Supercritical Speed Research Articles

Stability of steady-state solutions of Jeffcott rotor with varying rotational speed

AbstractThe behavior of flexible rotating systems with varying rotational speeds is essential in engineering applications. Analysis methods that consider linear dynamic models and many existing nonlinear analysis approaches assume constant rotational speed. These approaches are unsuited to study the dynamic interaction between driving torque and whirling motion in this class of applications. In this paper, an analysis of the stability and control of a Jeffcott rotor under varying operational conditions is presented. A nonlinear dynamic model of the system is formulated to enable a detailed stability and parametric analysis. A proportional-integral (PI) torque command is employed to achieve a steady-state rotational speed. Assuming constant lateral control effort, system equilibrium points and their stability characteristics as functions of the system’s parameters are analyzed. A control law that minimizes the lateral effort is derived. A feedback proportional lateral control strategy is introduced to enhance the system’s region of stability, particularly in the supercritical speed range. Finally, a simulation study is conducted to validate the analytical findings. Simulation results demonstrate the effectiveness of the proposed approach for defining stable operating conditions and improving system performance.

Analytical solution of ground vibration of a half-space induced by sub- and super-critical harmonic moving loads: Theory and application

This paper presents an analytical solution to the ground vibration of a half-space subjected to sub- and super-critical moving loads oscillating with the frequency f0 and distributed in different forms. By applying the contour integral combined with the residue theorem, the approximate values of the integrals for the ground response are determined, with a rigorous rationale provided for the exclusive consideration of the Rayleigh waves on the ground. The validity of the present solutions is verified against existing solutions, including Auersch's. Through the parametric analysis, it is observed that the frequency f0 of dynamic loading of the vehicle exhibits significant influence on the critical speed, the magnitude of ground vibrations and their rate of attenuation. In addition, the effect of rail irregularity on the ground vibration was included in some applications of the present theory for high-speed trains under the sub-and super-critical speeds with respect to the Rayleigh waves speed. It was shown that track irregularity can largely amplify the velocity and acceleration responses of the ground, albeit exerting a comparably little effect on the ground displacement. Besides, resonant vibrations are likely to occur when the train is accelerated to the critical speed of the soil.

Nonlinear modeling and vibration response analysis of rod fastening rotor-bearing system with curvic coupling

Rod fastening combined rotor system (RFCR) are particularly popular in gas turbine rotor systems. The bolted connection introduces a discontinuity in this RFCR, which has a substantial influence on system performance.Traditional models of RFCR tend to neglect the influence of the connection structure, especially failing to fully consider the contribution of the curvic coupling to the rotor dynamics. To address this problem, this study proposes a new nonlinear analytical model of the rotor system, which focuses on the impact of the curvic coupling. This study provides a detailed analysis of the structural parameters and load forces of the curvic coupling. Based on this analysis, a mechanical model is created to assess how bolt preload and torque impact the stiffness of the curvic connection. The stiffness matrix of the curvic coupling is derived by the gear stiffness analysis method, and the accuracy of the model is confirmed using the finite element method. Furthermore, the model also considers the viscous and sliding state between the curvic coupling, which effectively describes the damping effect of the curvic coupling. On this basis, the motion control equations for the RFCR are formed, and a dynamic model considering the curvic connection in the RFCR system is introduced. The results show that the curvic coupling results in a decrease in stiffness and the occurrence of nonlinear damping phenomena in the RFCR.The RFCR exhibits a significant stiffness loss compared to the continuous rotor system, resulting in a decrease in the critical rotor speed. The nonlinear damping is mainly excited at high friction coefficient conditions and may lead to a self-excited vibration component in the rotor response at supercritical speeds. The negative impact of structural discontinuities on rotor dynamic performance can be reduced by increasing the bolt preload, decreasing the friction coefficient, and adjusting the amount of imbalance. This study offers a theoretical foundation for predicting curvic coupling-induced instability and serves as a reference for enhancing rotor system stability.

Self-excited vibration analysis of spline rotor systems considering parallel and angular misalignments

To address the instability phenomenon of aerospace splines at supercritical speed, the law of self-excited vibration (SEV) phenomenon of rotor systems considering parallel and angular misalignments is systematically investigated. The finite element model (FEM) of a rotor system is developed, incorporating spline internal friction damping. The spline is modeled by a nonlinear time-varying stiffness theory derived from the slicing method and the penalty function method. The parallel and angular misalignments are considered in the nonlinear time-varying stiffness model. The effects of the misalignment on critical speed (CS) and SEV of systems are analyzed. Moreover, the validity of the proposed spline rotor dynamic model is established through a comparison between simulation and experimental results.

Friction-induced nonlinear dynamics in a spline-rotor system: Numerical and experimental studies

Excessive vibrations induced by intrinsic internal friction in spline joints critically impact the reliability and safety of high-speed rotating machinery. To address this concern, this paper presents experimental and numerical research on a flexible rotor system subjected to frictional contact within the spline joint. Experiments are conducted on a scaled rotor test rig designed to emulate the tail rotor drive shaft of helicopters, aiming to explore vibration behaviors under various operating conditions. A reduced-order mathematical model based on the test rig, considering the multi-interface friction of the spline joint and deformation coordination with a flexible coupling, is formulated. The numerical study, which incorporates both stability analysis and transient nonlinear dynamics, is then presented to quantitatively evaluate the occurrence conditions and excitation mechanisms leading to friction-induced instabilities. A parameter study is further conducted to elucidate the roles of various parameters in shaping self-excited vibration behaviors. The results reveal that the cross-coupling of frictional forces within the spline joint acts as the driving force of instabilities at supercritical speeds, with the resulting self-excited responses dominated by the frequency of the first critical speed. The study also highlights the necessity of integrating experimental and numerical analysis to better predict and understand the self-excited vibration phenomena in the spline-shaft systems.

Supercritical Operation of Bearingless Cross-Flow Fan

This paper presents a decoupled bearingless cross-flow fan (CFF) that operates at a supercritical speed, thereby increasing the maximum achievable rotational speed and fluid dynamic power. In magnetically levitated CFF rotors, the rotational speed and fan performance are limited by the bending resonance frequency. This is primarily defined by the low mechanical bending stiffness of the CFF blades, which are optimised for fluid dynamic performance, and the heavy rotor magnets on both rotor sides, which add significant mass but a minimal contribution to the overall rotor stiffness. This results in detrimental deformations of the CFF blades in the vicinity of the rotor bending resonance frequency; hence, the CFF is speed-limited to subcritical rotational speeds. The novel CFF rotor presented in this study features additional mechanical decoupling elements with low bending stiffness between the fan blades and the rotor magnets. Thus, the unbalance forces primarily deform the soft decoupling elements, which enables them to pass resonances without CFF blade damage and allows rotor operation in the supercritical speed region due to the self-centring effect of the rotor. The effects of the novel rotor design on the rotor dynamic behaviour are investigated by means of a mass-spring-damper model. The influence of different decoupling elements on the magnetic bearing is experimentally tested and evaluated, from which an optimised decoupled CFF rotor is derived. The final prototype enables a stable operation at 7000 rpm in the supercritical speed region. This corresponds to a rotational speed increase of 40%, resulting in a 28% higher, validated fluid flow and a 100% higher static pressure compared to the previously presented bearingless CFF without decoupling elements.

Dynamic modelling and parametric analysis of a bolted joint rotor-bearing system considering stick-slip behavior at mating interface

Bolted joint structure is widely utilized in the aeroengine compress rotor systems with convenience in assembly and maintenance. Due to the inherent structure discontinuity and stick-slip behavior at the mating interface, stiffness softening and frictional energy dissipation would be generated at the mating interface, resulting in complex vibration features of the rotor system. In this work, to describe the stick-slip behavior at the mating interface of the rotor system, an Iwan-based model is introduced to describe the contact state of asperities at the multi-scale level. The corresponding model identification method is proposed based on the partial hysteresis curve of the bolted joint. After that, a dynamic model of the bolted joint rotor system is established, and its dynamic response is obtained using the Runge–Kutta method. The effect of preload and friction coefficient on the nonlinear vibration of the bolted joint rotor is evaluated in detail based on the dynamic model. The result indicates that the stiffness softening would lead to the critical speed decreasing and aggravating the vibration response of the rotor system, while the frictional energy dissipation would restrain the response amplitude and induce the self-excited vibration within the super-critical speed range. The higher preload or friction coefficient has little effect on the motion state of the rotor system within the sub-critical speed range and makes system vibration stable between the 1st- and 2nd-order critical speed while unstable at the super-critical speed range. This research helps understand the vibration features of bolted joint rotor systems under the condition of stick-slip behaviors at the mating interface.

An exploration of the super-critical speed dynamic behaviours of the pantograph-catenary system

The train operating speed is traditionally limited to lower than 0.7 times the critical speed of the catenary. The material’s strength limits the tensile forces of the tension wires, preventing the further improvement of the train speed. A novel super-critical speed operating strategy of the pantograph catenary system (PCS) is proposed, using the relatively low tensile values of the catenary to make the pantograph run at a speed ratio larger than 1. The analytical results of a string under a moving load show the theoretical possibility of the super-critical speed operating strategy in moving load problems. The finite element method (FEM) is adopted for the model of the catenary. The differences between the simple catenary and the stitched catenary running at the super-critical speed regime are compared. Then, the tensile forces designation of the stitched catenary with different speed ratio values beyond 400 km/h is studied. Finally, the dynamic behaviours of the PCS reaching the super-critical speed regime are discussed, consisting of the continuous acceleration process and the transition process from the high-tension section to the low-tension section. The proposed super-critical speed operating strategy may be promising for the next-generation high-speed PCS.

Enhanced mixed boundary for modeling infinite domain in 2.5D soil vibration analysis

In this paper, an enhanced mixed 2.5D boundary for treating the infinite domains of the half-space is proposed. Unlike the perfectly matched layer (PML) that has a fixed outer boundary, the proposed method replaces the fixed outer boundary by infinite elements (IFE) to allow waves to propagate uninterruptedly to infinity. Such an approach not only improves the absorption capacity of the PML, but reduces its dependence on the attenuation function. The proposed boundary in the stretched coordinates is formulated in detail, for which the criteria for selecting the reference wavenumbers for multi types of waves are deliberately analyzed. A comprehensive parametric study for assessing the accuracy of the proposed boundary is presented, where the superiority of the proposed boundary relative to the pure PML and IFE is demonstrated. Particularly, the optimal ranges of parameters of the proposed boundary are identified. For illustration, the proposed boundary is applied to analyzing the frequency and time domain responses of a surface train moving at sub- and super-critical speeds. The results indicate that the proposed method can accurately predict the wave propagation characteristics, including the Mach cone at super-critical speeds.

Nonlinear dynamics of a vertical pipe subjected to a two-phase, solid-liquid internal flow

In this paper, a numerical study is carried out on a simply supported pipe model which conveys a two-phase, solid-liquid flow. First, a system of nonlinear, partial differential equations of dimensionless, two-phase, solid-liquid flow pipeline is established. Then, the above mentioned partial differential equation system is discretized by the Galerkin method of finite order to obtain a system of ordinary differential equations. Finally, the natural frequency characteristics of the corresponding linear system are solved to approximate the stability range of the nonlinear system, the structural response characteristics of which are solved by the Newton-Raphson method. The dimensionless aspect ratio, the dimensionless liquid-phase volume, and the dimensionless gravity coefficient are chosen to analyze the impact on the critical velocity of buckling and structural nonlinear response characteristic in this research. The research results indicate that a change to the above parameters will cause the critical speed to change in a linear or non-linear form. Moreover, at subcritical speeds, the vibrations are damped; at critical speeds, steady-state vibrations with extremely small amplitudes with a single frequency are seen; and at supercritical speeds, the pipeline model will produce buckling in different directions according to initial values.

Operating characteristics of supercritical carbon dioxide high speed tilting pad bearings considering inertia effect

Some degree of inertia effect on the bearing film is inevitable for high density supercritical carbon dioxide (S-CO2) working as a lubricant in high speed rotor operating conditions. This paper aims to reveal the influence of inertia effect on static and dynamic performances of the S-CO2 tilting pad bearings. Numerical results demonstrate that inertia effect on the lubricating film of the top pad is more pronounced. The peak pressure of all pads is increased at a high reduced Reynolds number compared with the inertialess condition, while the influence of inertia effect on the dynamic coefficient is weak. The theoretical model including inertia effect presented in this study can more accurately evaluate the performance of high speed S-CO2 tilting pad bearings.

Experimental Study of Coupled Torsional and Lateral Vibration of Vertical Rotor-to-Stator Contact in an Inviscid Fluid

Diagnosis of faults in a rotor system operating in a fluid is a complex task in the field of rotating machinery. In an ideal scenario, a forced shutdown due to rotor-stator contact failure would necessitate the replacement of the rotor or stator. However, factors such as time constraints, economic considerations, and the aging of infrastructure make it imprudent to abruptly shut down machinery that can still be safe to operate. The purpose of this paper is to present an experimental study that validates the theoretical results of the dynamic behavior and friction detection using the wavelet synchrosqueezing transformation (WSST) method for recurrent rotor-stator contacts in a fluid environment, as presented in a previous study. The investigation focused on the analysis of whirl orbits, shaft deflection, and fluctuation frequency during passage through critical speeds. The WSST method was used to decompose the dynamic responses of the rotor in the supercritical speed zone into several supercomponents. The variation of the high-frequency component was studied based on the fluctuation of the instantaneous frequency (IF) technique. Additionally, the fast Fourier transform (FFT) method, in conjunction with the WSST technique, was used to calculate the variation in the amplitude of high-order frequencies in the vibration signal spectrum. The experimental study revealed that the split in resonance caused by rubbing effects is reduced when the rotor and stator interact with an inviscid fluid. However, despite the effects of elasticity and fluid boundaries generating self-excitation at low frequencies and uneven motion due to stator clearance, the experimental results were consistent with the theoretical analysis, demonstrating the effectiveness of the contact detection method based on WSST.

Unbalance position of aeroengine flexible rotor analysis and identification based on dynamic model and deep learning

The rotor system design is gradually becoming lighter as the thrust-weight ratio of the new generation of aeroengines improves, making the power turbine rotor structure more slender and flexible, and it is difficult to judge the axial distribution of unbalance. Meanwhile, the engine rotor’s working speed is above the second critical speed and even exceeds the third-order speed. The small imbalance of the rotating shaft will cause large vibration, coupled with the influence of mode shape, resulting in difficulty in rotor dynamic balance, low efficiency and poor effect of the flexible rotor dynamic balance. Therefore, finding out the unbalance position is of great significance to determine the balance plane during rotor dynamic balance, and improve the balance efficiency and effect of flexible rotor. Firstly, the paper established the dynamic model of power turbine rotor, analyzed the vibration response law of rotor with unbalance at different positions, and found out the sensitive position of rotor unbalance. Then, a power turbine rotor unbalance position identification method, based on deep learning, one-dimensional convolution neural network (1D-CNN) is proposed. The multiple measured response data of the rotor with different unbalance positions are input into 1D-CNN for training. Finally, at subcritical speed, critical speed and supercritical speed, the accuracy of 1D-CNN method for rotor unbalance position identification is 99.5%, 99.5%, and 99.0% respectively, which verifies the effectiveness, feasibility of 1D-CNN method for rotor unbalance position identification, and provides a new solution for intelligent identification of aeroengine rotor unbalance position.

SELECTIVE SURFACE HARDENING OF GEAR MECHANISM SHAFT BY ROBOTIC LASER 3D SYSTEM

A robot-based laser 3D hardening method is proposed as a finishing operation to increase the wear resistance of the metal end-products. The laser hardening process of the surface layer of the products by changing its structure is one of the most effective methods of selective surface hardening. Thermal hardening of metals and alloys by laser radiation is based on local heating of a surface area under the influence of radiation and subsequent cooling of this surface area at a supercritical speed due to heat removal into the inner layers of the metal. The used robot-based laser 3D system (FANUC industrial robot and SCANLAB scanning optics) allows processing the surfaces of any complexity and geometry, including the shafts of the gear mechanism of the seed drill. It was found that the development and improvement of the technological process of manufacturing steel shafts of the gearbox of the seed drill is an urgent technological problem due to the rapid retirement of the shaft, which leads to the expenditure of time and money for its replacement. The most heavily loaded sections of the shaft were pre-estimated using the SolidWorks Simulation software package. A high-power TruDisk 8002 solid-state disk laser with a maximum laser power of 8 kW and a wavelength of laser radiation of 1030 nm is used for high-quality laser surface treatment of the AISI 1066 steel shaft of the gear mechanism. The laser surface heat treatment was carried out using a constant power strategy (continuous mode), varying the laser power in the range of 1.35–2.25 kW. Based on the Fe-Mn two-component state diagram, the critical points of the temperature of complete austenization of the studied steel were previously predicted taking into account the chemical composition of the material. Additionally, the energy density values of the laser beam with a diameter of 1 mm on the working surface was estimated. The results showed that the surface hardness of the shaft was about 2.5 times higher compared to the untreated surface. The working ranges of laser heat treatment parameters of the gearbox shaft were established to increase the hardening intensity (100–150%) of the responsible areas.

Experimental Investigation of the Dynamic Response of a Sliding Bearing System under Different Oil Pressure Levels

A sliding bearing system performs a critical function in a rotating machinery system and provides support for the rotating components. In this paper, a single-rotor experimental rig and a dual-rotor sliding bearing experimental rig were independently established, and the dynamic response of the test systems are investigated. To study the effect of oil pressure levels on the dynamic response of the test systems under the single-rotor system and the dual-rotor system, the vibration response of single-rotor and dual-rotor systems ware analyzed by different oil inlet pressures to study the effect of oil pressure levels on the dynamic response of the test systems under the single-rotor system and the dual-rotor system. Among them, the single rotor system under a large oil pressure system is mostly periodic and more stable. While under different inlet oil pressure conditions applied in the experiments, the dynamic motion of the dual-rotor bearing system is relatively stable in both the subcritical and supercritical speed regions. However, the oil whirl and oil whip of the dual-rotor system were observed at the critical speed, which causes the system into unstable multiperiodic motions. In general, this paper can give some insight into the dynamic response of the sliding bearing system influenced by different oil pressure levels.

Thermohydrodynamic investigation for supercritical carbon dioxide high speed tilting pad bearings considering turbulence and real gas effect

A global heat balance method for supercritical carbon dioxide (S-CO2) high speed tilting pad bearings was developed, in which the real gas effect, variable thermodynamic properties, and turbulence effect were considered. The bearing characteristics can be obtained by the partial derivative method embracing dynamic variations of complete variables. Then thermohydrodynamic lubrication mechanisms for S-CO2 bearings are studied, and results indicate that the real gas effect of thermal compressibility is significant. The influence of the thermal effect on the static and dynamic characteristics of bearings is caused by changes in density, specific heat, and thermal expansion coefficient rather than viscosity, which is totally different from oil bearings. In general, the influence of thermohydrodynamic lubrication on static characteristics of S-CO2 tilting pad bearings is more obvious than that on dynamic coefficients. For example, the maximum deviation of damping coefficients of a bearing with parameters in this research at a certain speed range is 8%, and that of the load capacity at the corresponding speed is 10.7%. However, the influence of the thermal effect on dynamic coefficients is varied at different rotating speeds.

Speed Oscillations of a Vehicle Rolling on a Wavy Road

Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential-algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

Strong oblique shock waves in granular free-surface flows

Strong oblique shock waves of granular flow are a steady-state solution formed when a granular free-surface flow deflects around a wedge-shaped obstacle at a supercritical speed, but they do not usually occur because their formation requires specific conditions to be satisfied downstream of the shock wave. This paper discusses the method of generating the strong oblique shock wave in a laboratory experiment and numerical simulation. The experiment is conducted on a plexiglass chute inclined at an angle to the horizontal, in which a dry granular material is released from a hopper at the top of the chute to form a channelized flow that passes a wedge at a downslope location. In order to generate a strong oblique shock wave, a second gate is established at the downstream of the wedge to control the material to flow out only at the designed time and height. Such a granular flowing process is simulated with a depth-averaged granular flow model, where the above two-gate system is mirrored into the inlet and outlet boundaries, respectively. The formation of the strong oblique shock is investigated through the transient solution of the flow field, and a good agreement is observed between the experiment and the simulation. Then, the steady-state solution of the interaction between the weak and strong oblique shocks is analyzed in the experiment and simulation. This result can be regarded as the third solution of granular shock because it can be formed by just changing the opening time of the second gate. With the dramatic change in flow thickness and velocity across the strong oblique shock, the bulk inertial number, used to quantify the rheological relation of granular materials, becomes extremely small, but it does not seem to affect the behavior of the flow discussed in this paper.

A CFD Study of the Resistance Behavior of a Planing hull in Restricted Waterways

The demand for high-speed boats that operating near to shoreline is increasing nowadays. Understanding the behavior and attitude of high speed boats when moving in different waterways is very important for boat designer. This research uses a CFD (Computational Fluid Dynamics) analysisto investigate the shallow water effects on prismatic planing hull. The turbulence fl ow around the hull was described by Reynolds Navier Stokes equations RANSE using the k-ɛ turbulence model. The free surface was modelled by the volume of fl uid (VOF) method. The analysis is steady for all the ranges of speeds except those close to the critical speed range Fh=0.84 to 1.27 due to the propagation of the planing hull solitary waves at this range. In this study, the planing hull lift force, total resistance, and wave pattern for the range of subcritical speeds, critical speeds, and supercritical speeds have been calculated using CFD. The numerical results have been compared with experimental results. The dynamic pressure distribution on the planing hull and its wave pattern at critical speed in shallow water were compared with those in deep water. The numerical results give a good agreement with the experimental results whereas total average error equals 7% for numerical lift force, and 8% for numerical total resistance. The worst effect on the planing hull in shallow channels occurs at the critical speed range, where solitary wave formulates.

Numerical Investigation of the Resistance of a Zero-Emission Full-Scale Fast Catamaran in Shallow Water

This paper numerically investigates the resistance at full-scale of a zero-emission, high-speed catamaran in both deep and shallow water, with the Froude number ranging from 0.2 to 0.8. The numerical methods are validated by two means: (a) Comparison with available model tests; (b) a blind validation using two different flow solvers. The resistance, sinkage, and trim of the catamaran, as well as the wave pattern, longitudinal wave cuts and crossflow fields, are examined. The total resistance curve in deep water shows a continuous increase with the Froude number, while in shallow water, a hump is witnessed near the critical speed. This difference is mainly caused by the pressure component of total resistance, which is significantly affected by the interaction between the wave systems created by the demihulls. The pressure resistance in deep water is maximised at a Froude number around 0.58, whereas the peak in shallow water is achieved near the critical speed (Froude number ≈ 0.3). Insight into the underlying physics is obtained by analysing the wave creation between the demihulls. Profoundly different wave patterns within the inner region are observed in deep and shallow water. Specifically, in deep water, both crests and troughs are generated and moved astern as the increase of the Froude number. The maximum pressure resistance is accomplished when the secondary trough is created at the stern, leading to the largest trim angle. In contrast, the catamaran generates a critical wave normal to the advance direction in shallow water, which significantly elevates the bow and creates the highest trim angle, as well as pressure resistance. Moreover, significant wave elevations are observed between the demihulls at supercritical speeds in shallow water, which may affect the decision for the location of the wet deck.