Time-varying Mesh Research Articles

Spiral Bevel Gears: nonlinear dynamic model based on accurate static stiffness evaluation

In the present paper non-linear dynamics of a spiral bevel gear pair with backlash are investigated in order to clarify the internal excitations of major importance from the vibration point of view: manufacturing errors in the teeth profile, teeth spacing errors, and elastic deformation of the teeth. In some conditions, like in the case of backside contact, the destructive effect of internal excitations can be intensified leading to complex dynamics; for such reasons here backside contacts and reverse rotation are investigated in detail using a nonlinear time-varying model. The effect of damping is investigated as well. A one-DOF model is developed in order to study the dynamic behavior; the resulting a nonlinear differential equation with time-varying mesh stiffness is solved via numerical integration based on an adaptive step-size implicit Runge-Kutta scheme. The dynamic response of the system is analyzed through time histories, phase portraits, bifurcation diagrams, and Poincaré maps. Results show that for small backlash values, the possibility of backside contact increases. Meanwhile, by increasing the backlash value, the amplitude vibration of the gear rotation rises as well. By comparing the dynamic response of the system with different damping ratios, the results show that higher damping effectively reduces gear vibration resonance, although the probability of unsteady response still exists.

An improved fault model of wind turbine gear drive under multi-stage cracks

Considering the real crack propagation paths using the fracture mechanics theory, a novel simulation dynamic model for the two-stage cracked gear drive of the wind turbine is established. The mesh characteristics (time-varying mesh stiffness, contact stress) are obtained with the finite element method. The mesh stiffness under the single-stage and multi-stage cracks is imported into the two-stage cracked gear drive dynamic model to solve the dynamics. The results of the simulation and experiment are performed to verify the proposed dynamic model. The dynamic characteristics and statistical indicators under different cracks are discussed. The results reveal that the crack has a significant influence on the mesh stiffness of the single-tooth contact zone. A rotating periodic impact is found on the time-domain response for the cracked gear drive. Under the single-stage crack, the components of rotating sideband frequencies also appear in the other-stage healthy gear drive. The gear acceleration amplitudes under the multi-cracks are larger than that under the single-stage crack. Therefore, the single-stage crack has an influence on the other-stage gear in the coupling system. This developed result can be applied well to vibration measurement and fault diagnosis.

Mesh characteristic analysis and dynamic simulation of spur gear pair considering corner contact and tooth broken fault

Theoretically, the contact of a spur gear pair occurs on the line of action (LOA), but the corner contact may happen due to the tooth deformation, especially when there is a tooth broken fault. In this paper, based on the tooth contact analysis, the initial gaps considering tooth broken fault are accurately calculated. Meanwhile, the initial gaps are brought into the loaded tooth contact analysis program and yield the time-varying mesh stiffness (TVMS) and non-load transmission error (NLTE). By comparing with the finite element (FE) method, the meshing state of the gear pair considering various tooth broken heights is presented. The result shows that it is inaccurate to regard the mesh stiffness value as zero when the missing tooth profile should have contact since serious corner contact phenomenon will occur in this state. Moreover, by comparing the simulated dynamic response with the experimental results, this paper finds that the simulation result considering the corner contact effect is closer to the experimental result. This study can provide a more accurate model for the diagnosis of gear tooth broken faults.

Dynamic Simulation of Cracked Spiral Bevel Gear Pair Considering Assembly Errors

The tooth root crack fault is a common fault type of the spiral bevel gear pair (SBGP). Affected by the strong bearing capacity, the early crack fault of the SBGP cannot be found in time. In this study, a finite element (FE) model of the SBGP is established and assembled through the tooth contact analysis. The maximum tooth root stress is analyzed considering the variation of assembly errors. Meanwhile, this study simulates the tooth root crack fault of the bevel pinion with different crack degrees. The initial position of the crack is located where the maximum tooth root stress appears. The time-varying mesh stiffness (TVMS) of the SBGP considering different degrees of the pinion tooth root crack fault is obtained. The TVMS and the non-load transmission error are brought into a hybrid FE dynamic model, and steady responses are solved. Based on this, the sensitivities of various statistical indicators for identifying the tooth root crack fault of SBGP under the influence of assembly errors are verified. This paper can provide the necessary theoretical basis for the analysis and diagnosis of tooth root crack faults in the SBGP transmission system.

A time-variant dynamic model for compound epicyclic–cycloidal reducers

Compound Epicyclic–cycloidal reducers are widely used in robot joints. Although the individual components have been studied extensively, few studies address dynamics of compound reducers assemblies. In order to predict dynamic stiffness of robots more accurately, a time-variant reduced-order numerical model is constructed for compound reducers in the current study. Full Lagrangian formulation is used, with time-variant meshing stiffness of planetary and cycloid gears determined by finite element and geometric analyses, respectively. The proposed numerical model shows that modulation sidebands occur even in the absence of assembly error of the planetary gear, because the contact stiffness of cycloid gears is time-variant. Compared with experimental data obtained from actual compound reducer, the proposed model can predict the meshing frequency and its harmonics.

Multi-meshing-state and disengaging-proportion analyses of a gear-bearing system considering deterministic-random excitation based on nonlinear dynamics

Backlash induces gear-teeth separation or back-side tooth mesh (BTM), resulting in gear disengagement (GD), which heavily aggravates the dynamic performance and transmission quality of gear drivetrains. This study focuses on the disengaging proportion (DP) characteristics of a gear- bearing systems based on a multi-meshing-state (MMS) model. A new nonlinear dynamic model of gear-bearing system with MMS and deterministic-random external excitation is established considering the synergistic effect of backlash and non-integer contact ratio. A modified time-varying mesh stiffness with temperature stiffness is included in the model. Afterwards, the effect of the gear pair meshing vibration on bearing radial vibration is studied. The bifurcation and evolution of the MMS behavior are investigated changing the system parameters. An algorithm of DP is proposed in the time domain based on the established MMS model. The characteristics of DP are systematically discussed with the effects of dynamic response type, phase trajectory topology, bifurcation and chaos, and correlation between parameters. The results demonstrate that grazing bifurcation reduces the number of GD and chaos is prone to BTM. The disengaging proportion is highly dependent on the phase trajectory topology and generally increases from period-1 to quasi-periodic and then to chaotic responses. Bifurcation changes the variation trend of DP, thus there is a peak near period-doubling bifurcation and a trough near grazing bifurcation. The chaotic response leads to a jump in DP. The correlation between parameters considerably dominates the DP. This study provides a reference for revealing the mechanism of gear system disengagement and improving the dynamic performance.

An accurate and numerically efficient time-varying mesh stiffness model for modified straight bevel gear under load

Gear time-varying mesh stiffness (TVMS) plays an important role in the analysis accuracy of a gear transmission mechanism. In previous research, the TVMS of a modified straight bevel gear (SBG) was mainly obtained by the finite element method (FEM), whose computation time is usually long to ensure computational accuracy. By contrast, this work proposes a fast calculation method with high accuracy for the TVMS of a modified SBG. First, B-spline surfaces are introduced to reconstruct a modified SBG using discrete tooth surface points. Next, based on the tooth contact analysis and loaded tooth contact analysis, combined with energy conservation principle and infinitesimal method, a semi-analytical instantaneous single mesh stiffness (SMS) model for the modified SBG is derived. Then, using the additional deformation compatibility condition and principle of force balance, the calculation method of the TVMS of the modified SBG is obtained. Finally, a comparison of the mesh stiffness calculation results between the method of this study and that of FEM is completed. The results show that the semi-analytical TVMS model used in this study can be accurately and quickly solved.

Influence of Tooth Surface Wear and Nonlinear Contact Stiffness on Dynamic Responses of Helical Gears

Background: Tooth surface wear is inevitable in helical geared transmission. Consequently, the worn profile deviates from the ideal involute one. As a result, the structural stiffness of worn tooth and contact stiffness of tooth-pair are both changed. Methods: This work presents an improved calculation method for structural stiffness of worn teeth by combining slicing and potential energy method, considering non-uniform distribution of wear amount along the tooth surface. Then, a nonlinear contact stiffness model is employed to investigate the influence of wear on contact stiffness. Meanwhile, taking wear as one kind of profile deviation, the analytical model of time-varying mesh stiffness (TVMS) of helical gear pair is derived. Furthermore, governing equations with 6 degree-of-freedom are established and influences of wear on dynamic responses are revealed. Results: Results indicate that structural stiffness of worn teeth decreases but contact stiffness does not always keep increasing or decreasing. The fluctuation of dynamic transmission error with the nonlinear contact model is not as significant as that from the constant contact stiffness model. Conclusion: The approach presented in this work is suitable for condition monitoring of helical gears in view of long-term service

Develop and Verify Energy-Based Statistical Linearization Technique to Analysis Nonlinear Stochastic Vibration of a Spur Gear Pair

Abstract In this article, an energy-based statistical linearization method (SL) is proposed to simulate a nonlinear dynamic model of spur gear pair. The gear pair operates under combined deterministic and random loads, and both backlash and time-varying mesh stiffness are considered in the dynamic model. The equivalent linear function approximates the teeth backlash nonlinearity in the gear model. The energy-based linearization, which minimizes the error in potential energy between the original and equivalent linear systems, is used. Simulations are conducted on a gear pair, and the effect of the input torque on the dynamic response of the gear pair is then examined. The results demonstrate that for high input torque, the system operates in the linear range. For low input torque, the results are not similar to the original because the system became strongly nonlinear. Monte Carlo simulations were carried out to verify the accuracy of the presented method.

An accurate identification method of the position of a broken tooth in the joint steering gear of service robot

The gearbox of the small joint steering gear of a service robot is characterized by a compact structure, a large reduction ratio, and a special manufacturing and assembling technology. However, when a tooth in the gear is broken, the gear transmission can be still maintained, so it is difficult to identify the broken tooth accurately. In order to provide a theoretical basis for the identification of a broken tooth in the study, the influence of gear tooth fracture on the time-varying meshing stiffness in the transmission process was firstly analyzed theoretically and its influencing mechanisms on the time domain and frequency domain characteristics of transmission error (TE) were further analyzed. Then, the TE measurement test bench of the joint steering gear was designed. The signal-to-noise ratios and root-mean-square errors of the TE data processed by wavelet noise reduction, Kalman filtering and combined wavelet-Kalman filtering algorithms were compared and analyzed. The analysis results showed that the joint algorithm had the best performance. Next, the TE data measured under multiple working conditions were processed with the joint filtering algorithm to obtain the time domain results and then the frequency domain results were obtained by Fourier transform. Whether there was a broken tooth in the joint steering gear could be judged based on the difference between the maximum and minimum values of the TE in the time domain. The broken-tooth gear shaft could be positioned based on the frequency of TE mutation waveform. Furthermore, combined with the frequency domain, the accurate identification of the position of the broken gear can be realized. Finally, the accurate positioning method of broken teeth was applied in the product inspection process. Broken teeth were successfully identified and located, thus proving the effectiveness of the identification method.

Influence of tooth root cracks on the mesh stiffness of asymmetric spur gear pair with different backup ratios

Gears are critical machine elements that transmit power and motion in diverse implementation fields. Over time, gears may produce a series of faults due to harsh operating conditions, fatigue, manufacturing errors, etc., leading to severe performance degradation. During the meshing process, the stiffness of a single tooth controls the load sharing, vibration, and noise characteristics of a geared system. An undetected fault could decrease the gear stiffness and thus may lead to a fatal breakdown, substantial economic losses, or even human casualties in safety-critical applications such as helicopters, high-speed trains, and wind turbines. Hence, the accurate quantification of the gear stiffness emerges as an important research area to obtain reliable gear designs. With this in mind, the asymmetric tooth concept offers superior bending strength, fatigue propagation life, and the ability to lessen vibration over the standard (symmetric) designs in applications where unidirectional loadings are predominant. This study investigates the influence of tooth root cracks on the single-tooth and meshing stiffness characteristics of the standard and asymmetric involute spur gears. To this end, the numerical crack propagation paths obtained in our previous works were introduced to the created 3D CAD geometries. Subsequently, the single tooth stiffness of both healthy and cracked (2 5%-50%-75%-100%) gears was calculated through the ANSYS® Workbench, and the time-varying mesh stiffness was obtained. The present study evaluated the effects of backup ratio and the tooth asymmetry on the spur gears’ meshing stiffness characteristics simultaneously and further expanded the scope of the research work. The results indicated that the single tooth stiffness and mesh stiffness could be improved by 35% and 22%, respectively, as the drive side pressure angle increased from 20° to 35°. It has been noted that the gear stiffness decreased as the crack level increased, while the increment of the backup ratio further increased the reduction in the stiffness. The findings could provide significant outputs for a better understanding of the influence of tooth asymmetry on the gear dynamics characteristics, life prediction, and early fault diagnosis.

A new tooth pitting modeling method based on matrix equation for evaluating time-varying mesh stiffness

Tooth pitting is the most common gear failure, and it is usually modeled as regular shapes. Although this modeling approach simplifies the calculation, it is far from the real pitting shape. In this study, a new tooth pitting modeling method is proposed based on matrix equation. The method can generate irregular shape pitting and can customize the depth and shape accuracy of pits easily, making it more suitable for practical situations. Considering the pitting propagation, the pits are distributed statistically on gear tooth surface. Four pitting degrees and three pitting types are modeled by using this method, and then the stiffness of the pitting models are evaluated by potential energy method. The results show that the shapes and distribution forms of pitting have great influence on the time-varying mesh stiffness. The effectiveness of the stiffness evaluation is verified by comparing with the iSO-6336 and finite element results.

Dynamic Analysis of Gear Rattling of a Certain Type of Dual-Clutch Transmission

In order to accurately explore the transmission rattling phenomenon and the influence of different factors on the dynamic characteristics of the gear rattling of the dual-clutch transmission under the condition of preselected gears, this paper establishes the gear rattling dynamics model of the transmission with the 1st gear without preselection and the preselected 4th gear, respectively; The model takes into account factors such as time-varying mesh stiffness, mesh damping, nonlinear oil film force, nonlinear backlash, and the drag torque generated by the clutch in the unengaged state. In addition, the feasibility of the dynamic model was verified by the bench test. On this basis, we took the gear meshing power and system power loss as quantitative indexes to analyze the influence of the preselected gear state and different parameters on the rattle vibration of the transmission. The results show that the pre-selected gear will not have a significant effect on the gears that have been rattled in the non-pre-selected state, and the torque fluctuation of the non-power flow shaft is aggravated by the influence of the transmission power flow branch and transmission ratio at different levels, which makes the overall rattling strength increase. In order to improve the transmission efficiency of the gear, the torque fluctuation of the input end of the system should be reduced as much as possible, and a larger lubricant viscosity can be appropriately selected, the inertia of the empty gear can be properly reduced, and the tooth clearance can be relaxed for selection.

Influence of 3D modification on dynamic characteristics of herringbone gearbox system and vibration reduction design

The vibration and noise of marine power systems mainly comes from gearbox in addition to engine, while gearbox vibration is mainly caused by internal excitation of gear transmission system (GTS) and transmitted to ship hull through the box. This paper proposes a vibration reduction method of 3D modification controlling the strength of vibration source and constrained damping structure restraining vibration transmission energy. For herringbone gear,The 3D modification form of tooth surface combining with two parabolas and one straight line is proposed and the modified tooth surface equation is derived based on the principle of forming grinding.Combined with tooth contact analysis (TCA) technology and loaded tooth contact analysis (LTCA) technology, three internal excitations of GTS after 3D modification are determined. Through force balance and node connection relationship between each element, the coupling dynamic model of gear-shaft-bearing with three internal excitations i.e., time-varying meshing stiffness, transmission error and meshing impact is established to calculate dynamic load of journal bearing.The load is loaded into the finite element model of the gearbox to obtain its vibration value and six symmetrical pressing plates are applied force to fix the foundation with adding the same constrained damping structure in the middle.The purpose of reducing gearbox vibration is realized by combining 3D modification technology with constrained damping structure.The results of numerical examples show 3D modification can effectively reduce the radial and axial vibration of the gearbox, while the constrained damping structure can availably cut down transverse vibration. Compared with modification, constrained damping structure is more obvious effect. This study will contribute to the in-depth study of vibration reduction measures of marine gearbox system under variable excitations, and analyze its vibration characteristics under different vibration reduction measures, so as to meet the requirements of its vibration reduction and realize safety and stability of ship hull.

Nonlinear characteristic analysis of gear rattle based on refined dynamic model

In order to explore the vibration response and the nonlinear behavior of gear rattle, a nonlinear dynamic model is developed in this paper considering the nonlinear oil film force, time-varying meshing stiffness, dragging torque and friction and gear backlash with time-varying characteristics. The time-varying meshing stiffness is calculated using the potential energy method. Considering the convolution and extrusion of the lubricant between the teeth, it is modeled as a nonlinear spring-damped element and a nonlinear oil film force is introduced. The drag torque is calculated numerically using fluid dynamics theory, friction principle, momentum conservation principle, and Bernoulli's principle. The model is verified by comparing the bench test results with the simulation results, and the parametric study is performed to investigate the nonlinear system characteristics by means of fast Fourier transform, Poincare map, and bifurcation diagram. The results show that as the excitation frequency increases, the system enters chaotic state under multiple frequency jumps. Excessive torque fluctuation may lead to severe rattling phenomenon of the gear pair and making the system enter a chaotic state; as the lubricant viscosity increases, the system shows periodic, bifurcation and chaotic behavior alternately, and when the viscosity is large enough, the high-frequency shock intensity decays more rapidly due to large drag torque, which is beneficial to reduce the percussion strength and prevent the system into chaos.

A Dynamic Wear Prediction Model for Studying the Interactions between Surface Wear and Dynamic Response of Spur Gears

Surface wear, as a major failure mode of gear systems, is an unavoidable phenomenon during the whole life of gears. It also induces other gear damages, such as fatigue cracks, surface pitting and spalling. Ultimately, those defects may result in the sudden failure of a gearbox transmission system, which can lead to a serious accident and unexpected economic loss. Therefore, it can provide huge cost and safety benefits to industries to monitor gear wear and predict its propagation. Gear wear raises the error rate of gear transmission systems, typically leading to improvements in dynamic loads, vibration, and noise. In return, the increased load conversely aggravates wear, creating a feedback cycle between dynamic responses and surface wear. For this purpose, a wear prediction model was incorporated into a tribo-dynamic model for quantitatively investigating how surface wear and gear vibration are mutually affected by each other. To obtain more precise dynamic responses, the tribo-dynamic model integrates the time-varying mesh stiffness, load-sharing ratio and friction parameters. To improve the computational efficiency and guarantee the calculation precision, an improved and updated wear depth methodology is constructed in the wear prediction model. This paper demonstrates the capability of the proposed dynamic wear prediction model in the investigation of the interaction effects between gear dynamics and surface wear, allowing for the development of improved gear wear prediction tools. The obtained results indicate that the surface wear impacts the dynamic characteristics, even with slight wear. In the initial stage of wear, the friction coefficient decreases slightly, largely due to the reduction in surface roughness; but the friction force increases because of the improved dynamic meshing force. Although the initial wear depth distributions of a pinion under dynamic and static conditions are similar, the wear depth distributions under dynamic conditions becomes significantly different compared to the those under static conditions with the wear process. The maximum wear depth of a pinion under dynamic conditions is about 1.6 times as the corresponding static conditions, when the wear cycle comes to 4 × 104. Similarly, the maximum accumulative wear depth of a pinion under dynamic conditions reaches 1.2 times of that under static conditions. Therefore, the proposed dynamic wear prediction model is more appropriate to be applied to the surface wear of gears.

Establishment and Validation of a Structural Dynamics Model with Power Take-Off Driveline for Agricultural Tractors

As off-road vehicles, in addition to field transportation, another vital function of agricultural tractors is to provide power for field machinery. Therefore, the dynamic performance of the power take-off (PTO) driveline directly affects the field reliability of tractors. Firstly, a torsional vibration coupled spatial dynamics model of the power take-off driveline is proposed according to the classical machine driveline dynamics and gear dynamics theory. In the dynamics model, the interactions among the vertical, lateral, and rotational motions of the driveline parts are fully included. The coupling vibrations from internal excitations (such as tooth surface friction, gear time-varying mesh, and engine pulse) and external excitations (such as field machinery load) are also considered. Secondly, the simulation results of the model are obtained using the numerical solving algorithm ode15s. The actual experiment is carried out on the indoor Tractor PTO Test Bench. Then, the model is verified by comparing the test results with the simulation results. Finally, the dynamic characteristics of the whole driveline are revealed under different drive modes, especially strong interactions between the driveline and field machinery in low-speed and heavy-load mode. The gear mesh forces and the root mean square (RMS) values of the acceleration amplitude for the main parts generally decrease gradually with the increase in the PTO rotation speed and the decrease in PTO torque. Furthermore, the model can be applied to reliability assessment, for instance, vibration, damage, and fatigue of off-road vehicles considering gear transmissions, particularly in a field working environment.

Meshing Characteristics of Spur Gears Considering Three-Dimensional Fractal Rough Surface under Elastohydrodynamic Lubrication

Taking the effect of actual surface topography under elastohydrodynamic lubrication (EHL) conditions on the contact state of gear pairs into consideration, a combination model with the analytical sliced method and two-dimensional (2D) EHL model is proposed to characterize the three-dimensional (3D) meshing characteristics of spur gears. Firstly, the surface topology of gears is tested by a surface profiler, which reflects that the topography of tooth surface accords with fractal characteristics. Thus, by adopting the Weierstrass–Mandelbrot (W-M) fractal function, the gear surface is characterized. Secondly, the numerical 2D EHL model with fractal roughness is established, and distributions of oil film pressure (OFP) and oil film thickness (OFT) at different meshing positions are obtained. Finally, considering the different topography distributions in the direction of face width, time-varying mesh stiffness (TVMS) is calculated based on the analytical sliced method. Thus, the influence of 3D surface topography can be considered. The Hertz contact stiffness is substituted by the time-varying lubricating oil film stiffness (OFS). The influences of tooth surface topography and lubricant film characteristics on meshing characteristics are investigated. The results show that the 3D rough tooth surface may be well characterized by a fractal function with random phase. Moreover, there is a great difference in the distribution of OFP and OFT between rough and smooth surfaces, which certainly influences the gear meshing characteristics.

Traveling wave resonance analysis of flexible spur gear system with angular misalignment

The aero-engine spur gear system is often designed as thin rim structure due to lightweight consideration, which leads to traveling wave resonance phenomenon when spur gear system has angular misalignment during operation. This paper mainly analyzes traveling wave resonance phenomenon of aero-engine angular misalignment thin rim spur gear system. Time-varying mesh stiffness (TVMS) of spur gear system is calculated by loaded tooth contact analysis (LTCA) method, angular misalignment is introduced into spur gear system, and axial component of TVMS is obtained by the projection method. Flexible gear foundations are simulated by 8-node shell element and rotational shafts are simulated by Timoshenko beam element, combined with axial stiffness, the dynamic model of flexible spur gear system is established. Simulation results are compared with the references and finite element (FE) model, which proves the accuracy of the proposed method. The analysis results of angular misalignment spur gear system traveling wave resonance phenomenon show that during the operation of spur gear system, angular misalignment produces axial excitation, which excite nodal diameter vibration of gear foundations, and traveling wave resonance phenomenon appears at nodal diameter vibration resonance speed. The effects of misalignment angle, web thickness and gear torque on spur gear system traveling wave resonance amplitude and resonance speed are discussed. The research of aero-engine angular misalignment spur gear system provides theoretical basis for structural design and fault diagnosis of aero-engine gear systems.

Influence of tooth crack parameters on bearing vibration signal of a geared rotor

Fault diagnosis is a mandatory process in modern industry management, because it may reduce costs due to unexpected failures that may occur in machines. It is proposed to study the dynamic effect in the rotor response of gear tooth crack caused by bending stress. As the bearings are the common location to install sensors as accelerometers, the response in their corresponding degrees of freedom is considered. A geared rotor model is assumed for the simulations. This model embraces a detailed finite element discretization, which contemplates five degrees of freedom per node, including torsional rotation. In addition, a time-varying mesh stiffness calculated by the potential energy method is adopted to represent the tooth contact. A crack model is assumed with three input parameters (initial position, propagation angle and depth). A design of experiments was set up to show the influence of each parameter on the system response for three different mean depths. The results analysis pointed out that for a precise parameter identification, all three parameters should be taken into account, because they affect the system response and they also present combined influence.