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 simple finite element analysis of axisymmetric vibration of annular and circular plates

The axisymmetric vibration of annular and circular plates, isotropic or polar orthotropic, is analysed by an axisymmetric finite element. The features of the present study are that: (i) the formulation of the finite element is based on elasticity theory and has no assumptions as in the conventional plate theory-based analysis, yet is still simple and ready for use, and (ii) the boundary conditions are satisfied exactly and the significant effect of boundary conditions on the vibration frequency is demonstrated. Comparisons with alternative solutions show the accuracy of the present approach and the inadequacy of conventional methods in dealing with the vibration of annular and circular plates with simply supported boundary conditions.     

A finite element beam analysis by interface matching

An alternative way of finite element beam analysis is presented. The beam deflection in an element is represented by the sum of general solution and particular solution. The general solution is approximated by using Hermite polynomials and the particular solution is obtained by applying zero boundary conditions at element boundaries. The inter-element stiffness matrices are obtained by requiring the continuity of moment and shear force across element boundaries. The inter-element stiffness matrices do not overlap each other to form the global stiffness matrix. The boundary conditions are explicitly specified. Numerical examples are provided for various boundary conditions and load conditions.

     

A mixed-hybrid finite element for three-dimensional isotropic helical beam analysis

This paper presents a two node finite element with six degrees of freedom per node, able to model the behaviour of a three-dimensional isotropic helical beam with two constant radii of curvature (curvature and torsion radii). The formulation, which includes the shear strain effects, is based on the assumed resultant forces hybrid approach. The resulting forces approximation verifies exactly the equilibrium equations. Numerical results of samples and distribution of generalized forces along the helical beam are presented. Comparison with other models indicates that the presented element gives exact solutions.     

A finite element approach of thin film lubrication in circular EHD contacts

So far, most of the numerical studies concerning elastohydrodynamic point contact lubrication have been realized using finite differences. This paper describes a finite element approach for the modelling of both, thick and thin film lubrication in circular contacts under isothermal elastohydrodynamic regime. Multigrid techniques are used to accelerate the convergence and reduce calculation time. The use of the finite element method (FEM) offers two advantages: first, the existence of high level ready-to-use Finite Element commercial software which reduces the time spent in implementing the method and second, the easier adaptability to different and various physical models such as lubricant rheology, starvation effects, thermal effects, non-Newtonian behaviour, etc. The numerical model is presented in detail and an example of its adaptation taking into account non-Newtonian effects is shown. Finally, a comparison is made between Newtonian and non-Newtonian solutions, exhibiting differences obtained when using both models, especially at high shear rates where the film thickness and the friction coefficients are overestimated by a Newtonian approach.     

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.     

A new beam finite element for the analysis of functionally graded materials

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams.     

A new hybrid-mixed composite laminated curved beam element

In this study, we piesent a new efficient hybrid-mixed composite laminated curved beam element The present element, which is based on the Helhnger-Reissnet vatiational principle and the first-order shear deformation lamination theory, employs consistent stress parameters coriespondmg to cubic displacement polynomials with additional nodeless degrees in order to lesolve the numerical difficulties due to the spurious constraints The stress parameters are eliminated and the nodeless degrees are condensed out to obtain the (6X6) element stiffness matrix The present study also incorporates the straightforwaid prediction of interlaminar stresses from equilibrium equations Seveial numencal examples confirm the superioi behavior of the present composite laminated curved beam element     

New rational interpolation functions for finite element analysis of rotating beams

A rotating beam finite element in which the interpolating shape functions are obtained by satisfying the governing static homogenous differential equation of Euler–Bernoulli rotating beams is developed in this work. The shape functions turn out to be rational functions which also depend on rotation speed and element position along the beam and account for the centrifugal stiffening effect. These rational functions yield the Hermite cubic when rotation speed becomes zero. The new element is applied for static and dynamic analysis of rotating beams. In the static case, a cantilever beam having a tip load is considered, with a radially varying axial force. It is found that this new element gives a very good approximation of the tip deflection to the analytical series solution value, as compared to the classical finite element given by the Hermite cubic shape functions. In the dynamic analysis, the new element is applied for uniform, and tapered rotating beams with cantilever and hinged boundary conditions to determine the natural frequencies, and the results compare very well with the published results given in the literature.     

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.     

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.     

In-plane vibration analysis of cantilevered circular arc beams undergoing rotational motion

Equations of motion of cantilevered circular arc beams undergoing rotational motion are derived based on a dynamic modeling method developed in this paper. Kane’s method is employed to derive the equations of motion. Different from the classical linear modeling method which employs two cylindrical deformation variables, the present modeling method employs a non-cylindrical variable along with a cylindrical variable to describe the elastic deformation. The derived equations (governing the stretching and the bending motions) are coupled but linear, so they can be directly used for vibration analysis. The coupling effect between the stretching and the bending motions, which could not be considered in the conventional modeling method, is considered in this modeling method. The effects of rotational speed, arc angle, and hub radius ratio on the natural frequencies of the rotating circular arc beam are investigated through numerical analysis.     

双圆弧齿轮传动的有限元热分析

综合利用传热学、摩擦学和齿轮啮合理论,建立双圆弧齿轮热分析计算机模型和有限元模型,运用有限元法,计算在稳定负荷下运转并处于热平衡状态的双圆弧齿轮本体温度和热变形,分析双圆弧齿轮温度场和热变形的变化规律。     

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 circularly curved multi-span Timoshenko beams by the pseudospectral method

The pseudospectral method is applied to the free vibration analysis of circularly curved multi-span Timoshenko beams. Each section of the beam has its own basis functions, and the continuity conditions at the intermediate supports as well as the boundary condition are treated as the constraints of the basis functions so that the number of degrees of freedom matches the number of the pseudospectral expansion coefficients. The computed natural frequencies are compared with those of existing literature, where it is shown that they are in good agreement. Numerical examples are provided for pinned-pinned, clamped-clamped and free-pinned boundary conditions for different numbers of sections and for different thickness-to-length ratios.     

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.     

Three-dimensional vibration analysis of thick, tapered rods and beams with circular cross-section

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, tapered rods and beams with circular cross-section. Unlike conventional rod and beam theories, which are mathematically one-dimensional (1-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the rods and beams are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four- digit exactitude is demonstrated for the first five frequencies of the rods and beams. Novel numerical results are tabulated for nine different tapered rods and beams with linear, quadratic, and cubic variations of radial thickness in the axial direction using the 3-D theory. Comparisons are also made with results for linearly tapered beams from 1-D classical Euler–Bernoulli beam theory.     

A novel modified analytical method and finite element method for vibration analysis of cable-driven parallel robots

Journal of Mechanical Science and Technology - Cable-driven parallel robots (CDPRs) are vulnerable to vibration due to the inevitable flexible properties of the cables. Thus, vibration analysis is...     

Large deflexion, geometrically non-linear finite element analysis of circular arches

This paper presents the results of an investigation into the geometrically nonlinear behaviour of circular arches. A curved element is used, based on satisfying the condition that the circumferential strain and change in curvature, rather than the displacements, should be simple independent functions of the co-ordinate axes. The analysis was carried out using the linearized incremental method based on the mid-increment stiffness in conjunction with the Newton-Raphson iterative technique. A comparison is first made between the results obtained with this element and those using a simple polynomial shape function. The behaviour of a range of arches is then considered and the results are compared and found to agree well with an analytical method. The results include the behaviour of deeper but still shallow arches which exhibit “looping” of the load-deflexion curve, and bifurcation of the equilibrium path into unsymmetric deflexions of the arch.A computer program was developed to allow any of the generalized degrees of freedom which are designated for incrementation to be expressed as either forces, to avoid failure at the vertical tangent to the load deflexion curves, or as generalized displacements to avoid failure at the horizontal tangent. The program also allows the quantity subjected to incrementation to be changed as necessary to follow complex load-deflexion equilibrium paths.     

Dynamic finite element method for generally laminated composite beams

A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.