Nonlinear vibration response of a complex aeroengine under the rubbing fault

Rolling bearing and squeeze film damper will introduce structural nonlinearity into the dynamic model of aeroengine. Rubbing will occur due to the clearance reduction design of the engine. The coupling of structural nonlinearity and fault nonlinearity will make the engine present rich vibration responses. This paper aims to analyze the nonlinear vibration behavior of the whole aeroengine including rolling bearing and squeeze film damper under rubbing fault. Firstly, the dynamic model of a turboshaft engine with nonlinear support and rubbing fault is established; The rolling bearing force, the oil film force and the rubbing force are introduced into a dual-rotor-casing model with six support points. Secondly, the linear part of the model is verified by the dynamic characteristics of the three-dimensional finite element model. Finally, the varying compliance vibration, the damping effect and the bifurcation mechanism are analyzed in detail in which the bearing clearance, speed ratio and rubbing stiffness are considered. Results show that the rubbing fault in the nonlinear support case will excite more significant varying compliance vibration in the low-speed region and expand the rotating speed range of the chaotic region in the high-speed region compared with that in the linear support case.

Design and characterization of a cervical phantom model for birth simulation training

In the midwifery and obstetrics, accurate diagnosis of labour progression is critical to ensure maternal and fetal wellbeing. In light of this, simulation systems have been designed to train practitioners. However, existing simulators lack a dynamic cervix to simulate the varying compliance of cervical ripening. In this thesis, characteristics of ex vivo human cervical tissue were studied and a pneumatically controlled compliant cervix phantom was designed. Cervical tissue exhibited strain of 0.50 for an applied stress of 325 kPa. The phantom demonstrated 35% effacement, 1 cm dilation and was capable of becoming 22 times softer. At maximum vacuum pressure, strain of 0.23 was achieved for an externally applied stress of 419 kPa. At atmospheric pressure, strain of 0.50 was achieved for an applied stress of 117 kPa. Results yield a deeper understanding of cervix characteristics and establishes the groundwork for the design of a dynamic human birth simulation training system.

Design and characterization of a cervical phantom model for birth simulation training

In the midwifery and obstetrics, accurate diagnosis of labour progression is critical to ensure maternal and fetal wellbeing. In light of this, simulation systems have been designed to train practitioners. However, existing simulators lack a dynamic cervix to simulate the varying compliance of cervical ripening. In this thesis, characteristics of ex vivo human cervical tissue were studied and a pneumatically controlled compliant cervix phantom was designed. Cervical tissue exhibited strain of 0.50 for an applied stress of 325 kPa. The phantom demonstrated 35% effacement, 1 cm dilation and was capable of becoming 22 times softer. At maximum vacuum pressure, strain of 0.23 was achieved for an externally applied stress of 419 kPa. At atmospheric pressure, strain of 0.50 was achieved for an applied stress of 117 kPa. Results yield a deeper understanding of cervix characteristics and establishes the groundwork for the design of a dynamic human birth simulation training system.

Electoral Engineering in New Democracies: Strong Quotas and Weak Parties in Tunisia

Bridging the literature on gender and politics, democratization, and political parties, this article investigates the causes of parties’ varying compliance with electoral quotas. Whereas research has so far focused on parties’ willingness to comply, this article sheds light on their ability to do so. It suggests that the more quotas parties have to comply with, and the more complex the quotas’ designs, the more difficult implementation becomes for the organizationally weak parties that we often encounter in new democracies. The argument is developed and substantiated in a comparative analysis of parties’ quota compliance in the 2018 Tunisian local elections. Although the Islamist party was able to comply fully with all quotas (for women, youth and people with disabilities), small secular parties lost a number of lists and state funding due to non-compliance. While the quotas were highly effective in securing group representation, they had repercussions on party and party system consolidation.

Mechanism and Characteristics of Global Varying Compliance Parametric Resonances in a Ball Bearing

Varying compliance (VC) is an unavoidable form of parametric excitation in rolling bearings and can affect the stability and safety of the bearing and its supporting rotor system. To date, we have investigated VC primary resonance in ball bearings, and in this paper other parametric VC resonance types are addressed. For a classical ball bearing model with Hertzian contact and clearance nonlinearities between the rolling elements and raceway, the harmonic balance and alternating frequency/time domain (HB–AFT) method and Floquet theory are adopted to analyze the VC parametric resonances and their stabilities. It is found that the 1/2-order subharmonic resonances, 2-order superharmonic resonances, and various VC combination resonances, such as the 1-order and 2-order summed types, can be excited, thus resulting in period-1, period-2, period-4, period-8, period-35, quasi-period, and even chaotic VC motions in the system. Furthermore, the bifurcation and hysteresis characteristics of complex VC resonant responses are discussed, in which cyclic fold, period doubling, and the second Hopf bifurcation can occur. Finally, the global involution of VC resonances around bearing clearance-free operations (i.e., adjusting the bearing clearance to zero or one with low interference) are provided. The overall results extend the investigation of VC parametric resonance cases in rolling bearings.

A Simplified Mathematical Model for the Analysis of Varying Compliance Vibrations of a Rolling Bearing

In this paper, a simplified approach in the analysis of the varying compliance vibrations of a rolling bearing is presented. This approach analyses the generation of vibrations in relation to two boundary positions of the inner ring support on an even and an odd number of the rolling element of a bearing. In this paper, a mathematical model for the calculation of amplitude and frequency of vibrations of a rigid rotor in a rolling bearing is presented. The model is characterized by a big simplicity which makes it very convenient for a practical application. Based on the presented mathematical model a parametric analysis of the influence of the internal radial clearance, external radial load and the total number of rolling elements on the varying compliance vibrations of rolling bearing was conducted. These parameters are the most influential factors for generating varying compliance vibrations. The results of the parametric analysis demonstrate that with the proper choice of the size of the internal radial clearance and external radial load, the level of the varying compliance vibrations in a rolling bearing can be theoretically reduced to zero. This result opposes the opinion that varying compliance vibrations of rolling bearing cannot be avoided, even for geometrically ideally produced bearing.

Control of Period-Doubling and Chaos in Varying Compliance Resonances for a Ball Bearing

Varying compliance (VC) is an inevitable parametrical excitation to rolling bearing systems due to time-varying stiffness from rolling element revolution. Period-doubling instability in the VC primary resonances of ball bearing is presented in many studies. Recently, this instability was demonstrated to be a probable indicator of occurrence of strong one to two internal resonances and chaotic motions, which has potential effects on the stability and safety of the bearing-rotor system. However, few studies have directly attempted to suppress this bifurcation instability. Here, a dynamic stiffness evaluating method is presented for assessing the threshold of the period-doubling and complex motions in VC primary resonances of ball bearings, where the elaborate evolution of the bifurcating process is obtained by harmonic balance and alternating frequency/time domain (HB-AFT) method and using Floquet theory. Our analysis indicates that by introducing certain additional stiffness, the period-doubling and corresponding subharmonic internal resonances can be suppressed. Besides, the evolution and mechanism of type I intermittency chaos in ball bearings will be clarified in depth. It is also shown that extensive chaotic motions for large bearing clearances (e.g., 40 μm) can vanish perfectly by action of additional stiffness.

Hysteretic Resonances and Intermittency Chaos in Ball Bearings

Ball bearings are essential parts of mechanical systems to support the rotors or constitute the revolute joints. The time-varying compliance (VC), bearing clearance and the Hertzian contact between the rolling elements and raceways are three fundamental nonlinear factors in a ball bearing, hence the ball bearing can be considered as a nonlinear system. The hysteresis and jumps induced by the nonlinearities of rolling bearings are typical phenomena of nonlinear vibrations in the rolling bearing-rotor systems. And the corresponding hysteretic impacts have direct effects on the cleavage derivative and fatigue life of the system components. Therefore, the behaviors of hysteresis and jumps are given full attentions and continued studies in the theoretical and engineering fields. Besides, many researchers have done a lot of calculations to depict the various characteristics of bifurcations and chaos in the rolling bearings and their rotor systems, but few researches have been addressed on the inherent mechanism of the typical intermittency vibrations in rolling bearings. With the aid of the HB-AFT (the harmonic balance method and the alternating frequency/time domain technique) method and Floquet theory, this paper will investigate deeply the resonant hysteresis and intermittency chaos in ball bearings.

The Varying Compliance Resonance in a Ball Bearing Rotor System Affected by Different Ball Numbers and Rotor Eccentricities

The varying compliance (VC) vibration directly reflects the oscillation intensity of a rolling bearing, and it can be fully revealed in the VC resonance. Moreover, we define the bearing vibration intensity as the bearing vibration information in this paper. Besides the rolling element number of the bearing, the rotor eccentricity is also an inevitable influencing factor for the VC vibration. This paper focuses on the VC resonance characteristics in a ball bearing rotor system. An analytical model is established, and the vibration responses of the system are calculated in a large speed range with the consideration of different ball numbers and different rotor eccentricities. The theoretical results show that the VC vibration is clearer in low-speed range where the VC resonance exist, while it is suppressed in high-speed range. In general, the intensity of the VC resonance decreases with the increase of ball numbers and is not sensible to the rotor eccentricities in low-speed range. Finally, a ball bearing rotor experiment system is setup, the VC resonance is clearly detected, and the high-quality bearing vibration information is obtained. The experimental results qualitatively agree with the theoretical results.