Euler-Bernoulli Beam Problems Research Articles

On the energy decay of a nonlinear time-fractional Euler–Bernoulli beam problem including time-delay: theoretical treatment and numerical solution techniques

On the energy decay of a nonlinear time-fractional Euler–Bernoulli beam problem including time-delay: theoretical treatment and numerical solution techniques

Selection of Radial Basis Functions for the Accuracy of Meshfree Galerkin Method in Rotating Euler-Bernoulli Beam Problem.

In this work, the radial basis function approximations are used to improve the accuracy of meshfree Galerkin method. The method is applied to the free vibration problems of non-rotating and rotating Euler–Bernoulli beams. The stiffness and mass matrices are derived by using conventional methods. In this meshfree method, only six nodes are considered within a single sub-domain. The parameters are varied for different approximations; the results are obtained with different approximations and found accurate. Two new basis function have been developed which are relatively accurate than conventional basis function: the first new basis function is obtained by multiplication of linear function to radial basis function and second new basis function is obtained by multiplying cubuic radial basis function to Gaussian radial basis function. The first few modes show same result that is available in literature using finite element method and higher modes are found very accurate as well. The result are found to be more accurate for first three modes of non-rotating and rotating Euler–Bernoulli beams where the cantilever beam boundary conditions are used; the first three modes do not change with the change in the parameter c of radial basis function.

Error propagation dynamics of velocimetry-based pressure field calculations (2): on the error profile

A recent study investigated the propagation of error in a velocimetry-based pressure (V-pressure) field reconstruction problem by directly analyzing the properties of the pressure Poisson equation (Pan et al 2016 Meas. Sci. Technol. 27 084012). In the present work, we extend these results by quantifying the effect of the error profile in the data field (shape/structure of the error in space) on the resultant error in the reconstructed pressure field. We first calculate the mode of the error in the data that maximizes error in the pressure field, which is the most dangerous error (called the worst error in the present work). This calculation of the worst error is equivalent to finding the principle mode of, for example, an Euler–Bernoulli beam problem in one-dimension and the Kirchhoff–Love plate in two-dimensions, thus connecting the V-pressure problem from experimental fluid mechanics to buckling elastic bodies from elastic mechanics. Taking advantage of this analogy, we then analyze how the error profile (e.g. spatial frequency of the error and the location of the most concentrated error) in the data field coupled with fundamental features of the flow domain (i.e. size, shape, and dimension of the domain, and the configuration of boundary conditions) significantly affects the error propagation from data to the reconstructed pressure. Our analytical results lend to practical applications in two ways. First, minimization of error propagation can be achieved by avoiding low frequency error profiles in data similar to the worst case scenarios and error concentrated at sensitive locations. Second, small amounts of the error in the data, if the error profile is similar to the worst error case, can cause significant error in the reconstructed pressure field; such a synthetic error can be used to benchmark V-pressure algorithms.

New non-destructive method for testing the strength of cement mortar material based on vibration frequency of steel bar: Theory and experiment

Timely and accurately obtaining the strength of pouring material, e.g., concrete, cement mortar, is of great significance for engineering construction. In this paper, a non-destructive, economical and accurate strength detection method that suites for on-site using is proposed for the steel bar cement mortar material. The method based on the relationship between the vibration frequency of the steel bar and the properties of the mortar material, which is obtained by solving the Euler-Bernoulli beam problem. Both Particle Flow Code (PFC) software simulation (calibrated) and Split Hopkinson pressure Bar experiment on test samples of cement mortar and steel bar were performed to verify the theoretically obtained relationship. Studies on samples of various aggregate ratio further confirmed such correspondence. Results show that the dynamic stiffness of the cement mortar material dominates the calculation of the vibration frequency of steel bar, while the combined effect of the density, length, elastic modulus, inertia moment of the steel bar can be safely ignored. A single-valued mapping relation exists in between the dynamic stiffness coefficient and the Uniaxial Compressive Strength (UCS) of the cement mortar sample, i.e., increased dynamic stiffness coefficient with increasing UCS. Both experimental and predicted results showed a linear relationship between the vibration frequency of the steel bar and the strength of the mortar material. Fitted linear relations were proposed with coefficients depending on sample size and aggregate ratio and might serve as a good indicator for the strength of the mortar material. Further studies on the effect of internal defects of the mortar materials as well as on samples of more size and aggregate ratio are required to make the proposed method a practical tool.

A New Construction of Boundary Interpolating Wavelets for Fourth Order Problems

In this article we introduce a new mixed Lagrange–Hermite interpolating wavelet family on the interval, to deal with two types (Dirichlet and Neumann) of boundary conditions. As this construction is a slight modification of the interpolating wavelets on the interval of Donoho, it leads to fast decomposition, error estimates and norm equivalences. This new basis is then used in adaptive wavelet collocation schemes for the solution of one dimensional fourth order problems. Numerical tests conducted on the 1D Euler–Bernoulli beam problem, show the efficiency of the method.

Aeroservoelastic modelling and control of a slender anti-air missile for active damping of longitudinal bending vibrations

Slender anti-air missiles experience longitudinal bending in supersonic flight and yet their autopilots are designed under the rigid-body assumption. Such autopilot design employs notch filters to remove the modal frequencies of the elastic airframe but this approach limits the autopilot bandwidth. In this paper, aeroservoelastic modelling and control of the ASTER 30 missile is proposed to enable autopilot design with extended bandwidth.The aeroservoelastic model combines missile flight dynamics, actuator dynamics and airframe elasticity, the latter focusing on longitudinal bending treated as a continuous Euler–Bernoulli beam problem. The beam is discretised leading to a nodal model and the modal analysis is then performed. The modal model is expressed in the state-space form and its order is reduced to enable optimal sensor placement and active damping control. The aeroservoelastic model of the ASTER 30 missile is further refined for control purposes by optimally choosing actuator inputs together with the number and position of sensors to be mounted on the missile airframe. Once these choices are made, several variants of active vibration damping control are proposed and analysed in order to enable an extended bandwidth for the autopilot by countering the airframe deformation measured by these sensors.

Dynamic analysis of Euler–Bernoulli beam problems using the Generalized Finite Element Method

This work presents dynamic analyses of one-dimensional bar and Euler–Bernoulli beam problems with Generalized Finite Element Method (GFEM). Enrichment monomials are trigonometric and exponential functions. A beam free vibration problem is analyzed to assess the element’s robustness and efficiency. Next, an elastodynamic analysis of a bar is performed using several enrichment levels. Finally, a dynamic elastoplastic analysis of a beam problem is carried out. Error measures and nonlinear strains are estimated. Results from GFEM are compared to results from a conventional FE formulation to show GFEM’s level of efficiency in solving Euler–Bernoulli beam elastic as well as elastoplastic dynamic problems.

A nonlinear planar beam formulation with stretch and shear deformations under end forces and moments

A new nonlinear planar beam formulation with stretch and shear deformations is developed in this work to study equilibria of a beam under arbitrary end forces and moments. The slope angle and stretch strain of the centroid line, and shear strain of cross-sections, are chosen as dependent variables in this formulation, and end forces and moments can be either prescribed or resultant forces and moments due to constraints. Static equations of equilibria are derived from the principle of virtual work, which consist of one second-order ordinary differential equation and two algebraic equations. These equations are discretized using the finite difference method, and equilibria of the beam can be accurately calculated. For practical, geometrically nonlinear beam problems, stretch and shear strains are usually small, and a good approximate solution of the equations can be derived from the solution of the corresponding Euler–Bernoulli beam problem. The bending deformation of the beam is the only important one in a slender beam, and stretch and shear strains can be derived from it, which give a theoretical validation of the accuracy and applicability of the nonlinear Euler–Bernoulli beam formulation. Relations between end forces and moments and relative displacements of two ends of the beam can be easily calculated. This formulation is powerful in the study of buckling of beams with various boundary conditions under compression, and can be used to calculate post-buckling equilibria of beams. Higher-order buckling modes of a long slender beam that have complex configurations are also studied using this formulation.

Superconvergent isogeometric free vibration analysis of Euler–Bernoulli beams and Kirchhoff plates with new higher order mass matrices

A superconvergent isogeometric free vibration analysis is presented for Euler–Bernoulli beams and Kirchhoff plates. This method is featured by new higher order mass matrices. For the 1D Euler–Bernoulli beam problem, it is shown that a new higher order mass matrix can be directly established by optimizing a reduced bandwidth mass matrix. The reduced bandwidth mass matrix is designed based upon the consistent mass matrix and it contains adjustable parameters to be determined via maximizing the order of accuracy of the vibration frequency. As a result, 4th and 6th orders of accuracy are observed for the proposed quadratic and cubic higher order mass matrices, while their corresponding consistent mass matrices are 2nd and 4th order accurate, respectively. Thus the higher order beam mass matrices have more superior frequency accuracy simultaneously with smaller bandwidth compared with their corresponding consistent mass matrices. While for the 2D Kirchhoff plate problem, in order to compute arbitrary frequency in a superconvergent fashion, a mixed mass matrix is formulated through a linear combination of the consistent mass matrix and the reduced bandwidth mass matrix. Then by introducing the wave propagation angle, the higher order plate mass matrix can be rationally derived from the mixed mass matrix. It is proved that the optimal combination parameter for higher order mass matrix depends on the wave propagation angle. Consequently a particular higher order mass matrix can always be set up for a superconvergent computation of arbitrary vibration frequency. It is shown that the quadratic and cubic higher order mass matrices possess 4th and 6th orders of accuracy, which are two orders higher than those of the consistent plate mass matrices. The construction of higher order mass matrices and the analytical results for vibration frequencies are systematically demonstrated by a set of numerical examples.

Discrete systems behave as nonlocal structural elements: Bending, buckling and vibration analysis

It is shown herein that the bending, buckling and vibration problems of a microstructured beam can be modeled by Eringen's nonlocal elasticity model. The microstructured model is composed of rigid periodic elements elastically connected by rotational springs. It is shown that this discrete system is the finite difference formulation of a continuous problem, i.e. the Euler-Bernoulli beam problem. Starting from the discrete equations, a continualization method leads to the formulation of an Eringen's type nonlocal equivalent continuum. The sensitivity phenomenon of the apparent nonlocal length scale with respect to the bending, the vibrations and the buckling analyses is investigated in more detail. A unified length scale can be used for the microstructured-based model with both nonlocal constitutive law and nonlocal governing equations. The Finite Difference Method is used for studying the exact discrete problem and leads to tractable engineering formula. The bending behaviour of the microstructured cantilever beam does not reveal any scale effect in the presence of concentrated loads. This scale invariance is not a deficiency of Eringen's nonlocality because it is in fact supported by the exact discreteness of the microstructured beam. A comparison of the discrete and the continuous problems (for both static and dynamics analyses) show the efficiency of the nonlocal-based modelling for capturing scale effects. As it has already been shown for buckling or vibrations studies, small scale effects tend to soften the material in this case.

Finite Difference Approximations of Elasto-Plastic Bending Problems: Layered Approach and Error Estimates

The finite difference method is applied to derive approximate solutions for the elasto-plastic bending of Euler-Bernoulli beam problems. The investigations are restricted to a simple exemplary configuration, i.e. a straight cantilevered beam with constant rectangular cross-section and linear-elastic/ideal-plastic material properties, loaded by a constant distributed load. Only finite difference approximations of second-order accuracy are considered and special emphasis is given to the influence of the load step, the number of layers, and the number of nodes. Based on comparisons with the analytical solution, clear recommendations can be given on the required parameters to obtain a certain accuracy in the numerical approach.

Asymptotic frequencies of beams left fixed and right supported

The purpose of this paper is to construct the asymptotic for natural frequencies of the Euler-Bernoulli beam problem with non–uniform section using the method of Wentzel-Kramers-Brillouin (WKB) and find the secular equation for the eigenvalues a non–uniform beam embedded at one end and supported at the other end.

Exact geometry based quasi-conforming analysis for Euler-Bernoulli beam

The problem of quick analysis using exact geometry data was proposed by Hughes et al. and the isogeometric analysis framework was introduced as a solution. In this letter, the exact geometry concept is combined into the quasi-conforming framework and a novel method, i.e., the exact geometry based quasi-conforming analysis is proposed. In present method the geometry is exactly described by non-uniform rational B-spline bases, while the solution space by traditional polynomial bases. Present method combines the merits of both isogeometric analysis and quasi-conforming finite element method. In this letter Euler-Bernoulli beam problem is solved as an example and the results show that the present method is effective and promising.

On the dynamic response of beams with multiple geometric or material discontinuities

An analytic framework is developed for determining closed form expressions for the natural frequencies, mode shapes, and frequency response function for Euler–Bernoulli beams with any number of geometric or material discontinuities. The procedure uses a convenient matrix formulation to generalize the single discontinuity beam problem to beams with multiple step changes. Specifically, the multiple discontinuity beam problem is solved by analyzing the total structure as a series of distinct Euler–Bernoulli elements with continuity and compatibility enforced at separation locations. The method yields each respective section's eigenmode which may then be superpositioned to give the entire beam's mode shape and derivation of the frequency response function follows. Although the Euler–Bernoulli beam problem is demonstrated, any one-dimensional continuous structure is amenable to the prescribed analysis. Theoretical predictions are experimentally validated as well.

The use of He’s variational iteration method for obtaining the free vibration of an Euler–Bernoulli beam

This paper presents a way of using He’s variational iteration method to solve free vibration problems for an Euler–Bernoulli beam under various supporting conditions. By employing this technique, the beam’s natural frequencies and mode shapes can be obtained and a rapidly convergent sequence is obtained during the solution. The results obtained are the same as the results obtained by the Adomian decomposition method. It is verified that the present method is accurate and it provides a simple and efficient approach for solving vibration problems for uniform Euler–Bernoulli beams. A robust and efficient algorithm is also programmed using Matlab based on the present method, which can be easily used to solve Euler Bernoulli beam problems.

A spectral-Tchebychev technique for solving linear and nonlinear beam equations

This paper presents a spectral-Tchebychev technique for solving linear and nonlinear beam problems. The technique uses Tchebychev polynomials as spatial basis functions, and applies Galerkin's method to obtain the spatially discretized equations of motion. Unlike alternative techniques that require different admissible functions for each different set of boundary conditions, the spectral-Tchebychev technique incorporates the boundary conditions into the derivation, and thereby enables the utilization of the solution for any linear boundary conditions without re-derivation. Furthermore, the proposed technique produces symmetric system matrices for self-adjoint problems. In this work, the spectral-Tchebychev solutions for Euler–Bernoulli and Timoshenko beams are derived. The convergence and accuracy characteristics of the spectral-Tchebychev technique is studied by solving eigenvalue problems with different boundary conditions. It is found that the convergence is exponential, and a small number of polynomials is sufficient to obtain machine-precision accuracy. The application of the technique is demonstrated by solving: (1) eigenvalue problems for tapered Timoshenko beams with different boundary conditions, taper ratios, and beam lengths; (2) an Euler–Bernoulli beam problem with spatially and temporally varying forcing, elastic boundary, and damping; (3) large-deflection (nonlinear) Euler–Bernoulli beam problems with different boundary conditions; and (4) a micro-beam problem with nonlinear electrostatic excitation. The results obtained from the spectral-Tchebychev solutions are seen to be in excellent agreement with those presented in the literature.

A radial basis function approach in the meshless local Petrov-Galerkin method for Euler-Bernoulli beam problems

A meshless local Petrov-Galerkin (MLPG) method that uses radial basis functions rather than generalized moving least squares (GMLS) interpolations to develop the trial functions in the study of Euler-Bernoulli beam problems is presented. The use of radial basis functions (RBF) in meshless methods is demonstrated for C1 problems for the first time. This interpolation choice yields a computationally simpler method as fewer matrix inversions and multiplications are required than when GMLS interpolations are used. Test functions are chosen as simple weight functions as in the conventional MLPG method. Patch tests, mixed boundary value problems, and problems with complex loading conditions are considered. The radial basis MLPG method yields accurate results for deflections, slopes, moments, and shear forces, and the accuracy of these results is better than that obtained using the conventional MLPG method.

Unboundedness for the Euler–Bernoulli beam equation with a fractional boundary dissipation

We consider the Euler–Bernoulli beam problem with some boundary controls involving a fractional derivative. The fractional derivative here represents a fractional dissipation of lower order than one. We prove that the classical energy associated to the system is unbounded in presence of a polynomial nonlinearity. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity.