Non-uniform Euler Bernoulli Beams Research Articles

Flexural wave rainbow trapping effect in the periodic non-uniform Euler-Bernoulli beams and its application in energy harvesting

The rainbow trapping of elastic wave enables spatial frequency shunting and energy concentration phenomenon, which implies that the broadband vibration will forbid propagating forward and occur energy concentration at different positions of metamaterial structure. This paper proposes a novel metamaterial consisting of the periodic non-uniform Euler-Bernoulli beams with an arbitrary profile section to achieve the rainbow trapping effect of flexural wave and application in piezoelectric energy harvester. Firstly, the differential quadrature method is introduced to solve a partial differential equation with variable coefficients. The convergence of this method is systematically demonstrated, and the correctness of band structures is validated by comparing theoretical results calculated by the differential quadrature method with those from the finite element method for different profiles. Based on band structures analysis, the working mechanism of the flexural wave rainbow trapping effect is attributed to the group velocity modulation; thereby, the constructed periodic array of non-uniform beams achieves integration of specific-band vibration reduction and energy enhancement at specific positions. Secondly, simulations indicate that the resonance rainbow trapping frequencies demonstrate more intense concentration of flexural wave energy compared to the counterpart with initial frequencies. Finally, the resonance rainbow trapping phenomenon demonstrates superiority in vibration energy harvesting. Specifically, when piezoelectric films are pasted on specific-position with enhanced energy density, the maximum output voltage in the independent circuit connection reaches up to 2.30V for the resonance rainbow trapping frequency of 8752Hz and the PVDF film position of x = 670 mm. Furthermore, simulations illustrate that the maximum output voltage and power are up to 4.52V and 1894.12 nW for the Series-A circuit connection, respectively, which is approximately the sum of the individual maximum output electrical energy generated by independent circuit connection at positions A and B. The proposed metamaterial beams can offer new selection guidelines for high-performance vibration energy harvesters.

Alleviating vibrations along a harmonically driven nonuniform Euler–Bernoulli beam by imposing nodes

AbstractA passive approach is developed to quench excess vibration along a harmonically driven, arbitrarily supported, nonuniform Euler–Bernoulli beam with constant thickness (height) and varying width. Vibration suppression is achieved by attaching properly tuned vibration absorbers to enforce nodes, or points of zero vibration, along the beam. An efficient hybrid method is proposed whereby the finite element method is used to model the nonuniform beams, and a formulation based on the assumed modes method is used to determine the required attachment force supplied by each absorber to induce the desired nodes. Knowing the attachment forces needed to induce nodes, design plots are generated for the absorber parameters as a function of the tolerable vibration amplitude for each absorber mass. When the node locations are judiciously chosen, it is possible to dramatically suppress the vibration along a selected region of the beam. As such, sensitive instruments can be placed in this region and will remain nearly stationary. Numerical studies illustrate the application to several systems with various types of nonuniformity, boundary conditions, and attachment and node locations; these examples validate the proposed method to passively control excess vibration by inducing nodes on nonuniform beams subjected to harmonic excitations.

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 Vibrational Analysis of Non-uniform Double Eulerbernoulli Beams on a Winkler Foundation Using Laplace Differential Transform Method

In this study, the free vibrational analysis of non-uniform Euler-Bernoulli double beams on a Winkler foundation under simply supported and fixed-fixed boundary conditions is examined. The governing equation is solved using the Laplace Differential Transform Method, the combined form of the Laplace transform and differential transform technique (DTM). The accuracy of the method used is demonstrated by comparing the natural frequencies obtained using LDTM with previously published results available in the literature. It is discovered that for non-uniform double Euler-Bernoulli beams on a Winkler foundation with fixed-fixed end conditions, the natural frequencies are higher than those of simply supported end conditions. It is also observed that as the non-uniformity of the cross section of the beam increases, the natural frequencies reduce. Hence, it is suggested that the non-uniformity of the cross-section of the beam for a simply supported end condition be between 0 and less than 0.8. While the fixed-fixed end condition should have a value between 0 and 0.95. Journal of Engineering Science 14(1), 2023, 111-121

Closed-form solution for non-uniform Euler–Bernoulli beams and frames

Non-uniform elements, such as non-prismatic and those made from composite materials, are commonly used in structures. This is because the variation in their internal forces makes uniform elements less effective to withstand the external loads. This paper presents the formulation of the Green’s Functions Stiffness Method (GFSM) for the static analysis of non-uniform Euler–Bernoulli frames subjected to arbitrary external loads and bending moments. The GFSM is a mesh reduction method that combines the strengths of the Stiffness Method (SM) and the Finite Element Method (FEM) with those of Green’s functions, resulting in an analytic method for obtaining closed-form solutions of reticular structures. As a particular case, the GFSM is closely related to the Transcendental Finite Element Method (TFEM), which allows to obtain closed-form solutions from an implementation of the latter by only making changes at the post-processing stage. Two examples are provided with the analysis of a beam and a plane frame. Additionally, two appendices are included with the particularization of the GFSM for some specific non-uniform elements, and general stepped elements.

PDE Modeling and Tracking Control for the Flexible Tail of an Autonomous Robotic Fish

This article studies a single boundary regulator for the flexible tail of an autonomous robotic fish to implement complex oscillating body motions. The dynamic model of the flexible tail is derived by Hamilton’s principle, and conforms to the partial nonuniform Euler–Bernoulli beam with the uneven parameters. Then, a boundary control at the body–tail junction is proposed to manipulate the oscillation of the flexible tail, which is given in the form as a torque. The exponential stability of the error system is deduced by the integral Lyapunov synthesis. For further considering the boundary disturbance at the same point with control and the distributed disturbance, a disturbance observer is proposed. By appropriately choosing the designed parameters, the uniformly ultimate boundedness with disturbances is proved and the tracking error converges to a small neighborhood of 0. Finally, some simulations are presented to illustrate the effectiveness of the proposed control.

Free vibration analysis of non-uniform Bernoulli beam by using Laplace Adomian decomposition method

This paper discusses an investigation into the influence of the cone ratio and axial force on the vibration problem in a non-uniform cantilever Euler–Bernoulli beam. In the analysis, the governing equation for the non-uniform cantilever Euler–Bernoulli beam is solved using the Laplace Adomian decomposition method (LADM). The LADM is used to convert the governing equation into a characteristic equation of a non-uniform Euler–Bernoulli beam, and a simple calculation is performed to obtain the natural frequencies and corresponding modals. The obtained numerical results are verified using a comparison with the results reported in previous studies. The present study speeds up the convergent rate and the accuracy of calculation by comparing the results using the modified Adomian decomposition method (MADM) and differential transformation method (DTM). The main power and advantage of the LADM are providing an analytical approximation to a nonlinear differential equation without linearization, perturbation, approximation, and discretization, all of which lead to huge numerical computation. The numerical methods demonstrated that the natural frequency increases with increasing the rotating spring modulus and moving spring modulus, and the moving spring modulus has a greater influence on the natural frequency. The effects of the cone ratio and axial force are presented for non-uniform Euler–Bernoulli beams. The numerical results show that the LADM is a suitable technique for analyzing the behavioral characteristics of beams.

A new method for estimating the bending stiffness curve of non-uniform Euler-Bernoulli beams using static deflection data

The static Euler-Bernoulli beam is a simplified model from linear elasticity theory for small deformations, with which the stresses and deflections of slender beams can be determined very well. The simplicity of Euler-Bernoulli beam theory makes it still an important tool in the sciences, particularly in construction and mechanical engineering. The behavior and the deflection of a static Euler-Bernoulli beam is completely given by the bending stiffness curve. The bending stiffness curve of a real beam structure can be determined using measured data from a bending test. However, the determination of the bending stiffness curve for non-uniform beams leads to a so-called inverse problem, which is ill-posed. In this work, a robust and computationally effective method for determining the bending stiffness curve of a non-uniform Euler-Bernoulli beam based on measured data from a static bending test and the calculus of variations is presented. With the calculus of variations, the complex optimization problem of the inverse problem is transformed into a boundary value problem, which is then solved by collocation with lower computational costs. In addition, the new method does not require the second derivative of the beam deflection and the problem of determining the bending stiffness at the boundaries does not arise, as with the traditional method. The new method to determine the bending stiffness curve of non-uniform beams is demonstrated in several test examples.

Flexural waves in a periodic non-uniform Euler-Bernoulli beam: Analysis for arbitrary contour profiles and applications to wave control

The flexural wave in a periodic non-uniform Euler-Bernoulli beam with arbitrarily contoured profiles is studied by utilizing the power series expansion method. The convergence criterion that makes the power series expansion method applicable is also illustrated. The validation is carried out by comparing the theoretical results with that from the finite element analysis when the beam thickness varies in different forms. For a quadratic thickness variation, the first band gap evolution versus the structural parameter is investigated, based on which a flexural-wave-based low-pass filter for frequency shunting and a rectangular lens for energy focusing are designed. It is revealed in the frequency domain analysis that the flexural wave with a lower frequency can propagate further when it travels into the wave filter. The lens designed exhibits a good focusing phenomenon with the focusing size smaller than one wavelength, and has a good performance at a certain finite frequency range. The theoretical method and design scheme can provide effective guidance for the flexural wave control.

Modeling and Control of Tower Cranes With Elastic Structure

Due to their lightweight construction, large tower cranes are prone to structural vibrations that pose a major challenge for load sway damping control design and are the main reason for the lack of anti-sway systems in most modern tower cranes. Since neglecting the structural elasticity for feedback control design can lead to instability or poor performance, a detailed dynamic model is required that takes the structural elasticity into account. This contribution presents a control-oriented dynamic model of top-slewing tower cranes as a flexible multibody system assuming jib and tower as nonuniform Euler-Bernoulli beams and taking into account the dynamic couplings between steel structure, load motion, system drives, and rope. The distributed deformations are discretized using the Ritz method. For active sway damping, a two-degree-of-freedom control structure is designed consisting of a nonlinear flatness-based feedforward controller and a gain scheduled linear quadratic regulator. Experimental results at a full-scale tower crane validate the model accuracy and demonstrate the effectiveness of the proposed anti-sway control concept.

Adaptive Neural Boundary Control Design for Nonlinear Flexible Distributed Parameter Systems

This paper proposes an adaptive neural boundary feedback control design for a nonlinear flexible distributed parameter system described by a nonuniform wave equation containing an unknown nonlinear function in the boundary conditions. In the proposed design method, a radial-basis-function neural network (NN) with an adaptive weight update law is used to approximate the unknown nonlinear term. By using this adaptive NN, an adaptive boundary feedback controller is constructed to ensure that the resulting closed-loop system is practically exponentially stable. The closed-loop stability of the system is analyzed by the Lyapunov direct method and technique of integration by parts, and the update law of an NN weight vector is determined simultaneously. The closed-loop well-posedness analysis is also provided by applying the C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -semigroup approach. Moreover, the proposed design method is extended for a nonlinear flexible distributed parameter system modeled by a nonuniform Euler-Bernoulli beam equation containing an unknown nonlinear function in the boundary conditions. Finally, extensive numerical simulation results are given to demonstrate the performance of the proposed adaptive neural boundary control design method.

Quadratic B-spline finite element method for a rotating nonuniform Euler–Bernoulli beam

In this article, we solve the free vibration problem of a rotating non-uniform Euler-Bernoulli beam using the quadratic B-spline finite element method. The Galerkin method is used to obtain the weak form of the problem. The quadratic B-spline approximation provides the required continuity. This approximation yields the mass and stiffness matrices, which are half the size of the matrices obtained by the conventional finite element approximation. The resulting polynomials are quadratic while the Hermite polynomials are cubic. The mass and the stiffness matrix are derived. Results are matched with the published literature and compared with the conventional finite element results. A non-uniform approximation is used as well to observe an impact of the centrifugal force in numerical solution. Results are found to be more accurate with the non-uniform B-spline approximation for the first three natural frequencies of a rotating beam. Since the centrifugal force makes a difference to the first few modes, it is a desirable outcome.

Transverse vibration of nonuniform Euler-Bernoulli beams on bounded Time scales

In this article, we consider Euler-Bernoulli equation of transverse vibrations of nonuniform beams on bounded time scales $\mathbb{T}.$ We will give a description of all maximal dissipative, maximal accretive, self adjoint and other extensions of such operators.

Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

The Lindstedt-Poincaré method is applied to a nonuniform Euler-Bernoulli beam model for the free transverse vibrations of the system. The nonuniformities in the system include spatially varying and piecewise continuous bending stiffness and mass per unit length. The expression for the natural frequencies is obtained up to second-order and the expression for the mode shapes is obtained up to first-order. The explicit dependence of the natural frequencies and mode shapes on reference values for the bending stiffness and the mass per unit length of the system is determined. Multiple methods for choosing these reference values are presented and are compared using numerical examples.

Nonlocal fully intrinsic equations for free vibration of Euler–Bernoulli beams with constitutive boundary conditions

Fully intrinsic beam theory has presented a relatively new set of equations for modeling of one-dimensional structures. Although this method can be utilized as a useful tool for modeling beam-like structures, it is scale independent and cannot be accurately employed when the size of a structure becomes comparable with its internal characteristic length. In this paper, we develop nonlocal fully intrinsic equations for accommodating the small-scale effect and overcoming this limitation. The original fully intrinsic equations are thus reformulated using the nonlocal differential constitutive relations of Eringen. The so-called constitutive boundary conditions are considered. Here the generalized differential quadrature method is used for numerically solving the nonlocal fully intrinsic equations. Furthermore, applicability of the present formulations for free vibration of both uniform and nonuniform Euler–Bernoulli beams is discussed. Moreover, it is shown that the present formulation is accurate and useful for predicting the mechanical behavior of micro- and nanobeams.

Adaptive finite element analysis for damage detection of non-uniform Euler–Bernoulli beams with multiple cracks based on natural frequencies

PurposeThis study aims to develop an adaptive finite element method for structural eigenproblems of cracked Euler–Bernoulli beams via the superconvergent patch recovery displacement technique. This research comprises the numerical algorithm and experimental results for free vibration problems (forward eigenproblems) and damage detection problems (inverse eigenproblems).Design/methodology/approachThe weakened properties analogy is used to describe cracks in this model. The adaptive strategy proposed in this paper provides accurate, efficient and reliable eigensolutions of frequency and mode (i.e. eigenpairs as eigenvalue and eigenfunction) for Euler–Bernoulli beams with multiple cracks. Based on the frequency measurement method for damage detection, using the difference between the actual and computed frequencies of cracked beams, the inverse eigenproblems are solved iteratively for identifying the residuals of locations and sizes of the cracks by the Newton–Raphson iteration technique. In the crack detection, the estimated residuals are added to obtain reliable results, which is an iteration process that will be expedited by more accurate frequency solutions based on the proposed method for free vibration problems.FindingsNumerical results are presented for free vibration problems and damage detection problems of representative non-uniform and geometrically stepped Euler–Bernoulli beams with multiple cracks to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.Originality/valueThe proposed combination of methodologies described in the paper leads to a very powerful approach for free vibration and damage detection of beams with cracks, introducing the mesh refinement, that can be extended to deal with the damage detection of frame structures.

Free vibration of non-uniform Euler–Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method

• Disadvantage of Chebyshev Wavelets Method is eliminated. • Non-uniform Euler–Bernoulli beams are investigated by CWCM. • The validity and applicability of the proposed method are demonstrated by examples. • Accuracy of the CWCM is quite good even for small number of grid points. This paper proposes operational matrix of r th integration of Chebyshev wavelets. A general procedure of this matrix is given. Operational matrix of r th integration is taken as r th power of operational matrix of first integration in literature. But, this study removes this disadvantage of Chebyshev wavelets method. Free vibration problems of non-uniform Euler–Bernoulli beam under various supporting conditions are investigated by using Chebyshev Wavelet Collocation Method. The proposed method is based on the approximation by the truncated Chebyshev wavelet series. A homogeneous system of linear algebraic equations has been obtained by using the Chebyshev collocation points. The determinant of coefficients matrix is equated to the zero for nontrivial solution of homogeneous system of linear algebraic equations. Hence, we can obtain i th natural frequencies of the beam and the coefficients of the approximate solution of Chebyshev wavelet series that satisfied differential equation and boundary conditions. Mode shapes functions corresponding to the natural frequencies can be obtained by normalizing of approximate solutions. The computed results well fit with the analytical and numerical results as in the literature. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation method is quite good even for small number of grid points.

A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams

A systematic procedure is developed for studying the dynamic response of a rotating nonuniform Euler–Bernoulli beam with an elastically restrained root. To find the solution, a novel approach is used in that the fourth-order differential equation describing the vibration problem is first written as a first-order matrix differential equation, which is then solved using the power series method. The method can be used to obtain an approximate solution of vibration problems for nonuniform Euler–Bernoulli beams. Specifically, numerical examples are presented here to demonstrate the usefulness of the method in frequency analysis of nonuniform Euler–Bernoulli clamped-free cantilever beams. Results for mode shapes and frequency parameters were found to be in satisfactory agreement with previously published results. The effects of tapering, both equal and unequal, were investigated for both a cantilever wedge and cantilever cone.

Non-uniform Euler-Bernoulli beams under a single moving oscillator: An approximate analytical solution in time domain

Dynamic responses of simply supported non-uniform beams traversed by a moving oscillator are analysed in this paper. An approximate analytical method based on Rayleigh-Ritz (R-R) formulation is developed. The fundamental approximate mode obtained from R-R method is used in the present formulation to determine the responses of the beam and the oscillator. Effects of surface irregularities on the displacement and acceleration responses of the beam and the vehicle are also analysed. The results are compared with those obtained using Finite element method (FEM). A numerical example is provided to illustrate the validity of the present method which shows that the proposed method is simple, computationally more efficient compared to FEM and gives fairly good results. Though the single-mode approach used in the present paper is a classical one and numerous studies on the responses of uniform beams under moving loads have been reported in the past, its application to non-uniform beams (for which there does not exist any closed form expression for mode shapes) under a moving load, especially a moving oscillator, is presented for the first time.

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.