Eccentricity Error Research Articles

Analysis and Simulation of Spur Gear System Considering Geometric Eccentricity

Geometric eccentricity errors inevitably occur in gear systems, resulting in the center of mass and center of rotation not coinciding. During the gear meshing process, the gear center distance and gear tooth meshing condition will change with time. In this paper, a model considering geometric eccentricity is proposed and a finite element model of the gear pair is developed. Finally, the dynamic response under different eccentricity and initial eccentricity angle is discussed. Based on Ansys, the simulation calculation of the gear pair is carried out, and the equivalent force maps of the dangerous areas of the gear tooth flanks are obtained.

Measurement and optimization method for aero-engine rotors based on binocular multi-line laser sensing and virtual assembly

During the assembly of aero-engine rotors, eccentricity and tilt errors along the radial and axial direction result in uneven loading along the mating surface, which significantly impacts the operational lifespan of the rotor. This paper proposes a three-dimensional contour measurement system based on binocular multi-line laser sensing and conducts an aircraft engine rotor stacking virtual assembly technology system, to response the practical requirements of aircraft engine rotor unit production assembly engineering. We combine multi-line structured light active projection device with the binocular stereo vision model to construct the contour measurement system, which enhances the surface features of the projected object by adhering to the common constraint of structured line and stereo vision. Furthermore, the error propagation model for multi-stage rotor assembly to investigate the mechanism of error impact is established aiming to facilitate quantitative research on virtual assembly mechanisms. By manipulating the rotor phases and evaluating the coaxiality under various conditions, effective prediction of phase variation can be achieved, the multi-objective benchmark virtual assembly method based on the error propagation model of multi-stage rotor coaxiality is presented. The experimental results indicate that the proposed measurement system achieves the maximum disparity between the spatial data points procured by the proposed measurement system and the predetermined value of 2.1 μm, and the maximum disparity between the target plane’s average movement distance and the preset value of 3.7 μm. The precision provided by this system effectively meets the objective of assembly phase optimization for multi-stage rotors. In the empirical experiment conducted on the three-stage rotors, both systems yielded identical optimized phases: 0° for the first-stage rotor, 270° for the second-stage rotor, and 120° for the third-stage rotor, thereby validating the efficacy of the proposed measurement and optimization method for aero-engine multi-stage rotors with virtual assembly.

Analytical investigation on load sharing characteristics and speed difference of coaxial reverse closed differential herringbone gear transmission system with floating gear and errors

To investigate the influence of gear floating on the load sharing characteristics of the Coaxial Reverse Closed Differential Herringbone Gear Transmission System (CRCDHGTS) and the rotational speed difference between the upper and lower rotors, a dynamic Bending-Torsional-Axial-Pendular (BTAP) model of the CRCDHGTS was established using the centralized parameter method, which considers various excitation factors such as gear floating, errors, Time Varying Meshing Stiffness (TVMS), gyroscopic effect, and tooth friction. It considers the interaction between the closed-stage gear set and the differential-stage gear set, treating the herringbone gear as a symmetric helical gear connected through a receding slot. The dynamic model was solved using the Runge-Kutta method to obtain the dynamic meshing forces for each gear pair under single and combined floating modes. The Dynamic Load Sharing Coefficient (DLSC) of the system, which characterizes the Load Sharing Performance (LSP), was deduced. The load sharing characteristics of different floating modes were analyzed, as well as the influence of different floating displacement on the DLSC. The motion path of the gear floating was also determined. Additionally, the impact of manufacturing error and assembly error of each component on the DLSC under combined floating mode was analyzed. Finally, the influence of gear floating on the output rotation speeds of the upper and lower rotors of the system was investigated. The results indicate that both free-floating of the center gear and combined floating can effectively improve the LSP of the system. When the system adopts combined floating mode, the DLSC of inner and outer meshing changes between 0.91 and 1.09, demonstrating a significant improvement in the LSP. The DLSC of the system increases with the increase in error, with the eccentricity error having a greater impact on the DLSC compared to the assembly error. The optimal floating value for the sun gear is between 0.6 mm and 0.8 mm, while for the planetary gear, it is between 0.4 mm and 0.6 mm. The rotational speed difference between the upper and lower rotors can be controlled within 1r/min. These research findings provide a theoretical basis for further analysis of the dynamic stability and reliability of the system.

Mechanism of eccentricity influence on 3-DOF aerodynamic stability: New insights into instability evolution, energy harvesting, and vibration control

The inconsistency of mass center and elastic center, i.e., eccentricity, widely exists in various engineering structures and components, including iced conductors, wind turbine blades, airfoils, and cable-supported bridges, and can also be extended to the application of energy harvesting and vibration control. Although the eccentricity influence on aerodynamic stability has already been recognized as important, the underlying mechanism remains rather unclear. This study established a comprehensive framework of explicit eigenvalue solutions, offering in-depth insights into the aerodynamic stability mechanism in a vertical-horizontal-torsional 3-degree-of-freedom linear system under all possible frequency scenarios. These solutions elucidated the coupling mechanism of eccentricity, aerodynamic damping, aerodynamic stiffness, and structural frequencies. Remarkably straightforward criteria to predict the promoting/suppressing effect of eccentricity were proposed, consisting of eccentric angle and aerodynamic forces, which not only align with the traditional principles but also offer a broader application range in terms of structure types and wind speeds. Validations of the proposed solutions were performed for all frequency cases via numerical studies. For instability tendency evaluation, numerical studies on a thin plate showed that eccentricity has a significant impact due to the strong aerodynamic effect and large ratio of width/gyration radius, highlighting the importance of considering small eccentricity errors during experiments. Examination for energy harvesting revealed that a small eccentricity can dramatically enhance power generation efficiency, and the explicit solutions successfully explained the rules governing performance changes against frequency ratio, mass center position, rotation center position, etc. The analysis of a bundled conductor showed that the double pendulum can control galloping because it diminishes the eccentricity effect by shifting the eccentric angle. The proposed explicit solution framework provides a systematic view and new insights into the aerodynamic instability mechanism influenced by eccentricity, serving as a reference for the design and studies of various engineering structures, energy harvesting, and vibration control.

Dynamic analysis of planetary gear transmission based on Lagrange interpolation polynomials

Abstract This paper proposes a novel dynamic model considering manufacturing errors and eccentricity errors to analyze the dynamics of planetary gear transmission (PGT). The dynamic model is established based on the fractional-order calculus (FOC) and solved by an enhanced fourth-order Lagrange interpolation polynomials (LIP) method. Three numerical examples and the vibration experiments of planetary gear transmission are employed for verification. The comparison results indicate that the proposed solution method has higher solution accuracy and efficient than the existing algorithms in solving fractional equations, and the relative errors of the proposed solution method in three examples are 0.32%, 0.78% and 0.16%, respectively. The proposed dynamic model of PGT has better agreement with the experimentally measured signal compared with the integer-order dynamic model, and the maximum error and average error of the characteristic frequency amplitude between the proposed dynamic model and the measured signal are 4.76% and 3.57%, respectively. The proposed method contributes to the theoretical foundation for the signal monitoring of PGT.

Eliminating eccentricity error in measuring residual stresses via hole-drilling method using strain gauge rosette with five measuring grids: For thin plates using through-holes

In the measurement of residual stresses (RSs) via the hole-drilling method, the error induced by the eccentricity of the drilled hole is inevitable and non-negligible in some cases. In this study, both the eccentricity coordinates and RS state (two principal stresses and one principal angle) are considered unknowns that must be solved. A set of five equations is required to determine these five unknowns. Therefore, two additional measuring grids are added to the strain gauge comprising three grids, which is typically used in measuring RSs. Consequently, a novel strain gauge rosette with five measuring grids (SGR-5MG) is created. Subsequently, an algorithm associated with the SGR-5MG is developed to solve the five unknowns based on the Newton–Raphson method. The algorithm is elasticity-based and derived according to the layout of the SGR-5MG. Finally, the proposed method is verified numerically and experimentally. The results show that (1) the eccentricity error is up to 35.1% when the eccentricity reaches 0.05 D, where D represents the diameter of the gauge circle; (2) Using the proposed method, the error can be significantly reduced to 7.2%; (3) The elasticity-based proposed method cannot make the predicted results exactly converge to the actual stresses in predicting high stresses because of the plasticity deformation around the hole. The eccentricity error can be reduced significantly by using the proposed method based on the SGR-5MG.

A Prediction Model of Two-Sided Unbalance in the Multi-Stage Assembled Rotor of an Aero Engine

In rotating machinery with a multi-stage assembled rotor, such as is found in aero engines, any unbalance present will undergo unknown changes at each stage when rotating the assembly phases of the rotor. Repeated disassembly and adjustments are often required to meet the rotor’s residual unbalance specifications. Therefore, developing a prediction model of this two-sided unbalance for a multi-stage assembled rotor is crucial for improving the first-time assembly pass rate and assembly efficiency. In this paper, we propose a prediction model of the two-sided unbalance seen in the multi-stage assembled rotor of an aero engine. Firstly, a method was proposed to unify the mass feature parameters of each stage’s rotor into a geometric measurement coordinate system, achieving the synchronous transmission of geometric and mass feature parameters during the assembly process of the multi-stage rotor. Building upon this, a linear parameter equation of the actual rotation axis of the multi-stage rotor was established. Based on this axis, the mass eccentricity errors of the rotor were calculated at each stage, further enabling the accurate prediction of two-sided unbalance and its action phase in a multi-stage rotor. The experimental results indicate that the maximum prediction errors of the two-sided unbalance and its action phase for a four-stage rotor are 9.6% and 2.5%, respectively, when using this model, which is a reduction of 53.0% and 38.1% compared to the existing model.

Research on Accurate Detection Algorithm for the Cross-Section of Shale Gas Casing Deformation Pipe String Based on Laser Ranging

Under shear and non-uniform loads, the deformation of the section shape of a casing results in an irregular section, and the spatial continuity is poor. The change in the distance between the wall of the casing before and after stress is recorded to analyze the deformation of the casing, and the distance value is taken as the key characteristic of the casing. A large number of the key characteristic values of shale gas casing deformation can be obtained by using the circular traversal detection method. At the same time, this article focuses on the center deviation between the laser sensor axis and the pipe string axis, as well as on the disturbance problem during measurement. An eccentricity error correction algorithm is derived to correct the eccentricity error that occurs during the detection process, and then we use interpolation algorithms to draw cubic spline curves to improve detection accuracy. The test results show that the algorithm can effectively eliminate eccentricity errors in measurement and achieve the accurate measurement of deformation casing characteristic values, which can provide basic data for the study of a shale gas casing deformation mechanism.

Identification and Compensation Method of Unbalanced Error in Driving Chain for Rate-Integrating Hemispherical Resonator Gyro.

The accuracy of the signal within a driving chain for the rate-integrating hemispherical resonator gyro (RI-HRG) plays a crucial role in the overall performance of the gyro. In this paper, a notable and effective method is proposed to realize the identification and compensation of the unbalanced error in the driving chain for the RI-HRG that improved the performance of the multi-loop control applied in the RI-HRG. Firstly, the assembly inclination and eccentricity error of the hemispherical resonator, the inconsistent metal conductive film layer resistance error of the resonator, the coupling error of the driving chain, and the parameter inconsistency error of the circuit components were considered, and the impact of these errors on the multi-loop control applied in the RI-HRG were analyzed. On this basis, the impact was further summarized as the unbalanced error in the driving chain, which included the unbalanced gain error, equivalent misalignment angle, and unbalanced equivalent misalignment angle error. Then, a model between the unbalanced error in the driving chain and a non-ideal precession angular rate was established, which was applicable to both single channel asynchronous control and dual channel synchronous control of the RI-HRG. Further, an unbalanced error identification and compensation method is proposed by utilizing the RI-HRG output with the virtual precession control. Finally, the effectiveness of the proposed method was verified through simulation and experiments in kind. After error compensation, the zero-bias instability of the RI-HRG was improved from 3.0950°/h to 0.0511°/h. The results of experiments in kind demonstrated that the proposed method can effectively suppress the non-ideal angular rate output caused by the unbalanced error in the driving chain for the RI-HRG, thereby further improving the overall performance of the RI-HRG.

Automatic compensation system for eccentricity error of absolute optical encoder.

Eccentric error is a vital part of high-precision optical encoder error. An automatic error compensation system is designed to lower the eccentric error of the encoder. On the periphery of the fan-shaped code path of the traditional encoder disk, a set of radial code paths is drawn. This radial code path is composed of several concentric circles with alternating light and dark lines. The direction of the radial code path is perpendicular to the direction of the fine code path. When the encoder rotates, the eccentricity of the encoder disk is measured by the moiré fringe signal output from the radial code channel. Based on the eccentricity error compensation algorithm, the eccentricity error of the encoder disk is compensated in real time to enhance the accuracy of the encoder. The experimental results of an encoder show that the mean square error of the encoder before the eccentricity error compensation is 21.25 arc seconds, and it is 3.66 arc seconds after compensation by this algorithm. The algorithm can significantly compensate the error caused by the eccentricity of the encoder and greatly improve the accuracy of the encoder.

Analytical method for time-varying meshing stiffness and dynamic responses of modified spur gears considering pitch deviation and geometric eccentricity

Pitch deviation and geometric eccentricity are almost inevitable in gear transmission systems. Tooth profile modification (TPM) is a common method used to improve gear contact properties. However, the coupling interaction between the tooth profile deviation introduced by TPM and manufacturing errors is complex. There is a lack of an analytical model that can clarify the dynamic behaviors of the modified gear pair affected by multiple manufacturing errors, specifically pitch deviation and geometric eccentricity. To fill this gap, a comprehensive analytical gear mesh model is proposed considering TPM, pitch deviation, and geometric eccentricity. The calculation formulas for the time-varying meshing stiffness (TVMS) and load sharing factor (LSF) are derived considering the iterative change of gear pair meshing parameters with phase under the influence of multiple manufacturing errors. Then, the translation-torsion coupling dynamic model of the spur gear pair based on the lumped mass method is established to investigate the dynamic transmission error (DTE) and vibration characteristics. Results show that TPM is an effective method for improving the comprehensive stiffness of a gear pair within the meshing range containing pitch deviation, but it reduces the meshing phase that includes only geometric eccentricity. Compared with the unmodified gear pair, the suppression effect of changing the modification length on the maximum DTE and vibration acceleration is more significant than that of the modification amount. TPM increases the overall fluctuation threshold range of DTE for gear pairs with only geometric eccentricity. It is expected that the proposed method can provide a theoretical basis for exploring vibration suppression in gear pairs with geometric eccentricity and pitch deviation errors.

Effects of Eccentric Errors on Sliding Velocity and Accumulative Wear Depth of Planetary Roller Screw Mechanism

Abstract The eccentric errors of planetary roller screw mechanism (PRSM) affect the contact kinematic characteristics and exacerbate the cumulative wear of thread pairs, which will reduce its transmission accuracy and reliability. Hence, it is significant to investigate the cumulative wear of PRSM with the eccentric errors. In this work, based on the conjugate surface contact condition and Archard wear theory, the sliding speed and cumulative wear depth of PRSM with eccentricity errors are modeled, respectively. The effects of eccentric errors on the sliding velocities and cumulative wear depth of PRSM are investigated. It has been determined that the eccentric errors of the screw and nut do not cause the sliding motion in the nut–roller contact region (NRCR). However, there are noticeable variations in sliding velocity and cumulative wear depth in the screw–roller contact region (SRCR). The eccentric errors of the screw and nut have a combined effect on the sliding velocity and wear between the screw and roller. This investigation and finding can provide a valuable reference for the processing and assembly of PRSM.

Influence of eccentricity error on the orbit of a two-stage double-helical compound planetary gear train with different mesh phasing configurations

The eccentricity error is an unavoidable error during the manufacturing and mounting process and has a significant influence on the response of gear systems, especially for planetary gear train. First, based on the most basic quasi-static model, pressure angle, contact ratio and meshing stiffness are considered to be the time-varying parameters due to the eccentricity error. Then, a new quasi-static model of a two-stage compound planetary gear train with eccentricity error is established, and the accuracy of the established model is verified by analyzing the orbit result of healthy gear. Furthermore, the influence of eccentricity error on the response in the time-domain (orbit) under various error combinations is studied and discussed. Finally, it is found that the orbit of the component with eccentricity error is very different from that without eccentricity error, and the formation law of orbit for gear with eccentricity error is revealed in this paper. The results of the quasi-static model could provide a fundamental understanding to the dynamic model for the studied compound planetary gear train.

Camera calibration based on lightweight fan-shaped target detection and fitness-distance-balance chaotic marine predators algorithm

Camera calibration represents a critical stage in visual measurement, directly impacting the measurement precision. Addressing the limitations of checkerboard and circular targets, resulting in low calibration accuracy and eccentricity errors, this study introduces a camera calibration method that relies on lightweight fan-shaped target detection and Fitness-Distance-Balance Chaotic Marine Predators Algorithm(FDBCMPA). Starting with the improvement of camera calibration targets by refining and designing fan-shaped targets, this paper aims to tackle eccentricity errors through hardware-based solutions. In the process of extracting target feature points, a lightweight object detection algorithm is devised to capture individual fan-shaped pattern positions from images, transitioning from global multi-shape feature detection to localized single-pattern feature extraction. This approach mitigates the distortions inherent in traditional checkerboard targets during global linear feature imaging and partially alleviates interference from background environments and similar features. Following the initial camera calibration, the FDBCMPA is employed to further optimize the calibration outcomes, thereby boosting the calibration method's performance. Experimental findings highlight that the enhanced object detection method in this study attains detection accuracy comparable to mainstream methods, with distinct advantages in detection speed, parameters, and FLOPs, rendering it more suitable for use in camera calibration. Additionally, the FDBCMPA demonstrates improvements in convergence results compared to MPA, PSO, TACPSO, Chimp, and other methods. The final camera calibration results demonstrate an impressive reprojection error of 0.0738 pixels through the utilization of the proposed camera calibration method introduced in this paper, and substantiate the practicality and effectiveness of the calibration method proposed in this study.

A Transit Tilt and Offset Errors Calibration Method for Improving Laser Tracker Measurement Accuracy Based on the Telecentric Measurement System

Laser trackers are instruments used to measure the three-dimensional coordinates of objects with high precision. It is necessary to study calibration methods to identify geometric errors of laser trackers so as to compensate for errors and improve measurement accuracy. Among the offsets, tilts, and eccentricity errors in the system, the transit tilt and offset errors play important roles and have an essential effect on the measurement accuracy, which need to be calibrated and compensated for. Current methods for detecting geometric errors between the transit and standing axes are complex and time-consuming. In this paper, a fast and novel calibration method is proposed, which can be easily operated with high precision. Two test rods are coaxially mounted on the ends of the transit axis to extend the hidden transit axis. Then, a telecentric measurement system is used to record the projection images of the test rods and identify the position of the transit axis. The transit offset and tilt errors were thus obtained by comparing the central axis positions of the two test rods before and after the standing axis rotated 180 degrees. Moreover, a numerical analysis method was proposed based on the least square circle fitting method to calibrate the installation eccentricity errors of the test rods. Experimental results verify that the measurement accuracy of the laser tracker can be improved after error compensation using the calibrated transit offset and tilt errors. The measurement error of the laser tracker can be reduced from 161 µm to about 73 µm after compensation at a distance of 5 m, while the error can be decreased from 143 µm to about 108 µm at a distance of 10 m.

Analysis of Vibration and Noise in Electric Drive System Under Resolver and Motor Rotor Coupling Eccentricity

Eccentricity in a permanent magnet synchronous motor (PMSM) can cause additional electromagnetic forces and resolver error, which in turn can lead to an aggravation of vibration and noise in the electric drive system. In this paper, the additional electromagnetic forces and the resulting vibration and noise under the conditions of the three-dimensional (3-D) static and dynamic motor eccentricities, the resolve error, and the coupling between these faults, are analytically derived and analyzed, with special emphasis on the coupling effect of the motor eccentricity and resolver error. The results of the present theoretical analyses agree well with those from the finite element simulations and the experimental measurements. It can be concluded from the results that the coupling between the 3-D motor eccentricity and the resolver error will induce additional electromagnetic forces with much more diverse spatial orders and frequencies.

On the design of a cup anemometer performance simulator. From the wind speed to the output data.

This paper describes some important aspects of the development of a cup anemometer performance simulator. This tool aims to simulate the different phenomena which affect the different subsystems of the anemometer, in order to develop different strategies to improve the accuracy of the data obtained through post-processing or design changes. Bearing in mind that the cup anemometer is the most widely used wind speed sensor in the wind energy sector (to control wind generators and to analyse the future economic revenue of wind farms in certain locations), the relevance of the present study should be highlighted. The software is designed in Simulink® and is divided in 4 modules representing the different subsystems of an anemometer from the actual wind speed to the output data. The first two modules of the simulator are detailed in this paper, which consist of the Rotation Rate Module and the Pulse Generating Module. The first module outputs the anemometer’s rotor angular position with time, which depends on the wind speed of the anemometer. Oscillations of the rotation rate caused by steady state harmonic accelerations are taken into account. The Pulse Generating Module simulates the optoelectronic system of the anemometer and creates the pulse train signal based on the rotor’s angular position, taking into account the manufacturing and eccentricity errors of the system. The architecture of the modules is detailed, and related with the phenomena which is intended to simulate.

Geometric error compensation method using the Laser R-test

Traditional methods for measuring geometric errors in machine tools, including interferometry and Double Ball Bar (DBB), are known to be expensive and time-intensive. Consequently, a non-contact calibration system called the “Laser R-test” has been developed. This innovative system is designed to measure both position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs) efficiently. Since its development in 2000, this tool has been instrumental in analyzing eccentricity errors, angular position errors, and simultaneous trajectory errors. Through extensive research, it has been determined that the total error in a five-axis machine tool can be controlled to below 40 µm after compensating for eccentricity parameters and angular position errors. However, reducing this error to below ± 10 µm is challenging, primarily due to wobble errors in the orientation of the rotary axis without compensating. In this study, a new methodology based on Laser R-test and Rodrigues’ rotation formula has been developed to establish a PIGE model of rotary axis. Based on the methodology, the 8 PIGEs can be analyzed by measuring 5 coordinate positions. The compensation of 8 PIGEs in the rotary axis is completed within 30 min using the inspection path. Compatibility with ISO-10791–6 standards for BK1, BK2, and BK4 path tests is confirmed, validating the compensation effects. A precision of below ± 10 µm is achieved, with inspection time reduced by over 50%. This system can complete multiple errors by simply using the different paths. This greatly reduces the setup time for future users, enhancing its commercial applicability.

Analysis of Dynamic Characteristics of Rotor Sail Using a 4DOF Rotor Model and Finite Element Model

The interest in wind-assisted ship propulsions (WASPs) is increasing to improve fuel efficiency and to reduce greenhouse gas emissions in ships. A rotor sail, one of the typical WASPs, can provide auxiliary propulsive force by rotating a cylinder-shaped structure based on the Magnus effect. However, due to its huge rotating structure, a meticulous evaluation of the influence on the ship structure and dynamical stability of the rotating structure should be conducted in the design stage. In this respect, an analysis of the rotating structure for a 30 m height and 3 m diameter rotor sail was conducted in this study. First, a 4DOF (four-degree-of-freedom) model was derived to simplify the dynamics of the rotor sail. Using the 4DOF model, natural frequencies for four low-order modes of the rotor sail were calculated, and frequency responses at support points were predicted. Next, a comparison and validation with the finite element model of the rotor sail were carried out. For the 1st and 2nd natural frequencies, a difference of approximately 0.3 Hz was observed between the 4DOF model and the finite element model, confirming the effectiveness of the 4DOF model for low-order modes. In analysis with changes in the bearing supporting stiffnesses, it was verified that lower support bearings have a significant impact on rotor dynamics compared to upper support bearings. Vibration response at the upper support was also confirmed through frequency response analysis caused by imbalance at Thom disk and mid-plate. Additionally, when estimating the eccentricity of the Thom disk as imbalance, a limit of eccentricity error could be set as 24 mm. The presented modeling procedures and analysis results can be references during early design stage of a novel rotor sail structure.

Elastohydrodynamic lubrication analysis with eccentric errors for planetary roller screw mechanism

The lubrication characteristics of planetary roller screw mechanism (PRSM) with eccentric errors are investigated in this work. And the formulas of entrainment velocities with eccentric errors are derived. The film thickness and pressure distributions of screw-roller (SR) and nut-roller (NR) thread pairs are analyzed. It finds that the film thickness of SR thread pair is larger than that of NR thread pair. Moreover, the effects of eccentric errors on comprehensive meshing radius, entrainment velocities and lubrication characteristics of SR and NR thread pairs are discussed. The results show that the eccentric errors have significant effects on lubrication characteristics of PRSM.