Diagnostics of closing cracks in rodlike elements at nonlinear resonances by the method of variation of the asymmetry of driving forces

The vibrations of elastic bodies with closing cracks are essentially nonlinear. As a specific feature of these vibrations, we can mention the manifestation of so-called nonlinear effects, in particular, sub- and superharmonic resonances and the nonlinearity of vibrations at these resonances. We propose a method for the diagnostics of cracks based on the variation of the asymmetry of driving forces. The static component of the concentrated harmonic driving force changes the state of a crack: it becomes partially or completely open or closed. Moreover, the degree of nonlinearity of vibrations for any nonlinear resonance varies from the maximum level (in the absence of static component) to its almost complete absence in the case where the crack becomes open or closed in the course of vibrations under the action of the static component. The proposed method enables one to detect the presence of cracks without any preliminary data on the analyzed object in the intact state.     

Approximate Analytical Determination of Vibrodiagnostic Parameters of Non-Linearity of Elastic Bodies Due to the Presence of a Closing Crack. Part 1. Available and Proposed Methods of Solution

The paper discusses the available analytical methods and the results of investigation of vibrations of elastic bodies with a bilinear asymmetric characteristic of restoring force that simulates the behavior of a local discontinuity in a material in the form of a closing fatigue crack. An approximate analytical method is put forward for the determination of vibrodiagnostic parameters of the vibration process of the nonlinear system under study in the range of weak superharmonic resonances.     

Resonant Acoustic Nonlinearity of Defects for Highly-Efficient Nonlinear NDE

In this paper, the effect of local defect resonance (LDR) on the nonlinear ultrasonic responses of defects is studied and applied for enhancement of sensitivity of nonlinear NDE. Unlike the resonance of the whole specimen, the LDR provides an efficient energy pumping from the wave directly to the defect and causes an efficient generation of the higher harmonics and wave mixing even at moderate input signals. At higher levels of excitation, a combined effect of LDR and nonlinearity results in qualitatively new “nonclassical” features characteristic of the nonlinear and parametric resonances. The resonant nonlinear defects demonstrate threshold dynamics of instable vibrations, hysteresis, super- and subharmonic resonances. Under nonlinear LDR conditions nearly total input energy can be converted into higher harmonic or subharmonic vibrations of the defect. This proposes nonlinear LDR application as an extremely efficient and sensitive mode for nonlinear imaging and NDE.     

简谐激励力作用下悬垂缆线的谐波共振

本文研究在简谐激励力作用下的悬垂缆线的谐波共振。用Hamilton原理导出悬垂缆线面内运动的非线性偏微分方程。通过假设悬垂缆线的挠度曲线，运用Galerkin方法将偏微分方程转化为常微分方程。用多尺度法研究悬垂缆线的超谐波共振和次谐波共振，得到了系统的定常周期解，平均方程和幅频曲线。研究了非线性对幅频曲线的影响和定常运动的稳定区域。     

Phase increment analysis of damped Duffing oscillators

A phase increment method is introduced to construct the response curves for the damped Duffing oscillator in primary, superharmonic, and subharmonic resonances. Non-linear parameters can be arbitrarily large. The algorithm is numerically stable. All resonance response curves are constructed in a unified manner. Closed loop curves are obtained in subharmonic resonances as opposed to open ended ones predicted by the perturbation method. Higher order resonances are constructed without difficulties. Loops are also observed in superharmonic resonances when non-linearity is not small.     

Approximate analytical method for determining the vibration-diagnostic parameter indicating the presence of a crack in a distributed-parameter elastic system at super- and subharmonic resonances

An approximate method for calculating the vibration-diagnostic parameter indicating the presence of a crack in an elastic distributed-parameter system at super- and subharmonic resonances is considered. The method involves the finding of the non-linearity characteristic of an elastic system based on the analysis of its forced vibrations in the undamaged state and the use of results of the approximate analytical determination of parameters of nonlinearity for vibrations 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 at the above-mentioned resonances.     

The effect of damping and force application point on the non-linear dynamic behavior of a cracked beam at sub-and superresonance vibrations

The finite element model of an elastic body with a closing crack is presented. Using this model, we show that the non-linear distortions of vibrations at superharmonic and subharmonic resonances are the high sensitive indicators of crack presence. The intensity of these indicators manifestation substantially depends on the level of damping in the system and on the load application point. __________ Translated from Problemy Prochnosti, No. 5, pp. 61–67, September–October, 2006. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2006, Kiev, Ukraine).     

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.     

Stochastic and chaotic sub- and superharmonic response of shallow cables due to chord elongations

The paper deals with the non-linear response of shallow cables driven by stochastically varying chord elongations caused by random vibrations of the supported structure. The chord elongation introduces parametric excitation in the linear stiffness terms of the modal coordinate equations, which are responsible for significant internal subharmonic and superharmonic resonances. Under harmonically varying support motions coupled ordered or chaotic in-plane and out-of-plane subharmonic and superharmonic periodic motions may take place. If the harmonically varying chord elongation is replaced by a zero-mean, stationary narrow-band random excitation with the same standard deviation and center frequency, qualitatively and quantitatively completely different modes of vibration are registered no matter how small the bandwidth of the excitation process is. Additionally, the stochastic excitation process tends to enhance chaotic behavior. Based on Monte Carlo simulation on a reduced non-linear two-degree-of freedom system the indicated effects have been investigated for stochastic subharmonic resonance of order 2:1, and stochastic superharmonic resonances of orders 1:2 and 2:3. By analyzing the responses for two chord elongation processes with almost identical auto-spectral density function, but completely different amplitudes, it is shown that the indicated qualitative and quantitative changes of the subharmonic resonance primarily are caused by the slowly varying phase of the stochastic excitation. The superharmonic stochastic responses are dominated by random jumps between a single mode in-plane and a coupled mode attractor, which are caused by the variation of the amplitude of the random excitation. Such jumps do not occur in the subharmonic response, because the single mode in-plane attractor is unstable.     

受机械力作用金属／陶瓷功能梯度板的超谐共振研究

研究机械力作用下金属／陶瓷功能梯度薄板3次超谐共振问题．按照功能梯度薄板的非线性动力学方程，得到金属／陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统3次超谐共振近似解并进行数值计算。分析阻尼、激励、几何尺寸等参数对系统3次超谐共振幅频响应曲线的影响．     

电机端盖超谐共振分析

根据电机端盖的结构特点, 将端盖抽象为圆环板。按照弹性力学理论建立了圆环板的振动方程。基于伽辽金方法, 引入振型试函数, 得到5 种不同边界条件端盖的非线性振动方程。应用多尺度法得到系统超谐共振近似解。计算了5 种边界条件下超谐共振的幅频响曲线。分析了半径、激励对系统超谐共振响应曲线的影响。     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 4. Analysis of distortion of harmonicity of vibration cycle of beams with closing transverse cracks

Using an asymptotic method of nonlinear mechanics and concepts of linear fracture mechanics, we derive analytical expressions that relate the constant component and amplitudes of even harmonics, and the ratio of maximum deviations and half-periods of displacements of opposite signs in a vibration cycle of prismatic beams to the length of a closing transverse crack. The analytical solution is compared to the data of experimental determination of these parameters in bending vibrations of rectangular cantilever beams with an edge transverse crack. Analysis of the results of analytic and experimental investigations has shown that harmonic analysis of the strain waveform of beam-type elements in resonance vibrations can be used as an efficient diagnostic method for detecting fatigue damage such as closin cracks. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 1, pp. 5–12, January–February, 2000.     

Dynamic Acousto-Elasticity in a Fatigue-Cracked Sample

Dynamic acousto-elasticity (DAE) provides a unique way to observe nonlinear elastic features over an entire dynamic stress cycle including hysteresis and memory effects, detailing the full nonlinear behavior under tension and compression. This supplemental information cannot be observed with conventional nonlinear ultrasonic methods such as wave frequency mixing or resonance measurements, since they measure average, bulk variations of modulus and attenuation versus strain level. Where prior studies have employed DAE in volumetrically nonlinear materials (e.g., rocks, bone with distributed micro-crack networks), here we report results of DAE on the application to a single localized nonlinear feature, a fatigue crack, to characterize the nonlinear elastic response in regions of the crack length, tip, and undamaged portions of an aluminum sample. Linear wave speed, linear attenuation and third order elastic moduli (i.e., nonlinear parameters) each indicate a sensitivity to the presence of the crack, though in unique manners. The localized nature of the DAE measurement and its potential for quantifying all of the third order elastic constants makes it a promising technique for both detecting cracks, as well as providing quantitative information on the effect of the cracks on the material integrity.     

结构-地基相互作用水平与扭转的超亚谐振动

基于多线性随动强化模型求得了基础和地基土之间的水平刚度与摇摆刚度，在此基础上建立了结构，地基非线性相互作用体系模型，利用能量法得到了结构水平侧移和扭转相耦合的振动方程，采用多尺度法研究了结构，地基相互作用体系的超谐和亚谐共振，比较了结构与地基非线性因素给体系反应带来的影响，揭示了发生超谐和亚谐振动时水平振动和扭转振动耦合的不同规律，为工程抗震提供了理论依据。     

Comparative analysis of nonlinear resonances of a mechanical system with unsymmetrical piecewise characteristic of restoring force

We present results of a numerical comparative analysis of superharmonic resonances of the order 2/1-1/5 and subharmonic resonance of the order 1/2 of a mechanical single degree-of-freedom vibrating system with unsymmetrical characteristic of restoring force at different ratios of system rigidities in half-cycles and under the conditions of considerable change in vibration damping level in the system. __________ Translated from Problemy Prochnosti, No. 2, pp. 72–87, March–April, 2007.     

Jeffcott转子碰摩故障试验研究

针对工种实际中遇到的机组转子摩故障开展了试验研究,通过大量的试验,观测到了工作转子速低于一阶临界转速时,转子定子局部碰摩引起的倍频振动,高于一阶临界转速时,局部碰摩引起的分频振 及某些转速内出现的异频伪共振现象,试验民其于碰摩力模型和仿真结果定性一致,试验还表明,异频伪共振是高速工作的转子发生碰摩时的主要振动成分,这一结论对诊断和抑制高速转子的异常超标振动有积极意义。     

复合材料层合板的非线性组合共振特性及分岔

考虑几何非线性项和阻尼的影响, 给出了四边简支的正交各向异性矩形层合板在两项横向简谐激励作用下的非线性振动微分方程, 利用伽辽金法导出了相应的达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解, 得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论, 得到了解的稳定性判定条件。通过数值算例, 分析了不同参数对系统组合共振及其分岔特性的影响。结果表明, 随着调谐参数、板厚度、阻尼系数以及激励力等参数的改变, 系统存在多幅值现象、滞后现象和跳跃现象, 出现不稳定解, 且在某些参数点处具有运动性态发生变化的分岔特性, 表现出较为复杂的动力学特性。      

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 2. Bending vibrations, analytical solution

We discuss the possibility of efficient application of the harmonic analysis of strain cycles formed in an elastic rod in the process of natural or resonance vibrations to the evaluation of the degree of fatigue damage. Fatigue damage is simulated by discontinuities of the material resulting in different values of the modulus of elasticity in the processes of tension and compression. In determining the potential strain energy of a rod in the presence of a crack on the basis of linear fracture mechanics, this model enables us to deduce expressions for relative changes in the stiffness of the rod in the presence of edge and surface cracks. By using the asymptotic method of nonlinear mechanics, we find the second approximation to the solution of the differential equation of vibrations of the rod whose nonlinearity is explained by the effect of “breathing” of the discontinuity formed in the material. It is shown that the discontinuity of the material caused by fatigue damage is responsible both for changes in the natural frequency of vibrations of the rod and for significant contributions of the constant component and even harmonics to the spectrum of strain cycles in the rod. The values of the constant component and the amplitudes of even harmonics (mainly of the second one) may serve as an efficient parameter for the detection of fatigue damage to materials. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 9–22, November–December, 1998.     

纵横向耦合梁的谐波响应分析

本文用谐波增量平衡法分析纵横向耦合梁的谐波响应。梁的动力方程由伽辽金法转化成含有二次和三次非线性项的常微分方程，求得了简谐载荷作用下梁的横向基谐波、超谐波响应和纵向基谐波、倍频基谐波及倍频超谐波响应，并发现了谐波响应的倒峰值现象。本文为求解含有二次和三次非线性振动系统的谐波响应提供了一套理论分析方法。