impact of the small end of the shell are discussed. The analysis is based on the general first-order linear shell theory of Sanders with all pertinent variables expanded into Fourier series in circumferential direction, resulting in decoupled sets of ordinary differential equations. Finite-difference approximations to these differential equations are then solved by matrix techniques. Theoretical values compare well with experimental results for the angle of oblicity β = 0, 5, 10, and 15 deg, the range of the experimental work.

Thin cylindrical shells under surface torques

Introduction I a previous paper the bending problem of an infinitely long cylinder loaded with concentrated equal and opposite forces, acting at the ends of a vertical diameter, was discussed. The loading function in this case was presented by a Fourier integral in the longitudinal direction and by a Fourier series in circumferential direction. The integral representation has the advantage that the boundary conditions are automatically taken care of, and no subsequent determination of Fourier coefficients is necessary. The Fourier coefficients and the undetermined function in the Fourier integral are determined simply from the loading condition. The radial displacement was then obtained from the solution of the eighth-order differential equation. In the present paper, the shearing stress-resultant of an infinitely long cylindrical shell subjected to two equal and opposite torques acting about the radial axis on the surface of the shell, as shown in Fig. la, has been analyzed. The solution of this problem was achieved by replacing the torque with two equal and opposite forces acting about the same axis at an infinitely small distance apart. The moment produced by the pair of forces is of the same magnitude and sign as the torque. In the case of a plate, the shearing stresses produced by two equal and opposite forces acting perpendicularly to either axis are identical. However, in the present problem, the phenomenon is quite different because of the effect of the curvature of the shell on the displacements. For this reason, in the present investigation, a combination of two pairs of forces with moments are introduced to replace the applied torque. The forces are so arranged that one pair of forces, equal and opposite in magnitude, act at an infinitely small distance apart in the direction of x axis and the other pair of forces under the same condition in the direction of s axis. The loading distribution functions under consideration may be presented by a combination of a Fourier series and a Fourier integral along the circumference and the generatrix, respectively.