Eccentricity Error Research Articles

A systematic error modeling and separation method for the special cylindrical profile measurement based on 2-dimension laser displacement sensor

The aim of this study is to improve aeroengine multistage rotor performance and to reduce the incidence of failures. Measuring the form error of seam allowance connecting cylindrical surface accurately is critical to achieve rotor optimal stack assembly stage-by-stage. In this paper, compared to the traditional cylindrical profile measurement model, a more comprehensive measurement model was built based on a 2-dimensional line laser sensor measurement technique, a model in which the component eccentricity error e, the sensor lateral offset error d, the sensor forward tilt error θx, the sensor lateral tilt error θy, and the rotary table tilt error γ were gradually modeled and separated. The particle swarm optimization algorithm was adopted to solve the model error parameters. The residual error simulation was performed to observe the effect of different levels of offset errors on the measurement results. The aeroengine rotor seam allowance surface measurement experiment was performed to verify the validity of the method model. We can conclude that the sensor lateral offset error d was 3.214 µm, sensor forward tilt error θx was 12.754″, sensor lateral tilt error θy was 10.365″, and rotary table tilt error γ was 2.146″. The cylindricity error value was 3.701 µm. Compared with the traditional cylindrical profile measurement method, the measurement accuracy of cylindricity error was improved by 1.768 µm. The proposed method can improve the measurement accuracy significantly of multistage rotors in the aeroengine measurement process; besides, it can also be extended to the measurement of other geometric form errors.

Design of high accuracy cylindrical profile measurement model for low-pressure turbine shaft of aero engine

Coaxiality and cylindricity are the important geometric parameters of the low-pressure turbine (LPT) shaft. And the measurement accuracy of coaxiality and cylindricity directly affect the rotary characteristics of the aero engine. Therefore, a cylindrical profile measurement model with five systematic errors is designed to improve the coaxiality and cylindricity measurement accuracy of the low-pressure turbine shaft in this paper, in which eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt are considered. Besides, the influence of systematic error and the shaft radius on the residual error for the stepped low-pressure turbine shaft is analyzed as well. The evaluation results of coaxiality and cylindricity are obtained based on different measurement strategies and models. In order to verify the effectiveness of the cylindrical profile measurement model with five systematic errors in the paper, a rotary measuring instrument with high precision is built. Compared with the traditional cylindrical profile measurement model with two systematic errors, the measurement accuracy of the coaxiality and cylindricity by the cylindrical profile measurement model with five systematic errors proposed in this paper are improved by 2.9 μm and 8.18 μm, respectively in the condition of the optimal measurement strategy for the LPT shaft with large radius. The proposed method is suitable for small probe radius and large eccentricity error, probe offset error, geometric axis tilt error and guide rail tilt error, especially for the LPT shaft with large radius. The proposed method can be applied to error separation and tolerance allocation for multistage rotor.

Load Sharing Performance of Herringbone Planetary Gear System with Flexible Pin

To study the load distribution mechanism of the power-split transmission, and improve the carrying capacity of the herringbone star gear transmission system, the dynamic model of the planetary gear system for a geared turbofan engine with the flexible pin was established based on the node finite element method. The model considered the comprehensive influence of meshing error, time-varying meshing stiffness, bearing supporting stiffness and meshing phase. The dynamic load sharing performance of the system were studied in the equivalent meshing error, and the influence of the error incentives and structure of pins on the load sharing characteristics was analyzed. The results indicate that the vibration track of the center gears deviates from the origin because of the eccentricity error. The load sharing performance of the system can be improved by improving gears’ manufacturing precision. Moreover, the influence of manufacturing errors is larger than that of assembling errors. The flexible pin improved by Montestruc has the best load sharing performance among the four types of pin models designed in the study. The load sharing performance declines with an increase of in input speed and a decrease in input power.

Effect of installation errors on beveloid gears’ precision ground by cone-shape worm wheel

Both the mathematical model of a cone-shape worm wheel and the mathematical model of cone-shape worm wheel grinding beveloid gears with installation errors of a gear workpiece and a cone-shape worm wheel are developed. The effect of the installation errors including gears’ incline error, eccentric error, mounting angle error of a cone-shape worm wheel and coupling error of installation errors mentioned on gears’ surface deviation is investigated. The sensitivity of tooth surface deviation to installation errors mentioned is analyzed. The results show that the helix slope deviation, helix total deviation and tooth cumulative pitch deviation are more sensitive to gears’ incline error. The tooth cumulative pitch deviation, profile total deviation, profile slope deviation are more sensitive to gears’ eccentric error. The effect of gears’ incline error and eccentric error on cumulative pitch deviation and individual cumulative pitch deviation is inverse which lead that coupling error including gears’ incline error and eccentric error causes smaller cumulative pitch deviation and individual cumulative pitch deviation compared to gears’ incline error or eccentric error. Moreover Effect of the mounting angle error of cone-shape worm wheel on gears’ surface deviation is very limited.

A study on tooth profile modification of cycloid planetary gear drives with tooth number difference of two

In the previous study, the authors showed the results that the cycloid gear pair with tooth number difference of two does not only increase the contact ratio and decrease the maximum shared load of the individual contact tooth pair, but also reduce the bearing loads of the cranks slightly. However, stress concentration occurs on the tip segment of cycloid teeth, if no suitable flank modification is available. The aim of the paper is to propose a new design approach for flank modification of cycloid profile to improve the contact characteristics. Besides the conventional methods for flank modification, i.e. equidistant offset and shifting offset modification, a new flank modification with multiple variables, so-called free profile modification, is proposed in the paper. Only the cycloid planetary stage with two cycloid discs are considered in the study to simplify the analysis. Two analysis approaches, tooth contact analysis and loaded tooth contact analysis, are developed for evaluation of the modified profile accordingly. A method based on the geometrical property of cycloid profile, named “tracing circle method”, is employed to determine exactly the contact points and the other kinematic characteristics of drives. The developed LTCA approach based on the influence coefficient method is used to calculate the contact stress of the modified cycloid tooth pair with considering the stiffness of supporting bearings and the influence of the eccentric error of the cycloid disc. The variation of shared load, the contact stress, the bearing load and the torque distribution are analysed in the paper to evaluate the various flank modification methods. Some useful design guidelines, summarized from the systematic analysis results, are also given in the paper.

A New Method of Angle Measurement Error Analysis of Rotary Encoders

In this article, a method of error source analysis and detection to improve the angle measurement accuracy of rotary encoders in precision measuring instruments is proposed. The angle measurement error caused by the installation eccentricity of the grating disk and the radial error motion of the rotating shaft is analyzed, and the error model is built. The method of measuring the radial error motion is introduced, and the visual system and image processing technology is proposed to detect the eccentricity. The verification experiment by the use of an autocollimator and a polygon is carried out. The residual error after comparison within ±6″ accounts for 9% of the angle measurement error. The proposed error model is verified, and the angle measurement error can be predicted if the installation eccentricity and radial error motion are known.

An evaluation method of gear profile deviations based on the consideration of installation errors

Gear elemental deviations, such as profile deviations, are important items for gear accuracy evaluation and generally measured on a Gear Measurement Center or Coordinate Measuring Machine. There are installation errors when the gear installed on the shaft positioned in the center before measurement. Therefore, the evaluation method of gear profile deviations considering installation errors was introduced to measure gear profile deviations without the precise adjustment of installation errors in the study. In this method, installation errors were described as the installation eccentricity error and the installation wobbling error. Then, the installation error model, the coordinate transformation model and the tooth profile deviation evaluation method were established. The gear base circle center was calculated with the tooth profile measurement points. Then, the tooth profile measurement points under the machine coordinate system were transformed into the coordinates in the work-piece coordinate system. After that, the profile deviations were evaluated according to ISO 1328-1:2013. The measurement experiments were carried out on a gear measurement center under different installation errors. The application scope of the method was discussed by simulation. The range of the total profile deviation obtained with this method was 0.51 μm under four groups of the installation conditions, whereas the value obtained with GMC was 3.17 μm. Experimental results showed that this method could significantly decrease the influence of installation errors for measuring gear profile deviations under the rough positioning conditions of product gears.

Influence mechanism of multi-coupling error on the load sharing characteristics of herringbone gear planetary transmission system

The load sharing characteristics of the herringbone planetary transmission system are a key indicator for evaluating the bearing stability and reliability of each planet gear in the transmission system. The value of the load sharing coefficient is closely related to the manufacturing error and the assembly error in gear processing and assembly process. Therefore, it is indispensable to study the influence mechanism of these errors on the load sharing characteristics of the transmission system. Nevertheless, researches on the multi-coupling transmission error to the load sharing characteristics of herringbone planetary transmission system did not receive enough attention, but much of the single errors. Based on the centralized parameters theory and the Lagrange method, this research establishes a dynamic model for herringbone planetary transmission system and creatively proposes a study of multi-coupling error which consists of eccentric error, tooth profile error, stagger angle and assembly error. This work shows that the changing regulation of the load sharing characteristics with any one of the above errors is different. However, the load sharing characteristics become worse with the increase of multi-coupling error, in which the eccentric error plays a main role. Therefore, the error control should focus on eccentric error.

Investigation on the influence of work holding equipment errors on contact characteristics of face-hobbed hypoid gear

This paper proposed the accurate mathematical model considering work holding equipment errors based on the three-face cutter differing from typical commercial software. The simulated flow included manufacturing issues from industry applications have been established. Quasi-static loaded tooth contact analysis was used to discuss the contact characteristic and gear geometry with existence of radial and angular eccentric errors. In addition, the correspondence between transmission error and gear geometry was established. The method of separating the long wave (low frequency) and short wave (high frequency) from total transmission error has been proposed. Results show that the effect of angular eccentric error on gear geometry is more sensitive than that of radial eccentric error. The gear geometry deviations present periodic changes because of variation of pitch error. And radial eccentric error cannot influence the position and shape of contact pattern. On the contrary, with existence of angular eccentric error, the contact area moves to the heel side and then backward. Furthermore, the length of contact pattern is firstly decreased and then increased in one period. The transmission error is sensitivity for both two eccentric errors and the peak to peak value of transmission error increase obviously.

An algorithm for roundness evaluation of crankshaft main journal

The roundness of crankshaft main journal is a key parameter which affects the performance of automobile engine. When the crankshaft comprehensive measuring machine is used to measure roundness error, the result of roundness is deviated because the workpiece is eccentric installed. In this paper, an technique of eccentric error separation is proposed to separate and correct the eccentric error existing in the original measurement data. On this basis, the least square evaluation algorithm based on polar coordinate is used to evaluate the roundness error of crankshaft main journal. The result shows that this method can effectively eliminate the installation eccentricity introduced in the measurement results and has high accuracy.

Pose error correction in offline three-dimensional profile measurement system for large-diameter aspheric mirror.

This paper proposes an offline measurement method for a large aspheric mirror during its grinding stage. A measurement method and mathematical model to realize a three-dimensional (3D) profile measurement for large-diameter aspheric mirrors are proposed, based on a cylindrical coordinate system. The measurement error induced by the alignment error is examined based on the error analysis of the measurement and workpiece coordinate systems. The corresponding pose error obtained through the nonlinear least squares method is eliminated to improve the measurement accuracy. To evaluate the entire surface profile error, a 3D profile model is established. The total uncertainties of the proposed measurement system are also estimated. Comparison experiments for a 300-mm-diameter K9 glass workpiece are conducted on UPFM900 and Taylor Hobson PGI 3D profile instruments, and the profile accuracy is found to be improved from 15.3 to 7.12μm after eccentricity and tilt error elimination.

On the estimation of absolute grating groove density and inter-grating groove density errors of laser pulse compression gratings

In this paper, experimental studies on estimation of absolute groove density of gratings and inter-grating groove density errors are reported with typical detector limited accuracies of ±0.23 lines mm−1 and ±0.005 lines mm−1, respectively at groove density of ~1740 lines mm−1 of holographic laser pulse compression gratings. A simple single detector based optical set-up with fixed optical elements to avoid mechanical eccentric errors, if any, due to goniometric movement of a rotatory stage, has been proposed to estimate absolute groove density of gratings. A modified Fizeau or a modified Michelson interferometer based optical set-up has been used to estimate inter-grating groove density errors of gratings. Various gratings from different manufacturers were examined for their absolute groove densities and inter-grating groove density errors.

Development of Bearingless Induction Motors and Key Technologies

In order to effectively suppress rotor periodic synchronous vibration caused by the rotor mass eccentricity and sensor detection error of bearingless induction motor (BIM), a rotor vibration suppression control strategy with improved repetitive control is proposed based on the analysis of the problems existing in traditional repetitive control. On the basis of the vector control model of the BIM system, a simplified suspension control model is derived. The fractional delay filter is used to optimize the repetitive control internal model. Based on the stability conditions of the control system, the design method of repetitive control compensation function in high, middle and low frequency region is given in detail, and the simulation and experimental system of BIM is built. The results show that the vibration suppression control strategy with the improved repetitive control can effectively suppress the vibration of the rotor in steady state and improve the suspension accuracy of the BIM.

Research on Dynamic Load-Sharing Characteristics of Two-Stage Asymmetric Star Gear System

It is well known that the asymmetric gears of the transmission system enhances the load carrying capacity of system, but there is no relevant conclusion about the dynamic load-sharing characteristics of the asymmetric gear transmission system. In order to analyze the dynamic performance of an asymmetric gear system, taking two-stage asymmetric helical gear star drive system as the research object, the load-sharing model of the two-stage asymmetrical helical star gear system is established by the centralized mass method and Newton's second law. The load sharing characteristic of the star gear is studied about the floating member and the tooth profile pressure angle. The influence of eccentricity error, load and input speed on the load sharing performance of the system is analyzed. Based on ADAMS virtual prototype model, through the simulation analysis of the dynamic meshing force of gears, the load sharing coefficient of each star gear is calculated, compare the results with the MATLAB solution to verify the correctness of the theory.

Modelling of transmission accuracy of a planetary roller screw mechanism considering errors and elastic deformations

Based on the motion principle of a planetary roller screw mechanism (PRSM) converting the rotary motion into linear motion, a calculation model of the transmission error under free load is established considering the error and clearance, as well as a transmission error model under different working loads considering the elastic deformation. The error is mainly concerned with PRSM manufacturing errors and assembly errors, the theoretical transmission error curve obtained by measuring the accumulative error and flank angle of the thread, the coaxial error of the screw thread, and the deviation error and eccentric error. Besides, an error analysis and measurement of the experimental system is performed to verify that the system meets the experimental requirements and is conducive to obtain accurate experimental results. Lastly, the transmission error experiment of the PRSM prototype under free load and different working loads are further conducted. The experimental data of the transmission error are compared with the analytical solution presented in this paper, and the results show that the difference is 20.1% under the free load condition and is less than 15% under different working loads.

Flexible calibration method for line-scan cameras using a stereo target with hollow stripes

With the advantages of ultrahigh resolution and acquisition rate, stereo line-scan sensors are gradually being developed for three-dimensional (3D) measurements. Calibration is crucial for line-scan cameras. However, viewing targets of line-scan cameras from different angles cause eccentricity errors. This paper presents a flexible calibration method for line-scan cameras to solve the eccentricity error problem. The calibration quality is improved by the accurate extraction of one-dimensional (1D) points on line-scan images, and the corresponding 3D points in space. Through the analysis of the eccentricity error in 1D image point extraction when utilizing targets with solid stripes, a stereo target with hollow stripes is designed. An algorithm is described to calculate the 3D points of intersection between the centerlines of hollow lines on the target and the viewing plane of the camera. A robust optimization is proposed to enhance the stability of parameter estimation. Experiments are performed to compare the re-projection error between the cross-ratio-based method and proposed method. A simulation is also performed to estimate the combined uncertainty. The comparative results of the experiments verify that the proposed method is accurate, low cost, and easy to implement.

A coaxiality measurement method by using three capacitive sensors

This paper describes a method of determining the coaxiality errors between suspended coil and outer magnet yoke by using three capacitive sensors. The coaxiality error sources including the cylindrical error of the outer yoke of the magnet, the movement trajectory straightness error of the lifting stage and the tilt error of the suspended coil are identified. Three capacitive sensors are used to measure the coaxiality eccentric errors. The theoretical model and the mathematical relationship between the outputs of three capacitive sensors and the coaxiality errors are established. The method has an advantage that the third radial redundancy sensor can be used to eliminate the influence of the form and the size. And the experiments are conducted to measure the coaxiality errors.

Research on Eccentric Error Compensation of Aviation Gear Grinding Installation

Research on Eccentric Error Compensation of Aviation Gear Grinding Installation

Ambiguous relative orbits in sequential relative orbit estimation with range-only measurements

This paper describes a manifold of ambiguous spacecraft relative orbits that arise in sequential relative orbit estimation. The development herein assumes linear relative dynamics, a circular reference orbit, and range-only measurements. Using a formulation based on relative orbit elements, the ambiguous orbits are categorized into two cases: mirror orbits, which conserve the size and shape but transform the orientation of the true relative orbit, and deformed orbits, which both distort the shape and change the orientation. A special case, that of central ambiguous relative orbits, which are geometrically symmetric relative to the chief's local-vertical-local-horizontal frame is also discussed. The multiplicity of mirror ambiguous orbits, deformed ambiguous orbits and central ambiguous orbits are shown to be three, four and infinity, respectively. Numerical results using an extended Kalman filter are provided to confirm the existence of these ambiguous orbits. Furthermore, the observability is studied analytically with a nonlinear observability criterion using Lie derivatives. It is also shown by numerical results that the inclusion of nonlinearities in the filter model can help resist the tendency of an extended Kalman filter to converge to the ambiguous relative orbits. Finally, the persistence of these ambiguous orbits under unmodeled chief eccentricity error and J2 perturbation is studied.

Mesh stiffness calculation of cycloid-pin gear pair with tooth profile modification and eccentricity error

Cycloid speed reducers are widely used in many industrial areas due to the advantages of compact size, high reduction ratio and high stiffness. However, currently, there are not many analytical models for the mesh stiffness calculation, which is a crucial parameter for the high-fidelity gear dynamic model. This is partially due to the difficulty of backlash determination and the complexity of multi-tooth contact deformation during the meshing process. In this paper, a new method to calculate the mesh stiffness is proposed including the effects of tooth profile modification and eccentricity error. The time-varying mesh parameters and load distribution of cycloid-pin gear pair are determined based on the unloaded tooth contact analysis (TCA) and the nonlinear Hertzian contact theory, allowing accurate calculations of the contact stiffness of single tooth pair and the torsional stiffness of multi-tooth pairs. A detailed parametric study is presented to demonstrate the influences of tooth profile modification, applied torque and eccentricity error on the torsional mesh stiffness, loaded transmission error, Hertzian contact stiffness and load sharing factor. This model can be applied to further study the lost motion and dynamic characteristics of cycloid speed reducer and assist the optimization of its precision, vibration and noise levels.