Contact-impact analysis of geared rotor systems

In this paper, a finite element formulation, used to analyze the contact-impact behavior of geared rotor systems coupled with the rotational, lateral, and axial vibrations between gears at high rotational speeds, has been developed. A gear impact element to model the contact-impact behavior between gears has been developed and its numerical method is discussed. A relative displacement measurement idea has been proposed to measure vibration parameter for contrast experiment in high rotational geared system. The equations of motion are derived and solved iteratively during each time increment until the unbalanced force decrease to an acceptable tolerance level. Based on the proposed method, an analysis program, GEARS, has been developed. The contact-impact behavior of geared rotor systems is analyzed especially under high rotational speed condition as numerical examples, which are demonstrated to show the effectiveness of the proposed method.     

Dynamic behavior analysis of cracked rotor

In the present study, the additional slope is used to consider the crack breathing, and is expressed explicitly in the equation of motion as one of the inputs to produce the bending moment at the crack position. Inversely, the additional slope is calculated by integrating on the crack region based on a fracture mechanics concept. The response of a cracked rotor is formulated based on the transfer matrix method. The transient behavior due to the crack breathing is considered by introducing a ‘moving’ Fourier-series expansion concept to the additional slope. The time-varying harmonic components of the additional slope are used to calculate the harmonic responses. The application considered is a general rotor model composed of multiple shafts, disks and cracks, and resilient bearings at both ends. Verification analysis is carried out for a simple rotor model similar to those found in the literature. Using the additional slope, the cracked rotor behavior is explained by the crack depth and rotation speed increase. It is shown that region on the crack front line having the dominant stress intensity factor value moves from the central area to both ends, as the crack depth increases. The result matches well with the crack propagation pattern shown in a bench mark test in the literature. Whirl orbits near the critical and sub-critical speed ranges of the rotor are discussed. It is shown that there exists some speed range near the critical speed, where the temporary whirl direction reversal and phase shift exist. When an unbalance is applied, the peculiar features, such as the whirl direction reversal and phase shift, disappear.     

An analytical study of rotor dynamics coupled with thermal effect for a continuous rotor shaft

This paper presents an analytical analysis of a continuous rotor shaft subjected to universal temperature gradients. To this end, an analytical model is derived to investigate the generic thermal vibrations of rotor structures. The analytical solutions are obtained in a rotating frame and include parameters related with both the thermal environment and the rotor dynamic structures. This provides an insight into the mechanisms for the rotor thermal vibration. Furthermore, numerical results based on the analytical solutions are given. An index denoting the temperature gradients is proposed for the occasions with nonlinear cross-sectional temperature distributions. Finally, the factors influencing the thermal vibrations are analyzed. The results show that the thermal vibration is affected by many factors including the shaft size, rotational speeds, heating locations, critical speed, etc. Moreover, it is investigated how the convection coefficient and the heat conductivity influence the thermal vibrations in order to provide an insight into the management of thermal vibrations from the perspective of thermal aspects.     

New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis

The actual breathing mechanism of the transverse breathing crack in the cracked rotor system that appears due to the shaft weight is addressed here. As a result, the correct time-varying area moments of inertia for the cracked element cross-section during shaft rotation are also determined. Hence, two new breathing functions are identified to represent the actual breathing effect on the cracked element stiffness matrix. The new breathing functions are used in formulating the time-varying finite element stiffness matrix of the cracked element. The finite element equations of motion are then formulated for the cracked rotor system and solved via harmonic balance method for response, whirl orbits and the shift in the critical and subcritical speeds. The analytical results of this approach are compared with some previously published results obtained using approximate formulas for the breathing mechanism. The comparison shows that the previously used breathing function is a weak model for the breathing mechanism in the cracked rotor even for small crack depths. The new breathing functions give more accurate results for the dynamic behavior of the cracked rotor system for a wide range of the crack depths. The current approach is found to be efficient for crack detection since the critical and subcritical shaft speeds, the unique vibration signature in the neighborhood of the subcritical speeds and the sensitivity to the unbalance force direction all together can be utilized to detect the breathing crack before further damage occurs.     

An insight into the breathing mechanism of a crack in a rotating shaft

The time history of local flexibilities associated with a breathing crack in a rotating shaft is the concern of this paper. Considering quasi-static approximation, the deflections of a circular cross-section beam presenting a crack of different depths, due to bending or torsion loads are analyzed with the aid of a refined nonlinear contact-finite element procedure in order to predict accurately the time-variant flexibility of the fractured shaft. This method predicts the partial contact of crack surfaces, and it is appropriate to evaluate the instantaneous crack flexibilities. The bending load is applied in several aperture angles, in order to simulate a rotating load on a fixed beam. Results obtained for the rotating beam can then be used for the analysis of cracked, horizontal axis rotors. The effect of friction is also considered in the cracked area. Portions of crack surfaces in contact are predicted, the direct and the cross-coupled flexibility coefficients are calculated by applying energy principles. The numerical results compared with relevant previously published results, show high consistency.     

Influence of load torque on stability of rotor driven by flexible shaft

The influence of the load torque on the stability of a symmetric rotor, driven by a flexible shaft, is studied. Both linear and angular displacements of the rotor are considered. The analysis—which is approximate, and with the deflection, the damping and the load torque assumed to be small—shows that the main destabilizing effect of the load torque is due to the transverse moments acting while the rotor is inclined. The reaction pattern at the ends of the shaft, determined by means of the Kirchhoff equations, indicates that the semitangential mode of loading (conservative) is operative.     

Analysis on cohesive crack opening displacement considering the strain softening effect

In order to investigate the crack propagation in quasi-brittle materials like rock, ceramic and concrete, Hillerborg and his co-researchers abstracted the fracture process zone in front of a stress free crack in terms of a “fictitious crack zone”. On the fictitious crack zone, cohesive stresses distribute following a given softening relationship of stress vs. crack opening. Based on the polynomial or power series expression of cohesive crack opening displacement, the relationship of the cohesive stress vs. the crack opening displacement is established using elastic theory and integral equation, and some unknown physics variables are obtained using variation approach. The calculation results gained in this paper are close to the experimentally test ones.     

Dynamic optical visualization on the interaction between propagating crack and stationary crack

Dynamic interactions between the propagating crack and the static crack in PMMA material are studied by combining high-speed Schardin camera with optical caustic method. A series of dynamic optical bifocal patterns (the specimen-focused image and the off-focused image) around the propagating crack tip and the static crack tip are recorded for PMMA thin strip which contains two collinear-edge-cracks subjected to tensile loading, the variations of the caustic diameter and the distortion of the caustic shape are revealed due to the influence of local stress singularity at the crack tip. Interactions between the moving crack and the static crack are analyzed by means of the evolution of dynamic fracture parameters. The influence of crack interaction on fracture parameters is discussed based on both a K-dominance assumption and a higher order transient crack-tip expansion. These results will be useful to the evaluation of dynamic properties and the design of structures in the cracked polymer material.     

Coupled lateral and torsional vibration characteristics of a speed increasing geared rotor-bearing system

The goal of this study was to examine the coupled vibration characteristics of a turbo-chiller rotor-bearing system having a bull-pinion speed increasing gear, using a coupled lateral and torsional vibration finite element model of a gear pair, and to provide the mechanism of the characteristic changes. The investigations were systematically carried out by comparing the uncoupled and coupled natural frequencies and their mode shapes with varying gear mesh stiffness, taking into account rotating speeds, and by comparing the strain energies of the lateral and torsional vibration modes. The results show that some modes may yield coupled lateral and torsional mode characteristics when the gear mesh stiffness increases over a certain value and, in addition, that their associated dominant modes may be different from their initial modes, i.e., a given dominant mode may change from an initial torsional one to a lateral one or vice versa.     

Order of the phase transition of a twodimensional rotor system

A twodimensional rigid lattice of rotors with one rotational degree of freedom is studied in molecular field approximation and by molecular dynamics. The system undergoes a phase transition from an orientationally ordered low temperature structure to the orientationally disordered crystal (ODIC) also called the plastic high temperature phase. As a parameter of the interaction is varied the MFA-solution shows a changeover from a continuous to a discontinuous transition. I have deliberately performed a computer simulation of the system in the microcanonical ensemble where it is possible to fix the energy at an arbitrary level — even right within the step occuring at a first order transition- and to measure the temperature. The surprising result is a smaller derivative of the energy versus temperature relation than expected for a first order phase transition.     

Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6×6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3×3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.     

The influence of crack breathing and imbalance orientation angle on the characteristics of the critical speed of a cracked rotor

By analyzing the limitations of weight dominance and by taking the complicated whirl of the rotor into account, general equations of motion have been developed in case of a Jeffcott rotor with a transverse crack. The angle between the crack direction and the shaft deformation direction is used to determine the closing and opening of the crack, allowing one to study the dynamic response without assuming weight dominance. Using the new equations, the dynamic response of a cracked rotor near its critical speed has been computed via a numerical method to investigate the influence of nonlinear breathing of the crack and that of the imbalance orientation angle β on the stability, critical speed and peak response of the rotor. The results show that nonlinear breathing can improve the stability of a rotor in contrast to a rotor with an open crack, and, with a reversed imbalance (70°<β<270°), that it can reduce the vibration response in contrast to an uncracked rotor. The basic characteristics of a cracked rotor near its critical speed are similar to those of an uncracked rotor. The critical speed can be determined by measuring the rotation of the center of gravity. The critical speed of a cracked rotor is located between the natural frequencies of the fully open crack and those of the fully closed crack and depends on the imbalance orientation angle. Its value is lowest at β≈90° and highest at β≈270°. The peak in the response at the critical speed is mainly determined by the imbalance orientation angle. At β≈0° and 180°, the peak corresponds to the maximum and minimum response, respectively.     

Identification of unbalance in a rotor bearing system

Model based methods for fault identification in rotating systems are gaining importance for the last three decades due to their ability to identify both location and severity of the fault. Model based methods are of different types. Among them, equivalent loads minimization method is one method. In this method, fault is identified in a rotor bearing system by minimizing difference between equivalent loads estimated in the system due to the fault and theoretical fault model loads. This method has a limitation that the error in identified fault parameters increases with decrease in number of measured vibrations. Thus a comprehensive methodology for fault identification with minimum error even in case of fewer measured vibrations is attempted in the present work. Two different approaches: equivalent loads minimization and vibration minimization method are applied for the identification of unbalance fault in a rotor system. Unbalance fault is identified using proposed methods by measuring transverse vibrations at only one location.     

Dynamic behaviour of the icosahedral Al——Pd——Mn quasicrystal with a Griffith crack

The dynamic response of an icosahedral Al--Pd--Mn quasicrystal with a Griffith crack to impact loading is investigated in this paper. The elastohydrodynamic model for the wave propagation and diffusion together with their interaction is adopted. Numerical results of stress, displacement and dynamic stress intensity factors are obtained by using the finite difference method. The effects of wave propagation, diffusion and phonon--phason coupling on the quasicrystal in the dynamic process are discussed in detail, where the phason dynamics is explored particularly.     

分数阶Newton-Leipnik系统的动力学分析

依据分数阶线性系统的稳定性理论,研究了具有双重混沌吸引子的Newton-Leipnik系统取不同分数阶时的动力学行为.研究表明该系统具有逆向Hopf分岔过程,即随着阶数的下降,分数阶Newton-Leipnik系统由双重混沌吸引子突变为单吸引子,其动力学行为将由混沌态历经短暂的周期态后收敛于稳定的平衡点.     

Theoretical and nonlinear behavior analysis of a flexible rotor supported by a relative short herringbone-grooved gas journal-bearing system

This paper considers the bifurcation and nonlinear behavior of a flexible rotor supported by a relative short herringbone-grooved gas journal bearing system. A numerical method is employed to a time-dependent mathematical model. A finite difference method with successive over relation method is employed to solve the Reynolds’ equation. The system state trajectory, Poincaré maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and journal centers in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and quasi-periodic response of the rotor and journal centers. It further shown the dynamic behavior of this type of system varies with changes in bearing number and rotor mass. The results of this study contribute to a better understanding of the nonlinear dynamics of herringbone-grooved gas journal bearing systems.