Vibration Analysis Of Euler-Bernoulli Beams Research Articles

Flexural-torsional-coupled vibration analysis of Euler-Bernoulli beam by using the differential transform method

This study introduces an analytical solution method which is called the differential transform method (DTM) to analyze the free vibration response of a beam which features material coupling between flapwise bending and torsional vibrations. The governing differential equations of motion are solved by applying DTM. The mode shapes and the natural frequencies are obtained and plotted for bending and torsional behavior, separately. Application of the model solved by DTM is investigated for a typical cantilever aircraft wing when flexural and torsional motions are coupled and the bending-torsion coupling effect is ignored, separately, using a numeric computing environment, MATLAB.

Free vibration analysis of Euler-Bernoulli beams with non-ideal clamped boundary conditions by using Padé approximation

Using the non-ideal boundary condition model, which is a linear combination of ideal simply supported and ideal clamped boundary conditions, the equation governing the free vibration of Euler-Bernoulli beams produces nonlinear rational functions that relate natural frequencies with the weighting factors of the non-ideal boundary condition model. The natural frequencies in practice are numerically computed by using a standard root-finding method with suitable initial guesses. In the present study, the nonlinear rational functions are approximated using Pade approximants to get analytical formulations of natural frequencies as functions of the weighting factors. Numerical examples are provided for cantilever and beams clamped at both ends with non-ideal boundary conditions. The formulas in most cases are accurate enough to get the natural frequencies up to two-digit accuracy. Those approximations can be easily utilized as starting values in the root finding method to avoid ambiguities in selecting initial guesses.

Application of the two-parameter single shooting method for natural vibration analysis of Euler-Bernoulli beams

The paper presents a new method of calculating the natural frequencies and modes of Euler-Bernoulli beams. The formulated algorithm is based on the fourth order homogeneous ordinary differential equation of motion that allows different support conditions to be considered. The idea of the method is to replace a two-point boundary value problem with an initial value problem. Distinct from the classic approach, where only one unknown parameter is considered, the method enables problems with two parameters to be solved. The proposed two-parameter version of the single shooting method is verified on the example of a non-prismatic simply supported beam and a prismatic propped cantilever beam resting on Winkler foundation and loaded by axial force. The obtained results confirm the correctness and high effectiveness of the presented method.

Dynamic interfacial fracture of a double cantilever beam

Assessment of the energy release rate (ERR) of layered material structures with account for dynamic and vibration effects is important for understanding and predicting fracture behavior in various engineering applications. In this work, the pure-mode-I interfacial fracture behavior of a symmetric double cantilever beam (DCB) under constant-rate opening displacement is studied using a dynamics and vibration analysis of Euler-Bernoulli beams, and the ERR is derived. Furthermore, a ‘dynamic factor’ that quantifies the dynamic effect in relation to the static component of ERR is defined. The resulting expressions are relatively short, mathematically elegant and convenient-to-use by engineers and researchers, which increases their usefulness. It is found that the dynamic factor is a function of the characteristic time only, and that this is an intrinsic property of DCB structures. An approximate method is also proposed to predict the crack extension. Predictions of ERR and crack extension are in good agreement with results from numerical results with finite-element method (FEM) simulations. Using only the first vibration mode is sufficient to capture the amplitude and frequency of ERR variation predicted by the FEM. Using higher-order vibration modes causes divergence in the amplitude of ERR oscillation; this is due to the limitation of Euler-Bernoulli beams in vibration analysis.

Accurate assessment of natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by adaptive generalized finite element method

Purpose – The purpose of this paper is devoted to present an accurate assessment for determine natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by an adaptive generalized finite element method (GFEM). The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. Design/methodology/approach – The variational problem of free vibration is formulated and the main aspects of the adaptive GFEM are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform Euler-Bernoulli beams are checked. The application of this technique in a frame is also presented. Findings – The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. The GFEM, which was conceived on the basis of the partition of unity method, allows the inclusion of enrichment functions that contain a priori knowledge about the fundamental solution of the governing differential equation. The proposed enrichment functions are dependent on the geometric and mechanical properties of the element. This approach converges very fast and is able to approximate the frequency related to any vibration mode. Originality/value – The main contribution of the present study consisted in proposing an adaptive GFEM for vibration analysis of Euler-Bernoulli uniform and non-uniform beams and frames. The GFEM results were compared with those obtained by the h and p-versions of FEM and the c-version of the CEM. The adaptive GFEM has shown to be efficient in the vibration analysis of beams and has indicated that it can be applied even for a coarse discretization scheme in complex practical problems.

Vibration Analysis of Euler-Bernoulli Beams Partially Immersed in a Viscous Fluid

The vibrational characteristics of a microbeam are well known to strongly depend on the fluid in which the beam is immersed. In this paper, we present a detailed theoretical study of the modal analysis of microbeams partially immersed in a viscous fluid. A fixed-free microbeam vibrating in a viscous fluid is modeled using the Euler-Bernoulli equation for the beams. The unsteady Stokes equations are solved using a Helmholtz decomposition technique in a two-dimensional plane containing the microbeams cross sections. The symbolic software Mathematica is used in order to find the coupled vibration frequencies of beams with two portions. The frequency equation is deduced and analytically solved. The finite element method using Comsol Multiphysics software results is compared with present method for validation and an acceptable match between them was obtained. In the eigenanalysis, the frequency equation is generated by satisfying all boundary conditions. It is shown that the present formulation is an appropriate and new approach to tackle the problem with good accuracy.

Vibration of a circular beam with variable cross sections using differential transformation method

In this paper, an application of differential transformation method (DTM) is applied on free vibration analysis of Euler-Bernoulli beam. This beam has variable circular cross sections. Natural frequencies and corresponding mode shapes are obtained for three cases of cross section and boundary conditions. MATLAB program is used to solve the differential equation of the beam using DTM. Comparison of the obtained results with the previous solutions proves the accuracy and versatility of the presented problem.

Transverse vibration analysis of Euler-Bernoulli beam carrying point masse submerged in fluid media

In the present paper, an analytical method is developed to investigate the effects of added mass on natural frequencies and mode shapes of Euler-Bernoulli beams carrying concentrated masse at arbitrary position submerged in a fluid media. A fixed-fixed beams carrying concentrated masse vibrating in a fluid is modeled using the Bernoulli-Euler equation for the beams and the acoustic equation for the fluid. The symbolic software Mathematica is used in order to find the coupled vibration frequencies of a beams with two portions. The frequency equation is deduced and analytically solved. The finite element method using Comsol Multiphysics software results are compared with present method for validation and an acceptable match between them were obtained. In the eigenanalysis, the frequency equation is generated by satisfying all boundary conditions. It is shown that the present formulation is an appropriate and new approach to tackle the problem with good accuracy.

A transfer matrix method for free vibration analysis of Euler-Bernoulli beams with variable cross section

To perform vibration analysis, many structures could be modeled entirely or partially as beams with variable or constant cross section. In this study, the differential equations for free bending vibrations of straight beams with variable cross section are solved using Bessel’s functions. Then, the general equations for one-step conical beams are used together with corresponding equations of cylindrical beams to model multi-step beams with various boundary conditions. The natural frequencies of a cantilever conical beam, calculated using its derived field matrix, were shown to be close to those obtained experimentally. In addition, the influence of geometrical factors upon the value of fundamental frequencies is investigated for both cantilever and clamped-clamped conical beams. It was shown numerically that changing a cantilever cylindrical beam into a conical beam by reducing the diameter of the free end increases its fundamental natural frequency even though its rigidity decreases. This property was not observed in a clamped-clamped conical beam. Finally, the effectiveness of a proposed transfer matrix on the calculation of natural frequencies for a multi-step beam was verified. It is believed that the current method could help in the design of various structures, including beams with variable cross section.

Vibration analysis of Euler-Bernoulli beams in multiple steps and different shapes of cross section

Stepped beams with elastic end supports have been extensively investigated due to their importance in structural engineering fields, including active structures, structural elements with integrated piezoelectric materials, shaft-disc system components, turbomachinery blades and many other structural configurations. Considering the importance of the use of discontinuous structures in the engineering, the authors propose a mathematic modeling which advantage is that the results are independent of the degree of mesh refinement. The analysis is based on the classical Euler-Bernoulli beam theory. In comparison with the published literature on the transverse vibration of single cross section change beams, there are relatively few works covering beam vibration when there is more than one change in the beam cross section. In the present study, the natural frequencies and the mode shapes of beams with variable geometry or material discontinuities are investigated. The mode shapes of a beam with multiple step changes in cross section are discussed theoretically and experimentally. Numerical results obtained by Euler-Bernoulli beam theory are compared with experimental results.

On the Use of Differential Transformation Method for Free Vibration Analysis of Euler-Bernoulli Beams with General Elastically End Restraints

On the Use of Differential Transformation Method for Free Vibration Analysis of Euler-Bernoulli Beams with General Elastically End Restraints

Transverse Vibration Analysis of Euler-Bernoulli Beams Carrying Concentrated Masses with Rotatory Inertia at Both Ends

Transverse vibration analysis is presented for Euler-Bernoulli beams carrying concentrated masses and taking into account their rotatory inertia at both ends. The dimensionless eigenfunctions for the problems are first obtained using the differential equations of motion and considering translational and rotatory springs at both ends. A numerical technique, the Newton–Raphson algorithm, is then used to solve vibration eigenfunctions of the beams. Finally, the influences of different non-dimensional parameters on frequencies are discussed.

A Semianalytical Method for Nonlinear Vibration of Euler-Bernoulli Beams with General Boundary Conditions

This paper presents a new semianalytical approach for geometrically nonlinear vibration analysis of Euler-Bernoulli beams with different boundary conditions. The method makes use of Linstedt-Poincaré perturbation technique to transform the nonlinear governing equations into a linear differential equation system, whose solutions are then sought through the use of differential quadrature approximation in space domain and an analytical series expansion in time domain. Validation of the present method is conducted in numerical examples through direct comparisons with existing solutions, showing that the proposed semianalytical method has excellent convergence and can give very accurate results at a long time interval.

Free vibration analysis of Euler-Bernoulli beam with double cracks

In this paper, the influence of two open cracks on the dynamic behavior of a double cracked simply supported beam is investigated both analytically and experimentally. The equation of motion is derived by using the Hamilton’s principle and analyzed by numerical method. The simply supported beam is modeled by the Euler-Bemoulli beam theory. The crack sections are represented by a local flexibility matrix connecting three undamaged beam segments. The influences of the crack depth and the position of each crack on the vibration mode and the natural frequencies of a simply supported beam are analytically clarified for the single and double cracked simply supported beam. The theoretical results are also validated by a comparison with experimental measurements.

미분변환법과 일반화 미분구적법을 이용한 탄성 지반상의 열림 균열을 가진 Euler-Bernoulli 보의 진동 해석

The main purpose of this paper is to apply differential transformation method(DTM) and generalized differential quadrature method(GDQM) to vibration analysis of Euler-Bernoulli beam with open cracks on elastic foundation. In this paper the concepts of DTM and GDQM were briefly introduced. The governing equation of motion of the beam with open cracks on elastic foundation is derived. The cracks are modeled by massless substitute spring. The effects of the crack location, size and the foundation constants, on the natural frequencies of the beam, are investigated. Numerical calculations are carried out and compared with previous published results.