Gear Dynamic Model Research Articles

Modeling Method of Meshing Stiffness of High Speed Spur Gears

As one of the most important dynamic excitation sources, the gear time-varying mesh stiffness is the key parameter of gear system dynamic model. So how to calculate the gear mesh stiffness accurately is of great importance. Based on Euler beam theory, this paper proposes a primitive calculation algorithm (PCA) to calculate the time-varying mesh stiffness of spur gears. A simplified gear model is developed based on nonlinear strain-displacement relationships, and a finite element program is used to simulate the gear's meshing stiffness. The rationality of this method was verified through finite element analysis. The results indicate that mesh stiffness plays a crucial role in controlling mesh deformation. In view of this, this study lays a theoretical foundation for further analysis of the vibration and noise of gears.

Study on meshing and dynamic characteristics of herringbone gear pair considering tooth surface wear and modification

Due to errors and tooth surface wear of herringbone gear is inevitable. And tooth surface modification is the key technology to improve gear performance. Therefore, this paper studies the influence of tooth surface uneven wear and modification on herringbone gear pair's meshing and dynamic characteristics with errors. Firstly, herringbone gear pair meshing characteristic analysis model based on discrete tooth surface is established. Secondly, herringbone gear pair bending-torsion coupling dynamic characteristic analysis model considering friction is established, and the dynamic frictional force analysis process is given. Thirdly, combined with Archard's wear theory, the tooth surface uneven wear amount is obtained. The tooth surface uneven wear amount is substituted into meshing matrix as a tooth surface deviation. Finally, the meshing and dynamic characteristics of herringbone gear pair considering tooth surface wear and modification are studied by using the established analysis model of meshing characteristics and dynamic characteristics. The calculation results show that without tooth surface modification, tooth surface wear will deteriorate the meshing and dynamic characteristics rapidly. However, reasonable tooth surface modification can effectively improve meshing and dynamic characteristics of gear pair, so that the gear pair can still maintain stable transmission under wear condition.

Exploring Fatigue Crack Initiation in Helical Gears: Impact of Carburization and Frictional Influences

A computational model is presented to analyze the fatigue crack initiation period in helical gears, incorporating considerations for heat treatment via carburization and friction effects. The aim is to determine the number of stress cycles necessary for the initiation and growth of initial cracks, while investigating the impact of dynamic behavior. This study employs a dynamic gear model with two degrees of freedom in torsion, developed using MATLAB, and incorporates the Papadopoulos criterion for fatigue analysis. The computational results are benchmarked against the strain-life method. The findings underscore the significant influence of the friction coefficient between surfaces, heat treatment, and dynamic loads on the growth of initial fatigue cracks. For instance, with a friction coefficient (μ) of 0.5, the initial crack forms on the tooth surface (y=0) in both methods, with the (ε-N) method suggesting 28-65 cycles and the Papadopoulos criterion indicating 101-156 cycles. The latter accounts for the greater influence of residual stresses. Additionally, untreated gears exhibit a minimum number of loading cycles for initial crack appearance at 1.07 × 104 cycles. In contrast, carburized gears demonstrate an extended lifespan, requiring 1.45 × 105 loading cycles for crack initiation in our example. This emphasizes the efficacy of carburization in enhancing gear durability.

Torsional behaviors of motor rotor of permanent magnet semi-direct drive system for aerial passenger device

Considering the influence of motor shaft torsional damping, the torsional dynamics equations of a permanent magnet semi-direct drive rotor system in an aerial passenger device are derived. The derivation relies on the coupling between the mathematical model and the gear dynamics model of the permanent magnet motor, and the Lagrange–Maxwell equations are used for streamlined analysis. To protect the motor under extreme working conditions, the influence of supercritical Hopf bifurcation caused by different torsional dampings of the motor shaft on the torsional vibration of the motor rotor is investigated. Based on the Hurwitz criterion, the influence of the linear control gain and nonlinear gain of the washout filter on the system bifurcation point and the limit cycle amplitude of the system is analyzed. Then, from the perspective of the system instability and oscillation caused by friction damping change of the drive motor and the complex electromechanical coupling behavior in the permanent magnet semi-direct drive cutting drive system of the aerial passenger device, the system stability domain is defined. The results can be applied to effectively suppress the torsional vibration of the shaft system in the permanent magnet semi-direct drive system of the aerial passenger device, providing a theoretical guidance for stable operation of the system.

Effect of tooth surface wear on dynamic characteristics of spur gear transmission system

The meshing stiffness and system dynamic characteristics caused by gear wear fault are studied. Firstly, potential energy method was used to model and solve the meshing stiffness of gears and the effect of wear on the meshing stiffness and static transfer error was analyzed. A 6-dof spur gear dynamics model was established considering various nonlinear factors, and the dynamic responses of the spur gear system under various wear degrees were analyzed by using time domain, spectrum, phase plane, and Poincare mapping. Finally, the visual test platform of the gear box is built, and the wear condition and dynamic response of the gear box under different wear cycles are analyzed. Results show that the meshing stiffness of gear decreases due to wear, and the meshing stiffness of double tooth decreases more than that of single tooth. In addition, the static transmission errors caused by wear changes periodically with meshing frequency. Wear causes the vibration response of gear system to become complex gradually, and the nonlinear of the system is enhanced, which changes from single period motion to quasi period motion. Results can provide theoretical support for gear wear reduction, life extension, vibration reduction, and noise reduction and lubrication improvement.

An impact-damping model of collision body with multi-point contact and external load, and its application in gear transmission

The damping term is an important component of the calculation model of the contact force of collision bodies because it simulates the energy dissipation of collision bodies during the contact-impact process. Therefore, an impact-damping model of a collision body with multi-point contact and external load is established by considering the restitution coefficient. The algorithm for solving this model is proposed based on iterative methods. Then, the accuracy of the established model is verified by the consistency between the calculated and tested results. The effect of multi-point contact and external load on the contact-impact process of the collision body is studied. The results show that the hysteresis damping factor increases from small to large when the collision body enters a stable state through repeated impacts. Further, the established damping model is applied to the dynamics model of a spur gear pair with backlash, extended teeth contact, and structure coupling effect of the gear body. The nonlinear dynamic characteristics, multi-state mesh, and teeth impact of the system are studied through the root-mean-square of transmission error, bifurcation diagrams with multi-mapping sections, and contact forces. The calculation results of the gear dynamics model with the impact-damping model are more consistent with the tested results than the calculation results of the gear dynamics model with the traditional damping model. Meanwhile, the connection between system resonance, response coexistence, teeth disengagement, and teeth impact is revealed.

Principal-feature-guided degradation trend prediction algorithm based on gear fault dynamics model

The long-term remaining useful life (RUL) prediction of gears is one of the key technologies for gear failure prediction and health management, which has significant practical significance for maintenance decision-making in the traditional mechanical industry. However, due to the lack of sufficient failure data, most RUL prediction methods face significant challenges when predicting gear degradation trends under different fault conditions. The purpose of this article is to solve the dependency of predictive models on data. The paper proposes a principal-feature-guided degradation trend prediction algorithm based on the gear fault dynamics model. Based on the degradation mechanisms under different failure modes, this paper establishes a high-fidelity faulty gears dynamic model. The accuracy of the dynamic model for two failure modes has reached 90% and 88%. The model can provide adequate failure data for RUL faulty gear prediction. Furthermore, a principal-feature-guided Transformer-TCN-ATT (TTA) algorithm was proposed to achieve trend prediction of faulty gears. The prediction algorithm introduces a principal feature reinforcement to reduce the impact of high-frequency fluctuations on feature extraction. Compared with the various typical methods, the proposed algorithm improves prediction accuracy by 76% and 77% for two failure modes.

Mechanism investigation on gear vibration-cavitation caused by tooth-pair lubricated contact

Elastohydrodynamic contact of gears under the influence of vibration induces cavitation and jet impact. To predict the evolution of cavitation and the degree of cavitation erosion in the gear contact region, the multi-phase flow cavitation model and gear dynamic model are coupled for simulating the cavitation effect in the gear lubrication region and the jet stresses at the tooth boundary under different load torques and rotational speeds. Moreover, the proposed model is confirmed to be valid through comparing to experimental results. The cavitation and jet impact intensity increase and mainly distribute at the outlet of tooth contact under high speed. Load torque mainly affects the cavitation oscillation, but there exists an expectation value that makes both the amplitude and intensity of the cavitation fluctuation and erosion low.

Effect of Turbulent Wind Conditions on the Dynamic Characteristics of a Herringbone Planetary Gear System of a Wind Turbine

Due to complex environmental factors, the gear transmission systems of wind turbines are continuously affected by large torque load excitation with periodic and random properties. This paper shares the load-sharing and dynamic characteristics of a herringbone planetary gear system applied in a wind turbine. The gear dynamic model is established using a typical lumped parameter method, in which the nonlinear transmission errors of the gear pairs and left and right-side coupling stiffness of the herringbone gears are included. With the help of the blade element momentum theory, the precise calculation of the hub load of the wind turbine, which is the external excitation of the gear system, is implemented, in which the wind shear, tower shadow, turbulent effect, and tip loss correction are taken into consideration. The nonlinear dynamic characteristics of the system are obtained using the Runge-Kutta method and then discussed. The results show that the turbulent effect plays a major role in the impact on the load-sharing characteristics, and a reasonable set of the support stiffness of rotational components can improve the load-sharing characteristics of the system. The purpose of this research is to provide some useful references in numerical modelling and methods for designers and researchers of wind turbine transmission systems.

Prediction of mechanical efficiency of spur gear pairs based on tribo-dynamics model of multi-tooth meshing

The prediction of load-dependent losses of gear pairs is a topic of constant concern, which depends largely on the modelling of friction coefficient in tooth meshing. Although the current load-sharing based models can handle the friction coefficient in mixed elastohydrodynamic lubrication regimes, dynamics behaviour of each tooth pair or the time variation of direction of normal contact load is not taken into consideration. This study proposes a new method for modelling efficiency of spur gear pairs in transient operating conditions by coupling the multi-tooth meshing model with the friction coefficient model. The friction coefficient is modelled through load-sharing function, and the gear dynamics model is established with regard to alternate meshing of single-double tooth pair and time variation of direction of normal load and mesh stiffness. The model is validated by comparing the calculated results of friction coefficients and load-dependent losses with experimental data in published literatures, showing good agreement. At the end, the efficiency model is applied to two types of gear pairs to investigate the influence of tooth shapes and operating parameters on the load-dependent losses of gear pairs.

A novel method for ring-planet gear mesh force identification via SVD-based Kalman filter

Dynamic mesh forces are the primary vibration excitations of the planetary gear transmission systems. However, they are difficult to measure directly, and conventional indirect measurement methods are typically only applicable to specific planetary gear sets with special measuring equipment. This study proposes a more widely applicable method for identifying the mesh forces of ring-planet gear pairs by utilizing vibration signals. To this end, a ring gear dynamic model is established based on the elastic theory of the thick ring in the form of a space state system. The state vector of the model is constructed from parameters of ring gear vibration responses, while the input state vector is derived from parameters of dynamic mesh forces. A singular value decomposition (SVD)-based Kalman filter is employed to provide a stable solution for the estimation of state vector and input vector simultaneously. A mesh force indirect measurement method based on tooth fillet strain signals collected by strain gauges is adopted to evaluate the accuracy of identification results. Results of dynamic experiments conducted in a planetary gear set test rig demonstrate that compared with measured results, the proposed method can obtain acceptable estimated mesh forces of ring-planet gear pairs by using vibration signals.

Dynamics analysis of spur gears considering random surface roughness with improved gear body stiffness

In previous calculation of gear mesh stiffness, the gear body is typically treated as a deformable elastic ring, and the stiffness of the gear body is determined using Sainsot's empirical formula. In this paper, a new calculation method for determining the gear body stiffness based on the analytical finite element-potential energy method is proposed, and its feasibility is verified by the finite element method. A new nonlinear dynamic model for a spur gear system is established, taking into account the time-varying backlash and random surface roughness. This model aims to investigate the dynamical responses of the gear system under various parameters, including rotational speed, shaft hole radius and surface roughness. The results indicate that the proposed method for calculating the stiffness of the gear body is significantly more accurate than Sainsot's empirical formula, as it closely aligns with the finite element method. This leads to a more precise calculation of the time-varying mesh stiffness. The gear dynamics model incorporating random surface roughness exhibits a heightened degree of realism and complexity in its dynamic phenomena. This study provides a valuable reference for the future development of dynamic gear systems.

Vibration characteristics analysis of flexible helical gear system with multi-tooth spalling fault: Simulation and experimental study

Gear systems operating at high speeds and under heavy loads are susceptible to continuous multiple tooth surface spalling. The time-varying meshing stiffness (TVMS) model of helical gear pair with multi-tooth spalling fault is established by combining image recognition method and loaded tooth contact analysis (LTCA) method to target the actual form of the spalling fault, and its effectiveness was verified through finite element method (FEM). The flexible helical gear dynamic model is established by utilizing 8-node shell model and Timoshenko beam model to simulate the gear foundations and shafts, the TVMS in multi-tooth spalling fault condition is introduced as the excitation. The mapping relationship between the meshing process, TVMS and vibration response under the influence of rotational speed, spalling surface morphology scale, spalling depth and location of spalling occurrence is analyzed by combining simulation and experimental results. As the rotational speed increases, the specific meshing frequency coincides with the natural frequency of the gear system, leading to the phenomenon of superharmonic resonance. Continuous multi-tooth spalling fault results in continuous time-domain shocks and the meshing frequency modulation of the rotational frequency of the gear with spalling fault. In the absence of bottoming out, the increase in spalling surface morphology scale exerts a more pronounced influence on the vibration characteristics of the helical gear system compared to the increase in spalling depth. The research results can provide theoretical basis for gear system fault diagnosis.

An Improved Coupled Dynamic Modeling for Exploring Gearbox Vibrations Considering Local Defects

Gearbox is a key part in machinery, in which gear, shaft and bearing operate together to transmit motion and power. The wide usage and high failure rate of gearbox make it attract much attention on its health monitoring and fault diagnosis. Dynamic modeling can study the mechanism under different faults and provide theoretical foundation for fault detection. However, current commonly used gear dynamic model usually neglects the influence of bearing and shaft, resulting in incomplete understanding on gearbox fault diagnosis especially under the effect of local defects on gear and shaft. To address this problem, an improved gear-shaft-bearing-housing dynamic model is proposed to reveal the vibration mechanism and responses considering shaft whirling and gear local defects. Firstly, an eighteen degree-of-freedom gearbox dynamic model is proposed, taking into account the interaction among gear, bearing and shaft. Secondly, the dynamic model is iteratively solved. Then, vibration responses are expounded and analyzed considering gear spalling and shaft crack. Numerical results show that the gear mesh frequency and its harmonics have higher amplitude through the spectrum. Vibration RMS and the shaft rotating frequency increase with the spalling size and shaft crack angle in general. An experiment is designed to verify the rationality of the proposed gearbox model. Lastly, comprehensive analysis under different spalling size and shaft crack angle are analyzed. Results show that when spalling size and crack angle is lager, RMS and the amplitude of shaft rotating frequency will not increase linearly. The dynamic model can accurately simulate the vibration of gear transmission system, which is helpful to gearbox fault diagnosis.

Study on nonlinear vibration and primary resonance characteristics of helicopter face gear-planetary gear coupling transmission system

In order to explore the dynamic response and primary resonance characteristics of the gear transmission system of the helicopter main reducer. The face gear-planetary gear torsion-translation vibration dynamics model is established, and the time-varying meshing stiffness, gear backlashes, gear eccentricity error, and friction are considered. The four-order variable-step Runge–Kutta was used to solve the dynamic response of the system, and the nonlinear dynamic characteristics of the system were described by time domain diagram, phase trajectory plane, Poincaré map, 3D frequency spectrum, and bifurcation diagram. Additionally, using the face gear-spur gear system's transmission as an example, the multi-scale method is used to analyze the face gear-spur gear system's primary resonance characteristics. The influence of load fluctuation amplitude, meshing damping, and meshing stiffness on its amplitude-frequency characteristics is also investigated. It is found that when external load excitation frequency changes, the vibration response of the whole system changes synchronously, which exhibits periodic, multi-periodic, and chaotic motion. In addition, according to the primary resonance analysis, the system's stability will be harmed by increasing the external load fluctuations’ amplitude, lowering the meshing damping, and raising the meshing stiffness.

Crack detection of a modified high contact ratio gear

For the purpose of vibration-based fault detection and to avoid gear failure consequences it is necessary to investigate the possibility of early fault detection. A reduction in the gear time-varying mesh stiffness (TVMS) due to the existence of a crack can be used to detect and evaluate tooth damage. The reduction in TVMS has an impact on the system’s dynamic response signal, and then the used fault detection indicators are affected. The current article sheds light on the fault detection possibility of high contact ratio (HCR) gears as compared to those standard ones of low contact ratio (LCR). The influence of modifying the gear tooth addendum dimension for obtaining a high contact ratio on the possibility of early fault detection is studied. Twenty different crack cases are examined by using both the LCR and HCR design cases. For every crack case, the mesh stiffness is evaluated and introduced to the used gear dynamic model for obtaining the system’s dynamic response. The TVMS evaluation is presented for both the LCR and HCR gear cases. Four fault detection indicators, namely the RMS, kurtosis, peak value and crest factor are applied as different measures to enhance the research outcome. The results show that the studied indicators will have less increase when the crack size increases in the HCR design than in the LCR one. Therefore, the HCR design will be less sensitive or more difficult to detect from the fault detection perspective. Lowering the amount of signal noise is applied to the HCR case to examine the impact on enhancing the obtained indicators. The four applied indicators show different behaviours with reducing noise.

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.

Dynamic modeling of spur gear system under marine ship heaving-pitching motion

Sea waves often result in multi-freedom motions of ships, and affect the stability of marine gear systems. This study presents a novel dynamic modeling method for spur gear system considering the influence of marine ship heaving-pitching motion and investigates the spur gear dynamics. An excitation transmission path from wave to gear is illustrated and incorporated into the gear dynamic model. The hydrodynamic force is analyzed and the hull heaving-pitching motion is solved using the New Strip Method (NSM). Considering the time-varying meshing stiffness, transmission error, and tooth side clearance, the motion differential equation of the spur gear system impacted by the heaving-pitching coupling motion of the hull is established. The influences of key wave and navigation parameters on gear dynamics are analyzed. The results show that ship heaving-pitching motion causes additional stiffness and damping excitation, and leads to typical nonlinear phenomena of the spur gear system in a marine ship, such as period, quasi-period, and chaos, and causes a sharp increase in the meshing impact amplitude up to a maximum of 36.81%. The changes in wave frequency, the hull's heading angle, and hull's speed have a significant impact on gear dynamics, whereas the wave amplitude has less of an impact.

Nonlinear dynamic modeling and analysis of spur gears considering dynamic contact state under misalignment errors

Traditional gear dynamic models take the mesh stiffness obtained under static conditions as an internal excitation. Under practical conditions, misalignment errors are unavoidable and will affect the contact state and mesh stiffness. The contact state and mesh stiffness under misalignment errors are dynamic force-dependent (DFD) since they are also affected by the violently fluctuating dynamic meshing force (DMF). Consequently, developing a dynamic model based on statically obtained mesh stiffness is inappropriate under misalignment errors. This study presents a torsional dynamic model of spur gears considering the DFD mesh stiffness and contact state change caused by misalignment errors. The mesh stiffness and torsional dynamic model of the proposed model are jointly solved. Parametric studies are conducted to reveal the differences in the dynamic responses between the proposed and traditional models under different misalignment errors. The dynamic responses of spur gears in resonance and off-resonance regions show that, compared with the traditional model, the proposed model under misalignment errors is more likely to undergo non-periodic and chaotic motion due to contact loss. In addition, misalignment errors lead to a significant increase in the maximum value and oscillation amplitude of the DMF of the proposed model.

Study on gear meshing power loss calculation considering the coupling effect of friction and dynamic characteristics

Gear transmission system is widely used in many mechanical equipment to achieve power transmission and rotational speed changes. Meshing power loss is one key index to assess the performance of a gear system, which directly influences heat generation and lubrication, etc. Over the past decades, many computational models of gear meshing power loss have been proposed to support gear design and optimization. However, the coupling effect between the gear friction and dynamic characteristics are usually ignored, including time-varying meshing stiffness (TVMS) and dynamic meshing force (DMF), which are not conducive to accurate calculation and improvement of meshing power loss. Therefore, an improved gear meshing power loss calculation method is proposed to settle this problem in this paper. With the proposed method, the friction calculation model is established considering the effect of DMF and surface roughness based on mixed elastohydrodynamic lubrication (EHL). Then the TVMS is obtained with the friction force along the meshing line and DMF was taken into consideration. On the basis of modelling of friction and TVMS, a six degrees-of-freedom (DOF) gear dynamic model, as well as a power loss iteration calculation process are established and studied. The effects of gear surface manufacturing quality and loads on power loss are analyzed. Furthermore, a gear meshing power loss experimental setup is constructed to verify the effectiveness of the proposed method. The results show that this method is in better agreement with the experimental results than the traditional method which does not consider the coupling effects.