Dynamic Transmission Error Research Articles

Dynamic Characteristics Analysis of the DI-SO Cylindrical Spur Gear System Based on Meshing Conditions

The dual-input single-output (DI-SO) cylindrical spur gear system possesses advantages such as high load-carrying capacity, precise transmission, and low energy loss. It is increasingly becoming a core component of power transmission systems in maritime vessels, aerospace, marine engineering, and construction machinery. In practical operation, the stability of the DI-SO cylindrical spur gear system is influenced by complex excitations. These excitations lead to nonlinear vibration, meshing instability, and noise, which affect the performance and reliability of the entire equipment. Hence, the dynamic performance of the DI-SO cylindrical spur gear system is thoroughly investigated in this research. The impact of excitations and nonlinear factors on the dynamic characteristics was investigated comprehensively. A comparative analysis of the gear system was conducted by establishing a bending–torsional coupling vibration analysis model under synchronous and asynchronous meshing conditions. Nonlinear factors such as periodic time-varying meshing stiffness, meshing damping, friction coefficient, friction arms, load sharing ratio, comprehensive transmission error, and backlash were considered in the proposed model. Then, the effect laws of excitations and nonlinear factors such as meshing frequency, driving load fluctuation, backlash, and comprehensive transmission error were analyzed. The results indicate that the dynamic characteristics exhibited staged stable and unstable states under different meshing frequencies and meshing conditions. At the medium-frequency meshing stage (0.96 × 104~1.78 × 104 Hz), alternating phenomena of multi-periodic, quasi-periodic, and chaotic motion states were observed. Moreover, the root mean square value (RMS) of the dynamic transmission error (DTE) in the asynchronized gear system was generally higher than that in the synchronized gear system. It was found that selecting the appropriate meshing condition could effectively reduce the amplitude of the DTE. Additionally, the dynamic performance could be significantly improved by adjusting control parameters such as driving load fluctuation (0~179 N), backlash (0.8 × 10−4~0.9 × 10−4 m), and comprehensive transmission error (7.9 × 10−4~9.4 × 10−4 m). The research results provide a theoretical guidance for the design and optimization of the DI-SO cylindrical spur gear system.

Dynamic reduction technique for nonlinear analysis of spur gear pairs

In this study the dynamic response of a spur gear pair is analyzed using a novel nonlinear approach. The actual rolling motion and engagement of the system is simulated using a set of reduced order models obtained in a pre-processing phase using the minimal amount of master degrees of freedom without loss of accuracy or generality. The flexibility of the gear bodies is included by a refined finite element model, and no geometry simplification is introduced while also retaining all nonlinearity sources. To reduce the computational cost the time-varying mesh stiffness is also pre-computed and used depending on the instantaneous loading conditions. Contact loss is also taken into account, and reconnection events are treated as vibro impacts. The results are compared against high quality and demanding experimental results with a computational cost several orders of magnitude lower than models with similar accuracy. Different loading conditions are investigated during the sweep-up and down maneuvers. Mainly, the dynamic transmission error is analyzed, showing remarkable agreement with the test campaign’s results. Different nonlinear phenomena such as hysteretic jumps and sub- and super-harmonic resonances are correctly predicted by the proposed model in terms of both frequency and amplitude. This method allows quick and accurate nonlinear analyses overcoming current limitations and is open to further complications to include other components and effects.

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.

Study on the Parameter Influences of Gear Tooth Profile Modification and Transmission Error Analysis

Gear transmission systems are widely used to transfer energy and motion and to guarantee the accuracy of the entire machine system. The modification technique is a common method that improves the gear profile and reduces the transmission error. Based on the parametric model, a modified gear can be established for the evaluation of static and dynamic characteristics. The influences of profile modification parameters and gear parameters are investigated while changing the rules of different kinds of factors. Based on sensitive parameters, a two-stage profile modification curve is proposed to improve the performance of gear pairs. Thus, considering the time-varying mesh stiffness and backlash, a novel, dynamic modified gear model is established to analyze the dynamic performance, such as the dynamic transmission error. Based on the proposed curve, the range and amplitude of the transmission error can be decreased. Additionally, the vibration displacement and noise can be reduced to improve the running characteristics.

Dynamic characteristics analysis of gear tooth modification in a three-stage gear system

Gears serve as vital elements in high-efficiency mechanical transmissions, playing a pivotal role in vehicles. Employing gear modification has the capacity to significantly enhance the dynamic properties of the Planetary Gear Transmission (PGT) systems. In this study, the investigation has been directed toward a three-stage PGT model system with a comprehensive consideration of various vibration-related factors that affect the dynamic behavior of the system. These factors are encompassed by Time Varying Mesh Stiffness (TVMS), torsion and bending forces experienced by connecting shafts, dynamic Transmission Error (TE), impact of gear meshing eccentric load forces, inherent flexibility in the supporting shaft structure, and influence of torque and power fluctuations originating from a multi-cylinder engine. The estimation of TVMS is carried out for both external-external and external-internal teeth meshing pairs, utilizing the Potential Energy Method (PEM). Subsequently, this study employs the lumped parameter method to establish a model with a specific objective of investigating the lateral-torsional-coupling behavior within adopted PGT model system. The lateral vibration displacement characteristics of the PGT model system is deeply analyzed, revealing the influence of speed on lateral vibration and displacement frequency domain characteristics through a 3D waterfall diagram. Furthermore, a Tooth Profile Modification (TPM) method is applied utilizing two distinct parameters namely [Formula: see text] TPM parameter and [Formula: see text] TPM parameter. The objective is to mitigate vibrations and evaluate the performance of the adopted PGT model after employing each TPM parameter through both TVMS and Root Mean Square (RMS).

Accuracy characteristics of the testing bench for dynamic transmission error

A high-precision testing bench is required for accurately assessing the dynamic Transmission error (TE) of precision reducers. However, the relationship between the testing bench’s accuracy and deviations in testing conditions has not been determined, leading to an inaccurate evaluation of its metrological performance. In this study, we examined the accuracy characteristics of the testing bench. We analyzed the structure of the bench and identified the main error sources that affect its accuracy, including mechanical system deviation, sensor measurement deviation, and testing condition deviation. In addition, we studied the mechanism of these error sources and proposed a method to convert drive speed deviation and load torque deviation into angle deviation by using a three-dimensional characterization model of lost motion. Based on the expression of TE, we proposed an evaluation method for the accuracy of the testing bench. Furthermore, we presented the methods for calculating the error sources and proposed a method for determining the TE of adjacent units based on the specific coupling structure. Finally, we conducted practical tests to validate the effectiveness of the proposed method in evaluating the accuracy characteristics of the testing bench. We found that the relative test error between quasi-static TE and dynamic TE is small, and more accurate test results are obtained when applying an appropriate load.

Dynamic performance analysis of a gear transmission system due to cavitation in gerotor pump

In this work, a dynamic model considering time-varying mesh stiffness and backlash was proposed and a test rig was built to investigate the vibration behavior of the idle gear set in a gerotor pump. The effects of input shaft speed, torque fluctuation, and gear tooth backlash on the dynamic performance of gerotor pump were analyzed according to the numerical and experimental results. Results demonstrate that when the torque fluctuation is neglected, the dynamic transmission error of drive gear is dominated by the first three mesh harmonic components; however, the RMS values of vibration acceleration have high frequency characteristics. When the torque fluctuation is considered, the dynamic responses of gerotor pump are modulated by the torque fluctuation. The effect of tooth backlash on the dynamic performance of gerotor pump is limited and mostly depended on the torque fluctuation condition. The gerotor pump noise is dominated by the low frequency component around the mesh frequency. The vibration energy can be divided into two parts in the frequency domain. In the low frequency band, the vibration energy is caused by the mechanical vibration. However, in the high frequency band, the cavitation phenomena may cause the concentration of the vibration energy and resonant of pump structure.

Multi-objective optimization model of transmission error of nonlinear dynamic load of double helical gears

Abstract In order to address the impact of reduced transmission stability and reliability caused by volume reduction on the quality of gear transmission, this article proposes a multi-objective optimization model for nonlinear dynamic load transmission errors of double helical gears. This study aims to introduce a multi-objective design method for gear transmission, using the volume and smooth reliability of helical gears as objective functions, and establish a multi-objective optimization design mathematical model for helical cylindrical gear transmission. In order to solve this multi-objective optimization problem, we utilized the optimization toolbox in the scientific calculation software MATLAB with examples. The results show that after the joint optimization design of volume and coincidence degree, it is calculated that the volume after the joint optimization design is still 2.2624 × 107 mm3, and the coincidence degree is 5.9908. After rounding, the design result is m n = 3 , Z 1 = 31 , β = 20 ∘ , Ψ d = 1.2 {m}_{{\rm{n}}}=3,{Z}_{1}=31,\beta ={20}^{\circ },{\Psi }_{{\rm{d}}}=1.2 . The optimization design results show that the joint optimization design with the minimum volume and the maximum coincidence as the objective function can reduce the volume and improve the output stability of the helical gear.

Investigation of High-Speed Dynamic Transmission Error Testing Using Gear Strain

The difficulties of testing the dynamic transmission errors of gears in complex environments, such as high-speed operations or situations with oil contamination, and the limited variety of available testing methods have been recognized as issues which this study endeavors to tackle. In this study, a testing method for evaluating high-speed dynamic transmission errors of spur gears through the use of strain sensors is proposed. To reduce the interference of environmental noise on testing signals, physical measures, such as the use of copper foil to shield signal wires and the grounding of data acquisition equipment, were implemented during the testing process. Utilizing wavelet decomposition to distinguish between the high- and low-frequency components of the testing signals, the transmission error of gears during high-speed operation was calculated. After confirming the feasibility of the stress–strain stiffness approach in gear transmission error testing using the magnetic grid method, tests on modified and non-modified gears at various speeds and loads were carried out. The two types of test data were processed and evaluated to determine the effect of speed and load on gear dynamic transmission error. It was possible to conduct research on the testing technique of gear dynamic transmission errors utilizing strain sensors, which provides a new, fast, simple, and practical testing strategy for gear transmission error testing.

Design for manufacturing of a spur gears profile modification based on the static transmission error for improving the dynamic behavior

This study presents a novel iterative algorithm for obtaining an improved profile modification for spur gears based on minimizing the peak-to-peak static transmission error. Two improved profile modifications, obtained with different initial constraints applied to the tooth profile, were calculated and characterized. The static behavior of the gears was analyzed using a finite element model, by which the modifications were compared considering the maximum contact pressure and peak-to-peak static transmission error. The dynamic behavior of the gears was analyzed using a lumped parameter model. The dynamic overload and the dynamic transmission error were evaluated under several working regimes. The grinding wheel profiles used to realize the proposed profile modifications and the total volume of material removed were calculated using a CAD model. The results showed that the proposed modifications lead to a significant improvement in the static and dynamic behavior of the transmission by reducing both the maximum dynamic overload and meshing noise. The profiles of the grinding wheel were found to be accurately represented by polynomial functions, thus a feasible manufacturing process can be obtained. The total volume of material removed was lower than that produced by commonly adopted profile modifications, thus reducing material waste and increasing the productivity of the manufacturing process compared to standard profile modifications.

Nonlinear dynamics of spur gear systems with time-varying misalignment errors

The mesh stiffness excitation of conventional dynamic models for misaligned spur gear systems is obtained under constant misalignment errors, which inevitably neglects the additional fluctuation of mesh stiffness caused by the influence of dynamic response on misalignment errors. The uneven load distribution and contact state alteration resulting from misalignment errors will produce nonlinear tilting moments that cause pendular vibration in gears. The pendular vibration of the gears, in turn, contributes to the time-varying property of misalignment errors. In this study, misalignment errors consist of an initial constant component that may originate from mounting errors and a time-varying component caused by the pendular vibration of gears. Subsequently, a nonlinear coupled lateral-torsional-axial-pendular vibration model for spur gear systems with time-varying misalignment errors (TVME) is constructed. Parametric analyses reveal that the dynamic behavior of the proposed TVME model and the conventional model differ due to the presence of an initial misalignment error. In contrast to the conventional model, the proposed model exhibits richer nonlinear characteristics, mainly in the high-speed zone, and is more prone to sub-harmonic resonance and contact loss owing to the time-varying property of misalignment errors. At 11,400 r/min, the proposed model undergoes sub-harmonic resonance, and its root-mean-square (RMS) value of dynamic transmission error (DTE) increases from 0.27 to 1.15 and 2.96 as the initial misalignment error increases from 0 to 0.005 ° and 0.01 ° , whereas the corresponding values of the conventional model remain almost unchanged at 0.27.

Effects of Optimal Tooth Microgeometry Modifications on Static and Dynamic Transmission Errors of Hybrid Spur Gear Drivetrains

A hybrid gear concept that combines a metallic outer rim of gear teeth with a composite web to reduce drivetrain weight was evaluated for impact of tooth microgeometry modifications on transmission error. Control of transmission error through tooth microgeometry modification is important for control of noise and vibrations generated by a drivetrain. The added flexibility of hybrid gears over steel gears brings to question the performance of hybrid over conventional gears relative to their dynamic transmission error and resulting noise levels. Previously developed drivetrain models featuring hybrid spur gears were used to determine optimal tooth microgeometry modifications that minimized peak-to-peak transmission error. Static and dynamic transmission errors were then calculated using the optimal microgeometries and compared to results for a similarly optimized all-steel drivetrain. From the results, it appears that the use of hybrid gears will not negatively affect vibration performance for low- and medium-speed applications, as hybrid gear models predicted similar transmission errors to their all-steel counterparts. At higher speeds, drivetrains featuring hybrid gears were predicted to have significantly different transmission errors, but whether this difference was an improvement or detriment is design and speed-dependent. Therefore, careful design is necessary for high-speed hybrid gears.

An improved time-varying mesh stiffness calculation method and dynamic characteristic analysis for helical gears under variable torque conditions

Based on the slice theory and numerical calculation methods, the time varying meshing stiffness (TVMS) calculation method of the gear pair was constructed for the helical gear pair of a electric continuously variable transmission (ECVT), and the TVMS of the helical gear were calculated under different constant and time-varying torque in this study. The obtained stiffness values were introduced into the established dynamics model of helical gear system, and the influence of changed TVMS, resulting from the variable speed and torque, on the nonlinear dynamic characteristics of gear pair was analyzed using the Runge-Kutta method. The results show that the proposed TVMS calculation method is very effective in computing the TVMS of helical gear pair under time-varying condition. The TVMS and dynamic transmission error (DTE) increase as the torque increase, and vice versa. Meanwhile, the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors at high speed and at low speed when torque is low, whereas not appear at steady high torque under low speed range. This study offers a method for gaining the TVMS of helical gear pair to analyze the dynamic characteristics of gear pair under actual working condition, and the influence of torque that change with the speed ratio on the dynamic performance of a gear system should be considered in the industrial applications.

Vibration characteristics of electric drive system considering rotor-step skewing

Electric vehicle has been gaining popularity in the worldwide market. The drive system for an electric vehicle usually consists of a permanent magnet synchronous motor (PMSM) and a two-stage gear system. The vibration characteristics of electric drive system is of great importance to the noise and durability performance. Recently, rotor-step skewing has drawn attention as a means to reduce the vibration response for the PMSM, since it can effectively reduce the torque ripple as shown in the previous studies. However, the coupling relationship between the torque ripple and the gear transmission system has not been studied thoroughly. To explore the influence of torque ripple on the motor-gear system when considering rotor-step skewing, this paper proposes a coupled electro-mechanical system model to study the vibration characteristics of electric drive system. The effect of rotor-step skewing on cogging torque and torque ripple of PMSM, dynamic transmission error (DTE) of gear transmission system, and vibration characteristics of electric drive system is studied. The simulation results show that the torque ripple has a significant effect on the DTE and the vibration responses of the electric drive system. The rotor-step skewing not only effectively suppress the torque ripple, but also optimize the DTE harmonics induced by the torque ripple.

Mechanism analysis for GDTE-based fault diagnosis of planetary gears

Planetary gear transmission systems are widely utilized in production and daily life equipment, and their fault diagnosis is crucial. The most common tool in this field is the vibration monitoring. In this study, a different tool based on global dynamic transmission error (GDTE) is investigated from a mechanism analysis perspective. The aim is to explore the dynamic responses of planetary gears with different pitting degrees, establish the mapping relationships between GDTE and gear damages, and reveal the mechanism of GDTE-based fault diagnosis. To ensure the accuracy of dynamics model, a novel pitting modeling method based on matrix equations is proposed, which makes the growth process of pitting more physically grounded and the model for irregular pitting on tooth surfaces more practical by considering the termination during random expansion of pitting. Meanwhile, the matrix equation-based pitting model and potential energy method-based time-varying meshing stiffness calculation can be perfectly matched to ensure the precision of subsequent dynamic response analysis. More practical pitting model and accuracy stiffness ensure that the planetary gear dynamics model outputs more precise dynamic responses. Simple time and frequency domain analysis performed on GDTE show that GDTE contains rich information about gear health status. The mapping relationships between GDTE and gear pitting damages are revealed and validated by multi-body dynamics methods. GDTE provides a promising tool for gear condition monitoring and fault diagnosis.

Analysis of Dynamic Mesh Stiffness and Dynamic Response of Helical Gear Based on Sparse Polynomial Chaos Expansion

This paper presents an efficient method for obtaining the dynamic mesh stiffness and dynamic response of a helical gear pair. Unlike the traditional dynamic model that utilizes a time-dependent sequence, the mesh stiffness using the presented method is updated according to the gear displacement vector at each sub-step of the numerical calculation. Three-dimensional loaded tooth contact analysis (3D LTCA) is used to determine the mesh stiffness, and a surrogate model based on sparse polynomial chaos expansion (SPCE) is proposed to improve the computational efficiency, which is achieved by reducing the number of coefficients in the polynomial chaos expansion (PCE) model though a quantum genetic algorithm. During the calculation, the gear displacement vector at each sub-step is converted into the changes in center distance, misalignment angle, and mesh force, which are then introduced into the SPCE model to update the mesh stiffness for subsequent calculations. The results suggest that the SPCE model exhibits high accuracy and can significantly improve the computational efficiency of the PCE model, making it suitable for dynamic calculations. Upon updating the mesh stiffness during the dynamic calculation, the mesh stiffness declines, the dynamic transmission error (DTE) increases, and the frequency components of the responses change significantly.

Effect of hybrid metal-composite gear on the reduction of dynamic transmission error

Effect of hybrid metal-composite gear on the reduction of dynamic transmission error

Dynamic Modeling and Analysis of an RV Reducer Considering Tooth Profile Modifications and Errors

Due to their advantages of compact size, high reduction ratio, large stiffness and high load capacity, RV reducers have been widely used in industrial robots. The dynamic characteristics of RV reducers in terms of vibratory response and dynamic transmission error have a significant influence on positioning accuracy and service life. However, the current dynamic studies on RV reducers are not extensive and require deeper study. To bridge this gap, a more effective and realistic lumped parameter dynamic model for RV reducers is developed, considering the tooth profile modification of cycloid gears and system errors. Firstly, for an efficient solution, the equivalent pressure angle and equivalent mesh stiffness of the cycloid–pin gear pair are introduced in the dynamic model based on the loaded tooth contact analysis. Secondly, the differential equations of the system are derived by analyzing the relative displacement relationships between each component, which are solved using the Runge–Kutta method. With this, the effects of errors such as machining errors, assembly errors and bearing clearances on the dynamic behaviors and transmission precision are investigated by comparison to quantify or qualify their influence. This research is helpful in characterizing the multi-tooth mesh and dynamic behavior, and revealing the underlying physics of the RV reducer.

Dynamic characteristics analysis of gear transmission and its support bearings of high-speed train on the curve

The gear transmission in high-speed trains is crucial for transmitting the traction torque. Its dynamic characteristics impact the system's reliability and safety. Additionally, the gear transmission and its support bearings can be affected by wheel-rail interactions, particularly on curves with larger centrifugal forces. To address this, a spatial vehicle-track coupled dynamics model that considers the complete power transmission path and mechanical structure of the gear transmission is proposed. The model considers factors such as time-varying mesh stiffness, dynamic transmission error (DTE), time-varying support stiffness, and internal interactions of the gearcase bearing. Results indicate that when a high-speed train passes through a curve, the derailment coefficient and lateral force of the wheelset are less on the right curve due to the lateral component of the mesh force, but the opposite occurs on the left curve. The centrifugal effect affects the contributions of the lateral and vertical motions of the gear pair, which are balanced by the torsional motions of the components in the power transmission path. Additionally, the internal interactions between the rolling elements and races are intensified due to track irregularity and gear engagement, but the centrifugal effect of the pinion can be ignored.

Effects of tooth root cracks on vibration and dynamic transmission error responses of asymmetric gears: A comparative study

Gears are significant machine elements used in various industrial applications. An undetected fault in a gear transmission system may lead to a fatal breakdown and, thus, severe economic losses or even human casualties. From this standpoint, the present study developed a numerical method to detect root crack damages in symmetric and asymmetric involute spur gears. To achieve this end, the single tooth stiffness values were calculated for healthy and cracked (25%-50%-75%-100%) gears, and then the time-varying mesh stiffness was obtained. A six-degree-of-freedom dynamic model of a single-stage gear mechanism was developed to collect vibration and Dynamic Transmission Error (DTE) signals. The efficacy of vibration and DTE responses in fault detection was compared through six statistical indicators. The results indicate that the vibration signals were more effective than DTE in detecting tooth root cracks. It was observed that the statistical indicator changes significantly increase with the increment of the drive side pressure angle from 20° to 30° for the backup ratios where the root crack propagates along the tooth, thus making fault detection easier. The findings could provide significant outputs for a better understanding of the influence of tooth asymmetry on gear dynamics characteristics and early fault diagnosis.