Dynamic analysis of a geared rotor system considering a slant crack on the shaft

The vibration problems associated with geared systems have been the focus of research in recent years. As the torque is mainly transmitted by the geared system, a slant crack is more likely to appear on the gear shaft. Due to the slant crack and its breathing mechanism, the dynamic behavior of cracked geared system would differ distinctly with that of uncracked system. Relatively less work is reported on slant crack in the geared rotor system during the past research. Thus, the dynamic analysis of a geared rotor-bearing system with a breathing slant crack is performed in the paper. The finite element model of a geared rotor with slant crack is presented. Based on fracture mechanics, the flexibility matrix for the slant crack is derived that accounts for the additional stress intensity factors. Three methods for whirling analysis, parametric instability analysis and steady-state response analysis are introduced. Then, by taking a widely used one-stage geared rotor-bearing system as an example, the whirling frequencies of the equivalent time-invariant system, two types of instability regions and steady-state response under the excitations of unbalance forces and tooth transmission errors, are computed numerically. The effects of crack depth, position and type (transverse or slant) on the system dynamic behaviors are considered in the discussion. The comparative study with slant cracked geared rotor is carried out to explore distinctive features in their modal, parametric instability and frequency response behaviors.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

DYNAMICS OF A TWO-CRACK ROTOR

The effect of the presence of the single transverse crack on the response of the rotor has been a focus of attention for many researchers. In the present work a simple Jeffcott rotor with two transverse surface cracks has been studied. The stiffness of such a rotor is derived based on the concepts of fracture mechanics. Subsequently, the effect of the interaction of the two cracks on the breathing behavior and on the unbalance response of the rotor is studied. When the angular orientation of one crack relative to the other is varied, significant changes in the dynamic response of the rotor are noticed. A special case of practical importance of a two-crack rotor is one when one of the cracks is assumed to remain open always whereas the other can breathe like a fatigue crack. This simulates a transverse crack in an asymmetric rotor. Effect of orientation of the breathing crack with respect to the open crack on the dynamic response is studied in detail. The results of the present study will be useful in diagnosing fatigue cracks in real rotors, which invariably have some asymmetry.     

RANDOM VIBRATION OF A ROTATING BLADE WITH EXTERNAL AND INTERNAL DAMPING BY THE FINITE ELEMENT METHOD

The finite element model is employed to investigate the mean-square response of a rotating blade with external and internal damping under stationary or non-stationary random excitation. The blade is considered to be subjected to white-noise and earthquake excitations. The effects of rotational speed, external and internal damping on the mean-square response are studied. It is found that the mean-square response decreases quickly when the external and internal damping increases within some scope. Moreover, the increment of rotational speed will reduce the mean-square response of a rotating blade. It is also found that the mean-square response decreases when the low natural frequency of base decreases. Inversely, the mean-square response increases when the high natural frequency of base (natural frequencies of base are over the first natural frequency of blade) decreases. The reliability of a rotating blade subjected to stationary or non-stationary excitations is also obtained.     

Chaotic dynamics of a whirling pendulum

A physical system is considered consisting of a rigid frame which is free to rotate about a vertical axis and to which is attached a planar simple pendulum. This system has “one and a half” degrees of freedom due to the fact that the frame and pendulum may freely rotate about the vertical axis, i.e., conservation of angular momentum holds for the “ideal”, or unperturbed, system. Using a Hamiltonian formulation we reduce the unperturbed equations of motion to a conservative planar system in which the constant angular momentum plays the role of a parameter. This system is shown to possess one or two sets of homoclinic motions depending on the level of the angular momentum. When this system is perturbed by external excitations and dissipative forces these homoclinic motions can break into homoclinic tangles providing the conditions for chaotic motions of the horseshoe type to exist. The criteria for this to occur can be formulated using a variation of Melnikov's method developed for slowly varying oscillators [1, 2]. For the present problem, the angular momentum becomes a slowly varying parameter upon addition of the disturbances. These ideas are used to rigorously prove the existence of chaotic motions for this system and to compute, to first order, global bifurcation parameter conditions. Since two types of homoclinic motions can occur, two different chaotic modes of motion can result and physical interpretations of these motions are given. In addition, a limiting case is considered in which the system becomes a single degree of freedom oscillator with parametric excitation.     

Stability Analysis of an Inverted Pendulum Subjected to Combined High Frequency Harmonics and Stochastic Excitations

Stability of vertical upright position of an inverted pendulum with its suspension point subjected to high frequency harmonics and stochastic excitations is investigated. Two classes of excitations, i.e., combined high frequency harmonic excitation and Gaussian white noise excitation, and high frequency bounded noise excitation, respectively, are considered. Firstly, the terms of high frequency harmonic excitations in the equation of motion of the system can be set equivalent to nonlinear stiffness terms by using the method of direct separation of motions. Then the stochastic averaging method of energy envelope is used to derive the averaged Ito stochastic differential equation for system energy. Finally, the stability with probability 1 of the system is studied by using the largest Lyapunov exponent obtained from the averaged Ito stochastic differential equation. The effects of system parameters on the stability of the system are discussed, and some examples are given to illustrate the efficiency of the proposed procedure.     

The simulation of random vibration response by discrete frequency testing

The paper presents the theoretical basis for an analogue method whereby the random response of a structure may be inferred from measurements of its harmonic response when the excitations are replaced by discrete frequency forces or imposed motions. In the elementary case of a single random input, the method takes a particularly simple form. In the general multi-input case it is shown that the method may be used directly only when all the inputs are fully coherent. Otherwise a superposition technique involving multiple applications of sets of harmonic excitations is required. It is considered that the method may be of practical use as a tool in vibration testing and that, in addition, the underlying analysis contains results of considerable interest on the structure of multi-dimensional stationary random processes.     

利用半经典微扰理论研究H－H_2O的非弹性碰撞—H_2O的转动、振动激发

利用半经典微扰理论计算了在Ｈ－Ｈ２Ｏ的非弹性碰撞中的Ｈ２Ｏ分子的转动、振动激发，并将计算结果与实验结果进行比较。计算中采用了Ｓｃｈａｔｚ和Ｅｌｇｅｒｓｍａ的半经验势能面，水分子的势函数，包括了谐振和非谐振力函数，在考虑了振动和转动耦合的情况下，通过半经典微扰理论来确定水分子末态的振动量子数。在Ｌｏｖｅｊｏｙ等人最近进行的实验中，当热氢原子入射时，水分子的转动和振动被激发了，而且发现水分子的转动角动量有垂直于水分子平面优先激发的趋势。我们的计算也得到出了相同的结果。     

Detecting cracked rotors using auxiliary harmonic excitation

Cracked rotors are not only important from a practical and economic viewpoint, they also exhibit interesting dynamics. This paper investigates the modelling and analysis of machines with breathing cracks, which open and close due to the self-weight of the rotor, producing a parametric excitation. After reviewing the modelling of cracked rotors, the paper analyses the use of auxiliary excitation of the shaft, often implemented using active magnetic bearings to detect cracks. Applying a sinusoidal excitation generates response frequencies that are combinations of the rotor spin speed and excitation frequency. Previously this system was analysed using multiple scales analysis; this paper suggests an alternative approach based on the harmonic balance method, and validates this approach using simulated and experimental results. Consideration is also given to some issues to enable this approach to become a robust condition monitoring technique for cracked shafts.     

Simplified models of the vibration of mannequins in car seats

A simplified two-dimensional modelling approach to predict the vibration response of mannequin occupied car seats about a static settling point is demonstrated to be feasible. The goal of the research is to develop tools for car seat designers. The two-dimensional model, consisting of interconnected masses, springs and dampers is non-linear due to geometric effects but, under the excitations considered, the model behaviour is linear. In this approach to modelling, the full system is initially broken down into subsystems, and experiments are conducted with subsystems to determine approximate values for the stiffness and damping parameters. This approach is necessary because of the highly non-linear behaviour of foam where stiffness changes with compression level, and because the simplified model contains more structure than is necessary to model the relatively simple measured frequency response behaviour, thus requiring a good initial starting point from which to vary parameters. A detailed study of the effects of changing model parameters on the natural frequencies, the mode shapes and resonance locations in frequency response functions is given, highlighting the influence of particular model parameters on features in the seat-mannequin system's vibration response. Reasonable qualitative as well as good quantitative agreement between experimental and simulation frequency response estimates is obtained. In particular, the two-dimensional motions at the peaks in the frequency response, a combination of up and down and rotational behaviour is predicted well by the model. Currently research is underway to develop a similar model with non-linear springs, surface friction effects and viscoelastic elements, that predicts the static settling point, a necessary step to aid in the subsystem modelling stage in this dynamic modelling approach.     

Flexural vibration of non-uniform beams having double-edge breathing cracks

Flexural vibration of non-uniform Rayleigh beams having single-edge and double-edge cracks is presented in this paper. Asymmetric double-edge cracks are formed as thin transverse slots with different depths at the same location of opposite surfaces. The cracks are modelled as breathing since the bending of the beam makes the cracks open and close in accordance with the direction of external moments. The presented crack model is used for single-edge cracks and double-edge cracks having different depth combinations. The energy method is used in the vibration analysis of the cracked beams. The consumed energy caused by the cracks opening and closing is obtained along the beam's length together with the contribution of tensile and compressive stress fields that come into existence during the bending. The total energy is evaluated for the Rayleigh-Ritz approximation method in analysing the vibration of the beam. Examples are presented on simply supported beams having uniform width and cantilever beams which are tapered. Good agreements are obtained when the results from the present method are compared with the results of Chondros et al. and the results of the commercial finite element program, Ansys^©. The effects of breathing in addition to crack depth's asymmetry and crack positions on the natural frequency ratios are presented in graphics.     

The critical excitation and response of simple dynamic systems

A method for defining the critical excitations and responses of dynamic systems is examined. The critical excitations are those functions which maximize some response norm with respect to the constraints placed on the admissible excitations. A class of critical responses for linear, elastoplastic and hysteretic single degree of freedoms systems is studied, showing the frequency and amplitude relations for these solutions. For linear systems it is shown that the critical excitations producing either a maximum displacement response or maximum energy input are harmonic and derivable from the harmonically excited response functions for the same linear system. The critical excitations for elastoplastic systems, however, are not harmonic and at low frequencies the response is significantly larger than the harmonically excited response. The critical response solutions require higher multiple frequency components to exist. Both periodic and inelastic offset types of critical response are examined for a hysteretic, elastoplastic system and the response characteristics for these solutions are discussed.     

Theory of molecule-surface scattering at thermal and hyperthermal energies

A theoretical model of molecule-surface scattering is developed which includes energy and momentum transfers between the surface and projectile for both translational and rotational motions and internal mode excitation for the projectile molecule. The translation and rotation motions are treated in the classical limit, while a quantum treatment for internal vibrational mode excitation is used. The results of calculations are compared with recent high-precision measurements of the scattering of a beam of C(2)H(2) molecules from a clean, ordered LiF(001) surface at energies of up to nearly 1 eV. The calculated results for angular distributions and rotational excitations are in good agreement with experiment.     

Response of uni-modal duffing-type harvesters to random forced excitations

Linear energy harvesters have a narrow frequency bandwidth and hence operate efficiently only when the excitation frequency is very close to the fundamental frequency of the harvester. Consequently, small variations of the excitation frequency around the harvester's fundamental frequency drops its small energy output even further making the energy harvesting process inefficient. To extend the harvester's bandwidth, some recent solutions call for utilizing energy harvesters with stiffness-type nonlinearities. From a steady-state perspective, this hardening-type nonlinearity can extend the coupling between the excitation and the harvester to a wider range of frequencies. In this effort, we investigate the response of such harvesters, which can be modeled as a uni-modal duffing-type oscillator, to White Gaussian and Colored excitations. For White excitations, we solve the Fokker-Plank-Kolmogorov equation for the exact joint probability density function of the response. We show that the expected value of the output power is not even a function of the nonlinearity. As such, under White excitations, nonlinearities in the stiffness do not provide any enhancement over the typical linear harvesters. We also demonstrate that nonlinearities in the damping and inertia may be used to enhance the expected value of the output power. For Colored excitations, we use the Van Kampen expansion and long-time numerical integration to investigate the influence of the nonlinearity on the expected value of the output power. We demonstrate that, regardless of the bandwidth or the center frequency of the excitation, the expected value of the output power decreases with the nonlinearity. With such findings, we conclude that energy harvesters modeled as uni-modal duffing-type oscillators are not good candidates for harvesting energy under forced random excitations. Using a linear transformation, results can be extended to the base excitation case.     

基础水平运动的弹簧摆的周期解与稳定性

针对基础水平运动的弹簧摆的非线性动力学响应进行研究,利用拉格朗日方程建立了系统的动力学方程.将离散傅里叶变换、谐波平衡法以及同伦延拓方法相结合,对系统的周期响应进行求解,避免了传统方法计算中使用泰勒展开引起的小振幅的限制,与数值计算结果的对比表明该求解方法具有较高的精确度.利用Floquet理论分析了周期响应的稳定性,给出了基础运动振幅和频率对系统周期响应的影响.研究发现:对应某些基础频率和振幅,系统的周期响应可能发生Hopf分岔;利用数值计算研究了Hopf分岔后系统响应随基础频率和振幅的变化,发现系统出现了倍周期运动、拟周期运动和混沌等复杂的动力学行为.研究表明系统进入混沌的主要路径是拟周期环面破裂和阵发性.     

Dynamics and stability of rigid rotors levitated by passive cylinder-magnet bearings and driven/supported axially by pointwise contact clutch

A stable rotor—supported laterally by passive magnetic bearings and longitudinally by magnetic forces and a clutch—loses suddenly its contact to the clutch and executes abruptly longitudinal movements away from its original equilibrium position as a result of small increases in angular velocity. Such an abrupt unstable behaviour and its reasons are thoroughly theoretically as well as experimentally investigated in this work. In this context, this paper gives theoretical as well as experimental contributions to the problem of two dimensional passive magnetic levitation and one dimensional pointwise contact stability dictated by mechanical–magnetic interaction. Load capacity and stiffness of passive multicylinder magnetic bearings (MCMB) are thoroughly investigated using two theoretical approaches followed by experimental validation. The contact dynamics between the clutch and the rotor supported by MCMB using several configurations of magnet distribution are described based on an accurate nonlinear model able to reliably reproduce the rotor-bearing dynamic behaviour. Such investigations lead to: (a) clear physical explanation about the reasons for the rotor's unstable behaviour, losing its contact to the clutch and (b) an accurate prediction of the threshold of stability based on the nonlinear rotor-bearing model, i.e. maximum angular velocity before the rotor misses its contact to the clutch as a function of rotor, bearing and clutch design parameters.     

基于Matlab和ADAMS的光电稳瞄系统结构控制联合仿真

为了研究多轴多框架光电稳瞄系统的稳定精度并且避免推导其复杂的动力学微分方程,提出了利用Matlab和ADAMS进行光电稳瞄系统结构控制联合仿真的方法。研究了联合仿真模型的方位轴和俯仰轴单位阶跃响应,和在不同方向的随机线振动下的瞄准线惯性角速率,以及在2°、1Hz不同方向角扰动下瞄准线的稳定精度。仿真结果与理论定性分析吻合,表明联合仿真方法可行,可用于后续产品的开发。     

Structural damage assessment in a cantilever beam with a breathing crack using higher order frequency response functions

Vibration measurements offer an effective, inexpensive and fast means of non-destructive testing of structures and various engineering components. There are mainly two approaches to crack detection through vibration testing; open crack model with emphasis on changes in modal parameters and secondly, the breathing crack model focusing on nonlinear response characteristics. The open crack model based on linear response characteristics can identify the crack only at an advanced stage. Researchers have shown that a structure with a breathing crack behaves more like a nonlinear system, similar to that of a bilinear oscillator and the nonlinear response characteristics can very well be investigated to identify the presence of the crack. In the present study, the bilinear restoring force is approximated by a polynomial series and a nonlinear dynamic model of the cracked structure is developed using higher order frequency response functions. The effect of crack severity on the response harmonic amplitudes are investigated and a new procedure is suggested whereby the crack severity can be estimated through measurement of the first and second harmonic amplitudes.     

超流Fermi气体在非简谐势阱中的集体激发

研究了非谐振势中超流Fermi气体的集体激发. 基于一维超流流体动力学模型, 采用变分法获得了体系从分子Bose-Einstein凝聚端渡越到Cooper对凝聚端时系统的两个低能激发模, 即偶极模和呼吸模. 分析发现: 在整个跨越区偶极模和呼吸模都发生了频移现象, 且在BCS端频移更加显著. 进一步研究发现在不同驱动振幅激发下超流Fermi气体质量中心和宽度变化呈现出了复杂动力学特性, 由于非谐振势的贡献,超流Fermi气体两低能模发生耦合, 使宽度变化产生量子拍频现象, 且拍频频率随着驱动振幅的增加而增大. 这种非线性耦合对外部驱动的响应在幺正区尤其显著. 关键词： 超流Fermi气体 非谐振势 集体激发     

Relative performance of a vibratory energy harvester in mono- and bi-stable potentials

Motivated by the need for broadband vibratory energy harvesting, many research studies have recently proposed energy harvesters with nonlinear characteristics. Based on the shape of their potential function, such devices are classified as either mono- or bi-stable energy harvesters. This paper aims to investigate the relative performance of these two classes under similar excitations and electric loading conditions. To achieve this goal, an energy harvester consisting of a clamped-clamped piezoelectric beam bi-morph is considered. The shape of the harvester's potential function is altered by applying a static compressive axial load at one end of the beam. This permits operation in the mono-stable (pre-buckling) and bi-stable (post-buckling) configurations. For the purpose of performance comparison, the axial load is used to tune the harvester's oscillation frequencies around the static equilibria such that they have equal values in the mono- and bi-stable configurations. The harvester is subjected to harmonic base excitations of different magnitudes and a slowly varying frequency spanning a wide band around the tuned oscillation frequency. The output voltage measured across a purely resistive load is compared over the frequency range considered. Two cases are discussed; the first compares the performance when the bi-stable harvester has deep potential wells, while the second treats a bi-stable harvester with shallow wells. Both numerical and experimental results demonstrate the essential role that the potential shape plays in conjunction with the base acceleration to determine whether the bi-stable harvester can outperform the mono-stable one and for what range of frequencies. Results also illustrate that, for a bi-stable harvester with shallow potential wells, super-harmonic resonances can activate the inter-well dynamics even for a small base acceleration, thereby producing large voltages in the low frequency range.