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.     

Dynamic response of cracked rotor-bearing system under time-dependent base movements

Dynamic response of cracked rotor-bearing system under time-dependent base movements is studied in this paper. Three base angular motions, including the rolling, pitching and yawing motions, are assumed to be sinusoidal perturbations superimposed upon constant terms. Both the open and breathing transverse cracks are considered in the analysis. The finite element model is established for the base excited rotor-bearing system with open or breathing cracks. Considering the time-varying base movements and transverse cracks, the second-order differential equations of the system will not only have time-periodic gyroscopic and stiffness coefficients, but also the multi-frequency external excitations. An improved harmonic balance method is introduced to obtain the steady-state response of the system under both base and unbalance excitations. The response spectra, orbits of shaft center and frequency response characteristics, are analyzed accordingly. The effects of various base angular motions, frequency and amplitude of base excitations, and crack depths on the system dynamic behaviors are considered in the discussions.     

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.     

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.     

Low-lying Collective Modes of a 1D Dipolar Quantum Gas in an Anharmonic Trap

The low-lying collective modes of an anharmonically confined one-dimensional ultracold dipolar Bose gas are investigated by using time-dependent variational method. The frequencies of dipole mode and breathing mode are obtained analytically in the full crossover regime from a Tonks-Girardeau gas to a dipolar density wave state. We find that the frequency shifts caused by quartic distortion of the potential can be amplified by interparticle interaction and the collapse and revival of the collective excitations can be observed in the anharmonic trap.     

A generator co-ordinate approach to the breathing mode

In the harmonic approximation of the generator co-ordinate method (GCM) with the Skyrme interaction we show that the frequency of the breathing mode is essentially the same as the semiclassical result. The accuracy of the harmonic approximation is tested in ¹⁶O. Also, a comparison is made between the adiabatic time-dependent Hartree-Fock (ATDHF) theory and an anharmonic version of the GCM.     

Rigid finite element model of a cracked rotor

The article introduces a new mathematical model for the cracked rotating shaft. The model is based on the rigid finite element (RFE) method, which has previously been successfully applied for the dynamic analysis of many complicated, mechanical structures. In this article, the RFE method is extended and adopted for the modeling of rotating machines. An original concept of crack modeling utilizing the RFE method is developed. The crack is presented as a set of spring–damping elements of variable stiffness connecting two sections of the shaft. An alternative approach for approximating the breathing mechanism of the crack is introduced. The approach is simple and allows one to intuitively and systematically prepare and analyze the model of a cracked rotor.The proposed method is illustrated with numerical and experimental results. The experiments conducted for the uncracked free–free rotor as well as the numerical results obtained with other software confirm the accuracy of the RFE model. The numerical analysis conducted for a set of cracked rotors has shown that, depending on the eccentricity and its angular location, the breathing behavior of the crack may take different forms. In spite of this, the frequency spectra for different cracks are almost identical.Due to its simplicity and numerous advantages, the proposed approach may be useful for rotor crack detection, especially if methods utilizing the mathematical model of the rotor are applied.     

Control of giant breathing motion in c60 with temporally shaped laser pulses

Femtosecond laser pulses tailored with closed-loop, optimal control feedback were used to excite oscillations in C60 with large amplitude by coherent heating of nuclear motion. A characteristic pulse sequence results in significant enhancement of C2 evaporation, a typical energy loss channel of vibrationally hot C60. The separation between subsequent pulses in combination with complementary two-color pump-probe data and time-dependent density functional theory calculations give direct information on the multielectron excitation via the t(1g) resonance followed by efficient coupling to the radial symmetric a(g)(1) breathing mode.     

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.     

The vibration signature of chordal cracks in a rotor system including uncertainties

The aim of this paper is to investigate the effects of the presence of a transverse crack in a rotating shaft under uncertain physical parameters in order to obtain some indications that might be useful in detecting the presence of a crack in rotating system. The random dynamic response of the cracked rotor is evaluated by expanding the changing stiffness of the crack (i.e. the breathing mechanism) as a random truncated Fourier series. To avoid the use of the Monte Carlo simulations (MCS), an alternative procedure that is based on a combination of the Harmonic Balance Method and the Stochastic Finite Element Method (SFEM) using the Polynomial Chaos Expansion (PCE) is proposed. So the response of the Fourier components of the cracked rotor is expanded in the polynomial chaoses. The random dynamic response obtained by applying this procedure is compared with that evaluated through numerical integration based on the Harmonic Balance Method and the Monte Carlo simulations.     

Dynamic behavior of cracked rotor subjected to multisine excitation

It is widely known that a developing shaft crack manifests itself in the appearance of nonlinear effects resulting in higher harmonics in a vibration spectrum. However, such symptoms are characteristic not only for developing shaft cracks but also for other malfunctions such as a shaft bow, a coupling misalignment, etc. That is why novel shaft crack detection methods introduce a specially designed diagnostic force applied to the shaft in order to amplify the particular symptoms of the crack. Most often a simple harmonic force is used for such purposes, yet the results may not be reliable.     

Modeling and simulation of vibrational breathing-like modes in individual multiwalled carbon nanotubes

We study the collective vibrational breathing modes in the Raman spectrum of multiwalled carbon nanotubes (MCNTs). First, a bond polarization theory and the spectral moment's method (SMM) are used to calculate the non-resonant Raman frequencies of the breathing-like modes (BLMs) and the tangential-like ones (TLMs). Second, the Raman active modes of MCNTs are computed for different diameters and numbers of layers. The obtained low frequency modes in MCNTs can be identified to each single-walled carbon nanotubes. These modes that originate from the radial breathing ones of the individual walls are strongly coupled through the concentric tube–tube van der Waals interaction. The calculated BLMs in the low-frequency region are compared with the experimental Raman data obtained from other studies. Finally, special attention is given to the comparison with Raman data on MCNTs composed of six layers.     

Tribocharging in a rotating shaft–oil–lip seal system for different engine base oils,additive, and their blends

The paper presents the results of experiments on tribocharging of different mineral, semisynthetic, and synthetic base oils and their blends with additives. An antiwear additive ZDDP (zinc dialkyldithiophosphate) is tested when added to base oils in different percentages by weight. Experiments are carried out on the specially built experimental facility to be a simplified model of an engine crankcase in whose interior an earthed metal shaft rotates at given angular velocities. The potential of a stiffening ring of a lip seal is measured directly with an electrometer. The potential is a measure of tribocharging in a rotating shaft–oil–lip seal system, friction junction, and especially between both interfaces: shaft surface–oil and oil–lip of a lip seal. The experimental results are presented in the form of some characteristics that are relationships of the potential induced in the stiffening ring with oil's temperature for different angular shaft's velocities and additives, contents. The oil's temperature ranges from 60 to 110 °C and is controlled automatically. The angular velocities of a shaft used are 500 and 1500 rpm. The additive contents in the blends with different base oils are 0.1 and 0.2%. The pure base oils of all the types and some lip seals are also examined. Moreover, the influence of an external DC electric field applied between the earthed shaft and the stiffening ring on the braking torque of the shaft is examined for a range of temperatures of the pure oils and their blends with the additive used, angular velocities, and additive contents. The electric field is produced while applying the high DC voltage of both polarities between the shaft and the ring. The absolute value of the voltage is in a range from 500 to 1500 V.     

Influence of additives blended with motor base oils on the braking torque under an auxiliary external DC electric field

The paper presents the results of experiments upon the influence of tribocharging of PAO and PAG synthetic motor base oils blended with different additives—friction modifiers (FM) and antiwear agents (AW)—and the effect of an external DC electric field on the braking torque. The experiments are carried out in a rotating shaft–oil–lip seal system which represents a specially built experimental facility to be a simplified model of an engine crankcase in the interior of which a metal shaft rotates. The research is especially aimed at the braking torque of a rotating shaft sealed with a lip seal and a possibility of reduction in the torque under external DC electric fields. DC voltage is applied between the stiffening ring of lip seal and a rotating, earthed shaft. The braking torque of rotating shaft is measured as a function of the oil–additive blend's temperature, the shaft's angular velocity, and the absolute value of the external DC voltage. In general, it is found that an external DC electric field causes the braking torque to change with the increasing DC voltage. The change depends on the additives and base oils used in their blends which in turn causes the torque to increase in the case of the PAO–additive blends or to decrease for the PAG–additive blends.     

Analysis of a non-contact magnetoelastic torque transducer

Results are presented for the performance of a magnetoelastic torque transducer that converts a torque-induced strain in a non-magnetic shaft into changes in a measurable magnetic field. The magnetic field is generated by a thin magnetostrictive layer that is coated onto the circumference of the shaft. The layer is magnetized and has an initial residual strain. The magnetization within the layer rotates in response to changes in the strain which occur when the shaft is torqued. The magnetic field produced by the layer changes with the magnetization and this can be sensed by a magnetometer to monitor the torque on the shaft. In this paper, a phenomenological theory is developed for predicting the performance of the transducer. The theory can be used to predict the magnetic field distribution of the transducer as a function of the physical properties of the magnetic coating, its residual strain, and the applied torque. It enables rapid parametric analysis of transducer performance, which is useful for the development and optimization of novel non-contact torque sensors.     

Phonon and thermal properties of achiral single wall carbon nanotubes

A detailed theoretical study of the phonon and thermal properties of achiral single wall carbon nanotubes has been carried out using force constant model considering up to third nearest-neighbor interactions. We have calculated the phonon dispersions, density of states, radial breathing modes (RBM) and the specific heats for various zigzag and armchair nanotubes, with radii ranging from 2.8 Å to 11.0 Å. A comparative study of phonon spectrum with measured Raman data reveals that the number of Raman active modes for a tube does not depend on the number of atoms present in the unit cell but on its chirality. Calculated phonon modes at the zone center more or less accurately predicted the Raman active modes. The radial breathing mode is of particular interest as for a specific radius of a nanotube it is found to be independent of its chirality. We have also calculated the variation of RBM and G-band modes for tubes of different radii. RBM shows an inverse dependence on the radius of the tube. Finally, the values of specific heat are calculated for various nanotubes at room temperature and it was found that the specific heat shows an exponential dependence on the diameter of the tube.     

Influence of body position on breathing and its implications for the evaluation and treatment of speech and voice disorders

This paper examines how breathing differs in the upright and supine body positions. Passive and active forces and associated chest wall motions are described for resting tidal breathing and speech breathing performed in the two positions. Clinical implications are offered regarding evaluation and treatment of breathing behavior in clients with speech and voice disorders.     

Collective oscillations of vortex lattices in rotating Bose-Einstein condensates

The complete low-energy collective-excitation spectrum of vortex lattices is discussed for rotating Bose-Einstein condensates by solving the Bogoliubov-de Gennes equation, yielding, e.g., the Tkachenko mode recently observed at JILA. The totally symmetric subset of these modes includes the transverse shear, common longitudinal, and differential longitudinal modes. We also solve the time-dependent Gross-Pitaevskii equation to simulate the actual JILA experiment, obtaining the Tkachenko mode and identifying a pair of breathing modes. Combining both approaches allows one to unambiguously identify every observed mode.     

Magnetostatic waves in a time-dependent medium

Propagation of magnetostatic waves (MSW) in planar magnetic structures with time-dependent parameters is studied theoretically and experimentally for the case where the structure is based on ferrite films. The time-dependent parameter is chosen to be the magnetizing field, which is taken to be uniform in space and slowly varying in time. Asymptotic perturbation theory is used to obtain an equation for the slowly varying amplitude and a relation for the phase, which describes the time behavior of the frequency and envelope of an MSW pulse. The time behavior of the frequency spectrum of an MSW propagating in an iron-yttrium garnet film is studied experimentally for different forms of modulation of the magnetic field. Space-time focusing of MSW pulses is studied for the case of a time-dependent field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 54–66, November, 1988.