Time-varying Mesh Stiffness Research Articles

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.

Research on nonlinear dynamic performance of the central bevel gear transmission system in aero-engine with complex excitation

For the requirement of the vibration control and fault diagnosis of the central bevel gear transmission system in an aero-engine, this paper proposes a 12-degree-of-freedom dynamic model for the first time in order to investigate nonlinear dynamic behavior of the central bevel gear transmission system considering the internal and external excitations. The dynamic model is developed by the use of lumped parameter modeling. Time-varying meshing stiffness, gear backlash and gear transmission error are included in the internal excitations, the input power of the central bevel gear transmission system, the speed fluctuation of the driven shaft and the high-pressure rotor unbalance excitation are included in the complex external excitations. Using the bifurcation diagram, the largest Lyapunov exponent, time domain diagram, frequency domain diagram, phase diagram and Poincaré map, the effects of input power change, driven shaft speed fluctuation excitation and high-pressure rotor unbalance excitation on the complex nonlinear dynamic behavior of the central bevel gear transmission system are analyzed. The increase of input power, reduction of speed fluctuation and control of high-pressure rotor unbalance are beneficial to expand the stable area of the system.

Rattle Dynamic Model and Vibration Behaviors of Noncircular Face Gear

Abstract Noncircular face gear (NFG) is a new type of variable transmission ratio gear mechanism. Based on nonlinear dynamic theory of gear, the rattling behavior of the NFG under multifrequency excitation was studied. The generalized transmission ratio function of the NFG with closed pitch curve was constructed by Fourier series. The nonlinear relative displacement between teeth was described by an analytical expression during the process of time-varying transmission. Considering the factors of time-varying instantaneous center, meshing stiffness, transmission error, and backlash, a nonlinear dynamic model of the gear was established with the rotational angle as the independent variable. The dynamical responses were calculated by Runge–Kutta method to analyze the effects of the input speed, load torque, transmission ratio coefficient, and error on rattling behaviors. The results show that under the action of combined excitation of time-varying instantaneous center and meshing stiffness, the rattle of noncircular gear is more likely to occur. As the increase in input speed, error, and range of transmission ratio variation or the decrease of load, the NFG appears no rattling, unilateral, and bilateral rattle successively and the rattling intensity increases accordingly. The rattling threshold speed can be improved by increasing the accuracy and load of gear or decreasing the range of the transmission ratio.

An improved mesh stiffness calculation model for cracked helical gear pair with spatial crack propagation path

As one of the most common failure modes for helical gear system, the gear crack in helical gear pair is studied, the improved spatial crack propagation paths are modelled for helical gear pair based on helical gear crack failure mode in actual working condition, in the direction along tooth width, the crack initiates in one end face of gear, and it propagates to tooth addendum or opposite end of the gear, meanwhile, along crack depth direction, the crack propagation path is modelled into straight line or curved line. Furthermore, the limiting line, to characterize the crack effect, is simulated as straight line or curved line. Unlike previous studies, the transverse and axial gear foundation stiffness are modified due to gear crack. In addition to transverse and axial gear tooth stiffness. Based on the improved mesh stiffness calculation method for cracked helical gear pair, the influence of crack parameters and crack types on the time varying mesh stiffness of helical gear pair are studied, and the analytical calculated results are validated against the simulated results of FE (finite-element) models.

Analysis of Nonlinear Vibration Characteristics of the Concentric Face-Gear Split-Torque Transmission System

The concentric face-gear split-torque transmission system (CFGSTTS) is a new type of transmission that has significant applications in helicopter main gearboxes. To study the influence of various parameters on the dynamic characteristics of the CFGSTTS, a 23-degree-of-freedom translation-torsion nonlinear dynamic model was established based on the lumped parameter theory. The model includes tooth backlash, error excitation, time-varying meshing stiffness with meshing phase difference, meshing damping, and elastic support deformation. The excitation conditions for the time-varying meshing stiffness of face-gear pairs were calculated based on the strain energy theory. The bifurcation characteristics of the system with different parameters were obtained by the nonlinear dynamics numerical analysis method. The research shows that the system exhibits rich vibration response characteristics at different rotating speeds. The amplitude of the vibration displacement in the system bifurcation diagram increases significantly with the increase of the tooth backlash and input torque, whereas the amplitude decreases constantly with the increase of the meshing damping. The critical rotational speed at which chaotic motion occurs increases significantly with increasing input torque and damping ratio but decreases with increasing tooth backlash. The bearing clearance has a weak influence on the vibration displacement amplitude of the system and the speed range of chaotic motion.

Bifurcation and Chaos Analysis of Gear System With Clearance Under Different Load Conditions

In the transmission process of gear system, the change of load will make the system in different states of motion, which affects the transmission efficiency of gear system. It is important to investigate the nonlinear dynamic characteristics of gear system under different load states. Using straight cylindrical gears as the object of study, the concentrated mass method is used to establish a dynamic model that takes into account nonlinear factors such as tooth side clearance, time-varying meshing stiffness and transmission errors. The differential equations of the system are solved by the Longe-Kutta method to obtain the bifurcation diagram, the maximum Lyapunov exponent diagram and the phase plane diagram of the gear system to analyze the effect of the meshing damping ratio on the dynamic characteristics of the system under different load states. The results show that the influence of the engagement damping ratio on the dynamic characteristics of the system is greater under light load conditions, showing different states of motion as the engagement damping ratio gradually increases. Under heavy load conditions, the effect of the engagement damping ratio on the dynamic characteristics of the system is small. Appropriately increasing the mesh damping ratio is beneficial to the gear system to avoid the chaotic zone and maintain a stable cyclic motion state. The results of the study provide a reference for the design of gear systems with variable loads.

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.

Nonlinear dynamic analysis of the central bevel gear transmission system in aero engine subjected to internal and external excitation

The research on the nonlinear dynamic characteristics of the central bevel gear transmission system was carried out for the requirement of the vibration control and fault diagnosis of the central bevel gear transmission system in an aero engine. This paper proposes the dynamic model for the first time in order to investigate nonlinear dynamic behavior of the central bevel gear transmission system, and the internal and external excitations are taken into account by using lumped parameter modeling. Time-varying meshing stiffness, gear backlash, and gear transmission error is included in the internal excitations, and the input power, the speed fluctuation of the driven shaft and the high-pressure rotor unbalance excitation are included in the external excitations. The effects of the external excitations on the complex nonlinear dynamic behavior of the system are studied. The results show that reasonable control of input power, speed fluctuations, and high-pressure rotor mass imbalance can improve engine vibration.

A New Resultant Vibration Acceleration Model of a Planetary Gear Train and Fault Response Analysis

A new resultant vibration acceleration model is proposed to reveal its signal characteristics more accurately in the healthy and faulty state. First, an analytical lateral-torsional coupled dynamic model is developed with consideration of time-varying mesh stiffness and damping, static transmission error, and gear backlash. Then, the effect of gear backlash and damping on the system is analyzed, and a numerical velocity signal in the healthy and faulty state is carried out. Considering the effects of transmission paths, the resultant acceleration signal along the vertical direction is constructed as the weighted vibration summation. This signal includes a vertical component of the vibration along the meshing lines of both sun-planet and ring-planet pairs. Moreover, it also contains the vertical component of the planet gears, sun gear, and planet carrier acceleration relative to their own supporting bearings. Finally, the simulation results from the resultant signal model are experimentally validated and analyzed in both time and frequency domains.

Dynamic Analysis of a High-Contact-Ratio Spur Gear System with Localized Spalling and Experimental Validation

The dynamic characteristics and tooth spalling fault features are studied for the high-contact-ratio spur gear bearing system. The bending torsional dynamic model is proposed in this study for the gear bearing system with an ellipsoid spalling fault. This model also considers time-varying meshing stiffness, tooth friction, fractal gear backlash, and comprehensive transmission error. The meshing stiffness of the system is evaluated using the potential energy method. The bifurcation diagram, time-domain waveform, Poincaré map, phase map, frequency spectrum, and related three-dimensional map are used as tools to analyze the system’s dynamic response qualitatively. The results reveal that the system’s motion with ellipsoid tooth spalling defect exhibits rich dynamic behavior. The response of the proposed dynamic model is consistent with experimental results in the frequency domain. Therefore, the developed dynamic model can predict the system’s vibration behavior with localized spalling fault. Hence, it could also provide a theoretical foundation for future spall defect diagnosis of the gear transmission system.

Dynamic characteristics of a straight bevel gear drive system considering multi-state meshing and time-varying parameters

In order to investigate the dynamic characteristics of the straight bevel gear drive system, its torsional-bending-axial dynamics model considered the multi-state meshing characteristics is established. The time-varying load distribution ratio and meshing stiffness of the system are calculated by micro element method according to the performance of the straight bevel gear. Three different Poincaré sections are defined to study the dynamic characteristics of the system. The mechanism of tooth disengagement and back-side contacting is studied according to the time history diagram of the total normal force. The nonlinear dynamics equation of the system is solved by the variable step fourth-order Runge-Kutta method when the bevel gear drive system in automobile reducer is taken as the research object. The transition process of system dynamic characteristics and the influences of the meshing frequency, load coefficient and the comprehensive transmission error are studied by using the time history diagram of the total normal force, phase portraits, Poincaré maps, bifurcation diagrams and the corresponded top Lyapunov exponent (TLE). The chaotic motion found in the research is unfavorable from a practical point of view which can be avoided by adjusting the value of parameters or the initial value of state variables in engineering. The research can provide a theoretical basis for the parameter design and dynamic characteristic control of straight bevel gear drive system.

Study on mesh stiffness and sensitivity analysis of planetary gear system considering the deformation effect of carrier and bearing

The mesh stiffness characteristics and tooth root crack propagation of planetary gear system not only have an important impact on its transmission error, but further affect its vibration and noise level and the bearing capacity of gearbox. It is of great theoretical and engineering significance to study them comprehensively. First, the stiffness-damping model of the planetary gear mechanism is obtained by simplification, and the meshing phase relationships between the studied gear pairs are gotten by theoretical calculation. Then, through the FE modeling, the time-varying mesh stiffness of the sun-planet substructure, ring-planet substructure, and single-branch sun-planet-ring substructure are calculated respectively, and the influence of carrier and bearing flexibility on mesh stiffness is further studied. The mesh stiffness prediction method of the single-branch substructure is proposed and verified, and the overall mesh stiffness of the planetary gear system is predicted and analyzed. Finally, a sensitivity analysis model is established to study the effect of crack propagation at the root of sun and ring on mesh stiffness, capacity, and NVH features. The results show that the sun tooth root crack has a more significant impact on the mean stiffness and bearing capacity, and the ring tooth root crack has a more dramatic influence on the peak-to-peak stiffness and the vibration and noise characteristics of the gear system.

Prediction of tribological and dynamical behaviors of spur gear pair considering tooth root crack

Variations in gear tooth meshing stiffness and tribological characteristics caused by tooth root cracks seriously affect the dynamic performance of gears, limiting the development of durable high strength gears. Therefore, the potential energy method was applied to obtain the time-varying meshing stiffness of the spur gear pair, and the finite element method was applied to verify the calculation results. Secondly, the load distribution of the gear with the cracked tooth root was derived using coordinated deformation conditions, and the effect of the tooth root crack on the tribological characteristics of the tooth surface was obtained based on the elastohydrodynamic lubrication (EHL) model. Finally, the tribo-dynamic behaviors of the gear under the action of the root cracks were investigated. The results indicated that root cracks change the load distribution ratio in the double-tooth meshing zone, which leads to corresponding changes in the tooth surface film thickness and friction coefficient. In addition, when the cracked teeth are engaged in meshing, the reduction in meshing stiffness causes vibration and shock, which causes the appearance of sidebands near the main meshing frequency. The amplitude of the dynamic meshing force increased with the increase in the degree of root cracks.

An improved analytical method for mesh stiffness calculation of helical gear pair considering time-varying backlash

The mesh stiffness characteristics of the gear pair have important influence on its transmission error (TE), and then affect the vibration and noise level of the gears. It is of great significance to develop a calculation method that can take both accuracy and efficiency into account. First, considering the difference between base circle radius and root circle radius, the potential energy method is improved to calculate the mesh stiffness of involute spur gear pair. Then, on the basis of mesh stiffness theory of spur gear and time-varying backlash effect, a new method for calculating the mesh stiffness of involute helical gear pair is proposed by improving the traditional slicing method. The analysis results of time-varying mesh stiffness curves show that, compared with traditional loaded tooth contact analysis (LTCA) method, the analytical method proposed in this paper is closer to finite element (FE) method, and the calculation cost is relatively low. Finally, the sensitivity analysis of the parameters affecting the mesh stiffness characteristics of helical gear pair is carried out, and the analysis results are discussed.

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.

Pitchfork and Hopf bifurcations of geared systems with nonlinear suspension in permanent contact regime

Gears are important mechanical parts with various industrial applications. Many researchers have investigated the complex nonlinear behavior of geared systems by studying the effect of time-varying mesh stiffness, clearance between the gears in mesh, radial clearance in the bearings, and bending of the supporting shafts. Most of these studies assume that the gear set operates under lightly loaded operational conditions, where the separation of the teeth in mesh occurs and the nonlinearity caused by the clearance between the gears in mesh has the major influence on the dynamic response of the system. Alternatively, in this work it is assumed that the transmitting load is great enough that gears in mesh do not separate, and consequently the clearance between the teeth does not participate in the dynamic response of the system. Then analytical and numerical techniques are used specifically to investigate the effect of the nonlinearity of the shafts on the dynamic behavior of the system. The results show that the nonlinear suspension has a significant influence on the creation of nontrivial equilibria and limit cycle within the parametrically unstable tongues which, for the right range of the parameters, can affect the rate of amplitude detonation and stabilization of the system.

Investigations on crack propagation and meshing characteristics of planetary gear train considering crack closure effect

The shape of spatial crack is more complex in practice and crack closure will occur due to deformation, which will result in partial contact of the crack surfaces. In previous researches, a simple linear plane assumption for the gear crack trajectory was usually made and the crack closure effect was ignored. Based on the principle of linear elastic fracture mechanics, a three-dimensional numerical model of planetary gear train crack propagation considering the initial crack location, ring gear backup ratio and constraint mode is built in this study. Considering the crack closure effect and the obtained crack trajectory, the meshing characteristics (including the time varying meshing stiffness (TVMS) and contact pressure) are investigated. The results indicate that when considering the crack closure effect, the reduction in the TVMS of the cracked gear decreases and the influence of the foundation crack closure effect on TVMS is greater than that of tooth crack closure effect. Appropriately constraining the outer rim of the thinner ring gear can help avoid catastrophic foundation crack, but the TVMS is more sensitive to the change in the backup ratio when only the pin support is restrained. The uneven distribution trend of cracked tooth contact pressure depends on the nonpenetrating crack type.

Nonlinear characteristics of a multi-degree-of-freedom wind turbine’s gear transmission system involving friction

In this study, a 42-degree-of-freedom (42-DOF) translation–torsion coupling dynamic model of the wind turbine’s compound gear transmission system considering time-varying meshing friction, time-varying meshing stiffness, meshing damping, meshing error and backlash is proposed. Considering the different meshing between internal and external teeth of planetary gear, the time-varying meshing stiffness is calculated by using the cantilever beam theory. An improved meshing friction model takes into account the mixed elastohydrodynamic lubrication to calculate the time-varying meshing friction. The bifurcation diagram of the excitation frequency is utilized to analyze the bifurcation and chaos characteristics of the system. Meanwhile, the nonlinear dynamic characteristics of the gear system are identified by using the time domain diagrams, phase diagrams, Poincare maps and amplitude–frequency spectrums of the gear system. The results show that the system has complex bifurcation and chaotic behaviors including periodic, quasi-periodic, chaotic motion. The bifurcation characteristics of the system become complicated and the chaotic region increases considering the effects of friction in the high-frequency region.

Vibration-Based Fault Diagnosis of Helical Gear Pair With Tooth Surface Wear Considering Time-Varying Friction Coefficient Under Mixed Elastohydrodynamic Lubrication Condition

Abstract Based on “slice method,” the improved time-varying mesh stiffness (TVMS) calculation model of helical gear pair with tooth surface wear is proposed, in which the effect of friction force that obtained under mixed elastohydrodynamic lubrication (EHL) is considered in the model. Based on the improved TVMS calculation model, the dynamic model of helical gear system is established, then the influence of tooth wear parameters on the dynamic response is studied. The results illustrate that the varying reduction extents of mesh stiffness along tooth profile under tooth surface wear, in addition, the dynamic response in time-domain and frequency-domain present significant decline in amplitude under deteriorating wear condition.