Shaft-bearing System Research Articles

Stability analysis of nonlinear synchronous vibration in an inclined rotor system supported by journal bearing with variational gravity

In vertical rotating shaft-bearing systems, synchronous vibration undergoes destabilization and stabilization owing to the self-excited vibration. Recently, these phenomena have been clarified numerically, experimentally, and analytically based on a newly proposed analytical method. In this method, a two-step linearization of the nonlinear journal bearing (JB) force and generalized eigenvalue analysis are proposed to directly determine the stability of the synchronous orbit. However, this method cannot be used directly for inclined-rotor systems with a large gravitational effect. Moreover, stability changes caused by gravity variations in inclined-rotor systems have not been fully investigated. This study extended the analytical method to one with weighted average dynamic coefficients to efficiently predict the stability changes of synchronous whirling vibrations in inclined rotor systems. The extended analytical method clearly clarifies gravity-induced stability changes in comparison with the numerical (shooting) method. Our analytical method enhances the ability of rotor dynamics software to calculate the stability thresholds of inclined rotor systems with gravity variations.

Functionality of Bearings in the Shafts of a Vertical-Axis Wind Turbine

The article contains a description of the design solutions proposed by the authors for a hybrid wind turbine bearing, in which the sliding part takes over the load to the turbine shaft after reaching the shaft rotation speed, ensuring hydrodynamic lubrication of the plain bearing and relieving the rolling bearing. This allows for low starting resistance of the power plant and ensures quiet operation during use. Two conceptual solutions of a hybrid bearing were presented, differing in the shape of the plain bearing journal. A mechanism for automatic switching of the load between a rolling and a plain bearing was developed. A solid simulation model of this mechanism was built in the Autodesk Inventor—Dynamic Simulation software Inventor Professional 2023 environment, and its operation was simulated. The results confirmed the usefulness of using this design in shaft-bearing systems of wind turbines with a vertical axis of rotation. Based on the simulation, the speed at which the thrust roller bearing will be released was determined. Technical parameters of a plain bearing with a spherical journal shape were calculated. The height of the lubrication gap and the shaft rotational speed at which the bearing load capacity index reaches a critical value were determined.

Active Control for Transverse Vibration of the Marine Stern Shaft-Bearing System with LQR Strategy

Active Control for Transverse Vibration of the Marine Stern Shaft-Bearing System with LQR Strategy

Vibration reduction and energy harvesting of the marine stern shaft-bearing systems with triboelectric generator

The stringent requirements for vibration reduction in marine propeller shaft are essential due to the increased effects of transverse vibration. In order to effectively minimize the energy consumption and realize the energy harvesting during the process of vibration control, a triboelectric generator is proposed for the marine stern bearing-shaft systems by a combination of vibration reduction and energy harvesting. The vibration reduction is achieved using an adjustable damping magnetorheological shock absorber. The energy harvesting is realized through contact electrification and electrostatic induction during the vertical contact-separation of electrode plate. The Poincaré surface of the shaft and bearing demonstrates that the proposed triboelectric generator can be applicable for the vibration reduction and energy harvesting. Meanwhile, the controlled responses of suspension travel, soleplate deflection, acceleration of shaft and bearing, and forces are subjected to analysis. The energy harvesting process encompasses the acquisition of resistor current, PDMS charge, open-circuit voltage and electrical energy of the system. Moreover, the influence of various model parameters on both vibration reduction rate and electrical energy harvesting are discussed. It is anticipated that a more optimal satisfaction in terms of vibration reduction and energy harvesting can be attained through the adjustment of model parameter.

Effects of the Shaft Speed on Stiffness and Damping Coefficients of Hydrodynamic Bearing-Shaft System under Variable Viscosity

The rotating shaft-hydrodynamic bearings systems operated with high speed and/or heavy load conditions expose serious rotor-dynamics instability problems due to characteristics of the supporting bearings. The stability and the dynamics of these systems, directly relate to the lubricant properties that are directly affect by the heat generation. In this study, the dynamic characteristics of a shaft-hydrodynamic journal bearing and its stability were investigated under variable viscosity. The equations of lubricant flow were derived by Dowson’s equation under variable viscosity, and the perturbation equations were obtained for 2 degrees-of-freedom system. The heat transfers between oil and the journal surface was modelled in a 3-dimensional energy equation, and the heat transfer on the journal structure was also modelled with heat conduction equation. An algorithm based on finite difference scheme with successive over relaxation method was developed to solve the theoretical models, simultaneously, and a serial simulation was performed to investigate the variations of the dynamic coefficients of the bearing-shaft system concerning the rotating speed for different radial clearance values. It was determined that the high speed increases the lubricant temperature, and so the static and dynamic performance characteristics decrease, moreover, this effect is more dominant for the smaller radial clearance.

Theoretical and In Situ High Speed Measurement Study on Friction-Induced Vibration in Water Lubricated Rubber Bearing-Shaft System

A water-lubricated rubber stern bearing causes frictional vibration and noise under extreme conditions, severely threatening the survivability and concealment performance of underwater vehicles. To investigate the mechanism of friction-induced vibration and noise, a four-degree-of-freedom nonlinear dynamic model that included bearing support vibration, frictional bearing vibration and torsional shaft vibration was proposed. The velocity-dependent Stribeck friction model described the dynamic friction characteristics between the rubber bearing and stern shaft. Nonlinear system stability was examined using the Lyapunov indirect method and the transient dynamic responses determined by the Runge-Kutta method. Meanwhile, a high-speed in situ measurement experimental study was adopted to explore the mechanism of frictional and torsional vibrations in a water-lubricated bearing-shaft system. Results showed that nonlinear friction excitation was the fatal cause of friction-induced vibration in bearing-shaft system. Under different friction excitations, the coupling vibrations formed between the bearing and bearing support were different, resulting in two main modes of friction-induced vibration: “Chatter” and “Squeal.” The frictional vibration of the bearing and torsional vibration of the stern shaft were tightly coupled. These trends and results might provide a theoretical basis for further research on frictional vibration and wear control in water-lubricated rubber bearing-shaft systems.

A Dynamic Model of Outer Race Defective Bearing Considering the Unbalanced Shaft-Bearing System With Experimental Simulation

Abstract In this study, the effects of the evolution of bearing outer race defect size and increase in speed on the vibration characteristics of a shaft-bearing system under unbalanced conditions are simulated and analyzed. A two degrees-of-freedom mathematical model is presented for a ball bearing used in an unbalanced shaft-bearing system. The contact stiffness between the races and the balls is considered as a series of springs is incorporated in the model. Hertzian contact deformation theory is used to obtain the contact stiffness. This model considers the contact deformation between the balls and the races, the additional displacement between the balls and the inner race due to radial clearance and due to defect geometry. The maximum possible radial displacement of the ball into the defect is calculated analytically using the groove radius, ball radius, and defect diameter. The rectangular function is used for modeling the defect. matlab codes are developed for modeling the bearing and for solving the differential equations of motion using the Runge–Kutta method. The vibration responses (peak and root-mean-square (RMS) values) obtained by modeling and by experimentation show similar vibration characteristics. The investigation shows that the values of statistical parameters initially increase with the increase in defect size and then decrease with a further increase in defect size. While peak and RMS increase with the increase in speed, crest factor and kurtosis decrease with the increase in speed. Peak is more sensitive for diagnosing spalls on outer race and its evolution. This study helps as an effective diagnosis of antifriction bearings having spalls on the outer race under unbalanced conditions.

Influence of Shaft Scratch on Static and Transient Behavior of Water-Lubricated Bearing

Abstract When sand enters the gap between the shaft and water-lubricated bearing, it will nick surfaces of shaft and bearing bush, and the scratch will appear. The variations of static and transient performance with the number and depth of scratches are studied in the paper. The results show that scratches have a significant effect on the critical load and critical speed of the transformation of bearing lubrication state. The existence of scratches reduces the critical load from elastohydrodynamic lubrication to mixed lubrication. The shaft with scratches vibrates more strongly than the shaft with no scratch at the moment of start-up. The contact area, contact time, and bush-burning probability are directly proportional to the number of scratches. Shaft center movement orbits under step load have the similar “L” shape, whether the shaft has scratch or not. But the scratched shaft has longer movement orbit and lower equilibrium point than the shaft with no scratch. This paper can provide a reference for structure design and service life evaluation of bearing-shaft system.

Analytical expressions of friction-induced self-excited vibration amplitudes of the marine rubber bearing-shaft system

By the analytical method, this paper investigates the mode-coupling friction-induced instability and the resulting stick-slip self-excited vibration of the marine stern water-lubricated rubber bearing-shaft system. The 2-DOF simplified theoretical model is reasonably developed to characterize stern water-lubricated rubber bearing-shaft system. The complex eigenvalue analysis is conducted to analyze the instability behaviors of the model. Results show the normal force and nonlinear stiffness often deteriorate the stability of the model. Subsequently, two types of analytical expressions for stick-slip vibration amplitude of the model are by the Krylov-Bogoliubbov-Mitropolsky method, which match well with the results from numerical integration. Then, parameter discussion about the influences of friction coefficient, damping, normal force and nonlinear contact stiffness on self-excited vibration amplitudes are performed. The results indicate that the bearing-shaft system can keep the steady-state static equilibrium or the oscillations with small amplitude if the proper parameter values of the variables are chosen, which will benefits controlling the friction-induced vibration of the water-lubricated rubber bearing-shaft system. The novelty of this work is to give analytical expressions for stick-slip self-excited vibration amplitude of the simplified model.

Lubrication analysis of hydrodynamic-static hybrid bearing with deep-shallow chambers considering thermal conduction

The oil film separates the shaft-bearing system to reduce the frictional force of contact surfaces with large temperature variations owing to thermal conduction. Multiple impact factors seriously affected the lubrication behavior of the bearing during actual working process. An applicable model regarding the lubrication analysis of a hydrodynamic-static hybrid bearing with deep-shallow chambers is proposed and validated with published literature. The lubrication behavior includes film pressure, load capacity, bearing stiffness and flow rate is numerically calculated with centered finite difference method by solving the Reynolds equation. The influence of film thickness and working temperature are considered, over a range of eccentricity ratio. Moreover, the temperature variation caused by thermal conduction is analyzed in details. To investigate the influence of thermal conduction on the lubrication behavior, the results show that the load capacity, bearing stiffness, friction power, and flow power significantly decline with the consideration of thermal conduction. This study proposes an accurate model that is helpful for the design of hydrodynamic-static hybrid bearings with deep-shallow chambers under low rotational speed, heavy load, and high temperature conditions.

A study on the influences of journal axial vibration on ship shaft stern bearing dynamic characteristics

PurposeThis paper aim to take the ship shaft stern bearing as the research object, and studies the influence of journal axial vibration on bearing dynamic characteristics under different misaligned angles and rotation speeds.Design/methodology/approachComputational fluid dynamics (CFD) and harmonic excitation method were used to build bearing unstable lubrication model, and the dynamic mesh technology was used in calculation.FindingsThe results indicate that journal axial vibration has a significant effect on bearing dynamic characteristics, like maximum oil film pressure, bearing stiffness and damping coefficients, and the effect is positively correlated with journal misaligned angle. The effect of shaft rotation speed and journal axial vibration on bearing dynamics characteristics are independent; they have no coupling. Bearing axial stiffness is mainly affected by the journal axial displacement, bearing axial damping is mainly affected by journal axial velocity and they are positively correlated with the misaligned angle. The influence of rotational speed on bearing axial stiffness and axial damping is not obvious.Originality/valueThis paper establishes the bearing dynamic model by CFD and harmonic excitation method with consideration of cavitation effect and analyzing the influence of journal axial vibration on the dynamic characteristics. The results are benefit to the design of ship propulsion shaft and the selection of stern bearing. Also, they are of great significance to improve the operation stability of the shaft bearing system and the vitality of the ship.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-11-2022-0337/

Harmonic analysis of a marine shaft-bearing suspension system and its semi-active control with PID strategy

A stable operating state of the marine shaft-bearing system is essential to minimize the vibration level and improve the transmission efficiency of the ship power system. A semi-active control with PID strategy is proposed through an adjustable damping magnetorheological shock absorber and the combination of the electromagnetic actuator based on the existing suspension system. The harmonic response of the system is calculated through transfer function with various suspension stiffness and damping. The controlled behaviors include suspension travel, soleplate deflection, sprung shaft and unsprung bearing, are obtained through Matlab/Simulink. The proposed controller is demonstrated to be applicable through robustness analysis by comparing with passive and LQR control based on the disturbance of pulse width modulation. The controlled behaviors with minor stiffness and significant damping are noticed to be more satisfied through the influence comparison of suspension parameters. The controlled performance includes reduction rate, overshoot rate and setting time with various values of control factors are discussed to prove the suitability. The semi-active control for the transverse vibration of the marine shaft-bearing system is thus realized based on the adjustment of control parameters.

Model-based low-speed rotation error prediction for the rigid shaft-bearing system considering the assembly deviation

The low-speed shaft-bearing system has been used in many navigation equipment, and its rotation error can directly affect the navigation accuracy. Currently, due to the emergence of super precision bearings, bearing accuracy has exceeded the overall accuracy level of the shaft-bearing system, and the other parts’ (such as shaft, housing) errors become the main factors to hinder the improvement of low-speed rotation accuracy. To solve this problem, this paper proposed a low-speed rotation error prediction method to determine the key tolerances, which can be used to improve the rotation accuracy of shaft-bearing system. Firstly, the formation mechanism of the low-speed rotation error is introduced. Secondly, the spatial pose kinematic model of the shaft-bearing system considering the assembly deviation is established in low-speed rotational movement. Thirdly, the coupled model of assembly deviation and static mechanics in shaft-bearing system is established. Fourthly, based on the above two models, the low-speed rotation error prediction method is proposed, and then verified with the measurement on a rigid shaft-bearing system. Finally, based on it, the effects of bearing installation errors (caused by the part’s tolerances) on the rotation error can be quantitatively analyzed to determine the key tolerances, which can be largely useful for improving the rotation accuracy of the shaft-bearing system.

Tribodynamic Modelling of High-Speed Rolling Element Bearings in Flexible Multi-Body Environments

This study presents a new flexible dynamic model for drive systems comprising lubricated bearings operating under conditions representative of electrified vehicle powertrains. The multi-physics approach importantly accounts for the tribological phenomena at the roller–race conjunction and models their effect on shaft-bearing system dynamics. This is achieved by embedding a non-linear lubricated bearing model within a flexible system level model; this is something which has not, to the authors’ knowledge, been reported on hitherto. The elastohydrodynamic (EHL) film is shown to increase contact deflection, leading to increased contact forces and total bearing stiffness as rotational speeds increase. Results show that for a 68 Nm hub motor operating up to 21,000 rpm, the input bearing EHL film reaches a thickness of 4.15 μm. The lubricant entrainment increases the roller–race contact deflection, causing the contact stiffness to increase non-linearly with speed. The contribution of the lubricant film leads to a 16.6% greater bearing stiffness at 21,000 rpm when compared to conventional dry-bearing modelling methods used in current multi-body dynamic software. This new methodology leads to more accurate dynamic response of high-speed systems necessary for the next generation of electrified vehicles.

Active control for stick-slip behavior of the marine propeller shaft subjected to friction-induced vibration

A stable rotational speed of the marine propeller shaft is essential to minimize the friction of the supporting bearings and to improve the efficiency of the power system. While, the friction induced vibration of the marine shaft-bearing system seriously affect the performance and reliability of the powertrain. This paper proposes an active control method to suppress the stick-slip behavior of the marine propeller shaft based on state observer. The dynamical response including relative velocity, power spectral density, dynamic friction coefficient and reaction force with various friction model are calculated. The active control with state feedback and internal model principle are applied to track the rotational speed of each segment of the whole shaft. The behavior of the state observer is analyzed through a comparison of actual value and observed value of the tracking for continuously varying speeds. The robustness of controlled system is investigated through comparison of overshoot rate and adjustment time with various control parameters. Moreover, a comparison of the observed velocity between present work and previous reference is conducted to validate the applicability of the proposed frictional model. The active control for friction-induced vibration of marine propeller shaft is thus realized based on the adjustment of control parameters.

Vibration transmission control of a flexible shaft-bearing system using active/passive support

An active/passive support is proposed to suppress the transmission of vibration from a flexible shaft-bearing system to its foundation. The active/passive support consists of passive isolators and electromagnetic suspension, in which the former bears the static load while the latter provides suspension force for displacement adjustment and vibration suppression. The dynamic model of the shaft-bearing-foundation system is built on the Rayleigh-Ritz method. Based on this model, the adjustment of stiffness, the transmissibility of vibration, the vibration characteristics of the coupled system and the effectiveness of active control are analyzed. An adaptive control method with subband filtering and disturbance reconstruction is adopted and the performance is evaluated. The effectiveness of the active/passive support is also verified by experiment. Numerical and experimental results demonstrate that the transmission of vibration via the active/passive support can be greatly suppressed and the vibration of the foundation can be reduced almost to the same level.

Failure Mode Detection and Validation of a Shaft-Bearing System with Common Sensors.

Failure mode detection is essential for bearing life prediction to protect the shafts on the machinery. This work demonstrates the rolling bearing vibration measurement, signals converting and analysis, feature extraction, and machine learning with neural networks to achieve failure mode detection for a shaft bearing. Two self-designed bearing test platforms with two types of sensors conduct the bearing vibration collection in normal and abnormal states. The time-domain signals convert to the frequency domain for analysis to observe the dominant frequency between these two types of sensors. In feature extraction, principal components analysis (PCA) combines with wavelet packet decomposition (WPD) to form the two feature extraction methods: PCA-WPD and WPD-PCA for optimization. The features extracted by these two methods serve as input to the long short-term memory (LSTM) networks for classification and training to distinguish bearing states in normal, misaligned, unbalanced, and impact loads. The evaluation arguments include sensor types, vibration directions, failure modes, feature extraction methods, and neural networks. In conclusion, the developed methods with the typical lower-cost sensor can achieve 97% accuracy in bearing failure mode detection.

Advanced method of including housing stiffness into calculation of gear systems

One of the central goals during the design of helical gear systems is the achievement of a well-distributed contact load in the gear mesh. An equal load distribution is a key factor for a high load carrying capacity, the economic use of materials and a long lifetime. Mesh misalignment can be caused by tooth deflections, manufacturing deviations or elastic deformation of the shaft-bearing system and the gearbox housing. Those deformations have to be taken into account during the design process of adequate tooth-flank geometry. Elastic deformations of gearbox housings can be significant, especially in the case of automotive applications with aluminium cases. This paper presents an advanced method of including housing stiffness into the calculation of gear systems. A validation of the approach is carried out by comparing the calculated deformations with measurements of a static test rig of a hypoid gearbox.Many calculation programs offer the opportunity to analyse the deformation behaviour of the shaft-bearing-housing system. Most of the components in these programs are described by analytic approaches. However, components that are geometrically more complex, like the housing or planet carriers cannot be represented as easily as that by analytic expressions. There are several alternatives to take into account the elasticity of those objects. One way is to model the stiffness of the bores using imported stiffness matrices. These matrices contain the elasticity of the bores itself as well as crossover influences between the bearings. The reduced stiffness matrices may be the result of a static reduction of the geometry using the finite element method (FEM). As state of the art, the reduction is mostly carried out at the centre points of the bearing bores. The proposed advanced method uses the static reduction of geometries on several points at the bores, distributed over the circumference. This approach offers a more detailed modelling of the elastic behaviour of complex geometries within the analytic deformation calculation of gear systems. To validate the advanced approach, the calculation results of the elastic deflections of the shaft-bearing-housing system is compared with measurements of a static test rig. In the course of these comparisons, the influence of different modelling methods of gearbox housings on the accuracy of the calculation results is discussed.

Experimental investigation of vibration damping capabilities of 3D printed metal/polymer composite sleeve bearings

Additive Manufacturing (AM) method enables to produce products easily, cheaply and quickly with more complex geometry compared to traditional production methods. In this context; Fused Deposition Modeling (FDM) is a widely used for AM method that requires a large number of process parameters. In this study, it is aimed to experimentally investigate the damping capabilities of the metal/polymer composite sleeve bearings printed using FDM. A total of 54 pieces (27 pairs) were printed from composite filaments such as nine pairs for each of PLA, PLA + 20% Bronze, and PLA + 20% Copper metal/polymer with different filling structures (Octogram spiral, Archimedian chords, and 3D Honey comb) and different occupancy rates (10, 30, and 50%), respectively. The experiments were carried out using the shaft-bearing system under the same operating conditions at a rotational speed of 900 revolution per minute (rpm), and vibration data was collected from the rotating shaft with proxy probes. Rotating shaft position is very important to determine journal position in bearing for understanding deterioration in the bearing. Bode and Orbit plots are used to detect the level of deterioration in the bearing. The value of bearing to shaft clearence can vary widely depending on application. Since the bearings were heavily loaded, they were compressed by radial load caused large clearence. The results showed that compressed bearing had significant influence on the stability of rotating shaft system and significant differences in damping capabilities of composite sleeve bearings with different filling structures and occupancy rate. Increasing occupancy rate decreases the vibration amplitude values in copper-reinforced sleeve bearing but increases in bronze-reinforced sleeve bearing. From the microstructure analysis, it has been observed that the vibration absorbation capability is better due to the more homogeneous distribution of the copper reinforced bearings than the bronze reinforced bearings. Also, vibration absorption capability of sleeve bearings with 3D Honeycomb filling structureis increased significantly proportion to the occupancy rate.

Impact factors on friction induced vibration of shaft-bearing system considering stick-slip behavior

An appropriate assessment of dynamical response of the marine propeller shaft is essential to enable optional power delivery to propeller and to minimize the dispensable mechanical friction of supporting bearings. The interaction behavior of the shaft-bearing system affects the stability of propulsion system seriously. An applicable numerical model regarding the friction induced vibration with velocity-dependent and stick-slip friction is proposed. The effect of smoothing factor, control factor, stick factor and slip factor which obtained with experimental data on the friction coefficient is analyzed separately. A cooperative Newton-Raphson & Newmark-β solving method is validated by experimental tests to be suitable to solve the proposed nonlinear model. To capture the nature of shaft-bearing interaction, the response of dynamic friction coefficient, relative velocity and reaction force with various model parameters are obtained. The residual of iteration which defined in the proposed solving method is also analyzed for each case to certificate the reliability of the numerical calculation. The stability analysis regarding various impact factor of the proposed shaft-bearing system is discussed with multiple scales method. An optimized design for propeller shaft and supporting bearing caused by friction induced by friction properties is thus realized based on proposed numerical model.