多频简谐激励下裂纹梁的非线性振动响应

主要对含裂纹梁在振动与超声波联合激励下所出现的非线性动力响应的机理和特性进行研究.将疲劳裂纹在外加激励下的状态简化为周期性张开-闭合的非线性过程,基于圣维南原理,采用有限元方法建立了含非对称疲劳裂纹梁的非线性数值分析模型.利用非线性输出频率响应函数(NOFRFs)概念,对裂纹梁在高-低频简谐激励下所出现的非线性动力响应特性的机理进行了解释.具体以悬臂梁为例,仿真分析了裂纹深度和裂纹位置等参数的变化对系统非线性动力响应特性的影响规律.     

一类非线性隔振器振动传递特性分析

非线性输出频率响应函数是由Volterra级数发展而来的一个新概念.对一类具有反对称阻尼特性的隔振器,通过该概念推导出了振动传递性与系统非线性参数之间的显式解析关系；进而系统地研究了非线性阻尼参数对隔振器的力传递性能和位移传递性能的影响.研究结果表明,虽然非线性隔振器在受正弦信号激励下会出现高次倍频分量,但对于其传递性...     

弹性预紧约束非线性对结构振动的影响

针对大型空间可展开结构中存在的关节非线性间隙约束问题,研究弹性预紧约束非线性对结构动态特性的影响.首先,基于轴向拉伸力学模型,构造了轴向拉伸关节三维梁单元.其次,使用Kelvin-Voigt接触及Coulomb摩擦模型,建立了粘弹性预紧约束下整体结构非线性动力学方程,继而分析了不同预紧约束下关节对结构非线性振动的影响.结果表明,在关节非线性约束下,结构振动传递特性的高频响应增加,低阶振动能量传递到高阶振动上.     

压电纤维复合材料层合壳的非线性动力学研究

本文对横向激励作用下的1-3型压电纤维复合材料层合壳进行了非线性动力学分析,并研究了压电特性对结构振动响应的影响.首先建立了压电纤维复合材料层合壳的非线性动力学方程,并且在已知的几何结构和材料特性基础上考虑了电场属性.然后根据位移边界条件,选择合适的振型函数,通过Galerkin方法将运动控制方程转化成两自由度的非线性常微分方程.通过数值模拟方法分析了横向激励和压电系数对压电纤维复合材料层合壳非线性振动特性的影响.通过波形图、三维相图、庞加莱图和分叉图等来研究壳体不同类型的周期和混沌运动.结果表明,外激励作用下结构存在复杂的非线性振动响应,同时压电参数对层合壳结构振动响应具有很强的调节作用.     

带边角裂纹悬臂Mindlin 板的振动特性研究

本文基于Mindlin板理论,应用Ritz法研究带边角裂纹Mindlin板的振动特性,分析了不同裂纹参数如裂纹位置,裂纹长度,裂纹角度对悬臂Mindlin板的固有频率和模态的影响.利用Ritz法求解固有频率和模态函数,本文构造了一个特殊的模态函数,其模态函数由两部分构成,一部分是用梁函数组合法得到的无裂纹理想完整矩形板的振型,另一部分是利用裂纹尖端奇异性理论,构造描述裂纹附近位移和转角不连续的角函数.通过高精度的数值计算软件Maple得出结果,并与有限元软件ANSYS分析的结果进行对比,验证本文计算结果的准确性.     

轴向可伸缩复合材料悬臂梁的非线性振动研究

以航空领域中可变体机翼的伸缩变形过程为研究对象,对可伸缩悬臂复合材料层合梁的时变非线性振动进行理论研究.建立可伸缩悬臂复合材料层合梁在外载荷作用下的非线性动力学模型;根据时变系数非线性动力学方程研究时变非线性振动特性.分析可伸缩悬臂复合材料层合梁在外伸与收缩变形过程中的非线性动力学特性.从数值结果上看:模型的外伸速度、飞行速度对振动的影响较大,初值对振动的影响较小.     

旋转粘弹性夹层梁非线性自由振动特性研究

对旋转粘弹性夹层梁的非线性自由振动特性进行了分析.基于Kelvin-Voigt粘弹性本构关系和大挠度理论,建立了旋转粘弹性夹层梁的非线性自由振动方程,并使用Galerkin法将偏微分形式振动方程化为常微分振动方程.采用多重尺度法对非线性常微分振动方程进行求解,通过小参数同次幂系数相等获得微分方程组,并通过求解方程组及消除久期项来获得旋转粘弹性夹层梁非线性自由振动的一次近似解.用数值方法讨论了粘弹性夹层厚度、转速和轮毂半径对梁固有频率的影响.结果表明:固有频率随转速增大而增大,随夹层厚度增大而减小,随轮毂半径的增大而增大.     

复模态分析超临界轴向运动梁横向非线性振动

近似解析研究了简支边界条件下超临界轴向运动梁横向非线性自由振动的固有频率和模态函数.采用复模态方法处理控制方程,一个积分偏微分方程.将Galerkin截断思想用于近似处理线性化方程,一个含空间依赖系数的常微分方程.给出了不同截断项数对固有频率的影响.基于8项截断,讨论了系统参数对模态函数的影响.     

变截面粘弹性旋转梁非线性参数振动研究

研究了变截面粘弹性旋转梁的非线性参数振动.基于Kelvin-Voigt粘弹性本构关系,考虑几何非线性建立了变截面粘弹性旋转梁的非线性振动方程,用Galerkin法将其转化为常微分方程.运用多重尺度法得到其幅频响应.用数值方法讨论了转速和轮毂半径对梁固有频率和幅频响应的影响.研究表明:不稳定域随轮毂半径、转速的增大而增大,随锥度的增大而减小.     

振动抑制与能量俘获专刊序

工程系统中梁结构经常处于各种激励的作用下,因而梁结构在这种环境下不可避免地发生着各种各样的强迫振动.在梁结构发生振动的过程中,其自身会受到的温度、湿度、电磁场、裂纹等众多内外部因素的影响,而众多内外部因素就构成梁的多物理场耦合环境.在多场耦合环境下,Green函数法作为一种解析方法在研究梁的多场耦合振动问题方面具有优势,有利于讨论力、电、热、裂纹等因素作用下梁的振动特性和多场耦合特性.Green函数法相比于模态叠加法,优点在于能够得到完整且精度较高的解析解,具有收敛性好,运算快的特点.本文主要阐述Green函数在梁的强迫振动、热力耦合振动、力电耦合振动、裂纹梁振动等研究问题上取得了大量的理论和工程研究成果.本文以裂纹为内因,热、力、电为外因进行分类,阐述了在内外因影响下梁的强迫振动问题Green函数解的研究现状,从而让读者进一步系统性的了解Green函数法在振动领域中的广泛应用,以及了解该方法本身的特色和优势奠定基础.     

Geometrically non-linear free vibrations of clamped-clamped beams with an edge crack

It is well known that a crack in a beam induces a drop in its natural frequencies and affects its modes shapes. This paper is a theoretical investigation of the geometrically non-linear free vibrations of a clamped-clamped beam containing an open crack. The approach uses a semi-analytical model based on an extension of the Rayleigh-Ritz method to non-linear vibrations, which is mainly influenced by the choice of the admissible functions. The general formulation is established using new admissible functions, called “cracked beam functions”, and denoted as “CBF”, which satisfy the natural and geometrical end conditions, as well as the inner boundary conditions at the crack location. Iterative solution of a set of non-linear algebraic equations is obtained numerically, which leads to the basic function contribution coefficients to the displacement response function. Then, an explicit solution is derived and proposed as an alternative procedure, simple and ready to use for engineering applications. Emphasis is made on the backbone curves, i.e. amplitude-frequency dependence, obtained for various crack depth, and the effect of the vibration amplitudes upon the non-linear mode shapes of a cracked beam is examined. The work is restricted to the fundamental mode in order to concentrate on the study of the influence of the crack on the non-linear dynamic response near to the fundamental resonance.     

Order reduction of non-linear differential-algebraic process models

A method for order reduction of non-linear differential-algebraic models arbitrary index is presented. The approach is a direct generalization of a method suggested by Pallaske in 1987 for the reduction of explicit differential equation models. It relies on an optimal orthogonal projection of the solution trajectories into a subspace of the original state space. A rigorous development of the reduction technique is given. Strong emphasis is on implementational issues such as the choice of tuning parameters for a particular problem. A theoretical and numerical evaluation of the method is provided. The case studies discussed include the reduction of a strongly non-linear catalytic fixed bed reactor model.     

Optimal vibration control of piezolaminated smart beams by the maximum principle

Active control of a vibrating beam using piezoelectric patch actuators is considered. The specific structure to be studied is an Euler–Bernoulli beam with piezoelectric actuators bonded to the top and bottom surfaces of the beam. The equation of motion includes Heaviside functions and their derivatives due to finite size piezo patches which provide the control force to damp out vibrations. Optimal control theory is formulated with the objective function specified as a weighted quadratic functional of the dynamic responses of the beam which is to be minimized at a specified terminal time. The expenditure of the control forces is included in the objective function as a penalty term. The optimal control law is derived using a maximum principle developed by Sadek et al. [1]. The maximum principle involves a Hamiltonian expressed in terms of an adjoint variable with the state and adjoint variables linked by terminal conditions leading to a boundary-initial-terminal value problem. The explicit solution of the problem is developed using eigenfunction expansions of the state and adjoint variables. The numerical results are given to assess the effectiveness and the capabilities of piezo actuation to damp out the vibrations.     

Dynamic response of a rectangular beam with a known non-propagating crack of certain or uncertain depth

In this paper the deterministic behaviour of a beam with a transverse on edge non-propagating crack is first studied. Moreover the stochastic setting pertaining the case in which the crack has an uncertain depth is investigated. The beam is discretized by finite elements in which a so-called closing crack model, with fully open or fully closed crack, is used to describe the damaged element. Once the mathematical model of the beam is defined, the dynamic response is evaluated by applying a numerical procedure based on the philosophy of structural systems with dynamic modification. In the stochastic case the improved perturbation method is modified in order to solve efficiently the stochastic non-linear differential equations.     

Divergence and flutter instabilities of nanobeams in moving state accounting for surface and shear effects

Transverse vibrations of elastically rested moving beam-like nanostructures accounting for surface effect are of high concern. The role of nonlocality on the free dynamic response of moving nanobeams has been revealed in recent years; nevertheless, the influence of the surface energy on the mechanical behavior of such elements has not been explained yet. In this paper, equations of motion of rested nanoscaled beams in the moving state are derived carefully via surface energetic-shear deformable beam models. Subsequently, the transverse vibrations of the nanostructure are evaluated using Galerkin-based assumed mode method. The explicit expressions of divergence velocities are obtained analytically, and these are successfully verified with the results of a numerical approach. The roles of crucial parameters on the first divergence velocity are addressed in some detail. Additionally, the stable and unstable regions are determined systematically and the influence of both surface energy and shear energy on the stability of moving nanostructure is discussed.     

Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures

Non-linear normal modes (NNMs) are used in order to derive accurate reduced-order models for large amplitude vibrations of structural systems displaying geometrical non-linearities. This is achieved through real normal form theory, recovering the definition of a NNM as an invariant manifold in phase space, and allowing definition of new co-ordinates non-linearly related to the initial, modal ones. Two examples are studied: a linear beam resting on a non-linear elastic foundation, and a non-linear clamped–clamped beam. Throughout these examples, the main features of the NNM formulation will be illustrated: prediction of the correct trend of non-linearity for the amplitude-frequency relationship, as well as amplitude-dependent mode shapes. Comparisons between different models—using linear and non-linear modes, different number of degrees of freedom, increasing accuracy in the asymptotic developments—are also provided, in order to quantify the gain in using NNMs instead of linear modes.     

Harmonic analysis of the vibrations of a cantilevered beam with a closing crack

In this article, the vibrational response of a cracked cantilevered beam to harmonic forcing is analysed. The study has been performed using a finite element model of the beam, in which a so-called closing crack model, fully open or fully closed, is used to represent the damaged element. Undamaged parts of the beam are modelled by Euler-type finite elements with two nodes and 2 d.f. (transverse displacement and rotation) at each node. Recently the harmonic balance method has been employed by other researchers to solve the resulting non-linear equations of motion. Instead, in this study, the analysis has been extended to employ the first and higher order harmonics of the response to a harmonic forcing in order to characterize the non-linear behaviour of the cracked beam. Correlating the higher order harmonics of the response with the forcing term the so-called higher order frequency response function (FRFs), defined from the Volterra series representation of the dynamics of non-linear systems, can be determined by using the finite element model to simulate the time domain response of the cracked beam. Ultimately the aim will be to employ such a series of FRFs, an estimate of which in practice could be measured in a stepped sine test on the beam to indicate both the location and depth of the crack, thus forming the basis of an experimental structural damage identification procedure.     

Torsional vibrations of rotors with transverse surface cracks

The torsional vibrations of a rotor with a transverse crack are investigated. The crack is modelled by way of a local flexibility matrix which is calculated analytically and measured experimentally. Good agreement between the two methods is obtained. The free vibration problem is solved and the three first eigenvalues are plotted versus the crack location and depth. The finite element method is used for the solution of any shaft system with a crack, using a modified stiffness matrix for the cracked element.     

Buckling, flutter and vibration analyses of beams by integral equation formulations

This paper presents a model for the investigation of buckling, flutter and vibration analyses of beams using the integral equation formulation. A mathematical formulation based on Euler–Bernoulli beam theory is presented for beams with variable sections on elastic foundations and subjected to lateral excitation, conservative and non-conservative loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebraic system related to internal and boundary unknowns. Eigenvalue problems related to buckling and vibrations are formulated and numerically solved. Buckling loads, natural frequencies and associated eigenmodes are computed. The corresponding slope, bending and shear forces can be directly obtained. The load-frequency dependence is investigated for various elastic foundations and the divergence critical loads are evidenced. Under non-conservative loads, a dynamic stability analysis is illustrated numerically based on the coalescence of eigenfrequencies. The flutter load and instability regions with respect to the elastic concentrated and distributed foundations are identified. Using the eigenmodes, numerically computed, non-linear vibrations of beams are investigated based on one mode analysis. The presented model is quite general and the obtained numerical results are in agreement with available data.