Time-Varying Mesh Stiffness Calculation and Dynamic Modeling of Spiral Bevel Gear with Spalling Defects

Time-varying mesh stiffness(TVMS)is a vital internal excitation source for the spiral bevel gear(SBG)transmission system.Spalling defect often causes decrease in gear mesh stiffness and changes the dynamic characteristics of the gear system,which further increases noise and vibration.This paper aims to calculate the TVMS and establish dynamic model of SBG with spalling defect.In this study,a novel analytical model based on slice method is proposed to calculate the TVMS of SBG considering spalling defect.Subsequently,the influence of spalling defect on the TVMS is studied through a numerical simulation,and the proposed analytical model is verified by a finite element model.Besides,an 8-degrees-of-freedom dynamic model is established for SBG transmission system.Incorporating the spalling defect into TVMS,the dynamic responses of spalled SBG are analyzed.The numerical results indicate that spalling defect would cause periodic impact in time domain.Finally,an experiment is designed to verify the proposed dynamic model.The experimental results show that the spalling defect makes the response characterized by periodic impact with the rotating frequency of spalled pinion.

Effect analysis of friction and damping on anti-backlash gear based on dynamics model with time-varying mesh stiffness

A nonlinear model of anti-backlash gear with time-varying friction and mesh stiffness was proposed for the further study on dynamic characteristics of anti-backlash gear. In order to improve the model precision, applied force analysis was completed in detail, and single or double tooth meshing states of two gear pairs at any timing were determined according to the meshing characteristic of anti-backlash gear. The influences of friction and variations of damping ratio on dynamic transmission error were analyzed finally by numerical calculation and the results show that anti-backlash gear can increase the composite mesh stiffness comparing with the mesh stiffness of the normal gear pair. At the pitch points where the frictions change their signs, additional impulsive effects are observed. The width of impulsive in the same value of center frequency is wider than that without friction, and the amplitude is lower. When gear pairs mesh in and out, damping can reduce the vibration and impact.

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.

Influence Predictions of Contact Effects on Mesh Stiffness of Face Gear Drives with Spur Gear

Mesh stiffness is one of important base parameters of face gear dynamic studies.However,a calculation solution of mesh stiffness of face gear drives is not to be constructed due to complex geometric flakes of face gear teeth.Thus,a calculation solution of mesh stiffness of face gear drives with a spur gear,which is based on the proposed equivalent face gear teeth and Ishikawa model,is constructed,and the influence of contact effects on mesh stiffness of face gear drives is investigated.The results indicate the mesh stiffness of face gear drives is sensitive to contact effects under heavy loaded operating conditions,specially.These contributions will benefit to improve dynamic studies of face gear drives.

Improved analytical model for mesh stiffness calculation of cracked helical gear considering interactions between neighboring teeth

As one of the most typical fault forms of the helical gear,the crack will change the dynamic excitation and further affect the dynamic behaviors of the transmission systems.Due to the complicated structure of the helical gears,the coupling effect between the neighboring loaded teeth is usually ignored in the mesh stiffness calculation,making it considerably overestimated especially in the case of the crack fault.An improved mesh stiffness calculation method of helical gear with spatial crack is proposed to make up this gap.The interactions between the loaded neighboring teeth induced by the gear body flexibility were considered to improve the calculation accuracy and applicability.Besides,the load distribution law for the engaged cracked tooth along the tooth width and profile can be obtained.The results indicated that the mesh stiffness of the multi-tooth engagement calculation using this model could be further improved compared with the traditional methods.Finally,the effects of the helix angle,crack depth,and crack propagation length on the mesh stiffness and load distribution were investigated using the proposed method.

Mesh stiffness evaluation of an internal spur gear pair with tooth profile shift

Tooth profile shift will change the thickness of gear teeth and a part of geometrical parameters of a gear pair, thus influencing its mesh stiffness and consequently the dynamic performances. In this paper, an analytical mesh stiffness calculation model for an internal gear pair in mesh considering the tooth profile shift is developed based on the potential energy principle. Geometrical representations of the tooth profile shift are firstly derived, and then fitted into the analytical tooth stiffness model of gears. This model could supply a convenient way for mesh stiffness calculation of profile shifted spur gears. Then, simulation studies are conducted based on the developed model to demonstrate the effects of tooth profile shift coefficient on the tooth compliances and the mesh stiffness of the internal spur gear pair. The results show that tooth profile shift has an obvious influence on the mean value, amplitude variation and phase of the mesh stiffness, from which it can be predicted that the dynamic response of an internal gear transmission system will be affected by the tooth profile shift.

Mesh stiffness calculation and vibration analysis of the spur gear pair with tooth crack,considering the misalignment between the base and root circles

An improved variable cross‐section cantilever beam model for evaluating the time‐varying mesh stiffness(TVMS)of the perfect gear tooth is developed in which the tooth number of driving gear is less than 42 and that of driven is more than 42.The TVMS obtained by the proposed method is compared with the result without considering the misalignment between the base circle and gear root.Four types of root crack models and changes inTVMS of 13‐crack levels are presented.The fault vibration characteristic of a single‐stage spur gear reducer with root crack is ana-lyzed and the correctness is qualitatively verified by the vibration signals of an experimental gearbox with crack or missing failure.The results presented in this paper are of great significance for a deep understanding of the possible causes of vibration and noise of gears and provide a theoretical foundation for the fault diagnosis of the gearbox.

Analysis of the Relationship Between the Meshing Stiffness and the Inherent Characteristics of Planetary Gear Transmission Mechanism

According to the relationship between the meshing stiffness and the inherent characteristics of a seven-speed three-row coupled planetary transmission mechanism,a equivalent concentrated mass dynamics model of the planetary transmission mechanism is established.The natural frequency of the planetary gear train at a specific gear is calculated and extracted.The relationship between the meshing stiffness of each row and the natural frequency of the system is analyzed,thereby avoiding possible resonance behavior by changing the meshing stiffness.These results show that the meshing stiffness,in its range of possible values,has nearly no effect on the low order natural frequency(<4.000.Hz),and that the time-varying meshing stiffness mainly affects the natural frequencies of the higher-and middle-order parts of the system.Changes of the natural frequencies lead to the change of the system's corresponding vibration mode,which will change the vibration situation of the system.

Simulation study on dynamics of cylindrical gear meshing with the evolution meshing stiffness

Simulation study on the cylindrical gear meshing with the evolution gear meshing stiffness is being done for better understanding the dynamic characteristics of the kinematics.With consideration of damping,bearing clearance and gear backlash nonlinearity,the dynamic model is set up and computed in MATLAB.The analysis about the relationship between the kinematic responses and the meshing stiffness are carried out.And the results showed that as the gear mesh stiffness is changed from small to large,the performance of the system is changed from the harmonic stable periodic motion to with one times,two times,four times,ending chaos of the stability of the bifurcation.The research results would have theoretical guidance value for the fault diagnosis in engineering.

Nonlinear Dynamic Analysis of Planetary Gear Train System with Meshing Beyond Pitch Point

Nonlinear dynamic analysis was performed on a planetary gear transmission system with meshing beyond the pitch point.The parameters of the planetary gear system were optimized,and a two-dimensional nonlinear dynamic model was established using the lumped-mass method.Time-varying meshing stiffness was calculated by the energy method.The model consumes the backlash,bearing clearance,time-varying meshing stiffness,time-varying bearing stiffness,and time-varying friction coefficient.The time-varying bearing stiffness was calculated according to the Hertz contact theory.The load distribution among the gears was computed,and the time-varying friction coefficient was calculated according to elastohydrodynamic lubrication(EHL)theory.The dynamical equations were solved via numerical integration.The global bifurcation characteristics caused by the input speed,backlash,bearing clearance,and damping were analyzed.The system was in a chaotic state at natural frequencies or frequency multiplication.The system transitioned from a single-period state to a chaotic state with the increase of the backlash.The bearing clearance of the sun gear had little influence on the bifurcation characteristics.The amplitude was restrained in the chaotic state as the damping ratio increased.

Spur Gear Tooth Pitting Propagation Assessment Using Model-based Analysis

Tooth pitting is a common failure mode of a gearbox. Many researchers investigated dynamic properties of a gearbox with localized pitting damage on a single gear tooth. The dynamic properties of a gearbox with pitting distributed over multiple teeth have rarely been investi- gated. In this paper, gear tooth pitting propagation to neighboring teeth is modeled and investigated for a pair of spur gears. Tooth pitting propagation effect on time-vary- ing mesh stiffness, gearbox dynamics and vibration char- acteristics is studied and then fault symptoms are revealed. In addition, the influence of gear mesh damping and environmental noise on gearbox vibration properties is investigated. In the end, 114 statistical features are tested to estimate tooth pitting growth. Statistical features that are insensitive to gear mesh damping and environmental noise are recommended.

Quasi-Static and Dynamic Behaviors of Helical Gear System with Manufacturing Errors

Time?varying mesh stiffness(TVMS) and gear errors include short?term and long?term components are the two main internal dynamic excitations for gear transmission. The coupling relationship between the two factors is usually neglected in the traditional quasi-static and dynamic behaviors analysis of gear system. This paper investigates the influence of short?term and long?term components of manufacturing errors on quasi?static and dynamic behaviors of helical gear system considering the coupling relationship between TVMS and gear errors. The TVMS, loaded static transmission error(LSTE) and loaded composite mesh error(LCMS) are determined using an improved loaded tooth contact analysis(LTCA) model. Considering the structure of shaft, as well as the direction of power flow and bearing location, a precise generalized finite element dynamic model of helical gear system is developed, and the dynamic responses of the system are obtained by numerical integration method. The results suggest that lighter loading conditions result in smaller mesh stiffness and stronger vibration, and the corresponding resonance speeds of the system become lower. Long?term components of manufacturing errors lead to the appearance of sideband frequency components in frequency spectrum of dynamic responses. The sideband frequency components are predominant under light loading conditions. With the increase of output torque, the mesh frequency and its harmonics components tend to be enhanced relative to sideband frequency components. This study can provide effective reference for low noise design of gear transmission.

Characteristic of involute slope modification of asymmetric spur gear

The meshing characteristic of asymmetric involute spur gear was studied, the equations of the geometric shape of the asymmetric gear for both sides were deduced, and the equations of contact ratio and the key points of contact were also obtained.Meanwhile, an involute slope modification method considering the effects of static transmission errors was proposed based on the meshing properties. The characteristic of the involute slope modification was analyzed by changing different modification parameters.The mesh stiffness and synthetic mesh stiffness of unmodified and modified asymmetric spur gears were investigated. Furthermore,the spectrums of synthetic mesh stiffness under different modification parameters were compared. Research results showed that the modification parameters influence the meshing performance of gear pairs, and the proposed modification method was feasible to improve the transmission performance of gear pairs with appropriate modification parameters.

Mesh excitations of spur gear considering strong correlation among tooth contact parameters,contact force,and tooth profile deviations

Gear mesh excitations are widely concerned in the dynamic studies of the gear transmission system.Meanwhile,intentional and unintentional tooth profile deviations often occur in gears.At present,the established calculation models of gear mesh excitations consider tooth profile deviations as displacement excitation.However,gear mesh excitations calculated by such models have reduced stability compared with the actual situation.Therefore,in this study,an improved analytical model of gear mesh excitations with tooth profile deviations is established.This established model considers tooth profile deviations,extended tooth contact,and the structure coupling effect of the gear body simultaneously.More importantly,the model considers the strong correlation among tooth contact parameters,contact force,and tooth profile deviations to better reflect the actual gear mesh.A calculation flowchart with a simple calculation method of contact forces is also proposed to calculate the gear mesh excitations.Finally,the effects of tooth profile deviations on gear mesh excitations are studied.The results show that the effects of tooth profile deviations on tooth contact position,the direction of contact force,and equivalent basic circle radii should be considered in the calculation of gear mesh excitations because of smaller system transmission errors,larger double-teeth meshing area,and slighter extended tooth contact.Tooth profile deviations also cause jumps in tooth contact position and time-varying mesh stiffness.Thus,our findings show that the proposed model can be used to calculate the gear mesh excitations more accurately when the tooth profile deviates greatly.

Dynamics of a Dual Power-split Transmission Based on Loaded Tooth Contact Analysis

In order to implement the dynamic characteristic of a dual power-split transmission, a dynamic me- chanics model is built. Firstly, according to the method of theoretical analysis of the tooth contact analysis （TCA） and loaded tooth contact analysis （LTCA）, the actual meshing process of each gear pairs is simulated, and the time-varying mesh stiffness excitations are obtained, which can improve the numerical precision. Second- ly, by using the lumped mass method, the bending-torsional coupling three dimensional dynamic model of the dual power-split transmission is established, and the identical dimensionless equations are deduced by elimina- ting the effect of rigid displacement and the method of dimensional normalization. Finally, by the method of the fourth order Runge-Kutta algorithm with variable step lengths, the responses of this system in a frequency domain and time domain are obtained, and the dynamic load change characteristics of each gear pairs are analyzed. The results show that the establishment, solution and analysis of the system dynamics model could provide a basis for the dynamic design, and have an important significance for the dynamic efficiency analysis and dynamic perform- ance optimization design of the dual power-split transmission.

Gear fault diagnosis using gear meshing stiffness identified by gearbox housing vibration signals

Gearbox fault diagnosis based on vibration sensing has drawn much attention for a long time.For highly integrated complicated mechanical systems,the intercoupling of structure transfer paths results in a great reduction or even change of signal characteristics during the process of original vibration transmission.Therefore,using gearbox housing vibration signal to identify gear meshing excitation signal is of great significance to eliminate the influence of structure transfer paths,but accompanied by huge scientific challenges.This paper establishes an analytical mathematical description of the whole transfer process from gear meshing excitation to housing vibration.The gear meshing stiffness(GMS)identification approach is proposed by using housing vibration signals for two stages of inversion based on the mathematical description.Specifically,the linear system equations of transfer path analysis are first inverted to identify the bearing dynamic forces.Then the dynamic differential equations are inverted to identify the GMS.Numerical simulation and experimental results demonstrate the proposed method can realize gear fault diagnosis better than the original housing vibration signal and has the potential to be generalized to other speeds and loads.Some interesting properties are discovered in the identified GMS spectra,and the results also validate the rationality of using meshing stiffness to describe the actual gear meshing process.The identified GMS has a clear physical meaning and is thus very useful for fault diagnosis of the complicated equipment.

Nonlinear parametrically excited vibration and active control of gear pair system with time-varying characteristic

In the present work, we investigate the nonlinear parametrically excited vibration and active control of a gear pair system involving backlash, time-varying meshing stiffness and static transmission error. Firstly, a gear pair model is established in a strongly nonlinear form, and its nonlinear vibration characteristics are systematically investigated through different approaches. Several complicated phenomena such as period doubling bifurcation, anti period doubling bifurcation and chaos can be observed under the internal parametric excitation. Then, an active compensation controller is designed to suppress the vibration, including the chaos. Finally, the effectiveness of the proposed controller is verified numerically.

Mesh relationship modeling and dynamic characteristic analysis of external spur gears with gear wear

Gear wear is one of the most common gear failures,which changes the mesh relationship of normal gear.A new mesh relationship caused by gear wear affects meshing excitations,such as mesh stiffness and transmission error,and further increases vibration and noise level.This paper aims to establish the model of mesh relationship and reveal the vibration characteristics of external spur gears with gear wear.A geometric model for a new mesh relationship with gear wear is proposed,which is utilized to evaluate the influence of gear wear on mesh stiffness and unloaded static transmission error(USTE).Based on the mesh stiffness and USTE considering gear wear,a gear dynamic model is established,and the vibration characteristics of gear wear are numerically studied.Comparison with the experimental results verifies the proposed dynamic model based on the new mesh relationship.The numerical and experimental results indicate that gear wear does not change the structure of the spectrum,but it alters the amplitude of the meshing frequencies and their sidebands.Several condition indicators,such as root-mean-square,kurtosis,and first-order meshing frequency amplitude,can be regarded as important bases for judging gear wear state.