Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems

The non-linear modal properties of a vibrating 2-DOF system with non-smooth (piecewise linear) characteristics are investigated; this oscillator can suitably model beams with a breathing crack or systems colliding with an elastic obstacle. The system having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of non-linear normal modes (NNMs) that are greater than the degrees of freedom. Since the non-linearities are concentrated at the origin, its non-linear frequencies are independent of the energy level and uniquely depend on the damage parameter. An analysis of the NNMs has been performed for a wide range of damage parameter by employing numerical procedures and Poincaré maps. The influence of damage on the non-linear frequencies has been investigated and bifurcations characterized by the onset of superabundant modes in internal resonance, with a significantly different shape than that of modes on fundamental branch, have been revealed.     

Nonlinear dynamics of a planar beam–spring system: analytical and numerical approaches

The multiple timescales method is applied to the exact partial differential equations of the planar motion of a hinged–simply supported beam with a linear axial spring of arbitrary stiffness. The forced-damped and free oscillations of the system around frequencies corresponding to nth natural bending mode are examined thoroughly and compared with numerical simulations as well as with already published results obtained by Lindstedt–Poincaré method. A special numerical technique using explicit finite element method to draw the frequency–response curves is appositely developed. The well-known jump phenomena between resonant and non-resonant branches, as well as superharmonic resonances, have been detected numerically.     

Determining Lyapunov spectrum and Lyapunov dimension based on the Poincaré map in a vibro-impact system

We develop a method to compute the Lyapunov spectrum and Lyapunov dimension, which is effective for both symmetric and unsymmetric vibro-impact systems. The Poincaré section is chosen at the moment after impacting, and the six-dimensional Poincaré map is established. The time between two consecutive impacts is determined by the initial conditions and the impact condition, hence the Poincaré map is an implicit map. The Poincaré map is used to calculate all the Lyapunov exponents and the Lyapunov dimension. By numerical simulations, the attractors are represented in the projected Poincaré section, and the Lyapunov spectrum is obtained. The multi-degree-of-freedom vibro-impact system may exhibit complex quasi-periodic attractors, which can be characterized by the Lyapunov dimension.     

Global dynamics of a harmonically excited oscillator with a play: Numerical studies

In this paper a harmonically excited linear oscillator with a play is investigated. Direct numerical simulation and numerical continuation techniques were employed to study the system behaviour. To conduct the numerical analysis, the system differential equations were transformed into the autonomous form and were then solved using our newly developed in-house Matlab-based computational suite ABESPOL [1]. The results are presented in form of trajectories and Poincaré maps on the phase plane, bifurcation diagrams and basins of attraction. The bifurcation analysis was supported by a path following method. The influence of each system parameter (except gap) on the system dynamics was studied in detail. The bifurcations known as interior crisis and boundary crisis were observed and discussed in this work. Notably, the parameter regions where various types of grazing induced bifurcations occurred were detected and investigated.     

Thermo-mechanical nonlinear dynamics of a buckled axially moving beam

The thermo-mechanical nonlinear dynamics of a buckled axially moving beam is numerically investigated, with special consideration to the case with a three-to-one internal resonance between the first two modes. The equation of motion of the system traveling at a constant axial speed is obtained using Hamilton??s principle. A closed form solution is developed for the post-buckling configuration for the system with an axial speed beyond the first instability. The equation of motion over the buckled state is obtained for the forced system. The equation is reduced into a set of nonlinear ordinary differential equations via the Galerkin method. This set is solved using the pseudo-arclength continuation technique to examine the frequency response curves and direct-time integration to construct bifurcation diagrams of Poincaré maps. The vibration characteristics of the system at points of interest in the parameter space are presented in the form of time histories, phase-plane portraits, and Poincaré sections.     

A cyclic steady-state method for fatigue crack propagation: Evaluation of plasticity-induced crack closure in 3D

The numerical study of plasticity-induced crack closure using the node-release technique presents many difficulties widely studied in literature. For instance various rules, proposed for overcoming mesh sensitivity, are challenged by more recent studies. This paper intends to propose and evaluate a numerical method for the investigation of crack propagation under fatigue loading, and particularly for the assessment of plasticity-induced crack closure in three-dimension. The method is an extension of the “steady-state method” to cyclic loadings. The steady-state method allows a direct computation (on a fixed mesh, without releasing nodes) of stress and strain fields around the crack tip and in the wake for a steady crack growth. The method is extended to simulate crack propagation under fatigue loading. Therefore it constitutes a valuable numerical tool for gaining insight into the physics of crack propagation, as it provides accurate mechanical fields around the crack tip and their relation with crack growth rate, various loading modes and parameters. The proposed method is also compared with the classical node-release technique. A very good agreement between the two methods is found. However the steady-state method needs much less mesh refinement and computational time. Following an analysis of some features of the fatigue crack, a discussion on a crack closure criterion is opened, and a reliable criterion for the determination of local crack closure is proposed.     

求解变分型积分方程的一种新型数值方法——有限变分法

将作者提出的多虚拟裂纹扩展法(MVCE法)拓展为求解变分型积分方程问题的一种新型数值方法——有限变分法(FVM)。它的基本思想是,给定有限个(N个)局部变分模式,将所求解的未知量用适当的方法离散化,针对这N个局部变分模式列出N个方程,求解N个未知系数,从而求得未知量。单一未知变量FVM的最终方程组的系数矩阵通常是一个对称的窄带矩阵,对角元是大数,有很好的数值计算性能。用FVM求解了三维I型裂纹前缘的应力强度因子(SIF)分布。利用基于FVM的通用权函数法计算程序,可以高精度和高效率地求解表面力、体积力和温度载荷共同作用情况下三维裂纹前缘SIF的分布及其时间历程。FVM可以被推广到更广泛的领域,是一个求解变分型积分方程问题的普遍适用的新型数值方法。     

基于扩展有限元的应力强度因子的位移外推法

针对平面裂纹问题,阐述了扩展有限元法的单元位移模式、推导了扩展有限元法的控制方程、介绍了特殊单元的数值积分技术.基于最小二乘法,建立了应力强度因子位移外推法的计算公式.利用MATLAB编写计算程序,对平面裂纹问题用扩展有限元法进行了计算.基于扩展有限元法的计算结果,分别利用位移外推法和相互作用积分法,对平面裂纹的应力强度因子进行了计算.计算结果表明,位移外推法比相互作用积分法能更方便和准确地计算平面裂纹的应力强度因子.     

Analysis of limit cycle oscillations of a typical airfoil section with freeplay

A typical airfoil section system with freeplay is investigated in the paper. The classic quasi-steady flow model is applied to calculate the aerodynamics, and a piecewise-stiffness model is adopted to characterize the nonlinearity of the airfoil section’s freeplay. There are two critical speeds in the system, i.e., a lower critical speed, above which the system might generate limit cycle oscillation, and an upper critical one, above which the system will flutter. Then a Poincaré map is constructed for the limit cycle oscillations by using piecewise-linear solutions with and without contact in the system. Through analysis of the Poincaré map, a series of equations which can determine the frequencies of period-1 limit cycle oscillations at any flight velocity are derived. Finally, these analytic results are compared to the results of numerical simulations, and a good agreement is found. The effects of freeplay value and contact stiffness ratio on the limit cycle oscillation are also analyzed through numerical simulations of the original system. Moreover, there exist multi-periods limit cycle oscillations and even complicated "chaotic" oscillations may occur, which are usually found in smooth nonlinear dynamic systems.     

Experimental evidence of bifurcating nonlinear normal modes in piecewise linear systems

A system with piecewise linear restoring forces, typical of damaged beams with a breathing crack, exhibits bifurcations characterized by the onset of superabundant modes in internal resonance with a significantly different shape than that of modes on a fundamental branch. A 2-DOF frame with piecewise linear stiffness is analyzed by means of an experimental investigation; the frame is forced by an harmonic base excitation and the operative modal shapes as well as the response amplitude are directly measured; the results are compared with numerical outcomes for different damping values. This study shows that the shapes and the frequencies of certain nonlinear normal modes (NNMs) of the related autonomous system strongly affect the forced response, in both the numerical and the experimental environments. Therefore, it is possible to match the NNM with the forced response of the system, leading to the prospect of identifying the severity and position of the damage from experimental tests.     

双频激励下Filippov系统的非光滑簇发振荡机理

旨在揭示含双频周期激励的不同尺度Filippov系统的非光滑簇发振荡模式及分岔机制. 以Duffing和Van der Pol耦合振子作为动力系统模型,引入周期变化的双频激励项,当两激励频率与固有频率存在量级差时,将两周期激励项表示为可以作为一慢变参数的单一周期激励项的代数表达式,给出了当保持外部激励频率不变,改变参数激励频率的情况下,快子系统随慢变参数变化的平衡曲线及因系统出现的fold分岔或Hopf分岔导致的系统分岔行为的演化机制.结合转换相图和由Hopf分岔产生稳定极限环的演化过程,得到了由慢变参数确定的同宿分岔、多滑分岔的临界情形及因慢变参数改变而出现的混合振荡模式,并详细阐述了系统的簇发振荡机制和非光滑动力学行为特性.通过对比两种不同情形下的平衡曲线及分岔图,指出虽然系统有相似的平衡曲线结构, 却因参数激励频率取值的不同,致使平衡曲线发生了更多的曲折,对应的极值点的个数也有所改变,并通过数值模拟, 对结果进行了验证.      

Nonlinear vibrations of a circular cylindrical shell with multiple internal resonances under multi-harmonic excitation

The nonlinear response of a water-filled, thin circular cylindrical shell, simply supported at the edges, to multi-harmonic excitation is studied. The shell has opportune dimensions so that the natural frequencies of the two modes (driven and companion) with three circumferential waves are practically double than the natural frequencies of the two modes (driven and companion) with two circumferential waves. This introduces a one-to-one-to-two-to-two internal resonance in the presence of harmonic excitation in the spectral neighbourhood of the natural frequency of the mode with two circumferential waves. Since the system is excited by a multi-harmonic point-load excitation composed by first and second harmonics, very complex nonlinear dynamics is obtained around the resonance of the fundamental mode. In fact, at this frequency, both modes with two and three circumferential waves are driven to resonance and each one is in a one-to-one internal resonance with its companion mode. The nonlinear dynamics is explored by using bifurcation diagrams of Poincaré maps and time responses.     

Trapped modes due to narrow cracks in thin simply-supported elastic plates

We consider an elastic plate of infinite length and constant width supported simply along its two parallel edges and having a finite length crack along its centreline. In particular, we look for and find trapped modes (localised oscillations) in the presence of the crack. An explicit wide-spacing approximation based on the Wiener–Hopf technique applied to incident wave scattering by semi-infinite cracks is complemented by an exact formulation of the problem in the form of integro-differential equations. An application of a Galerkin method for the numerical calculation of results from the latter method leads to a novel explicit ‘small-spacing’ approximation. In combination with the wide-spacing results this is shown to provide accurate results for all lengths of crack.     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid

The reflection and transmission of antiplane surface waves (Love waves) by a surface-breaking crack in a layered elastic solid is investigated. The crack is normal to the free surface, and breaks into the lower half-space solid. The formulation of the problem is reduced to a singular integral equation of the Cauchy type. In this equation, the unknown function, which is the slope of the crack-face displacement, is discontinuous at the interface between the two solids. It is shown that the magnitude of the discontinuity is related to the ratio of the shear moduli. A Gaussian numerical method is used to obtain the solution of the singular integral equation. At some distance from the plane of the crack, the wave motion is the superposition of a finite number of Love-wave modes. The amplitudes of these modes are readily evaluated in terms of the slope of the crack-face displacement. Curves are presented for the reflection coefficients corresponding to the first three modes and for the transmission coefficient as functions of the dimensionless frequency.     

含环向表面裂纹圆柱壳的波传播特性研究

研究了振动波在含有环向表面裂纹的无限长圆柱壳中的传播特性.圆柱壳体的振动用Flügge方程来描述.运用线弹性断裂力学的理论,考虑到裂纹的张开、滑移和撕裂3种模式以及它们相互之间的耦合,利用分布的线弹簧来模拟裂纹并建立了裂纹所在区域的局部柔度矩阵,得到由此引起的附加位移和壳体中内力之间的关系.在入射波已知的情况下,根据裂纹两侧区域的位移和内力的连续性条件得到了反射和透射波的幅值系数.分析了入射波通过裂纹后的透射、反射系数与激励频率和裂纹尺寸之间的关系.为基于振动功率流方法识别圆柱壳表面损伤提供了理论基础.     

An iterative approach for nonproportionally damped systems

Modal analysis of nonproportionally damped linear dynamic systems is considered. Dynamic response of such systems can be expressed by a modal series in terms of complex modes. Normally state-space based methods or approximate perturbation methods are necessary for the computation of complex modes. In this paper, an iterative method to calculate complex modes from classical normal modes for general linear systems is proposed. A simple numerical algorithm is developed to implement the iterative method. The new method is illustrated using a numerical example.     

Dynamic instability and steady-state response of an elliptical cracked shaft

An elliptical front crack has been found to be more accurate and realistic for modeling the transverse surface crack in rotating machinery compared with the widely used straight front crack. When the shaft rotates, the elliptical crack opens and closes alternatively, due to gravity, and thus, a “breathing effect” occurs. This variance in shaft stiffness is time-periodic, and hence, a parametrically excited system is expected. Therefore, the dynamic instability and steady-state response of a rotating shaft containing an elliptical front crack are studied in the paper. The local flexibility due to the crack is derived, and the governing equations of the crack shaft system are established using the assumed modes method. Utilizing the Bolotin’s method and harmonic balance method, the boundaries of two typical instability regions and maximum response amplitude of the cracked shaft could be computed numerically. The elliptical crack parameters (depth, shape factor and position) and damping are, respectively, considered and discussed for their effects on the dynamic behavior of the elliptical cracked shaft. Some research results might be helpful for the crack detection in rotating machinery.     

Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam

The three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam is investigated in this paper by means of two numerical techniques. The equations of motion for the longitudinal, transverse, and rotational motions are derived using constitutive relations and via Hamilton’s principle. The Galerkin method is employed to discretize the three partial differential equations of motion, yielding a set of nonlinear ordinary differential equations with coupled terms. This set is solved using the pseudo-arclength continuation technique so as to plot frequency-response curves of the system for different cases. Bifurcation diagrams of Poincaré maps for the system near the first instability are obtained via direct time integration of the discretized equations. Time histories, phase-plane portraits, and fast Fourier transforms are presented for some system parameters.     

EFFECT OF Z-PINS＇ DIAMETER, SPACING AND OVERLAP LENGTH ON CONNECTING PERFORMANCE OF CMC SINGLE LAP JOINT

The effect of through-thickness reinforcement by composite pins （Z-pins） on the static tensile strength and failure mechanisms of the joints made from ceramic matrix composite （CMC） is investigated. Overlap length of the single lap joint is 15 mm, 20 mm, 23 mm, 37 mm, and 60 mm, respectively. The experimental results indicate that the final failure modes of the joints can be divided into two groups, （a） the bond-line stops debonding until crack encounters Z-pins; and then the adherends break at the location of Z-pins, when overlap length is more than 20 mm; （b） the bond-line detaches entirely and Z-pins are drawn from adherends, when overlap length is equal to 15 mm. A simple efficient computational approach is presented for analyzing the benefit of through-thickness pins for restricting failure in the single lap joints. Here, the mechanics problem is simplified by representing the effect of the pins by tractions acting on the fracture surfaces of the cracked bond-line. The tractions are prescribed as functions of the crack displacement, which are available in simple forms that summarize the complex deformations to a reasonable accuracy. The resulting model can be used to track the evolution of complete failure mechanisms, for example, bond-line initial delamination and ultimate failure associated with Z-pin pullout, ultimate failure of the adherends. The paper simulates connecting performance of the single lap joints with different Z-pins＇ diameter, spacing and overlap length; the numerical results agree with the experimental results; the numerical results indicate enlarging diameter and decreasing spacing of Z-pins are in favor of improving the connecting performance of the joints. By numerical analysis method, the critical overlap length that lies between two final failure modes is between 18 mm and 19 mm, when Z-pins＇ diameter and spacing are 0.4 mm, 5 mm, respectively.