In-plane vibration analyses of circular arches by the generalized differential quadrature rule

In-plane free vibrations of circular arches are investigated using the generalized differential quadrature rule (GDQR) proposed recently. The Kirchhoff assumptions for thin beams are considered, and the neutral axis is taken as inextensible. Several examples of arches with uniform, continuously varying, and stepped cross-sections are presented to illustrate the validity and accuracy of the GDQR. The necessary domain decomposition technique is used for some cases. The obtained frequencies are compared with those calculated from a number of other approaches from the Rayleigh–Ritz, Rayleigh–Schmidt and Galerkin methods to the finite element technique and the cell discretization method. The GDQR frequencies are always greater than those obtained from the cell discretization method that produces the lower bounds to the exact results, and are also in agreement with the upper bounds to the exact results.     

An efficient approach for free vibration analysis of conical shells

In this paper, the global method of generalized differential quadrature (GDQ) is applied for the first time to study the free vibration of isotropic conical shells. The shell equations used are Love-type. The displacement fields are expressed as product of unknown functions along the axial direction and Fourier functions along the circumferential direction. The derivatives in both the governing equations and the boundary conditions are discretized by the GDQ method. Using the GDQ method, the natural frequencies can be easily and accurately obtained by using a considerably small number of grid points. The accuracy and efficiency of the GDQ method is examined by comparing the results with those in the literature and very good agreement is observed. The fundamental frequency parameters for four sets of boundary conditions and various semivertex angles are also shown in the paper.     

Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method

In this paper, the generalized differential quadrature (GDQ) method is used for the first time to study the effects of boundary conditions on the frequency characteristics of a thin rotating cylindrical shell. The present analysis is based on Love-type shell theory and the governing equations of motion include the effects of initial hoop tension and the centrifugal and coriolis accelerations due to rotation. The displacement field is expressed as a product of unknown smooth continuous functions in the meridional direction and trigonometric functions along the circumferential direction so that the three-dimensional dynamic problem may be transformed mathematically into a one-dimensional problem. Based on this approach, the results are obtained for the effects of the boundary conditions on the frequency characteristics at different circumferential wave numbers and rotating speeds and various geometric properties; the effect of rotating speed on the relationship between frequency parameter and circumferential wave number is also discussed. To validate the accuracy and efficiency of the GDQ method, the results obtained are compared with those in the literature and very good agreement is achieved.     

Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness

The free vibration analysis of laminated conical shells with variable stiffness is presented using the method of differential quadrature (DQ). The stiffness coefficients are assumed to be functions of the circumferential coordinate that may be more close to the realistic applications. The first-order shear deformation shell theory is used to account for the effects of transverse shear deformations. In the DQ method, the governing equations and the corresponding boundary conditions are replaced by a system of simultaneously algebraic equations in terms of the function values of all the sampling points in the whole domain. These equations constitute a well-posed eigenvalue problem where the total number of equations is identical to that of unknowns and they can be solved readily. By vanishing the semivertex angle (α) of the conical shell, we can reduce the formulation of laminated conical shells to that of laminated cylindrical shells of which stiffness coefficients are the constants. Besides, the present formulation is also applicable to the analysis of annular plates by letting α=π/2. Illustrative examples are given to demonstrate the performance of the present DQ method for the analysis of various structures (annular plates, cylindrical shells and conical shells). The discrepancies between the analyses of laminated conical shells considering the constant stiffness and the variable stiffness are mainly concerned.     

Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells

Based on the three-dimensional fundamental equations of anisotropic elasticity, a state equation with variable coefficients is derived in a unified matrix form. The free vibration of simply supported, fluid-filled cylindrically orthotropic functionally graded cylindrical shells with arbitrary thickness is then investigated. A laminate approximate model is employed which is suitable for an arbitrary variation of material constants along the radial direction. Numerical examples are presented and compared with existing results. The effects of related parameters on natural frequencies are discussed finally.     

Generalized differential quadrature for free vibration of rotating composite laminated conical shell with various boundary conditions

By using the generalized differential quadrature (GDQ) method, this paper presents the orthotropic influence of composite materials on frequency characteristics for a rotating thin truncated circular symmetrical cross-ply laminated composite conical shell with different boundary conditions. The present governing equations of free vibration include the effects of initial hoop tension and the centrifugal and Coriolis accelerations due to rotation. Frequency characteristics are obtained to study in detail the orthotropic influences. Effects of boundary condition, rotating speed, circumferential wave number and geometric property are also discussed. To ensure the accuracy of the present results by the GDQ method, comparisons are made with those available in open literature and very good agreements are achieved.     

Free vibration of multi-layered circular cylindrical shell with cross-ply walls,including shear deformation by using spline function method

The spline function technique is used to analyze the vibration of multi-layered circular cylindrical shells with cross-ply walls including first-order shear deformation theory. Both antisymmetric and symmetric cross-ply laminations are considered in this analysis. The governing equilibrium equations are obtained in terms of displacement and rotational functions. A system of coupled ordinary differential equations in terms of displacement and rotational functions are obtained by assuming the solution in a separable form. These functions are approximated by using Bickley-type splines of suitable order to obtain the generalized eigenvalue problem by applying point collocation techniques with appropriate boundary conditions. Parametric studies are performed to analyze the frequency response of the shell with reference to the material properties, number of layers, fiber orientation, thickness to radius ratio, length to radius ratio and circumferential node number. Reasonable agreement is found with existing results obtained by FEM and other methods. Valuable results are presented as graphs and discussed. This paper was recommended for publication in revised form by Associate Editor Maenghyo Cho Dr. K. K. Viswanathan was born in 1962 in Vellore District, India. He received his B.Sc. in Mathematics from University of Madras and M.Sc. in 1992 and Ph.D. in 1999 from Anna University, India. Later he was a Project Associate in Indian Institute of Science, Bangalore. He served as lecturer in Crescent Engg. College and as Asst. Professor in SRM University, India. He did his post doctoral research in Korea for three years. At present he serves as Professor in the Dept. of Naval Architecture, Inha University, Incheon, Korea. His research areas of interest includes vibration of plates, shells and the application of numerical techniques in Engineering problems. Dr. Kyung Su Kim was born in Korea in 1954. He is a professor in Naval Architecture and Ocean Engineering at Inha University, Korea. He obtained his B.Sc. degree in Naval Architecture and Ocean Engineering from Seoul National University, Korea, in 1981. He worked for KR (Korean Register of Shipping) from 1981 to 1983. He obtained M.Sc. degree in Naval Architecture and Ocean Engineering in 1986, and Ph.D. degree in Structural Mechanics in 1991 from Rheinisch — Westfaelische Technische Hoch-schule Aachen, Germany. From 1986 to 1992, he was a Post Doctoral Research Engineer of Engineering Research Institute at Rheinisch — Westfaelische Technische Hochschule Aachen. He was appointed as a professor of Inha University, Korea, in 1994. His major area of study is Impact and Fatigue Fracture. Dr. Jang Hyun Lee was born in Korea in 1969. Currently, he is an Assistant professor of the Department of Naval Architecture and Ocean Engineering at Inha University, Korea. He obtained his B.Sc., M.Sc. and Ph.D. degrees in Naval Architecture and Ocean Engineering from Seoul National University, Korea, in 1993, 1995 and 1999 respectively. From 1999 to 2002, he was a Post Doctoral Research Engineer of Engineering Research Institute at Seoul National University. He joined the Inha University in 2005 after holding the Chief Technology Officer at Xinnos for four years. His research interests include press forming of thick plates and shells, computational welding mechanics and Product Lifecycle Management.     

The differential quadrature method for irregular domains and application to plate vibration

By its very basis, the differential quadrature method may be applied to domains having boundaries oriented along the coordinate axes. In this paper, it is shown that quadrature rules may also be formulated for irregular domains using the natural-to-Cartesian geometric mapping technique. The application of the technique is demonstrated through the vibration analysis of thin isotropic plates of general quadrilateral and sectorial planforms.     

Non-linear free vibration analysis of laminated orthotropic cylindrical shells

A method to predict the influence of geometric non-linearities on the natural frequencies of an empty laminated orthotropic cylindrical shell is presented in this paper. It is a hybrid of finite element and classical thin shell theories. Sanders—Koiter non-linear and strain-displacement relations are used. Displacement functions are evaluated using linearized equations of motion. Modal coefficients are then obtained for these displacement functions. Expressions for the mass, linear and non-linear stiffness matrices are derived through the finite element method (in terms of the elements of the elasticity matrix). The uncoupled equations are solved with the help of elliptic functions. The frequency variations are first determined as a function of shell amplitudes and then compared with the results in the literature.     

Vibration of functionally graded cylindrical shells based on different shear deformation shell theories with ring support under various boundary conditions

In the present work, study of the vibration of thin cylindrical shells with ring supports made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. Effects of boundary conditions and ring support on the natural frequencies of the FGM cylindrical shell are studied. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which imposed a zero lateral deflection. The study is carried out using different shear deformation shell theories. The analysis is carried out using Hamilton’s principle. The governing equations of motion of a FGM cylindrical shells are derived based on various shear deformation theories. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin M. M. Najafizadeh received his BS degree in 1995 from Azad University (Arak) and the Ms Degree in 1997 from Azad University (Arak), and his Ph.D. degree in 2003 from Science and Research Branch Islamic Azad University (Tehran, Iran), all in mechanical Engineering. He is member of faculty in Islamic Azad University (Arak) since 1998. He teaches courses in the areas of dynamics, theory of plates and shells and finite element method. He has published more than 20 articles in journals and conference proceeding. Mohammad Reza Isvandzibaei received his Ms Degree from Azad University (Arak), and now he is the student of Ph.D. in university of Pune, (India) all in mechanical Engineering. He is member of faculty in Islamic Azad University (Andimeshk).     

Frequency analysis of multiple layered cylindrical shells under lateral pressure with asymmetric boundary conditions

Natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure are studied. The multiple layered cylindrical shell configuration is formed by three layers of isotropic material where the inner and outer layers are stainless steel and the middle layer is aluminum. The multiple layered shell equations with lateral pressure are established based on Love＇s shell theory. The governing equations of motion with lateral pressure are employed by using energy functional and applying the Ritz method. The boundary conditions represented by end conditions of the multiple layered cylindrical shell are simply supported-clamped（SS-C）, free-clamped（F-C） and simply supported-free（SS-F）. The influence of different lateral pressures, different thickness to radius ratios, different length to radius ratios and effect of the asymmetric boundary conditions on natural frequency characteristics are studied. It is shown that the lateral pressure has effect on the natural frequency of multiple layered cylindrical shell and causes the natural frequency to increase. The natural frequency of the developed multilayered cylindrical shell is validated by comparing with those in the literature. The proposed research provides an effective approach for vibration analysis shell structures subjected to lateral pressure with an energy method.     

Natural frequencies of fluid-filled transversely isotropic cylindrical shells

The free vibration of a simply supported, transversely isotropic cylindrical shell filled with compressible fluid is exactly analyzed based on three-dimensional elasticity. The displacement separation technique as well as the orthogonal series expansion method are employed. Bessel functions with complex arguments are directly used in the solution for the case of complex eigenvalues. Frequency equation of the coupled vibration problem is obtained by taking account of the effect of the inner fluid. Numerical results are finally presented and compared with those based on three typical shell theories.     

Free vibration of transversely isotropic piezoelectric circular cylindrical panels

This paper presents an exact three-dimensional free vibration analysis of a transversely isotropic piezoelectric circular cylindrical panel. The general solution for coupled equations for piezoelectric media that was recently proposed by Ding et al. (Int. J. Solids Struct. 33 (1996) 2283) is employed. By using the variable separation method, three-dimensional exact solutions are obtained under several boundary conditions. Numerical results are finally presented and compared with available data in literature. The results show the non-dimensional frequencies of the piezoelectric panel are bigger than that of the non-piezoelectric one.     

Linear free vibration analysis of cross-ply laminated cylindrical helical springs

A linear free vibration analysis of symmetric cross-ply laminated cylindrical helical springs is performed based on the first-order shear deformation theory. Considering the rotary inertia, the shear and axial deformation effects, governing equations of symmetric laminated helical springs made of a linear, homogeneous, and orthotropic material are presented in a straightforward manner based on the classical beam theory. The free vibration equations consisting of 12 scalar ordinary differential equations are solved by the transfer matrix method. The overall transfer matrix of the helix is computed up to any desired accuracy. The soundness of the present results are verified with the reported values which were obtained theoretically and experimentally. After presenting the non-dimensional graphical forms of the free vibrational characteristics of (0°/90°/90°/0°) laminated helical spring made of graphite-epoxy material (AS4/3501-6) with fixed–fixed ends, a non-dimensional parametric study is worked out to examine the effects of the number of active turns, the shear modulus in the 1–2 plane (G₁₂), the ratio of the cylinder diameter to the thickness (D/d), and Young's moduli ratio in 1 and 2 directions (E₁/E₂) on the first six natural frequencies of a uniaxial composite helical spring with clamped-free, clamped-simple, and clamped–clamped ends.     

Vibration of cylindrical shells with ring support

In this paper, a study on the vibration of thin cylindrical shells with ring supports is presented. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which imposed a zero lateral deflection. The study is carried out using Sanders' shell theory. The governing equations are obtained using an energy functional with the Ritz method. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.     

Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

Free vibration of two-side simply-supported laminated cylindrical panels via the mesh-free kp-Ritz method

In this paper, the free vibration of laminated two-side simply-supported cylindrical panels is analyzed by the mesh-free kp-Ritz method. In this latest form of the Ritz technique, the reproducing kernel particle estimation is employed in hybridized form with harmonic functions to approximate the two-dimensional displacement field. The present analysis is based on the energy functional of the classical thin-shell theory based on Love's hypothesis. To validate the accuracy of the results and the stability of the present method, convergence studies were carried out based on the influences of the support size and the number of nodes. Comparisons of the present results were also made with existing results available in the open literature and good agreement obtained. This study also examines in detail the effects of different curved-edge boundary conditions on the frequency characteristics of cylindrical panels.     

Frequency analysis of submerged cylindrical shells with the wave propagation approach

The wave propagation approach is extended to coupled frequency analysis of finite cylindrical shells submerged in a dense acoustic medium. Comparison of the results by the present method and numerical FEM/BEM has been carried out. A finite cylindrical shell enclosed with plate end-caps was modeled for coupled frequency analysis. The first eight coupled natural frequencies were obtained using SYSNOISE. These results are compared with the new method and good agreement has been found. With the new method the effects of shell parameters, m,n,h/R,L/R and boundary conditions, on the coupled frequencies are investigated. The coupled frequency curves are also u-shaped as the uncoupled frequency curves. As the circumferential mode n increases, the difference between the coupled and uncoupled frequencies reduces at small mode n, but increases after the n where the fundamental frequency is achieved. The effects of boundary conditions are significant at small mode n. Due to the coupling effects the transition of fundamental coupled frequency is earlier with the h/R ratios than that of the uncoupled case. Both the coupled and uncoupled frequencies increase with the h/R ratios, but at different rates which show the different coupling effects on the different wave modes. With the present method the calculation of coupled frequency of submerged cylindrical shells is relatively easy, quick but with good accuracy.     

A Vectorial-Wave Method for free and forced vibration analysis of extra thin cylindrical shells with boundary discrete damping

The Vectorial-wave method (VWM) is developed to study free and forced vibrations of cylindrical shells in the presence of dampers at supports. In modeling the issue, a circular cylindrical shell is considered with two ended supports, including separate springs and viscous dampers in the possible directions. Accordingly, based on Flügge thin shell theory and by considering the wave vectors going in the opposite direction along with the shell axis, reflection and transmission matrices are determined to satisfy the shell continuity as well as the boundary conditions. The proposed method is verified through comparing its results with the available literature and the numerical results calculated by Finite element method (FEM). Employing VWM, the viscous characteristics of the applied supports on natural frequencies of the shell are investigated. Furthermore, frequency responses of the shell, which are affected by point-load excitation, are obtained. Finally, the results show that several tandem resonance picks can be eliminated via accurate setting of the support damping.     

Free vibration analysis of spherical caps by the pseudospectral method

The pseudospectral method is applied to the axisymmetric and asymmetric free vibration analysis of spherical caps. The displacements and the rotations are expressed by Chebyshev polynomials and Fourier series, and the collocated equations of motion are obtained in terms of the circumferential wave number. Numerical examples are provided for clamped, hinged and free boundary conditions. The results show good agreement with those of existing literature. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Jinhee Lee received B.S. and M.S. degrees from Seoul National University and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from University of Michigan in 1992 and joined Dept. of Mechano-Informatics of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.