The non-linear dynamic response of the Euler-Bernoulli beam with an arbitrary number of switching cracks

In this study the non-linear dynamic response of the Euler-Bernoulli beam in presence of multiple concentrated switching cracks (i.e. cracks that are either fully open or fully closed) is addressed. The overall behaviour of such a beam is non-linear due to the opening and closing of the cracks during the dynamic response; however, it can be regarded as a sequence of linear phases each of them characterised by different number and positions of the cracks in open state. In the paper the non-linear response of the beam with switching cracks is evaluated by determining the exact modal properties of the beam in each linear phase and evaluating the corresponding time history linear response through modal superposition analysis. Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control has been enforced with the aim of establishing the minimum number of linear modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented in order to illustrate the efficiency of the proposed approach for the evaluation of the non-linear dynamic response of beams with multiple switching cracks. In particular, the behaviour under different boundary conditions both for harmonic loading and free vibrations has been investigated.     

Three-dimensional finite element modeling of a vibrating beam with a breathing crack

This paper develops a full three-dimensional finite element model in order to study the vibrational behavior of a beam with a non-propagating surface crack. In this model, the breathing crack behavior is simulated as a full frictional contact problem between the crack surfaces, while the region around the crack is discretized into three-dimensional solid finite elements. The governing equations of this non-linear dynamic problem are solved by employing an incremental iterative procedure. The extracted response is analyzed utilizing either Fourier or continuous wavelet transforms to reveal the breathing crack effects. This study is applied to a cracked cantilever beam subjected to dynamic loading. The crack has an either uniform or non-uniform depth across the beam cross-section. For both crack cases, the vertical, horizontal, and axial beam vibrations are studied for various values of crack depth and position. Coupling between these beam vibration components is observed. Conclusions are extracted for the influence of crack characteristics such as geometry, depth, and position on the coupling of these beam vibration components. The accuracy of the results is verified through comparisons with results available from the literature.     

Impulsive loading of ideal fibre-reinforced rigid-plastic beams—I Free beam under central impact

A theory was developed in [1] for the dynamical behaviour under transverse load of ideal fibre-reinforced beams (that is, beams which are inextensible in their longitudinal direction) which exhibit rigid-plastic mechanical response. This theory is here applied to the problem of a beam of finite length, free at both ends, which is struck centrally by a mass which subsequently adheres to the beam. The general solution for the motion of the beam is determined for a fairly wide class of non-linear strain-hardening laws. Simplified approximate solutions are derived for the cases of (a) a heavy striker, (b) a light striker and (c) low impact speed and/or slight strain-hardening.     

Linear and non-linear vibration and frequency response analyses of microcantilevers subjected to tip-sample interaction

Despite their simple structure and design, microcantilevers are receiving increased attention due to their unique sensing and actuation features in many MEMS and NEMS. Along this line, a non-linear distributed-parameters modeling of a microcantilever beam under the influence of a nanoparticle sample is studied in this paper. A long-range Van der Waals force model is utilized to describe the microcantilever-particle interaction along with an inextensibility condition for the microcantilever in order to derive the equations of motion in terms of only one generalized coordinate. Both of these considerations impose strong nonlinearities on the resultant integro-partial equations of motion. In order to provide an understanding of non-linear characteristics of combined microcantilever-particle system, a geometrical function is wisely chosen in such a way that natural frequency of the linear model exactly equates with that of non-linear model. It is shown that both approaches are reasonably comparable for the system considered here. Linear and non-linear equations of motion are then investigated extensively in both frequency and time domains. The simulation results demonstrate that the particle attraction region can be obtained through studying natural frequency of the system consisting of microcantilever and particle. The frequency analysis also proves that the influence of nonlinearities is amplified inside the particle attraction region through bending or shifting the frequency response curves. This is accompanied by sudden changes in the vibration amplitude estimated very closely by the non-linear model, while it cannot be predicted by the best linear model at all.     

Generic non-linear modelling of a bi-material composite beam with partial shear interaction

Composite members composed of two materials joined by shear connection find widespread use in engineering infrastructure, in both traditional practice and innovative applications. Studies in the literature dating back nearly 60 years have elucidated the mechanics of the behaviour of these composite structural members in which the solution for the slip at the interface between the materials was determined by solving a linear differential equation. However, these solutions are based on a linear formulation of the strain-displacement relationship, and in some applications this relationship must be represented in non-linear form, so that the second order effects in the member can be quantified correctly. This paper presents such a study for a composite member with two materials, being typical of a steel-concrete composite beam in structural engineering. It quantifies the restraint of the member ends by longitudinal and rotational elastic springs, so that the axial tension developed is a function of the transverse loading, material properties, cross-sectional properties and the restraint stiffness. The problem is treated using minimisation of the total potential stored in the two members, the elastic shear connection at their interface, the restraints at the ends and the work done by the transverse forces, for which the differential equations for the deformations can be determined from routine variational calculus. The non-linear equation of equilibrium relating the external loading to the internal actions is stated in closed form by invoking the static and kinematic boundary conditions for the member. The solution is compared with closed form treatments derived elsewhere, and a representative member is analysed so that the influences of the non-linearity, end restraint stiffness and degree of partial shear interaction on its behaviour can be examined.     

Non-linear interactions in imperfect beams at veering

Non-linear interactions in a hinged-hinged uniform moderately curved beam with a torsional spring at one end are investigated. The two-mode interaction is a one-to-one autoparametric resonance activated in the vicinity of veering of the frequencies of the lowest two modes and results from the non-linear stretching of the beam centerline. The excitation is a base acceleration that is involved in a primary resonance with either the first mode only or with both modes. The ensuing non-linear responses and their stability are studied by computing force- and frequency-response curves via bifurcation analysis tools. Both the sensitivity of the internal resonance detuning—the gap between the veering frequencies—and the linear modal structure are investigated by varying the rise of the beam half-sinusoidal rest configuration and the torsional spring constant. The internal and external resonance detunings are varied accordingly to construct the non-linear system response curves. The beam mixed-mode response is shown to undergo several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization.     

On the use of non-linear vibrations and the anti-resonances of Higher-Order Frequency Response Functions for crack detection in pipeline beam

The identification of new scientific challenges, as well as the increasing high-performance computing support, indicates that the benefits of applying novel nonlinear techniques for crack detection will continue to grow. So, significant effort has been invested in recent years to develop effective techniques to detect crack in mechanical structures. The objective of this paper is to discuss and propose a robust diagnostic of damage based on non-linear vibrational measurements with particular regard to the Higher-Order Frequency Response Functions. An important observation is that the appearances of the non-linear harmonic components and the emerging anti-resonances in Higher-Order Frequency Response Functions can provide useful information on the presence of cracks and may be used on an on-line crack monitoring system for small levels of damage. Efficiency of the proposed methodology is illustrated through numerical examples for a pipeline beam including a breathing crack.     

Quasi-static response of linear viscoelastic cantilever beams subject to a concentrated harmonic end load

This study evaluates the response of a uniform cantilever beam with a symmetric cross-section fixed at one end, and submitted to a lateral concentrated sinusoidal load at the free extremity. The beam material is assumed to be homogeneous, isotropic and linear viscoelastic. Due to the nature of the loading and the beam slenderness, large displacements are developed but the strains are considered small. Consequently, the mathematical formulation only involves geometrical non-linearity. It is also assumed that the beam is inextensible (neutral axis length is constant) and that inertial forces are negligible, i.e., dynamic effects are insignificant and the system can thus be modeled quasi-statically. The beam is therefore subject to oscillations caused by the sinusoidal time-dependent load, leading to a transient response until the material stabilizes and the system exhibits a periodic response, which can be conveniently described in the frequency domain. The time domain solution of this problem is elaborated by considering the quasi-static response for each time interval. The mathematical equations are presented in dimensional and dimensionless forms, and for the latter case, a numerical solution is generated and several case studies are presented. The problem is governed by a set of non-linear ordinary differential equations encompassing functions of space and time that relate the curvature, rotation angle, bending moment and geometrical coordinates. In this study, an elegant solution is deduced using perturbation theory, yielding a precise steady-state solution in the frequency domain with considerable computational economy. The solutions for both time and frequency domain methods are developed and compared using a case study for a series of dimensionless parameters that influence the response of the system.     

Efficient analysis of large-scale structural problems with geometrical non-linearity

The paper presents an efficient methodology for the analysis of large-scale structural problems with geometrical non-linearity. A finite element based tool is developed, taking advantage of the analytical formulation of the stiffness matrix of a beam element, which is explicitly separated in linear and non-linear terms. The methodology proposes the substitution of the typical Newton-type non-linear analysis procedure, by a series of incremental linear analyses and a set of ‘fictitious’ forces, replacing the non-linear effect. The proposed technique is demonstrated in several structural problems that exhibit geometrical non-linear behaviour, with satisfactory results. The method’s advantages on the analysis of large-scale non-linear problems are discussed, as well as the limitations and the further development that is required.     

Effects of a crack on the stability of a non-linear rotor system

The stability of a rotor system presenting a transverse breathing crack is studied by considering the effects of crack depth, crack location and the shaft's rotational speed. The harmonic balance method, in combination with a path-following continuation procedure, is used to calculate the periodic response of a non-linear model of a cracked rotor system. The stability of the rotor's periodic movements is studied in the frequency domain by introducing the effects of a perturbation on the periodic solution for the cracked rotor system.It is shown that the areas of instability increase considerably when the crack deepens, and that the crack's position and depth are the main factors affecting not only the non-linear behaviour of the rotor system but also the different zones of dynamic instability in the periodic solution for the cracked rotor. The effects of some other system parameters (including the disk position and the stiffness of the supports) on the dynamic stability of the non-linear periodic response of the cracked rotor system are also investigated.     

Effect of damping on the postcritical behaviour of autonomous non-conservative systems

Postcritical behavior of non-linear autonomous non-conservative systems is studied with special attention to the effects produced by the damping forces non-uniformly distributed among the natural modes (or the degrees of freedom). The objective is to find out, to what degree the well-known destabilization effect of damping primarily observed in linear systems affects the postcritical behaviour of a system if non-linearities are taken into account. As a bench model, an initially planar elastic panel is considered subjected to the supersonic gas flow. However, the qualitative conclusions may be extrapolated upon a wide class of phenomena in non-linear autonomous non-conservative systems such as airfoils and panel flutter, instabilities induced by the jet thrust or jet pressure, etc. The main conclusion confirms the already formulated statement that, from the rigorous viewpoint, all these phenomena are to be treated taking into account the initial complete positive damping, however small. There are no “paradoxes” neither in the linear nor the non-linear statement of a problem under the condition that the concept of stability is used in the proper sense. On the other hand, a strong effect is demonstrated by the ratio of partial damping factors on the postcritical behaviour. In general, the ratio of these factors influences the postcritical behaviour to a higher degree that their absolute magnitudes, at least if the latter are small or moderate.     

基于压电振动能量俘获的弯曲结构损伤监测研究

建立了含裂纹损伤的曲梁压电能量俘获系统在强迫振动下的动力学模型. 基于Prescott型压电曲梁力电耦合振动方程的解析解和裂纹截面处的连续性条件, 求解了含裂纹损伤的压电曲梁的格林函数. 根据线性叠加原理, 对含裂纹的力电耦合模型的系统方程解耦, 得到强迫振动下含裂纹损伤的曲梁压电俘能器的输出电压. 在得到模型的强迫振动解析解后, 提出逆方法检测结构中的裂纹损伤, 这一检测方法适用于处于振动状态下的结构. 在数值计算中, 令裂纹深度为零, 通过对比本文的解析解与现有文献中的解析解, 验证了本文解的有效性. 分别分析了含裂纹损伤的压电曲梁的电压响应与裂纹深度、裂纹位置、材料的几何参数以及阻尼之间的关系. 研究结果表明: 裂纹的存在对曲梁式压电俘能器的影响比直梁式更加复杂; 裂纹出现时, 损伤曲梁在健康曲梁的一阶频率值处一定会出现波动并被激励出二阶频率, 此时的二阶频率是开路中健康压电曲梁的一阶频率值; 通过对电压响应的检测可以确定的损伤裂纹的深度和在结构中出现的位置范围; 利用振动问题的解来检测压电曲梁的健康状况是可行且准确的.      

A novel statistical linearization solution for randomly excited coupled bending-torsional beams resting on non-linear supports

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique.

     

Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances

The non-linear behaviour of a slender beam carrying a lumped mass subjected to principal parametric base excitation is investigated. The dimension of the beam–mass system and the position of the attached mass are so adjusted that the system exhibits 3 : 1 internal resonance. Multi-mode discretization of the governing equation which retains the cubic non-linearities of geometrical and inertial type is carried out using Galerkin’s method. The method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and the stability of the system. The linear first-order perturbation results show new zones of instability due to the presence of internal resonance. For low amplitude of excitation and damping Hopf bifurcations are observed in the trivial steady-state response. The multi-branched non-trivial response curves show turning point, pitch-fork and Hopf bifurcations. Cascade of period and torus doubling, crises as well as the Shilnikov mechanism for chaos are observed. This is the first natural physical system exhibiting a countable infinity of horseshoes in a neighbourhood of the homoclinic orbit.     

An elastic–plastic crack bridging model for brittle-matrix fibrous composite beams under cyclic loading

A fibrous composite beam with an edge crack is submitted to a cyclic bending moment and the crack bridging actions due to the fibers. Assuming a general elastic-linearly hardening crack bridging model for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging actions are obtained from compatibility conditions. The elastic and plastic shake-down phenomena are examined in terms of generalised cross-sectional quantities and, by employing a fatigue crack growth law, the mechanical behaviour up to failure is captured. Within the framework of the proposed fracture mechanics-based model, the cyclic crack bridging due to debonding at fiber–matrix interface of short fibers is analysed in depth. By means of some simplifying assumptions, such a phenomenon can be described by a linear isotropic tensile softening/compressive hardening law. Finally, numerical examples are presented for fibrous composite beams with randomly distributed short fibers.     

Vibration control of a cantilever beam of varying orientation

We investigate the problem of suppressing the vibrations of a non-linear system with a cantilever beam of varying orientation subject to parametric and direct excitation. It is known that the growth of the response is limited by non-linearity. Therefore, vibration control and high-amplitude response suppressions of the first mode of a cantilever beam can be performed using a simple non-linear feedback law. This control law is based on cubic velocity feedback. The method of multiples scales is used to construct first-order non-linear ordinary differential equations governing the modulation of the amplitudes and phases. The stability and effects of different system parameters are studied numerically.     

带裂纹旋转柔性梁系统的刚柔耦合动力学研究

对具有刚柔耦合效应的带裂纹旋转柔性梁进行建模和动力学特性分析研究。采用晶格弹簧离散模型,利用无质量弹簧模拟梁上裂纹,通过考虑梁变形的二阶耦合项建立了带裂纹旋转柔性梁系统的一次近似耦合动力学控制方程。数值计算结果表明,裂纹的存在会使旋转柔性梁的固有频率降低,并且随着梁转速的增大,这种降低效应呈减弱趋势;值得注意的是,裂纹梁的固有频率与裂纹处的弯矩具有正相关关系。此外,裂纹的存在不仅会使转速变化阶段梁的末端位移响应增大,还会对转速稳定后梁的末端振荡产生显著的影响。     

Buckling of cracked composite columns

A crack in a structural element introduces a significant local flexibility which enhances the instability. Buckling of an edge-notched beam is studied for isotropic and anisotropic composites. The local compliance due to the presence of cracks in an anisotropic medium is formulated as a function of the crack-tip stress intensity factors and the elastic constants of the material. The general integration of the non-linear differential equations expressing the buckling model of an eccentrically loaded composite beam is derived for two different types of hinged supports ; namely freely approaching and fixed span. The effect of reducing rigidity on the load-carrying capacity and the post-buckling behavior of the beam is discussed and exemplary numerical solutions are provided. The study indicates that the instability increases with the beam slenderness and the crack length. In addition, the material anisotropy conspicuously reduces the load-carrying capacity of an externally cracked member.