In-plane free vibration analysis of curved Timoshenko beams by the pseudospectral method

The pseudospectral method is applied to the analysis of in-plane tree vibration of circularly curved Timoshenko beams. The analysis is based on the Chebyshev polynomials and the basis functions are chosen to satisfy the boundary conditions. Natural frequencies are calculated for curved beams of rectangular and circular cross sections under hinged-hinged, clampedclamped and hinged-clamped end conditions and the results are compared with those by transfer matrix method. The present method gives good accuracy with only a limited number of collocation points.     

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.     

Free vibration analysis of a hermetic capsule by pseudospectral method

A pseudospectral method is applied to the axisymmetric free vibration analysis of a hermetic capsule. The displacements and the rotation of a hermetic capsule are expressed by the Chebyshev expansions. The equations of motion are collocated to yield the system of equations in the hemispherical regions and the cylindrical region separately. The numbers of collocation points are chosen to be less than those of the expansion terms. The continuity conditions of deformations and stress resultants at the junctions serve as the constraints of the expansion coefficients. The set of algebraic equations is condensed so that the total number of the expansion terms matches the total number of degrees of freedom of the problem. Present method might be useful in the analyses of composite shell structures where different types of shells are joined together.     

Eigenvalue analysis of double-span timoshenko beams by pseudospectral method

The pseudospectral method is applied to the free vibration analysis of double-span Timoshenko beams. The analysis is based on the Chebyshev polynomials. Each section of the double-span beam has its own basis functions, and the continuity conditions at the intermediate support as well as the boundary conditions are treated separately as the constraints of the basis functions. Natural frequencies are provided for different thickness-to-length ratios and for different span ratios, which agree with those of Euler-Bernoulli beams when the thickness-to-length ratio is small but deviate considerably as the thickness-to-length ratio grows larger.     

A finite difference method for the free vibration analysis of stepped Timoshenko beams and shafts

A method based on the variational principles in conjunction with the finite difference technique is used to examine the free vibrational characteristics of Timoshenko beams and shafts. The interlacing grid technique is used to express the strain energy of nodal subdomains and the partial derivatives appearing in the functionals are replaced by the finite difference equations in terms of discrete displacement and rotational components. The developed technique is applied to dynamic analysis of uniform and nonuniform stepped thickness beams and shafts.     

Free lateral vibration analysis of double-span uniform beams

The establishment of the frequencies and modal shapes for the free lateral vibration of double-span uniform beams is discussed in detail. Data are provided which permit the immediate determination of the first five frequencies and modal shapes for the entire family of beams regardless of the location of the intermediate support.     

On the effect of shear coefficients in free vibration analysis of curved beams

We did a comparative study of shear coefficients in free vibration analysis of curved beams having circular and rectangular crosssections. Until recently, the shear coefficient k in Timoshenko beam theory has been studied by many researchers to include transverse shear deformation effect. To obtain more reliable numerical results, a higher-order hybrid-mixed curved beam element is formulated and programmed in MATLAB. The present numerical experiments show that k = 6(1 + v)² / (7 + 12v + 4v ²) is the best expression both for circular and rectangular cross-sections in the flexural vibration of curved beams.     

Free vibration analysis of beams with non-ideal clamped boundary conditions

A non-ideal boundary condition is modeled as a linear combination of the ideal simply supported and the ideal clamped boundary conditions with the weighting factors k and 1-k, respectively. The proposed non-ideal boundary model is applied to the free vibration analyses of Euler-Bernoulli beam and Timoshenko beam. The free vibration analysis of the Euler-Bernoulli beam is carried out analytically, and the pseudospectral method is employed to accommodate the non-ideal boundary conditions in the analysis of the free vibration of Timoshenko beam. For the free vibration with the non-ideal boundary condition at one end and the free boundary condition at the other end, the natural frequencies of the beam decrease as k increases. The free vibration where both the ends of a beam are restrained by the non-ideal boundary conditions is also considered. It is found that when the non-ideal boundary conditions are close to the ideal clamped boundary conditions the natural frequencies are reduced noticeably as k increases. When the non-ideal boundary conditions are close to the ideal simply supported boundary conditions, however, the natural frequencies hardly change as k varies, which indicate that the proposed boundary condition model is more suitable to the non-ideal boundary condition close to the ideal clamped boundary condition.     

Free vibration analysis of composite beams with two non-overlapping delaminations

An analytical solution to the free vibration of composite beams with two non-overlapping delaminations is presented. The delaminated beam is modeled as seven interconnected Euler-Bernoulli beams using the delaminations as their boundaries. The continuity and the equilibrium conditions are satisfied between adjoining beams. The analysis includes the differential stretching between the delaminated layers and the bending-extension coupling. The results of the present model agree well with the analytical and experimental data reported in the literature. Parametric studies show that the sizes and locations of the delaminations have significant effect on the natural frequencies and mode shapes. These results provide useful information in the study of the free vibration of delaminated composite beams.     

Effects of nearfield waves and phase information on the vibration analysis of curved beams

At high frequencies, energy methods such as the statistical energy analysis and the power flow analysis have been popularly used to predict the averaged responses of vibro-acoustic subsystems. Usually, these energy methods ignore flexural nearfield components and phase information, mainly for simplicity. Such assumptions sometimes lead to an erroneous conclusion, in particular for complex structures and at medium frequencies around the Schroeder cutoff frequency. This paper deals with the effects of nearfield waves and phase information at medium to high frequencies by using the ray tracing method (RTM). A curved beam and a coupled beam system were chosen as test examples, which exhibit the typical mode conversion between various types of travelling waves. Propagation of longitudinal, flexural, and torsional waves was studied based on the Euler-Bernoulli beam theory. Analyses of the spatial distribution of vibrational energy quantities revealed that the conventional RTM could mimic the overall trend of the traveling wave solution. However, the results varied smoothly in space due to the neglect of wave interference. By considering the phase information, local fluctuations of vibration energy could be correctly described. It was confirmed that the flexural nearfield plays a significant role near boundaries and junctions. It was also shown that the accuracy of the analysis depends mainly on the modal overlap factor. Similar to other high frequency methods, the results become close to the traveling wave solutions as the modal overlap factor increases. This paper was recommended for publication in revised form by Associate Editor Yeon June Kang Cheol-Ho Jeong received his M.S. and Ph.D. degrees from KAIST in 2002 and 2007, respectively. He is currently an assistant professor in the Department of Electrical Engineering at Technical University of Denmark in Denmark. His research interests include room acoustics, building acoustics, and structural acoustics. Jeong-Guon Ih earned M.S. and Ph.D. degrees from KAIST in 1981 and 1985, respectively. He is currently a full professor in the Department of Mechanical Engineering at KAIST in Daejeon, Korea. He serves as an Editor of the Applied Acoustics journal and the head vice-president of the Acoustical Society of Korea. His research interests include duct acoustics, vehicle noise/vibration control, theoretical and experimental modeling of vibro-acoustic fields and sources, product sound quality.     

Effect of axial force on free vibration of Timoshenko multi-span beam carrying multiple spring-mass systems

The situation of structural elements supporting motors or engines attached to them is usual in technological applications. The operation of machine may introduce severe dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli–Euler single-span beams carrying a number of spring-mass system and Bernoulli–Euler multi-span beams carrying multiple spring-mass systems are plenty, but that of Timoshenko multi-span beams carrying multiple spring-mass systems with axial force effect is fewer. This paper aims at determining the exact solutions for the first five natural frequencies and mode shapes of a Timoshenko multi-span beam subjected to the axial force. The model allows analyzing the influence of the shear and axial force effects and spring-mass systems on the dynamic behavior of the beams by using Timoshenko Beam Theory (TBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The calculated natural frequencies of Timoshenko multi-span beam by using secant method for non-trivial solution for the different values of axial force are given in tables. The mode shapes are presented in graphs.     

A finite thin circular beam element for out-of-plane vibration analysis of curved beams

A finite thin circular beam element for the out-of-plane vibration analysis of curved beams is presented in this paper. Its stiffness matrix and mass matrix are derived, respectively, from the strain energy and the kinetic energy by using the natural shape functions derived from an integration of the differential equations in static equilibrium. The matrices are formulated with respect to the local polar coordinate system or to the global Cartesian coordinate system in consideration of the effects of shear deformation and rotary inertias. Some numerical examples are analyzed to confirm the validity of the element. It is shown that this kind of finite element can describe quite efficiently and accurately the out-of-plane motion of thin curved beams. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Chang-Boo Kim received his B.S. degree in Mechanical Engineering from Seoul University, Korea in 1973. He then received his D.E.A., Dr.-Ing. and Dr.-es-Science degrees from Nantes University, France in 1979, 1981 and 1984, respectively. Dr. Kim is currently a Professor at the School of Mechanical Engineering at Inha University in Incheon, Korea. His research interests are in the area of vibrations, structural dynamics, and MEMS.     

A finite thin circular beam element for in-plane vibration analysis of curved beams

In this paper, the stiffness and the mass matrices for the in-plane motion of a thin circular beam element are derived respectively from the strain energy and the kinetic energy by using the natural shape functions of the exact in-plane displacements which are obtained from an integration of the differential equations of a thin circular beam element in static equilibrium. The matrices are formulated in the local polar coordinate system and in the global Cartesian coordinate system with the effects of shear deformation and rotary inertia. Some numerical examples are performed to verify the element formulation and its analysis capability. The comparison of the FEM results with the theoretical ones shows that the element can describe quite efficiently and accurately the in-plane motion of thin circular beams. The stiffness and the mass matrices with respect to the coefficient vector of shape functions are presented in appendix to be utilized directly in applications without any numerical integration for their formulation.     

Large amplitude free vibration analysis of Timoshenko beams using a relatively simple finite element formulation

Free vibration analysis of uniform isotropic Timoshenko beams with geometric nonlinearity is investigated through a relatively simple finite element formulation, applicable to homogenous cubic nonlinear temporal equation (homogenous Duffing equation). Geometric nonlinearity is considered using von-Karman strain displacement relations. The finite element formulation begins with the assumption of the simple harmonic motion and is subsequently corrected using the harmonic balance method. Empirical formulas for the non-linear to linear radian frequency ratios, for the boundary conditions considered, are presented using the least square fit from the solutions of the same obtained for various central amplitude ratios. Numerical results using the empirical formulas compare very well with the results available from the literature for the classical boundary conditions such as the hinged–hinged, clamped–clamped and clamped–hinged beams. Numerical results for the beams with non-classical boundary conditions such as the hinged-guided and clamped-guided, hitherto not studied, are also presented.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

Free vibration analysis of aboveground LNG-storage tanks by the finite element method

Recently, in proportion to the increase of earthquake occurrence-frequency and its strength in the countries within the circum-pan Pacific earthquake belt, a concept of earthquake-proof design for huge structures containing liquid has been growing up. This study deals with the refinement of classical numerical approaches for the free vibration analysis of separated structure and liquid motions. According to the liquid-structure interaction, LNG-storage tanks exhibit two distinguished eigenmodes, the sloshing mode and the bulging mode. For the sloshing-more analysis, we refine the classical rigid-tank model by reflecting the container flexibility. While, for the bulging-mode analysis, we refine the classical uncoupled structural vibration system by taking the liquid free-surface fluctuation into consideration. We first construct the refined dynamic models for both problems, and present the refined numerical procedures. Furthermore, in order for the efficient treatment of large-scale matrices, we employ the Lanczos iteration scheme and the frontal-solver for our test FEM program. With the developed program we carry out numerical experiments illustrating the theoretical results.     

Free vibration and dynamic response analysis by Petrov-Galerkin natural element method

In this paper, a Petrov-Galerkin natural element method (PG-NEM) based upon the natural neighbor concept is presented for the free vibration and dynamic response analyses of twodimensional linear elastic structures. A problem domain is discretized with a finite number of nodes and the trial basis functions are defined with the help of the Voronoi diagram. Meanwhile, the test basis functions are supported by Delaunay triangles for the accurate and easy numerical integration with the conventional Gauss quadrature rule. The numerical accuracy and stability of the proposed method are verified through illustrative numerical tests.     

Free vibration analysis of arbitrary shaped thick plates by differential cubature method

In this paper, a new numerical solution technique, the differential cubature method, is applied to solve the free vibration problems of arbitrary shaped thick plates. The basic idea of the differential cubature method is to express a linear differential operation such as a continuous function or any order of partial derivative of a multivariable function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. By using the differential cubature procedure, the governing differential equations and boundary conditions are transformed into sets of linear homogeneous algebraic equations. This is an eigenvalue problem, of which the eigenvalues can be calculated numerically. The subspace iterative method is employed in search of the free vibration frequency parameters. Detailed formulations are presented, and the method is examined here for its suitability for solving the vibration problems of moderately thick plates governed by Mindlin shear deformation theory. The applicability, efficiency and simplicity of the method are demonstrated through solving some example plate vibration problems of different shapes. The numerical accuracy of the method is ascertained by comparing the vibration frequency solutions with those of existing literatures.     

Free vibration analysis of joined conical-cylindrical shells by matched Fourier-Chebyshev collocation method

A free vibration analysis of joined conical-cylindrical shells based on the first order shear deformation theory is developed. Deflections and rotations are represented by the expansions of Chebyshev polynomials and Fourier series. Equations of motion are collocated to yield the system of algebraic equations. Boundary conditions and compatibility conditions are considered as side constraints, and the set of algebraic equations is condensed so that the number of degrees of freedom matches the number of expansion coefficients. Numerical examples are provided for a joined conical-cylindrical shell, a complete conical shell attached to a cylindrical shell and a hermetic can.     

Free vibration analysis of point-supported plates by vibration testing technique

Sweep sine-wave testing is applied for the identification of dynamic characteristics of a four-point-supported square plate with free edges. The idea behind the method is to make use of free vibration time-response data such as acceleration to determine the natural frequencies and associated mode shapes. To compare with the theoretical results, detailed experimental results have been obtained for the various support locations lying at specified positions along the plate diagonals. The natural frequencies and associated mode shapes for the first five modes have been predicted, and the variation of natural frequencies with various support positions have been analyzed for the first symmetric and antisymmetric modes. It is found that the experimental results are generally in reasonable agreement with the theoretical ones.