Application of the differential transformation method and variational iteration method to large deformation of cantilever beams under point load

Large deflection of a cantilever beam subjected to a tip-concentrated load is governed by a non-linear differential equation. Since it is hard to find exact or closed-form solutions for this non-linear problem, this paper investigates the aforementioned problem via the differential transformation method (DTM) and the variational iteration method (VIM), which are well-known approximate analytical solutions. The mathematical formulation is yielded to a non-linear two-point boundary value problem. In this study, we compare the DTM and VIM results, with those of Adomian decomposition method (ADM) and the established numerical solution obtained by the Richardson extrapolation in order to verify the accuracy of the proposed methods. As an important result, it is depicted from tabulated data that the DTM results are more accurate in comparison with those obtained by the VIM and ADM, which is one of the objectives of this article. Moreover, the effects of dimensionless end point load, ??, on the slope of any point along the arc length and the dimensionless vertical and horizontal displacements are illustrated and explained. The results reveal that these methods are very effective and convenient in predicting the solution of such problems, and it is predicted that the DTM and VIM can find a wide application in new engineering problems.     

联合载荷作用下悬臂梁大变形分析的打靶法

利用微积分基本知识，这里导出了集中力和任意横向分布载荷作用下悬臂梁的挠曲线微分方程。基于两点边值问题的打靶法，分别计算了两种联合载荷作用下悬臂粱的大变形。所得微分方程不仅具有普遍性，而且本文的计算方法对类似问题的解决具有一定的借鉴意义。     

The elastic stability of a thin cantilever beam under an articulated tip force—I: Statics analysis

Previous work on the buckling of thin elastic cantilever beams when subjected to tip forces applied by means of an articulated rod is here extended to cases when the axis of the rod is directed at any angle to the beam axis. The system is typical of engineering situations in which mechanical control circuits involve a bellcrank lever connected to an articulated rod, the remote end of which is loaded by a tube directionally guided in rollers. The function of the articulated rod is to permit geometric changes to take place in the mechanical system.A numerical solution is provided to the differential equation system by finite-difference approximations. The resulting equations are of a sufficiently simple form as to permit evaluation of buckling forces by desk calculation; alternatively a computer adaptation can be made.The results of the analysis are verified by an experimental investigation.     

The elastic stability of a thin cantilever beam under an articulated tip force—II: Kinetic analysis

Previous work on the buckling of thin elastic cantilever beams when subjected to tip forces applied by means of an articulated rod is here extended to cases when the axis of the rod is directed at any angle to the beam axis. The system is typical of engineering situations in which mechanical control circuits involve a bellcrank lever connected to an articulated rod, the remote end of which is loaded by a tube directionally guided in rollers. The function of the articulated rod is to permit geometric changes to take place in the mechanical system.A numerical solution is provided to the differential equation system by finite-difference approximations. The resulting equations are of a sufficiently simple form as to permit evaluation of buckling forces by desk calculation; alternatively a computer adaptation can be made.The results of the analysis are verified by an experimental investigation.     

Application of a LIPCA for the structural vibration suppression of an aluminum cantilever beam with a tip mass

Use of bare PZT as an actuator in the field of active vibration suppression may cause some drawbacks such as critical breaks in the installation process, short circuits in the host material and low fatigue performance. To alleviate these problems, we developed a new actuator called a lightweight piezocomposite actuator (LIPCA). The LIPCA has five layers: three glass-epoxy layers, a carbon-epoxy layer and a PZT layer. We implemented a LIPCA as an actuator to suppress the vibration of an aluminum cantilever beam with a tip mass. For the control algorithm in our test, we used positive position feedback. The filter frequency for this type of feedback should be tuned to the frequency of the target mode. The first three experimental natural frequencies of the aluminum cantilever beam agree well with the results of finite element methods. The effectiveness of using a LIPCA as an actuator in active vibration suppression was investigated with respect to the time and frequency domains, and the experimental results show that LIPCAs can significantly reduce the amplitude of forced vibrations as well as the settling time of free vibrations.     

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.     

The effect of deformation anisotropy on the plastic zone at the tip of a crack under biaxial loading

A theoretical analysis for investigating the form and dimension of the plastic zone in the tip of a crack in a plate made from a material with deformation anisotropy is presented. The anisotropy is explained by the hardening process during plastic deformation caused by loading along a straight-line trajectory until a crack is formed in the plate plane. The idea of this method is to take into account the second term in the Williams’ series representation in eigenfunctions for stress components for the 2D case. The contribution of the second term in the Cartesian coordinate system is independent of the distance from the crack tip for the K ₁ − T concept. It is shown that for the case of anisotropy, the dimension of the plastic zone decreases. The stress along the crack’s axis changes the plastic zone significantly.     

复杂载荷变刚度静不定梁通用力学模型及变形方程

对复杂载荷下变刚度静不定梁建立了一种通用力学模型，得出梁任一截面的变矩方程。通过对变矩方程的积分和积分后方程的递推，推导出梁任意截面的转角和挠度变形的一般方程，并给出工程计算实例。该方程易于采用计算机处理，为复杂载荷下变刚度静不定梁程序化求解提供基础。     

Structural damage identification of a cantilever beam using excitation force level control

Most studies in damage identification so far have concentrated on comparing modal parameters of a structure with an open crack with those of an intact structure. In this study, a new damage identification method for beam-like structures with a fatigue crack is proposed, which does not require comparative measurement on an intact structure but several measurements at different level of excitation forces on the cracked structure. The idea comes from the fact that dynamic behavior of a structure with a fatigue crack changes with the level of the excitation force. In other words, a beam with a real fatigue crack would behave as an intact beam at low excitation forces due to the crack closure. The 2nd spatial derivatives of frequency response functions along the longitudinal direction of a beam are used as the sensitive indicator of crack existence. Then, weighting function is employed in the averaging process in frequency domain to account for the modal participation of the differences between the dynamic behavior of beam with a fatigue crack at the low excitation and one at the high excitation. Subsequently, a damage index is defined such that the location and level of the crack may be identified. Finally, it is shown that damage identification method using the proposed damage index is very successful through experiment and finite element analysis.     

Combined analytical and numerical solution for an elastically supported Timoshenko beam to a moving load

A combined analytical and numerical method is presented to get a response for an elastically supported Timoshenko beam to a moving load. The analytical steady-state solution as a particular solution is established and summed with the numerically calculated homogeneous solution. A steady-state solution is sought analytically through the direct application of the Fourier transform to the moving step load. It is shown to be a compact formula composed of exponential and sinusoidal functions depending on the load velocity. The homogeneous solution is established numerically to cancel out the discontinuities and the inconsistent boundary and initial conditions of the steady-state solution. The discontinuities produced by the steady-state solution are removed using the physical characteristics related to the bar wave. Some response curves are shown to compare the beam motions at different load velocities.     

Identification of pseudo-natural frequencies of an axially moving cantilever beam using a subspace-based algorithm

This study focuses on extraction of frequency information of a linear time-varying system using free response data. Frequency information is obtained from the pseudo-modal parameters that were defined in a previous study. A subspace-based identification algorithm is introduced. An improved version is proposed to make the algorithm less sensitive to measurement noise. An axially moving cantilever beam is used as the experimental system. A dynamic model is presented to show that lateral vibration of the axially moving cantilever beam is governed by a linear time-varying model. A computer simulation is conducted to compare the true pseudo-modal parameters and approximate ones that can be identified using the improved algorithm. The experimental study focuses on the capabilities of the algorithm and the factors that affect the identification results. A method of grouping identified structural pseudo-natural frequencies is proposed. Limitations of the algorithm are discussed.     

Photocatalytic deposition of a gold nanoparticle onto the top of a SiN cantilever tip

We propose a new technique for depositing a gold nanoparticle onto the tip of a dielectric support. We employed the photocatalytic effect of titanium dioxide for the deposition. When the titanium dioxide immersed in a solution including gold ions is subject to optical exposure, the excited electrons in the conduction band reduce gold ions into gold metal. Illumination by an evanescent wave generated with a total reflection configuration limits the deposition region to the very tip. In experiments we obtained 100–300 nm gold particles on SiN cantilever tips for atomic force microscopes. The contrast of evanescent interference fringes measured by a near-field scanning optical microscope with this gold nanoparticle probe has proved to be higher than that with a non-deposited SiN probe by a factor of 1.5.     

Investigation on steady state deformation and free vibration of a rotating inclined Euler beam

The steady state deformation and infinitesimal free vibration around the steady state deformation of a rotating inclined Euler beam at constant angular velocity are investigated by the corotational finite element method combined with floating frame method. The element nodal forces are derived using the consistent second order linearization of the nonlinear beam theory, the d'Alembert principle and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The rotating element coordinates rotate about the hub axis at the angular speed of the hub. The equations of motion of the system are defined in terms of an inertia global coordinate system, which is coincident with a rotating global coordinate system rigidly tied to the rotating hub. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady state deformation and natural frequency of the rotating inclined beam.     

Dynamic analysis of composite beam subjected to harmonic moving load based on the third-order shear deformation theory

The response of an infinite Timoshenko beam subjected to a harmonic moving load based on the third-order shear deformation theory (TSDT) is studied. The beam is made of laminated composite, and located on a Pasternak viscoelastic foundation. By using the principle of total minimum potential energy, the governing partial differential equations of motion are obtained. The solution is directed to compute the deflection and bending moment distribution along the length of the beam. Also, the effects of two types of composite materials, stiffness and shear layer viscosity coefficients of foundation, velocity and frequency of the moving load over the beam response are studied. In order to demonstrate the accuracy of the present method, the results TSDT are compared with the previously obtained results based on first-order shear deformation theory, with which good agreements are observed.     

On the elastic solution of frictional contact problems using variational inequalities

This article is devoted to the development and implementation of a variational inequalities approach to treat the general frictional contact problem. Unlike earlier studies which adopt penalty methods for the solution of the corresponding variational inequalities, the current investigation uses quadratic programming and Lagrange's multipliers to solve the frictional contact problem and identify the candidate contact surface. The proposed method avoids the use of user-defined penalty parameters, which ultimately govern the convergence and accuracy of the solution. To establish the validity of the method, a number of test cases are examined and compared with existing solutions where penalty methods are employed.     

Experimental behaviour of a fixed ended beam under simulated uniformly distributed load

The behaviour of an aluminium beam of constant rectangular cross section which is fully restrained against translation and rotation and subjected to a simulated uniformly distributed load is investigated. It is shown experimentally and supported analytically that the axial end restraint has a significant stiffening effect on the post-elastic behaviour when the central plastic zone spreads towards the beam ends. The engineering bending theory which includes axial force effects gives an accurate prediction of the pre-yield behaviour and the rigid-plastic theories are shown to give a close approximation to the load/central deflection response for a real fixed ended beam in the plastic range.     

Verification of the performance of rotatable jig for a single cantilever beam method using the finite element analysis

The performance of new jig for single cantilever beam test method was verified by finite element analysis. Two types of jig were designed for a small specimen that had relatively short length compared to the width of cantilever; one was simple fixed jig and the other was specially designed rotatable jig. The rotatable jig has a rotatable seesaw which adjusts the experimental misalignments between the specimen and test machine. Among the three translational and three rotational misalignments, following three important factors were considered; rotation about x-axis, rotation about z-axis, and translation in y-axis. Adhesive layer was modeled by cohesive zone element, and crack propagation behavior and the deviation of energy release rate were investigated. The fixed jig showed undesired asymmetric crack propagation and large deviation of energy release rate when it had rotational misalignment about x-axis. However, the proposed new rotatable jig showed almost symmetrical crack propagation and small deviation of energy release rate regardless of misalignments. Rotational motion of the seesaw automatically compensated the rotational misalignment of the specimen. The rotatable jig also showed relatively small deviation of energy release rate compared with the fixed jig by the rotational misalignment about the z-axis. In contrast, the rotatable jig showed deviation of energy release rate by translational misalignments in the y-axis. However, the magnitude of the deviation was very small within the controllable range of experimental misalignment. In conclusion, it was found out that the proposed jig was appropriate for the measurement of adhesion of a small specimen by single cantilever beam method.

     

Mapping of lateral vibration of the tip in atomic force microscopy at the torsional resonance of the cantilever

Lateral vibration of the tip in atomic force microscopy was mapped at the torsional resonance of the cantilever by attaching a shear piezo element at the base of the cantilever or under the sample. Fixed frequency excitation and self-excitation of torsional motion were implemented. The lateral vibration utilized as measured by an optical lever was in the order of 10 pm to 3 nm, and its frequency approximately 450 kHz for a contact-mode silicon nitride cantilever. The amplitude and phase of the torsional motion of the cantilever was measured by a lock-in-amplifier or a rectifier and plotted in x and y as the sample was raster scanned. The imaging technique gave contrast between graphite terraces, self-assembled monolayer domains, silicon and silicon dioxide, graphite and mica. Changing contrast was observed as silicon islands oxidized in atmosphere, showing that the imaging technique can detect change in lateral tip mobility due to changes occurring near the surface. Torsional self-excitation showed nanometric features of self-assembled monolayer islands due to different lateral dissipation. Dependence of torsional resonance frequency on excitation amplitude, and contrast change due to driving frequency around resonance were observed.     

Measurement and correlation of high frequency behaviors of a very flexible beam undergoing large deformation

A correlation method of high frequency behaviors of a very flexible beam undergoing large displacement is presented. The suggested method based on the experimental modal analysis leads to more accurate correlation results because it directly uses the modal parameters of each mode achieved from experiment. First, the modal testing and the parameter identification method are suggested for flexible multibody dynamics. Due to the flexibility of a very thin beam, traditional testing methods such as impact hammer or contact type accelerometer are not working well. The suggested measurement with high speed camera, even though the test beam is very flexible, is working well. Using measurements with a high speed camera, modal properties until the 5th mode are measured. And After measuring each damping ratio until the 5th mode, a generic damping model is constructed using inverse modal transformation technique. It’s very interesting that the modal transformation technique can be also applied even in the ANCF simulation which uses the global displacement and finite slope as the nodal coordinates. The results of experiment and simulation are compared until the 5th mode frequency, respectively, by using ANCF forced vibration analysis. Through comparison between numerical simulation and experiment, this study showed that the proposed generic damping matrix, modal testing and parameter identification method is very proper in flexible multibody dynamic problems undergoing large deformation.     

齿轮根部沉切尖点啮合冲击速度分析

文章以啮合理论为出发点,考虑齿轮基节偏差,给出了沉切尖点啮合冲击的概念,推导出啮入与啮出沉切点冲击速度计算公式。