On determination of vibration characteristics of a beam with a closing crack in bending vibrations

We consider free vibrations of an elastic cantilever beam with an edge crack, which is simulated as a beam portion with a decreased moment of inertia. The weight of this portion is assumed to be constant, while the dimensions of the portion are determined by the energy criterion of equivalence. We propose an analytical approach to determination of natural frequencies and vibration modes, of a beam with an open or closing crack and to investigation of nonlinear distortions of the displacement wave and acceleration and deformation of various sections of a beam with a closing crack. The solution allows for the possibility that more than one vibration mode of a beam can be generated at the moment of the crack opening and includes the effect of the crack on the strain distribution in the beam volume. It is demonstrated that the approach we propose gives reliable relationships between various vibration characteristics of a beam and the crack parameters and makes it possible to solve an inverse problem of damage diagnostics. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 5–23, May–June, 2000.     

On determination of the natural frequency of transverse and longitudinal vibrations of a cracked beam. Part 1. analytical approach

We consider a perfectly elastic cantilever beam with a crack. The influence of the crack depth and location on the vibration mode of the beam is simulated by a local change in the cross section. This problem is solved by the Rayleigh method. Fairly simple formulas for the determination of the natural frequency of the first mode of bending and longitudinal vibrations of a cantilever beam with an open and closing crack are derived. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 35–44, March–April, 1999.     

Approximate analytical determination of vibrodiagnostic parameters of the presence of a crack in an elastic body under superharmonic resonance

We discuss an approximate analytical method for calculating the main parameter of nonlinearity of vibration of an elastic body with a closing crack, which is simulated by a single-degree-of-freedom system with an asymmetric bilinear characteristic of the restoring force, under a strong 2nd-order superharmonic resonance.     

负刚度吸振器对有限长弹性梁的抑振效果研究

考虑不同形式负刚度动力吸振器对有限长弹性简支梁动态响应的影响,提出并建立"弹性梁-负刚度动力吸振器"耦合系统动力学模型。基于模态叠加法,推导得到各阶模态对应幅频响应解析表达式。以弹性梁第1阶振动模态作为振动抑制目标,结合固定点理论和最大值最小化优化准则得到各类型动力吸振器的最优设计参数。以功率流作为振动控制效果的评价指标,建立"弹性梁-动力吸振器"耦合系统的导纳功率流理论模型。在此基础上,计算得到安装动力吸振器前后弹性梁的总功率流和净功率流,以及动力吸振器消耗的功率流,研究不同形式动力吸振器的振动抑制效果。最后,选择振动控制效果最显著的动力吸振器作为研究对象,针对部分主要设计参数展开研究。计算结果表明：在目标控制模态频率附近,负刚度动力吸振器对弹性梁动态响应的控制效果较好,且多个振动模态响应均被有效控制;当阻尼元件和负刚度元件同时接地对弹性梁动态响应的控制效果最佳;众多设计参数均存在最优值。     

Winkler地基上修正Timoshenko梁振动分析

利用Bernoulli-Euler梁理论建立的弹性地基梁模型应用广泛,但其在高阶频率及深梁计算中误差较大,利用修正的Timoshenko梁理论建立新的弹性地基梁振动微分方程,由于其在Timoshenko梁的基础上考虑了剪切变形所引起的转动惯量,因而具有更好的精确度。利用ANAYS beam54梁单元进行振动模态的有限元计算,所求结果与理论基本无误差,从而验证了该理论的正确性。基于修正Timoshenko梁振动理论推导出了弹性地基梁双端自由-自由、简支-简支、简支-自由、固支-固支等多种边界条件下的频率超越方程及模态函数。分析了弹性地基梁在不同理论下不同约束条件及不同高跨比情况下的计算结果,从而论证了该理论计算弹性地基梁的适用性。分析了不同弹性地基梁理论下波速、群速度与波数的关系。得到了约束条件和梁长对振动模态及地基刚度对振动频率有重要影响等结论。     

Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load

An analytical approach for evaluating the forced vibration response of uniform beams with an arbitrary number of open edge cracks excited by a concentrated moving load is developed in this research. For this purpose, the cracked beam is modeled using beam segments connected by rotational massless linear elastic springs with sectional flexibility, and each segment of the continuous beam is assumed to satisfy Timoshenko beam theory. In this method, the equivalent spring stiffness does not depend on the frequency of vibration and is obtained from fracture mechanics. Considering suitable compatibility requirements at cracked sections and corresponding boundary conditions, characteristic equations of free vibration response are derived. Then, forced vibration response is treated under a moving load with a constant velocity. Using the determined eigenfunctions, the forced vibration response may be obtained by the modal superposition method. Finally, some parametric studies are presented to show the effects of crack parameters and moving load velocity.     

考虑结构裂纹扩展的振动疲劳寿命计算方法

工程实践中任何结构都存在不同程度的裂纹损伤,振动激励下动响应与疲劳裂纹扩展之间互相耦合,直接影响结构振动疲劳寿命.为了考虑结构振动疲劳耦合效应对疲劳寿命的影响,提出了一种考虑结构裂纹扩展的振动疲劳寿命计算方法.分析时,通过建立若干个含不同长度裂纹的结构有限元模型模拟结构裂纹扩展,采用Paris方程分段计算结构振动疲劳裂纹扩展寿命,通过试验确定的固有频率降变化规律反推结构裂纹萌生寿命,最后累计得到结构疲劳总寿命.结论表明,仿真计算结果与试验结果比较吻合.     

A finite-element model for the study of vibration of a beam with a closing crack

We present a finite-element model to be used in a study of vibration of a beam with a closing crack. Some special features of numerical solution and methods for its fast realization are discussed. The results of experimental verification of the model are provided. __________ Translated from Problemy Prochnosti, No. 5, pp. 114–120, September–October, 2008.     

Uncoupled numerical method for fracture analysis

An uncoupled numerical method for the analysis of dynamic crack propagation is proposed. The approach consists of two main steps. Firstly, the internal stresses in the intact, unfractured, elastic body are calculated with the use of the finite element method. Firstly, the internal stresses in the intact, unfractured, elastic body are calculated with the use of the finite element method. In this calculation it is assumed that no cracks are present and that fracture does not occur. Secondly, a theoretical crack is initiated and possible crack paths are derived from the elastic stress data. The stress-intensity factors for the planar fracture modes I and II, for the anti-plane mode III, and for the bending modes 1 and 2 are calculated from the well-known, linearized expressions for arbitrary, slightly curved cracks in thin plate-like and shell-like structures. The direction and speed of crack propagation are determined from a dynamic fracture criterion based on the energy release rate. Several applications of the uncoupled numerical method are presented, concerning standard fracture specimens loaded by tensile forces and bending moments, a single-edge notched beam loaded by shear forces, and a three-dimensional cylindrical tube loaded by torsional moments. Good agreement with both experimental and numerical results from the literature has been obtained. The major advantages of the uncoupled approach are its ease-of-use and the limited computational effort. This revised version was published online in August 2006 with corrections to the Cover Date.     

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 2. Strong resonance

Using the approach proposed in Part 1, an approximate calculation of vibration parameters is made for an elastic body with a closing crack, in the region of a strong 1/2-order subharmonic resonance with the lower-harmonic amplitude of free vibration spectrum larger than the main amplitude of forced vibrations. __________ Translated from Problemy Prochnosti, No. 3, pp. 5–16, May–June, 2008.     

Vibration Diagnostics of a Damaged Inclined Cylindrical Shell

Free vibrations of an inclined cylindrical shell with a breathing crack are analyzed by the authors in view of the results of spectral diagnostics of transverse vibrations of a beam with a closing crack, which were obtained by A. A. Matveev and A. P. Bovsunovskii at the Institute of Problems of Strength of the National Academy of Sciences of Ukraine. Elastic characteristics of the shell are interpreted as a certain bilinear system. Data on the stiffness degradation due to the crack opening can be used for the selection of vibration damage indicators. We have obtained relationships between the relative stiffness degradation, crack depth, and crack location. It is demonstrated that the potential energy of tension of the shell median surface must be taken into account. We have found that the efficiency of detecting damages using vibration diagnostic methods improves with increasing shell thickness.     

基于小波分析的Timoshenko梁裂缝识别研究

采用等效转动弹簧代替梁内的不扩展横向裂缝,研究Timoshenko裂缝梁的横向振动特性,建立了一种与有限元分析相结合的、基于模态参数的小波分析识别Timoshenko梁内裂缝的方法。以一简支梁为例,通过建立含横向不扩展裂缝的Timoshenko梁的有限元模型,用Lanczos法对结构的模态进行了计算分析,求出了基本振型和转角模态。分别应用mexh小波和db小波为母小波对二者做小波变换,进行多尺度分析,通过小波系数模极大值位置识别出梁内的裂缝。并对识别结果进行对比,发现识别Timoshenko梁裂缝时,基于转角模态小波变换的方法对小波基、尺度的要求较低,变换后的小波系数线更为平滑,奇异性特征更为明显,故运用转角模态小波变换来识别Timoshenko梁裂缝,较之运用基本振型小波变换的方法更为方便、有效。该方法对Timoshenko梁裂缝识别的工程应用具有参考价值。     

Nanoscale vibration characterization of multi-layered graphene sheets embedded in an elastic medium

Detailed studies on the nanoscale vibration characteristics of multi-layered graphene sheets (MLGSs) that are embedded in an elastic medium are carried out using continuum-based modelling and Generalized Differential Quadrature (GDQ) method. Natural frequencies and their associated vibration modes of practical interest of single-layered and triple-layered graphene sheets, as well as general MLGSs that are embedded in an elastic medium are established. Numerical simulations are conducted to examine the effects of van der Waals (vdW) interactions, which are present as bonding forces between the layers, on nanoscale vibration natural frequencies and their mode shapes. The results show that for a general MLGSs embedded in an elastic medium, vibration modes can in general be classified into three families - lower classical synchronized modes which are independent of van der Waals forces and are somewhat sensitive to the surrounding elastic medium, middle van der Waals enhanced modes which are largely determined by the presence of van der Waals interactions and are hence less sensitive to the changes of the surrounding elastic medium, and higher mixed modes which are combinations of classical synchronized modes and van der Waals enhanced modes. Detailed characterizations of these modes from their derived mode shapes have been achieved for the typical case of an embedded triple-layered GSs, as well as general embedded MLGSs. Effects of Winkler modulus K_W, the shear layer modulus G_b, different boundary conditions, aspect ratio β and the number L of graphene layers on nanoscale vibration properties have been examined in detail. The results presented in this paper, for the first time, provide accurate and wholesome studies and characterizations on the interesting nanoscale vibration properties of multi-layered graphene sheets embedded in an elastic medium and the results obtained will certainly be useful to those who are concerned with the dynamics of embedded graphene sheets which are increasingly being deployed for various innovative engineering applications such as nano-electro-mechanical systems (NEMS).     

传递函数法在非局部弹性梁动力学分析中的应用

采用传递函数方法进行了非局部弹性梁的动力学分析。非局部弹性梁内一点的应力与梁某一区域内任意一点的应变均有关系。本文基于Eringen的非局部弹性积分型本构关系,采用幂指数型核函数,利用Laplace变换导出梁的四阶偏微分形式振动方程,通过定义状态向量,将控制方程化为一阶微分方程组,并采用传递函数方法进行了求解,针对两种边界条件给出了非局部弹性梁的固有频率和固有振型。结果表明,同阶频率下,非局部弹性梁的频率比局部梁的频率低,振型基本一致。     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 3. Analytical and numerical determination of natural frequencies of longitudinal and bending vibrations of beams with transverse cracks. Solution

Based on the concepts of linear fracture mechanics, we derive analytical expressions for the determinational of natural frequencies of longitudinal and bending vibrations of beams which are rectangular in cross section, are fixed in different ways, have variable ratios of the section height to the beam length, and have transverse cracks of various types. The results of the analytical solution are compared with those obtained by the finite-element method as well as with experimental data obtained by the authors and other scientists. The analytical solution under consideration is shown to be quite simple and provide a fairly good accuracy of the results obtained. Using a cantilever beam with one or two symmetrical edge cracks or a central through crack as an example, we consider the possible dependence of the relative change in natural frequencies of vibration on the relative crack length, crack location, and the vibration mode of a beam. We discuss the possible methods of evaluating the crack size and location from the results of experimental determination of the change in natural vibration frequencies of a cracked beam. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 19–31, July–August, 1999.     

波传播法分析两端固支轴向运动梁的横向振动

用波传播法分析研究了两端固支轴向运动梁的横向自由振动。应用相位封闭原则导出了系统的频率方程,并求出了固有频率和模态函数。所得结论和传统方法解带有边界条件的微分方程的结果一致。但是,波传播法可以从物理角度更好的解释系统振动的本质,这是传统解法所没有提及的。     

Numerical study of vibrations of a nonlinear mechanical system simulating a cracked body

The paper presents the results of a numerical study of forced and decaying vibrations of a system simulating a body with a closing crack under the action of various modes of a nonlinear restoring force and nonlinear viscous friction or the hysteresis-type energy dissipation. We obtained the general patterns of appearance of higher harmonics of the Fourier expansion of time dependence of vibration of a cracked body model. The sensitivity of the higher-harmonic method to the presence of a crack is compared with that of some other vibration damage indicators. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 65–80, November–December, 1999.     

Free vibrations of shells of revolution filled with a fluid

This paper describes the method of determining the normal modes of vibrations and natural frequencies of elastic shells of revolution with an arbitrary meridian, partially filled with a fluid. The modes of vibration of the shells with fluids are determined as a linear combination of the natural modes of vibration in vacuum. The solution of the problem of hydroelastic vibrations has been obtained using the methods of the boundary element (BEM) and the finite element (FEM). Numerical investigations of vibrations of hemi-spherical shells conveying fluid have been conducted and analyzed. Illustrative examples are provided to demonstrate the accuracy and efficiency of the developed numerical procedure.     

Non-orthogonal stress modes for interfacial fracture based on local plastic dissipation

Interfacial cracks have several features which are different from those of cracks in homogeneous materials. Among those, the loading mode dependency of interfacial toughness has been a main obstacle to the widespread utilization of interfacial fracture mechanics. In this study, plasticity-induced toughening of an interface crack between an elastic–plastic material and an elastic material is studied. A useful relationship between the plastic dissipation and the plastic zone size is derived via an effective crack length model. Non-orthogonal stress modes for interface cracks are proposed on the basis of the plastic dissipation mechanism and a mixed-mode criterion for interfacial crack growth is also proposed using these stress modes. The non-orthogonal stress modes are able to represent the asymmetric behavior, mode-dependent toughening and ε-dependency of interfacial crack growth.     

Vibration of cracked beams

A surface crack on a beam section introduces a local flexibility to the structural member. This alters the vibration response of the system, shifting its frequency and increasing its mobility. Strain energy concentration arguments lead to the development of a compliance matrix for the behavior of the beam in the vicinity of the crack. This matrix is used to develop the transfer matrix for the cracked beam element. The transfer matrix developed, was used to evaluate the dynamic characteristics of simple cracked beams experiencing free vibrations. Reasonant frequencies and vibration modes are considerably affected by the existence of moderate cracks. Results for simple beams are presented for different boundary conditions.