Vibration Of Euler-Bernoulli Beam Research Articles

Eigenvalues and eigenfunctions of nonuniform beams with a new method based on perturbation

By using a new method based on perturbation method and Fredholm theory, the eigenvalues and eigenfunctions of nonuniform Euler-Bernoulli beam vibration are solved analytically. The eigenvalues and eigenfunctions under boundary conditions are given for the tension beam problem which considers the variation of tension in vertical marine riser from top to bottom caused by gravity effect. The asymptotic analytical results are compared with classical results to verify the effectiveness of the new method based on perturbation method. The influence of the change of the bottom tension on the perturbation method is analyzed. The results show that: for each order of natural frequency of riser, the natural frequency will gradually increase with the increase of bottom tension, and the obtained natural frequency will become more and more accurate; when the bottom tension is constant, the new method based on perturbation method in this paper can give good accuracy for higher order natural frequency rather than lower order.

Free vibration and stability analysis of inclined pipes conveying gas-liquid slug flow

In this paper, the free vibration and stability of inclined pipes conveying gas-liquid slug flow are investigated, by considering the intermittency of slug flow and gravity effects of the system. A new lateral motion model has been established based on the stable slug flow dynamics model and the Euler–Bernoulli beam vibration model and is then truncated by the Galerkin approach. The natural frequencies, critical gas velocities, and responses to flow conditions are comprehensively investigated to show the basic dynamic characteristics of inclined pipes. The results show that the gravity effects caused by the inclined angle have a significant effect on the vibration characteristics. The system is less stable with lower critical gas velocity and natural frequency with the increase of inclined angle, due to the gravity-induced compression. For a given inclined angle, the system dynamic responses to superficial gas/liquid velocities are directly related to the mass of liquid per unit length. In addition, the fluctuation of natural frequency caused by the intermittency of slug flow decreases with the increase of inclined angle. This study contributes to understanding the complex dynamic behavior of inclined pipes conveying slug flow and promoting pipeline safety.

Natural characteristics for transverse vibration of Euler Bernoulli beams with variable end constraints

Abstract The transverse vibration of Euler Bernoulli beam with mass of ends and springs is studied. The exact frequency equation is derived and natural frequencies and the corresponding mode shapes are calculated. With the linearly increasing mass of ends, natural frequencies and the rate of frequency change of the beam system initially decrease sharply and then level out, which demonstrates that the beam system is transforming from the free beam to the pinned beam. When the springs are added at two tips, the natural characteristics of the beam are affected by mass of ends and spring stiffness. If the added mass has much lower magnitude than that of the beam, the stiffness of springs exerts major impact on the increase of natural frequencies. While the added mass of ends is increased to the same magnitude of the beam, the natural characteristics of the beam are determined by both the mass of ends and spring stiffness. As the growing magnitude of added mass, mass of ends performs a dominant role in decreasing the natural frequencies. Therefore, spring stiffness and mass of ends should be first considered to establish different dynamic models accurately.

Local/nonlocal mixture integral models with bi-Helmholtz kernel for free vibration of Euler-Bernoulli beams under thermal effect

We present predictive models of the free vibration of Euler-Bernoulli beams subjected to a uniformly thermal environment using two-phase local/nonlocal mixture theory of strain- and stress-driven types. The equation of motion and standard boundary conditions are derived via Hamilton's principle and the constitutive equation is expressed in local/nonlocal mixed integral form with bi-Helmholtz kernel. The temperature effect is taken as equivalent to a thermal load taking into account the two-phase nonlocal effects. After transforming the local/nonlocal mixture integral constitutive equation to differential form with four constitutive boundary conditions, the problem is solved numerically via a generalized differential quadrature method (GDQM). We provide a series of numerical examples in which we present predictions of the size-effects on beams with various boundary edges using different local/nonlocal mixture models. The influence of the two-phase thermal load on the natural frequencies of the beam is also examined. Furthermore, it can be observed that the normalized frequency value varies consistently with the increase of the order of the vibration mode.

Control of Vibrations of a Beam with Nonlocal Boundary Conditions

The control problem the first eigenvalues of the Euler-Bernoulli beam vibration with hinged and clamped fixings at both ends is considered. The control is based on Kanguzhin’s algorithm through integral perturbations of the boundary conditions of the original problem. Conditions for the boundary parameters for controlling the first eigenvalues are found.

Experimental investigation on vortex-induced force of a Steel Catenary Riser under in-plane vessel motion

A method to identify vortex-induced forces and coefficients from measured strains of a Steel Catenary Riser (SCR) undergoing vessel motion-induced Vortex-induced Vibration (VIV) is proposed. Euler–Bernoulli beam vibration equations with time-varying tension is adopted to describe the out-of-plane VIV responses. Vortex-induced forces are reconstructed via inverse analysis method, and the Forgetting Factor Least Squares (FF-LS) method is employed to identify time-varying vortex-induced force coefficients, including excitation coefficients and added mass coefficients. The method is verified via a finite element analysis procedure in commercial software Orcaflex. The time-varying excitation coefficients and added mass coefficients of an SCR undergoing vessel motion-induced VIV are investigated. Results show that vessel motion-induced VIV is excited at the middle or lower part of the SCR and in the acceleration period of in-plane velocity, where most of the excitation coefficients are positive, while during the deceleration period, the excitation coefficients becomes too small to excite VIVs. Parameter γ [1] has strong correlation with excitation coefficients. In addition, time-varying tensions contribute significantly to the variations of added mass coefficients under the condition that the ratio of dynamic top tension to pretension exceeds the range of 0.7–1.3. Moreover, chaotic behaviors are observed in vortex-induced force coefficients and are more evident with the increase of vessel motion velocity. This behavior may attribute to the randomness existing in in-plane velocity and its coupling with out-of-plane vibrations.

Dynamical shakedown analysis of high-rise tower structure

PurposeHigh-rise tower structures supported by side frame structure and viscous damper in chemical industry can produce plasticity under dynamic loads, such as wind and earthquake, which will heavily influence the long-term safety operation. This paper aims to systematically study the optimization design of these structures by free vibration and dynamic shakedown analysis.Design/methodology/approachThe transfer matrix method and Euler–Bernoulli beam vibration are used to study the free vibration characteristic of the simplified high-rise tower structure. Then the extended stress compensation method is used to construct the self-equilibrated stress by using the dynamic load vertexes and the lower bound dynamic shakedown analysis for the structure with viscous damper. Using the proposed method, comprehensive parametric studies and optimization are performed to examine the shakedown load of high-rise tower with various supported conditions.FindingsThe numerical results show that the supported frame stiffness, attached damper or spring parameters influence the free vibration and shakedown characters of high-rise tower very much. The dynamic shakedown load is lowered down quickly with external load frequency increasing to the fundamental natural frequency of the structure under spring supported condition, while changed little with the damping connection. The optimized location and parameter of support are obtained under dynamical excitations.Research limitations/implicationsIn this study, the high-rise tower structure is simplified as a cantilever beam supported by a short cantilever beam and a damper under repeated dynamic load, and linear elasticity for solid is assumed for free vibration analysis. The current analysis does not account for effects such as large deformation, stochastic external load and nonlinear vibration conditions which will inevitably be encountered and affect the load capacity.Originality/valueThis study provides a comprehensive method for the dynamical optimization of high-rise tower structure by combining free vibration and shakedown analysis.

Estudio de vibraciones en componentes estructurales con presencia de fisuras

Los componentes mecánicos y estructurales, a causa de la presencia de fisuras, son susceptibles de presentar fallas en servicio; ocasionando daños catastróficos e irreversibles. Este trabajo, se centra en el estudio de un método analítico de identificación de fisura basado en medición de frecuencias naturales de vibración. Se obtuvo, mediante la aplicación del cálculo de variaciones, la solución exacta del problema de vibraciones de vigas Euler Bernoulli de sección constante y material isótropo y homogéneo. Se desarrolló un método analítico de detección de fisuras en vigas. El mismo consiste en resolver el problema inverso de obtener la posición y profundidad de la fisura, a partir de conocer los 3 primeros valores del coeficiente adimensionales de frecuencias. Se realizó un análisis de sensibilidad del método para la condición articulado-articulado, considerando una incertidumbre máxima en la medición de la frecuencia del 2%. Los resultados presentados permiten determinar la capacidad de detección del método,para esa incertidumbre, para cualquier ubicación y profundidad.

Modal Analysis of Vibration of Euler-Bernoulli Beam Subjected to Concentrated Moving Load

This paper investigates the modal analysis of vibration of Euler-Bernoulli beam subjected to concentrated load. The governing partial differential equation was analysed to determine the behaviour of the system under consideration. The series solution and numerical methods were used to solve the governing partial differential equation. The results revealed that the amplitude increases as the length of the beam increases. It was also found that the response amplitude increases as the foundation increases at fixed length of the beam.

Transverse free vibration of Euler-Bernoulli beam with pre-axial pressure resting on a variable Pasternak elastic foundation under arbitrary boundary conditions

Transverse free vibration of Euler-Bernoulli beam with pre-axial pressure resting on a variable Pasternak elastic foundation under arbitrary boundary conditions

Modified Multi-level Residue Harmonic Balance Method for Solving Nonlinear Vibration Problem of Beam Resting on Nonlinear Elastic Foundation

Nonlinear vibration behavior of beam is an important issue of structural engineering. In this study, a mathematical modeling of a forced nonlinear vibration of Euler-Bernoulli beam resting on nonlinear elastic foundation is presented. The nonlinear vibration behavior of the beam is investigated by using a modified multi-level residue harmonic balance method. The main advantage of the method is that only one nonlinear algebraic equation is generated at each solution level. The computational time of using the new method is much less than that spent on solving the set nonlinear algebraic equations generated in the classical harmonic balance method. Besides the new method can generate higher-level nonlinear solutions neglected by previous multi-level residue harmonic balance methods. The results obtained from the proposed method compared with those obtained by a classical harmonic balance method to verify the accuracy of the method which shows good agreement between the proposed and classical harmonic balance method. Besides, the effect of various parameters such as excitation magnitude, linear and nonlinear foundation stiffness, shearing stiffness etc. on the nonlinear vibration behaviors are examined

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.

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

Nonlinear Free Transverse Vibration Analysis of Beams Using Variational Iteration Method

In this study, Variational Iteration Method is employed so as to investigate the linear and non-linear transverse vibration of Euler-Bernoulli beams. This method is a very powerful approach with a high convergence speed providing an analytical and semi-analytical solution to the linear equations and is able to be extended to present semi-analytical solution to the non-linear ones. In this method, firstly, Lagrange`s multiplier and Initial Function should be chosen. The suitable choice of these two elements would effectively affect the convergence speed. In this attempt, in addition to presenting a discussion on how to choose these two functions appropriately, the calculated frequencies in the non-linear state are compared with the available results in the literature, and the accuracy and convergence speed are studied, as well.

Optimal control of transverse vibration of Euler-Bernoulli beam with multiple dynamic vibration absorbers using Taguchi's method

Vibration absorbers are frequently used to suppress the excessive vibrations in structural systems. In this paper, an imposing nodes technique is applied for vibration suppression of Euler-Bernoulli beams subjected to forced harmonic excitations by means of multiple dynamic vibration absorbers. A procedure based on Taguchi's method is proposed to determine the optimum absorber parameters to suppress the vibration amplitude of the beams. Numerical tests are performed to show the effectiveness of the proposed procedure.

Nonlinear free and forced vibration of Euler-Bernoulli beams resting on intermediate flexible supports

This paper deals with the geometrically nonlinear free and forced vibration analysis of a multi-span Euler Bernoulli beam resting on arbitrary number N of flexible supports, denoted as BNIFS, with general end conditions. The generality of the approach is based on use of translational and rotational springs at both ends, allowing examination of all possible combinations of classical beam end conditions, as well as elastic restraints. First, the linear case is examined to obtain the mode shapes used as trial functions in the nonlinear analysis. The beam bending vibration equation is first written in each span. Then, the continuity requirements at each elastic support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. Numerical results are given to illustrate the effects of the support stiffness and locations on the natural frequencies and mode shapes of the BNIFS. The nonlinear theory is then developed, based on the Hamilton’s principle and spectral analysis. The nonlinear beam transverse displacement function is defined as a linear combination of the linear modes calculated before. The problem is reduced to solution of a non-linear algebraic system using numerical or analytical methods. The nonlinear algebraic system is solved using an explicit method developed previously (second formulation) leading to the amplitude dependent nonlinear fundamental mode of the BNIFS.

Comparative vibration study of EN 8 and EN 47 cracked cantilever beam

Since earlier days, most of the failures encountered by the structures or machines are mainly due to material fatigue. The dynamic behaviour of the beam may change when cracks begin to appear in it. Knowledge of these changes in the dynamic individualism is important in crack detection as well as in structure or in machine design. This paper deals with systematic study on the free vibration of Euler-Bernoulli beam containing open edge transverse cracks. In this study, two springs steel materials (EN 8 and EN 47) are considered. The effect of the top side cracks and bottom side cracks on the natural frequency of a cantilever beam is discussed. The natural frequency of a cracked case cantilever beam is investigated numerically using FE analysis software ANSYS. Experimental work is done by using DeweFRF to investigate the natural frequency of cracked beams for strong validation of the numerical results. The results of this study suggest that the average value of natural frequencies for all top side cracked beams are identical to the average value of natural frequency for all bottom side cracked beams. This is true for both EN 47 and EN 8. Hence, it is clear that the dynamic characteristic (natural frequency) is not changing, when same configuration of cracks is either on top or bottom side of the beam. The natural frequencies for EN 8 material are comparatively on higher side than EN 47 material for the same crack configurations. In most of the cracked cases, the damping effect of EN 47 is greater than EN 8. It is also found that as crack location increases at constant crack depth, then natural frequency increases. At the last location, as crack depth increases, natural frequencies almost remain same. It is observed that, the presence of top side crack and bottom side crack of the same configuration in the cantilever beam is not a function of natural frequency, when cantilever beam is of a square cross section.

On Newton-Raphson formulation and algorithm for displacement based structural dynamics problem with quadratic damping nonlinearity

Quadratic damping nonlinearity is challenging for displacement based structural dynamics problem as the problem is nonlinear in time derivative of the primitive variable. For such nonlinearity, the formulation of tangent stiffness matrix is not lucid in the literature. Consequently, ambiguity related to kinematics update arises when implementing the time integration-iterative algorithm. In present work, an Euler-Bernoulli beam vibration problem with quadratic damping nonlinearity is addressed as the main source of quadratic damping nonlinearity arises from drag force estimation, which is generally valid only for slender structures. Employing Newton-Raphson formulation, tangent stiffness components associated with quadratic damping nonlinearity requires velocity input for evaluation purpose. For this reason, two mathematically equivalent algorithm structures with different kinematics arrangement are tested. Both algorithm structures result in the same accuracy and convergence characteristic of solution.

Analytical Approximation of Nonlinear Vibration of Euler-Bernoulli Beams

In this paper, the Homotopy Analysis Method (HAM) with two auxiliary parameters and Differential Transform Method (DTM) are employed to solve the geometric nonlinear vibration of Euler-Bernoulli beams subjected to axial loads. A second auxiliary parameter is applied to the HAM to improve convergence in nonlinear systems with large deformations. The results from HAM and DTM are compared with another popular numerical method, the shooting method, to validate these two analytical methods. HAM and DTM show excellent agreement with numerical results (the maximum errors in our calculations are about 0.002%), and they additionally provide a simple way to conduct a parametric analysis with different physical parameters in Euler-Bernoulli beams. To show the benefits of this method, the effect of different physical parameters on the amplitude is discussed for a cantilever beam with a cyclically varying axial load.

Non-symmetric and chaotic vibrations of Euler-Bernoulli beams under harmonic and noisy excitations

In this paper we study non-linear dynamics of flexible Euler-Bernoulli beams subjected to harmonic load and white noise. We report that in the case of continuous mechanical systems like the studied Euler-Bernoulli beams the action of white noise yieldes novel and unexpected phenomena, i.e. symmetric transversal loads and symmetric boundary conditions imply non-symmetric beam vibration. In addition, regions of non-symmetric beam vibrations for two values of white noise intensity are given and the charts of regular and chaotic vibrations of the studied beams are presented.