Gear Mesh Stiffness Research Articles

Improved analytical method for gear body-induced deflections with tooth root crack considering structural coupling effect

There is no existing effective analytical method for calculating the mesh stiffness of spur gears with tooth root crack when the interactions between the neighboring teeth induced by gear body flexibility are considered, while this factor has been demonstrated to have a considerable effect on the mesh stiffness. In this paper, an improved method of spur gear mesh stiffness is proposed based on the theory of elastic mechanics for circular ring, which is capable of calculating the mesh stiffness with tooth root crack accurately. Then, the calculated gear body-induced stiffness results considering structural coupling effect with tooth root crack is validated by the finite element method. In addition, the effect of different hub hole radius and crack lengths on the stiffness caused by the structural coupling effect are investigated. The results indicate that the proposed analytical method is more accurate for spur gear mesh stiffness calculation with tooth root crack.

Research on gear mesh stiffness of helical gear based on combining contact line analysis method

A new analytical model of mesh stiffness of helical gears is proposed using a combination of contact line analysis with slice method, which can well predict the dimensionless time-varying mesh stiffness. Based on the derived formula, the influence of gear basic parameters on mesh stiffness fluctuation coefficient is studied. The simulation shows that the influence of the overlap ratio on the mesh stiffness fluctuation coefficient is greater than that of the transverse contact ratio. By designing reasonable helix angle and tooth width, the overlap ratio is close to an integer, and then the mesh stiffness fluctuation coefficient can obtain a local minimum value, which is an effective way to reduce the vibration and noise of gear system.

A general contact stiffness model for elastic bodies and its application in time-varying mesh stiffness of gear drive

As a critical element of time-varying mesh stiffness (TVMS), contact stiffness of a gear drive has been defined based on simplified Hertzian contact stiffness or semi-empirical nonlinear Hertzian contact stiffness in previous works. This study proposes a general contact stiffness model for elastic bodies through piecewise linear interpolation of contact pressure. The TVMS of a spur gear drive is determined through potential energy method and proposed contact stiffness model verified by Hertzian contact theory and finite-element method. Then, the influence of applied load on contact stiffness is studied, and the differences among proposed contact stiffness, simplified Hertzian contact stiffness, and nonlinear Hertzian contact stiffness are analyzed. Results show that contact stiffness increases with the applied load, and the TVMS based on the proposed contact stiffness model is the smallest among the three contact stiffness models. Effects of tooth width and input torque on the TVMS are further discussed. The TVMS becomes bigger with increased tooth width and input torque, but the increase rate decreases as tooth width or input torque increases. These findings indicate that reasonable matching of design parameters is beneficial for increasing load capacity and optimizing the dynamic performance of gear systems.

기어시스템에서 설계변수에 따른 전달오차와 물림강성 해석

The gear has a wide range of transmitted force as various gear ratios are possible using a combination of teeth. It can transmit power reliably and cause relatively little vibration and noise. For this reason, the application of reducers of electric vehicles is being expanded. Vibration noise generated from gears is propagated into the quiet interior of a vehicle, causing various claims. In most gear studies, transmission error has been pointed out as the main cause of vibration noise of gears. Transmission errors have various causes, including design factors, manufacturing factors, and assembly factors. In general, when predicting transmission error through finite element analysis, design factors play an important role without considering manufacturing factors or assembly factors. In this study, relationships among important design variables (gear module, compensation rate, load torque, and transmission error) in gear design were investigated using analytical and experimental methods. In addition, a method of predicting gear meshing stiffness through the predicted gear transmission error was proposed to obtain variation of meshing stiffness due to changes of gear design parameters.

Nonlinear dynamic analysis of high-speed gear pair with wear fault and tooth contact temperature for a wind turbine gearbox

This paper aims to reveal the relationship between the high-speed gear wear and nonlinear dynamics for a wind turbine gearbox considering the tooth contact temperature. The temperature and deformation of the tooth surface are calculated respectively using the Blok's flash temperature theory and thermal deformation formula, and the meshing stiffness caused by the tooth contact temperature is obtained using the Hertz theory. The wear depth and time-varying meshing stiffness of high-speed gears are calculated using the Archard's equation and potential energy method, respectively. The nonlinear dynamic model of the gearbox is established to study the characteristics, and the frequency domain values of gear wear are compared with experimental data. The results show that the tooth contact temperature makes the time-varying meshing stiffness decrease, then affecting the dynamics of the system, the stability decreases with the increase of comprehensive transmission error, and for the gearbox considering the contact temperature, the gear wear makes the chaotic motion occur in advance and its region increase obviously. The research provides a basis for the wear fault diagnosis and detection of gear transmission systems.

Dynamic modeling of the gear-rotor systems with spatial propagation crack and complicated foundation structure

Considering both spatial crack fault and complex foundation structure, the dynamic finite element model and multi-body dynamic model are established simultaneously to analyze the dynamic characteristics of gear-rotor systems. Firstly, the extended finite element method is used to simulate the three-dimensional crack propagation paths of spur gears under partial load. Then, based on the proposed three-dimensional loaded tooth contact analysis (3D-LTCA) method, the mesh stiffness of spur gears with the spatial crack and complex foundation is calculated and the results are verified by the finite element method. Finally, the dynamic finite element model and multi-body dynamic model of the gear-rotor system are established. The dynamic responses under different crack levels are analyzed. The advantages and disadvantages of these two dynamic models are evaluated. According to the above analysis, it is indicated that the 3D-LTCA can accurately consider the influence of complex foundation and spatial crack on mesh stiffness. The finite element model and multi-body dynamic model can effectively analyze the influence of crack fault and foundation structure on dynamic responses.

Investigation of Spur Gear Dynamics with Gear MeshImpacts Induced by Tooth Wear

Abstract Tooth wear is one of the most common faults in gear transmission systems. However, due to its limited influences on gear dynamics, the condition monitoring of gear wear experiences more challenges. To investigate the relationship between tooth wear and gear dynamics, this study develops an eight-degree of freedom dynamic model which takes into account the gear mesh impact induced by tooth wear. In this model, the gear mesh impact force induced by tooth wear was calculated as an extra input excitation for the gear dynamic model. Besides, a comparative study was carried out to explore the real reason that causes the poor dynamic behaviour due to the wear from two points of view: 1) the reduction in gear mesh stiffness; 2) the increase in gear mesh impact force. The simulation results show that the wear causes a slight reduction in gear mesh stiffness, thus leading to the decrease in the amplitude of gear mesh frequency. This is not consistent with the phenomena observed in the lab experiment which shows the increase in the gear vibrations. The second simulation of impact force demonstrates that the wear causes an increase in the impact force which further leads to the increase in the amplitude of gear mesh frequency. This study investigates the cause for the deterioration in gear dynamics induced by gear wear, which can benefit the early condition monitoring and fault diagnosis of the gear tooth wear.

Nonlinear vibration characteristics of gear-rotor-bearing system

Based on the Timoshenko beam element model in the finite element theory and the lumped mass method, the meshing dynamics equation of helical gears was established considering the time-varying meshing stiffness of helical gears, tooth side clearance, bearing support stiffness and other factors, and the nonlinear dynamics model of the gear-rotor-bearing system was established. Through the analysis of the dynamics model, meshing stiffness, backlash and the influence of bearing stiffness for the system vibration characteristics, studies have shown that the bearing stiffness for the purpose of this article studies the effect to the stability of the main reducer system transmission, vibration noise of the gear-rotor-bearing system parameter selection, and provides theory basis for diagnosis.

Dynamic modelling of two-stage gear systems with localized wear defects

Abstract Considering the profile deviation induced by localized wear defects, a dynamic model of two-stage gear systems is established. In the proposed dynamic model, the time-varying mesh stiffness of worn gears is calculated by the loaded tooth contact analysis method. The mesh stiffness and the non-loaded transmission error are regarded as the time-varying stiffness excitation and the displacement excitation, respectively. Based on the proposed dynamic model, the dynamic responses of the two-stage gear system with localized wear defects are acquired. The simulated vibration responses are compared to the measured results to test the effectiveness of the proposed dynamic model. It is found that the proposed model can yield similar fault feature with the experiment, which shows the validity of the proposed dynamic model. This paper can afford theoretical basis for the fault mechanism and dynamic characterization of localized wear defects.

A new effective mesh stiffness calculation method with accurate contact deformation model for spur and helical gear pairs

This paper proposes a new effective mesh stiffness calculation method for spur and helical gear pairs, which is explicitly formulated by an original FE-analytical slice model for tooth local contact deformation, efficient FE model for gear global deformation and mathematical programming approach for gear contact. The original FE-analytical slice model comprehensively integrates the advantages of FE method, Hertz contact theory and slice approach, which can address the displacement datum and elastic interaction of tooth slices effectively, so that tooth contact deformation is determined accurately. The geometry modeling of tooth slices is established and the pressure configuration in the contact region of each slice is prescribed by Hertz distribution. A specified FE programming is developed to derive the tooth contact flexibility matrix accurately. Thus, the mesh stiffness can be effectively calculated by using the proposed method with the FE-analytical slice model. The local contact deformation of gear tooth and mesh stiffness of gear pairs under ideal and error conditions are simulated. The accuracy and efficiency of the proposed method are validated against the ANSYS benchmark and common existing methods.

Effect of a Novel Tooth Pitting Model on Mesh Stiffness and Vibration Response of Spur Gears

The existence of pitting failure has a direct influence on the time-varying mesh stiffness (TVMS) and thus changes the vibration properties of the gears. The shape of pitting on the tooth surface is characterized by randomness and geometric complexity. The overlapping pitting shape has rarely been investigated, especially when the misalignment of gear base circle and root circle was considered. In this paper, the pitting shape is considered as approximately the union of several ellipse cylinders, in which the gear tooth is treated as a cantilever beam starting from the root circle. Then, the TVMS of perfect gear and that of gear with different pitting severity levels are solved by the potential energy method. The effect of pitting size on TVMS is discussed in detail. In addition, the vibration response in the frequency domain for the gear system is analyzed, and the effectiveness is qualitatively verified by comparing with the vibration signals of the experimental gearbox. The results indicate that the new pitting model overcomes the problem of ignoring the overlap between different pits and is more consistent with the actual situation. The presence of tooth pitting reduces the TVMS, and the complex sidebands appear around the gear mesh frequency and its harmonics. The proposed model can be used to predict the fluctuation of gear mesh stiffness when tooth pitting occurs, and the corresponding dynamic characteristics can provide the theoretical basis for gear condition monitoring and fault diagnosis.

A revised method to calculate time-varying mesh stiffness of helical gear

Time-varying mesh stiffness (TVMS) of gear plays vital role in analysing dynamic characteristic of gear transmission. So accurately evaluating the TVMS is important and essential. In this paper, a revised method to calculate the TVMS of helical gear is proposed. Based on slice method, the helical gear is sliced into pieces along the tooth width direction. The proposed method corrects the fillet foundation stiffness within multi-tooth in contact and considers the non-linearity and load-dependence of the Hertzian contact stiffness. The effect of the axial mesh force is considered. Finally, an equivalent helical gear model is established in ANSYS to study the mesh stiffness. The results show the proposed method has high effectiveness compared with FEM (finite element method).

A mathematical model for vibration analysis of a parallel hybrid electric bus powertrain

This paper presents a parallel hybrid electric bus equipped with an automated manual transmission (AMT) and a mathematical model of the hybrid powertrain is developed for vibration analysis. The powertrain dynamic model is established by a modular modelling approach. A detailed AMT dynamic model, considering gear time-varying meshing stiffness, shaft elastic deformation, and bearing elastic support, is incorporated into the powertrain dynamic model. The damping and gyroscopic effect of gear-rotor in the AMT are considered as well. The AMT dynamic model is validated by experiment data from the time and frequency domain comparisons. Finally, the parameter analysis of the dual-mass flywheel (DMF) is utilised to illustrate how to use the proposed powertrain dynamic model for vibration reduction. The influence of the DMF parameters on vibration responses of the system with varying engine rotation speed is investigated. This study provides a basis for further vibration control of the hybrid powertrain during the engine driving mode.

Dynamic modeling of central bevel gear transmission system in aero-engine

Aero-engine is the most critical part of an aircraft. As its key component, the central bevel gear transmission system is always designed only considering the meshing vibration and ignoring structural characteristics of shafts and bearings, and comprehensive vibration characteristics is not well addressed. The transmission system may suffer severe vibration. To accurately study the dynamic characteristics of the system, a model of a central bevel gear transmission system is developed considering the effects of bevel gear mesh stiffness, shaft stiffness, bearing stiffness, and unbalance excitation in this paper. It has been validated by finite element software ANSYS. On this basis, the influence of variation of bearing stiffness and mesh stiffness on the natural characteristics of the central bevel gear transmission system is investigated. The influence of unbalance on the radial vibration and gear meshing force is studied by the analytical calculation method. A case study is carried out. The results show that variation of stiffness within a certain range will cause dramatic changes in the natural characteristics of the system. The unbalance of the driven bevel gear shaft has an effect on the radial vibration and meshing force of the system. The proposed model can reflect radial vibration characteristics, axial vibration characteristics, torsional vibration characteristics, and some coupled natural characteristics of the system. This research provides a theoretical basis to optimize dynamic performance and reduce the vibration of the system and will be helpful in designing the central bevel gear transmission system.

Analytical calculation models for mesh stiffness and backlash of spur gears under temperature effects

The temperature has a great contribution to the mesh stiffness and backlash of the gear pair. Presence of thermal deformation caused by temperature will complicate the gear teeth interaction. In this paper, the thermal time-varying stiffness model and thermal time-varying backlash model are proposed with the consideration of tooth profile error and total thermo-elastic deformation consists of the teeth deformation, teeth contact deformation, and gear body-induced deformation. The key parameters of thermo-elastic coupling deformation affected by temperature are calculated. Based on the proposed models, the influencing mechanism of temperature on the tooth profile error, mesh stiffness, total deformation, and backlash are revealed. The effects of shaft radius and torque load on the thermal stiffness and thermal backlash are studied. The proposed thermal stiffness and backlash calculation model are proven to be more comprehensive and the correctness is validated.

Analytical models for thermal deformation and mesh stiffness of spur gears under steady temperature field

Thermal effect is an important cause of engineering failure of gear system. However, the lack of accurate model limits the research on the thermal behavior of gear transmission system. Here, based on the linear thermal expansion theory, involute profile theory and energy method, the thermal expansion model, the mesh stiffness model, the load sharing ratio model and the loaded static transmission error model of spur gears are proposed which comprehensively consider the thermal changes of material properties, theoretical involute, thermal profile error and actual contact positions. The corresponding finite element models are established to verify the accuracy of the proposed model and published models. The results indicate that the proposed model is accurate enough. The increase of temperature leads to the decrease of mesh stiffness, the unevenness of load sharing ratio and the increase of loaded static transmission error, accompanied by the phenomenon of earlier meshing. The hub bore radius has obvious influences on the thermal profile error and mesh stiffness, and the load torque also has apparent influences on mesh stiffness. The research provides supports for gear thermal effect analysis and gear system design.

Analysis of transmission box vibration characteristics under random road torsional excitation

Random road torsional excitation is a key excitation condition for transmission box vibration of tracked vehicles. In order to accurately analyze influences of random road torsional excitation on the vibration characteristics of the transmission box, a calculation method of this excitation for tracked vehicle is proposed based on the random expression of the roughness of standard road surface. Furthermore, random road torsional excitations under different road grades and vehicle speeds are simulated. With the finite element method and modal superposition method, the box body is discretized, and the elastic characteristics of the box body are characterized to explore the dynamics coupling mechanism of gear shafting and the box body. By considering bending-torsional coupling vibration of gear shafting under multi-source excitations, such as the fluctuated torque of engine and dynamic meshing stiffness of gears, dynamic coupling model of gear shafting and box body under random road torsional excitation is established. The dynamic response of the gearbox under random road torsion excitation is obtained by co-simulation with the variable step length Runge-Kutta method. Influences of different road grades, track preload and vehicle speeds on dynamic response characteristics of the gearbox are analyzed. Real vehicle road test scheme is designed to obtain surface acceleration response of the box body at different speeds on the cement road surface. Both test and simulation results are compared and analyzed to verify the accuracy of the simulation method, which provides a theoretical reference for dynamic optimization of the transmission box.

Nonlinear dynamic analysis of gear system in shearer cutting section under wear failure

Gear wear failure is one of the important failures of the gear system in the shearer cutting section. To reveal the influence mechanism of shearer cutting gear wear on the system dynamic characteristics, considering coupling factors such as time-varying meshing stiffness, dynamic gear clearance, internal error excitation, end load constraint and bearing radial clearance under wear failure, an improved dynamic model of shearer drive gear system is introduced to present an in-depth investigation of uniform wear of gear teeth effect. The dynamic meshing stiffness of gears under different degrees of wear is analysed. Furthermore, the bifurcation diagram is utilized to observe the motion state of the system experiencing different excitation frequencies, support damping as well as terminal loads. It is demonstrated that the gear surface wear could bring a change in gear dynamic transmission error, vibration impact state and amplitude, which is mainly manifested in increasing the unstable area and the vibration amplitude of the gear system, providing a method for monitoring and diagnosing of gear surface faults.

Dynamic modelling of the gear system under non-stationary conditions using the iterative convergence of the tooth mesh stiffness

This paper presents a new gear dynamic model for the transmission system under the non-stationary condition with the consideration of the coupling between the external load and the internal gear excitation. The gear mesh stiffness variation is found to be the main internal excitation and therefore, the effect of the varying external load on the gear mesh stiffness is evaluated firstly, which can then be incorporated into the gear dynamic model. When the gearbox is subjected to a wide range of external load, the mesh stiffness varies correspondingly and an iterative process is proposed to ensure the convergence of the gear dynamic simulation at each step. The dynamic responses from the model with and without the iterative process have been compared. It indicates that the proposed model can successfully incorporate the external-internal coupling effect in gearbox and further shows that the operating condition has a significant influence on the gear response. The proposed model can provide an effective tool to improve the accuracy of gear dynamic modelling and improve the accuracy of gear fault diagnostics under non-stationary conditions.

A new analytical model for evaluating the time-varying mesh stiffness of helical gears in healthy and spalling cases

Accurately evaluating the time-varying mesh stiffness (TVMS) of a mating gear pair is one of the most important aspects in gear vibration dynamic simulation. Existing work on estimating TVMS of helical gears with tooth spalls generally assumes the defects are in flat bottom shape which deviates from the spalling features as observed in curved bottom shape in practice. The cliff like spalling feature results in inaccurate calculation of TVMS of helical gears with such spalls. This paper proposes an advanced method to evaluate the TVMS of helical gears with tooth spalls with curved-bottom features. In the proposed method, the gear tooth is sliced along the center line of tooth spall in order to easily evaluate the equivalent cross-section area and area moments of inertia of the gear tooth which enables the proposed method adapts to tooth spall(s) with curved-bottom features in arbitrary position, angle and size. The proposed method corrects the foundation stiffness errors of existing methods and takes the influence of axial mesh forces into consideration. Its effectiveness on modelling tooth spalls with curved-bottom features for helical gears is validated by FEM (finite element method).