C0-continuous isoparametric Timoshenko beam element for rotating

This paper presents a C0-continuous isoparametric finite element for free vibration analysis of a rotating, tapered Timoshenko beam mounted on the periphery of a rotating rigid hub, at a setting angle with the plane of rotation. The finite element has three nodes and two degrees of freedom per node and employs modified shape functions for rotational displacement associated with the shear strain energy to avoid shear locking. To obtain a finite element equation of the generalized eigenvalue problem, Hamilton’s principle is applied to kinetic and potential energy expressions of a rotating Timoshenko beam with non-zero setting angle. The numerical solutions for various situations including variations of rotational speed, taper ratio, slenderness ratio, hub radius and setting angle are compared with other numerical results available in the literature whenever possible. The results show that the new 3-noded isoparametric element yields accurate results when compared to other numerical ones.     

Euler-Bernoulli梁单元的完整二阶位移场

从几何变形角度出发,使用插值理论推导了Euler-Bernoulli梁单元的完整的二阶位移场.使用三次Hermite插值函数建立了单元的横向一阶位移场,一次Lagrange插值函数构造了单元的轴向位移场,进而在单元的横向位移场和纵向位移场中包含了因单元截面转动而产生的附加位移,从而将Euler-Bernoulli梁单元的位移场表达为结点位移的二次函数,可用于杆系的非线性静、动力学分析.     

Dynamic characteristics of rotating pretwisted clamped-clamped beam under thermal stress

Effects of thermal stress on the vibration characteristics, buckling limit and critical speed of a rotating pretwisted beam clamped to rigid hub at a stagger angle were investigated. By considering the work done by thermal stress, the thermal influence on stiffness matrix was introduced in the dynamic model. The motion equations were derived based on Lagrange equation by employing three pure Cartesian deformation variables combined with nonlinear von Karman strain formula. Numerical investigations studied the modal characteristics of the beam. Numerical results calculated from a commercial finite element code and obtained with the present modeling method were in good agreement with the previous results reported in the literature. The combined softening effects due to the thermal stress and the rotation motion were observed. Furthermore, it is shown that the inclusion of thermal stress is necessary for blades operating under a high temperature field. Buckling thermal loads and the critical rotating speed were calculated through solving the corresponding nonlinear equations numerically, and some pertinent conclusions are outlined. It is also found that the peak value position of the first mode shape approaches to the tip of blade with the increment of rotating speed and hub radius. However, the variation in the environment temperature causes only a slight alteration in the mode shape.     

Flexural-torsional coupled vibration of slewing beams using various types of orthogonal polynomials

Dynamic behavior of flexural-torsional coupled vibration of rotating beams using the Rayleigh-Ritz method with orthogonal polynomials as basis functions is studied. Performance of various orthogonal polynomials is compared to each other in terms of their efficiency and accuracy in determining the required natural frequencies. Orthogonal polynomials and functions studied in the present work are : Legendre, Chebyshev, integrated Legendre, modified Duncan polynomials, the special trigonometric functions used in conjunction with Hermite cubics, and beam characteristic orthogonal polynomials. A total of 5 cases of beam boundary conditions and rotation are studied for their natural frequencies. The obtained natural frequencies and mode shapes are compared to those available in various references and the results for coupled flexural-torsional vibrations are especially compared to both previously available references and with those obtained using NASTRAN finite element package. Among all the examined orthogonal functions, Legendre orthogonal polynomials are the most efficient in overall CPU time, mainly because of ease in performing the integration required for determining the stiffness and mass matrices.     

Dynamic crack growth in TDCB specimens

Dynamic crack propagation in tapered double cantilever beam (TDCB) specimens is analysed via beam theory and the finite element method. Steady state and transient solutions of the energy release rate G are given for various load conditions. Finite element analysis is performed to obtain the dynamic G at given crack speed or the crack history for a given fracture toughness. The stress wave effects on the dynamic G are discussed. The beam solutions are compared with the finite element results and some experimental phenomena are explained.     

Sensitivity of fundamental mode shape and static deflection for damage identification in cantilever beams

Fundamental mode shape and static deflection are typical features frequently used for identification of damage in beams. Regarding these features, an interesting question, still pending, is which one is most sensitive for use in damage identification. The present study addresses the key sensitivity of these features for damage identification in cantilever beams, wherein these features are extremely similar in configurations. The intrinsic relation between the fundamental mode shape and static deflection is discussed, and in particular, an explicit generic sensitivity rule describing the sensitivity of these features to damage in cantilever beams is proposed. The efficiency of this rule in identifying damage is investigated using Euler-Bernoulli cantilever beams with a crack. The validity of the approach is supported by three-dimensional elastic finite element simulation, incorporating the potential scatter in actual measurements. The results show that the generic sensitivity rule essentially provides a theoretical basis for optimal use of these features for damage identification in cantilever beams.     

Dynamic plastic behavior of a free–free beam striking the mid-span of a clamped beam with shear and membrane effects considered

Recently Yu et al. (Int. J. Solids Struct. 38 (2001) 261) made a study on the dynamic behavior of a flying free–free beam striking the tip of a cantilever beam using the rigid, perfectly plastic (r-p-p) material model. Later, also based on the r-p-p material model Yang and Yu (Mech. Struct. Mach. 29 (2001) 391) analyzed another impact problem of a free rotating hinge beam striking a cantilever beam. Both of these studies ignored the finite deflection effects on the plastic behavior of the colliding beams. However if the free–free beam strikes a clamped beam, the influence of finite-deflections, or, geometric changes, must be retained in the governing equation if the maximum permanent transverse displacement of the clamped beam exceeds the corresponding beam thickness. The problem becomes more interesting since the deformation mechanisms of the beam system and the partitioning of energy dissipation in the beams are significantly different from those predicted by ignoring the influence of membrane forces. Accordingly the failure modes of the structure are different.In the present paper, a theoretical model based on the r-p-p material idealization is proposed to simulate the dynamic behavior when the mid-point of a translating free–free beam impinging on the mid-span of a clamped beam with the beam axes perpendicular to each other. The plastic behavior of the beam system is explored with shear sliding and finite deflection effects taken into account. The final deflection, the dissipation of energy within the two beams after impact and the influence of the structural and material parameters are discussed. It is shown that membrane force plays an important role during the response process, especially when the deflection is of the same order as the thickness of the clamped beam.     

一种分析旋转梁振动的梁柱有限元

针对旋转柔性梁的弯曲振动问题,提出了一种高精度的有限元——梁柱元。根据旋转梁弯曲时与梁柱相同的受力特点,推导了梁柱元弯曲变形的形状函数,采用传递矩阵方法导出了单元的刚度矩阵及相应的一致质量阵。证明了梁柱元刚度矩阵中各元素的泰勒展开式前两项正好是采用常规弯曲梁元形状函数得到的弹性刚度矩阵和几何刚度矩阵对应的元素的和,因而采用梁柱元分析旋转梁元弯曲振动问题具有很好的精度。对铰支和固支旋转梁振动分析的数值算例也证明了这个结论。     

Mechanical and dynamic characteristics of double and single beam cantilevers for MEMS manipulation

Micro and millimeter-scaled cantilever beams are commonly used to apply and measure microforce and manipulate micro-objects, biological cells, and tissues. In manipulating micro-objects and actuating micro-devices by cantilever beams, sudden application and release of forces are typical, and subject to static and dynamic modes of operation. Therefore, the cantilever’s mechanical behavior and vibration characteristics are vital since they are also used in force sensors and probes. The dynamic behavior of the double beam cantilever (DBC) in micro and millimeter-scaled is explored by varying the length without changing the stiffness and compared with the single beam cantilever (SBC). The dynamic attributes such as mode shape, natural frequency, resonance, and response under the impulse force and Coulomb friction are evaluated numerically. This study will assist in selecting the appropriate type and length of cantilevers in micro and millimeter-scale to manipulate micro-objects, biological cells and tissues, and use in MEMS sensors.

     

Vibration and reliability of a rotating beam with random properties under random excitation

In this paper, vibration and reliability of a rotating beam with random properties under random excitations are studied. The rotating beam is under a stochastic load modeled as a stationary white noise. The cross-sectional area, elasticity modulus, moment of inertia, shear modulus, damping coefficient, mass density and rotational speed are modeled as random variables. To develop the equations of motion, the finite element method and space state analysis are applied. In order to consider the randomness of properties, a second order perturbation method is used. The effects of rotational speed, setting angle, hub radius, variances of random properties, correlation of random variables and damping matrix forms on the vibration and reliability of rotating beams, are studied completely.     

旋转悬臂梁动力学的B样条插值方法

采用B样条插值方法对旋转悬臂梁的动力学特性进行研究。考虑柔性梁的纵向拉伸变形和横向弯曲变形,计入由于横向弯曲变形引起的纵向缩短,即非线性耦合项。利用B样条插值方法对柔性梁的变形场进行离散。采用Lagrange方程建立系统的动力学方程,并编制旋转悬臂梁动力学仿真软件。进行动力学仿真,将B样条插值方法的仿真结果与假设模态法、有限元法进行比较分析,验证了提出的方法的正确性,并表明B样条插值方法作为变形体离散法在柔性多体系统动力学中具有优良性能和应用价值。     

Repair of notched beam under dynamic load using piezoelectric patch

In this paper, the repair of a cracked beam under an external dynamic load employing the electro-mechanical characteristic of piezoelectric material to induce a local moment is presented. Conceptually, an external voltage is applied to actuate a piezoelectric patch bonded on the beam to effect closure of a crack so that the singularity at the crack tip under dynamic load may be decreased. Globally, this has the effect of altering the resonant frequency of the cracked beam towards that of the healthy beam, which is the criterion used for the repair. To demonstrate the repair methodology, a cantilever beam is used as an illustration, where the repair moment coefficient and the voltage required are mathematically derived. The relationship between repair moment coefficient, crack parameters and length of piezoelectric patch is investigated. The difference between the proposed repair criterion and an earlier published criterion for cracked beam under static load is also shown. A numerical example is used to study the effectiveness of the proposed repair methodology and its results are compared with those from 3-D finite element analyses using ABAQUS 6.4 as one means of verification.     

电流变夹层梁的动力稳定性分析

将电流变液等效为线性粘弹性材料，并假定在小变形情况下其储能模量和损耗因子与加在它上面的电场成正比，利用Hamilton原理和有限元方法建立电流变夹层梁的动力学方程。分析不同外加电场和厚度比情况下，电流变夹层梁的振动特性及动力稳定性。通过对单频轴向激励作用下电流变夹层悬臂梁的仿真计算显示，外加电场的增大能提高电流变夹层梁的刚度和阻尼损耗，减少不稳定区域的大小，而电流变层厚度的增加将使梁的固有频率降低，但提高了梁的稳定范围。表明合理设计电流变夹层梁可以有效抑制振动，提高系统的稳定性。     

Model study and active control of a rotating flexible cantilever beam

For a dynamic system of a rotating flexible cantilever beam, the traditional model assumes the small deformation in structural dynamics where axial and transverse displacements at any point in the beam are uncoupled. This traditional hybrid coordinate model is referred as the zero-order approximation coupling model in this paper, which may result in divergence to the dynamic problem of a flexible cantilever beam with a high rotational speed. In this paper, a first-order approximation coupling model is presented to analyze the dynamics of rotating flexible beam system, which is based on the Hamilton theory and the finite element discretization method. The proposed model for the system considers the second-order coupling quantity of the axial displacement caused by the transverse displacement of the beam. The dynamic characteristics of the rotating beam system when using the zero-order approximation coupling model are compared with those when using the first-order approximation coupling models through numerical simulations. In addition, the applicability of the two dynamic models for control design are studied by using the classical optimal control method. Simulation and comparison studies show that, for the case without control for the system, there exists big difference between the result using the zero-order approximation coupling model and that using the first-order approximation coupling model even for the case of small angular velocity of the system. The larger is the angular velocity, the bigger is the difference. Vibration frequency of the beam by using the first-order approximation coupling model is higher than that by using the zero-order approximation coupling model. When the angular velocity of the system is close to or is larger than the fundamental frequency of the beam without rotation motion, the zero-order approximation coupling results in a wrong result, while the first-order approximation coupling model is valid. For the case with control for the system, the applicability of the zero-order approximation coupling model can be much broadened. The critical angular velocity of the system for validity of the zero-order approximation coupling model is much larger than that without control for the system. The first-order approximation coupling model is available not only for the case of small angular velocity but also for the case of large angular velocity of the system, and is applicable to the cases with or without control for the system.     

Shape optimization of rotating cantilever beams considering their varied modal characteristics

The modal characteristics of rotating structures vary with the rotating speed. The material and the geometric properties of the structures as well as the rotating speed influence the variations of their modal characteristics. Very often, the modal characteristics of rotating structures need to be specified at some rotating speeds to meet their design requirements. In this paper, rotating cantilever beam is chosen as a design target structure. Optimization problems are formulated and solved to find the optimal shapes of rotating beams with rectangular cross section.     

Dynamic stability of a rotating shaft embedded in an isotropic winkler-type foundation

The dynamic instability of a rotating shaft subjected to axial periodic forces and embedded in an isotropic, Winkler-type elastic foundation is studied by the finite element technique. The equations of motion for such a rotating shaft element are formulated using deformation shape functions developed from the Timoshenko beam theory. The effects of translational and rotatory inertia, gyroscopic moments, bending and shear deformation are included in the mathematical model. The numerical results show that the elastic foundation can shift the regions of dynamic instability away from the dynamic load factor axis and thus reduces the sizes of these regions, whereas the effect of gryoscopic moments is to shift the boundaries of the regions of dynamic instability outwardly and, therefore, increases the sizes of these regions.     

Study of the tip mass and interaction force effects on the frequency response and mode shapes of the AFM cantilever

The vibrational characteristics of an atomic force microscope (AFM) cantilever beam play a key role in dynamic mode of the atomic force microscope. As the oscillating AFM cantilever tip approaches the sample, the tip–sample interaction force influences the cantilever dynamics. In this paper, we present a detailed theoretical analysis of the frequency response and mode shape behavior of a cantilever beam in the dynamic mode subject to changes in the tip mass and the interaction regime between the AFM cantilever system and the sample. We consider a distributed parameter model for AFM and use Euler–Bernoulli method to derive an expression for AFM characteristics equation contains tip mass and interaction force terms. We study the frequency response of AFM cantilever under variations of interaction force between AFM tip and sample. Also, we investigate the effect of tip mass on the frequency response and also the quality factor and spring constant of each eigenmodes of AFM micro-cantilever. In addition, the mode shape analysis of AFM cantilever under variations of tip mass and interaction force is investigated. This will incorporate the presentation of explicit analytical expressions and numerical analysis. The results show that by considering the tip mass, the resonance frequencies of the cantilever are decreased. Also, the tip mass has a significant effect on the mode shape of the higher eigenmodes of the AFM cantilever. Moreover, tip mass affects the quality factor and spring constant of each modes.     

Dynamics of an axially moving Bernoulli-Euler beam: Spectral element modeling and analysis

The spectral element model is known to provide very accurate structural dynamic characteristics, while reducing the number of degree-of-freedom to resolve the computational and cost problems. Thus, the spectral element model for an axially moving Bernoulli-Euler beam subjected to axial tension is developed in the present paper. The high accuracy of the spectral element model is then verified by comparing its solutions with the conventional finite element solutions and exact analytical solutions. The effects of the moving speed and axial tension on the vibration characteristics, wave characteristics, and the static and dynamic stabilities of a moving beam are investigated.     

An improvement EMD method based on the optimized rational Hermite interpolation approach and its application to gear fault diagnosis

A demodulation technique based on improvement empirical mode decomposition (EMD) is investigated in this paper. Firstly, the problem of the envelope line in EMD is introduced and the drawbacks of two classic interpolation methods, cubic spline interpolation method and cubic Hermite interpolation method are discussed; then a new envelope interpolation method called optimized rational Hermite interpolation method (O-EMD) is proposed, which has a shape controlling parameter compared with the cubic Hermite interpolation algorithm. At the same time, in order to improve the envelope approximation accuracy of local mean, the parameter determining criterion is put forward and an optimization with Genetic Algorithm (GA) is applied to automatic select the suitable shape controlling parameter in each sifting process. The effectiveness of O-EMD method is validated by the numerical simulations and an application to gear fault diagnosis. Results demonstrate that O-EMD method can improve the reliability and accuracy significantly compared with traditional EMD method.     

Elastic dynamic research of high-speed multi-link precision press considering structural stiffness of rotation joints

An elastic dynamic model of high-speed multi-link precision press considering structural stiffness of rotation joints was established by the finite element method. In the finite element model, rotation joint was established by four bar elements with equivalent stiffness, and connected link was established by beam element. Then, the elastic dynamics equation of the system was established, and modal superposition method was used to solve the dynamic response. Compared with the traditional elastic dynamic model with perfect constraint of the rotation joints, the elastic dynamic response value of the improved model is larger. To validate the presented new method of elastic dynamics analysis with stiffness of rotation joints, a related test of slider Bottom dead center (BDC) position in different speed was designed. The test shows that the model with stiffness of rotation joints is more reasonable. So it provides a reasonable theory and method for dynamic characteristics research of sucha multi-link machine.