Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness

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

Angular misalignment is a common error in gear transmission systems, significantly altering the contact and dynamic behaviors of the gear system. Conventionally, the time-varying mesh stiffness (TVMS) of helical gear pairs with angular misalignment is estimated under the assumption of a uniform load distribution along a fixed contact line. However, this simplification neglects the variations in contact line length and load distribution due to angular misalignment, thereby compromising the accuracy of TVMS predictions. To address this limitation, we propose a novel dual-iterative model for TVMS calculation in helical gears, accounting for the angular misalignment-induced contact line variation. This model combines traditional iterative calculations for gear slices with an additional iterative loop for the misalignment-induced contact line changes. Furthermore, it considers both axial and tangential meshing stiffnesses. The proposed dual-iterative TVMS model is verified by using the finite element (FE) method. Moreover, a six degree-of-freedom (DOF) dynamic model for misaligned helical gears is established to investigate the effects of misalignments on the vibration characteristics of the gear pair. An experimental setup utilizing a back-to-back helical gear test rig with adjustable angular misalignment is constructed to validate the proposed model. The experimental results exhibit a close alignment with the simulation predictions. It is concluded that the proposed model is suitable for estimating the TVMS and dynamic analysis of helical gear pairs with angular misalignment errors.

<div class="section abstract"><div class="htmlview paragraph">Motivated by accurate representations in gear dynamics models, this work analyzes the force-deflection relationship between spur gear pairs when the gear teeth have tooth root cracks. A finite element/contact mechanics approach is used to accurately capture the elastic deformations of the gear mesh incorporating kinematic gear motion; elastic deflections of the teeth, root, and blank; and elastic contact between the mating gear teeth. Tooth root crack damage of fixed sizes are analyzed, and the resulting static transmission error and mesh stiffness are calculated. These FE/CM model outputs are relatively insensitive to important gear crack geometry, including the initial crack location, the path it follows, and its final location. Crack-induced changes in static transmission error and mesh stiffness are driven by the remaining amount of the tooth that is healthy. Calculations of average-slope and local-slope mesh stiffness are included because both are used in gear dynamic models.</div></div>

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

In order to accurately describe the complex transmission systems and clarify the effects of internal and external excitation on the vibration characteristics for large complex transmission systems, a dynamics model considering gear meshing error and time-varying meshing stiffness is established at two-stage planetary gears systems. The coupling model is established by matrix set method for two-stage planetary gears and multistage spur gears transmission systems. The random load excitation verified by experiments is the external excitation in the system, and the excitation caused by the time-varying meshing stiffness and meshing error is the internal excitation. The research on the response of the vibration systems shows that the internal excitation caused by gear meshing stiffness and gear meshing error has important influence on the dynamic load behavior of spur gears transmission in complex systems. For the planetary system with heavy external load, the dynamic behavior mainly depends on the external load, and the influence of internal excitation on the dynamic meshing force is less than that of the external excitation for the planetary system. The external excitation mainly affects the low-frequency region of the dynamic meshing force for the system, and the internal excitation affects the high-frequency region of the dynamic force for the system. The frequency influence interval of the internal excitation is much larger than that of the external excitation.

Gear tooth spalling is a typical tooth surface defects in gear transmission. The accurate calculation of the time-varying mesh stiffness of the gear pair with spalling defect under mixed elastohydrodynamic lubrication (EHL) plays an important role in tooth damage prediction and fault feature analysis. In this work, an improved time-varying mesh stiffness (TVMS) model of spalling faulty gear in mixed EHL line contact considering the coupled effect of rough surface topography is established. The proposed model considers the combined effect of spalling defect, surface roughness topography, and lubrication on the contact characteristics of the meshing interface to obtain a revised contact stiffness, which is further used to determine the comprehensive TVMS. The cylindrical contact coefficient for curved meshing interface with spalling fault is derived and incorporated into the statistical micro-contact Greenwood and Williamson model (GW model) to obtain the asperity contact stiffness. The revised contact stiffness at the rough faulty gear interface is obtained from the parallel action of the asperity contact stiffness and oil film stiffness under load-sharing concept. The dynamic responses of the gear system are analyzed using the established mesh stiffness model through the six degrees of freedom translational-torsional dynamic model, in which the dynamic meshing force is calculated iteratively using the transit surface topography and geometrical parameters along the line of contact instead of a quasi-static meshing force. Effects of multi-parameters as surface roughness, spalling fault size, and lubricant on the mesh characteristics and dynamic responses of the gear system are further investigated and discussed.

The traditional method for calculating the time-varying mesh stiffness (TVMS) of cracked gears usually assumes that the root crack is solely located within the gear tooth. Accurate calculation becomes a challenge when the crack propagates into the gear body. In this paper, an improved model for calculating the TVMS of spur gears with crack propagation into the gear body is proposed based on a combined analytical finite element-potential energy method, and the accuracy of the proposed model is verified by the finite element method. A 5-degree-of-freedom (5-DOF) dynamic model with speed fluctuation is developed, taking into account the coupling shaft-bearing stiffness as the support stiffness. Subsequently, a dynamic simulation of a gear system with a gear body crack is conducted. The results indicate that the proposed analytical model can provide relatively accurate mesh stiffness for crack propagation into the gear body. The TVMS decreases as the crack length increases, the decrease in mesh stiffness becomes more significant when considering the influence of the crack on the gear body stiffness. The dynamic response obtained from the proposed dynamic model shows more pronounced pulse characteristics compared to the model that does not consider the influence of shaft stiffness. The low speed and heavy load conditions restrain the speed fluctuation of the gear system. The residual signal is more effective for identifying fault signals than the original signal. The proposed model can provide theoretical and methodological references for diagnosing faults in gear systems with crack propagation into the gear body.

<div class="section abstract"><div class="htmlview paragraph">A new capability to analyze the dynamic interaction of nonlinear hypoid gear mesh characteristics and time-varying bearing stiffness is proposed. Both backlash nonlinearity and time-varying mesh parameters, such as mesh stiffness, mesh point and line-of-action, are included in the nonlinear hypoid gear mesh model. The time-varying bearing stiffness behavior due to the changing orbital position of rolling elements is also modeled. A practical application is studied to reveal the dynamic characteristics of the complex interactions. Dynamic simulation results show that dynamic mesh force is relatively insensitive to the temporal variation in the bearing stiffness. On the other hand, the dynamic bearing loads are affected significantly by the time-varying bearing stiffness, especially in the case of heavy drive torque load without the occurrence of jump response phenomenon. However, for the case of light load condition assuming time-invariant mesh parameters, the dynamic bearing loads are insensitive to time-varying bearing stiffness. The analysis also reveals that the presence of appropriate backlash nonlinearity can stabilize the parametric resonance caused by the time-varying bearing stiffness for the case of light load condition assuming the mesh parameters are time-invariant.</div></div>

Studies on the planetary gear have attracted considerable attention because of its advantages, such as compactness, large torque-to-weight ratio, vibrations, and high efficiency, which have resulted in its wide applications in industry, wind turbine, national defense, and aerospace fields. We have established a novel dynamic model of the planetary gear transmission by using Newton’s theory, in which some key factors such as time-variant meshing stiffness, phase relationships, and tooth contact characteristics are considered. The influences of gear axial tipping, operating conditions, and the meshing phase on the contact characteristics and the dynamic characteristics were researched systematically. It was found that the contact area of the tooth surface was moved due to the axial gear tipping, which obviously affected the meshing stiffness. With the increase in the inclination angle of the sun gear, the meshing stiffness decreases, which produces an evident influence on the high natural frequency in the planetary transmission system. In terms of the dynamic characteristics of the system, the component of rotating frequency appeared in the dynamic meshing force of the sun gear and the planetary gear. Moreover, the floating track of the center wheel varied significantly and exhibited an oval distribution as the inclination angle of the sun gear changed. When the inclination angle of the sun gear increased, the rotating frequency component increased significantly, but the other meshing frequency components remained unchanged; meanwhile, the deformation of the floating track also increased. If the inclination angle of the sun gear changes, the vibration state of the system and the collision impact could become more serious, and the lifetime of the planetary transmission system will reduce. Furthermore, when the load was increased, we found that the gear-tooth contact zone transformed from line contact to surface contact, the meshing stiffness increased, the effect of high natural frequency on the planetary transmission system became more evident, but its low-order natural frequency remained stable. With regard to the dynamic characteristics of the system, the components of the major frequency at the external gearing remained unchanged, but the rotation frequency of the sun gear and the meshing frequency amplitude increased linearly with the increase in load. In conclusion, the variation in the meshing stiffness of the planetary gear system had minor impact on the low-order natural frequency, but had a significant impact on the high natural frequency of the planetary transmission system due to the phase variation of the gear.

NON-LINEAR DYNAMIC RESPONSE OF A SPUR GEAR PAIR: MODELLING AND EXPERIMENTAL COMPARISONS

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-DOF dynamic model is established for SBG transmission system. Incorporating the spalling defect into TVMS, the dynamic responses of spalled SBG are analysed. 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. Conflict of Interest Statement The authors declare no conflicts of interest.

Effects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system

The determination of meshing force and the load sharing ratio of gear teeth is critical to predict the dynamic behavior or the load capacity of gear transmissions. In the previous literature, the dynamic meshing force is usually calculated based on the traditional dynamic model, which ignores the different effects of the meshing characteristics of each pair of teeth on the dynamic behavior of the gear system. In this work, a new calculation method of dynamic meshing force is proposed based on the new dynamic model considering the meshing state of multiple pairs of teeth. The difference between the traditional calculation method and the new calculation method of dynamic meshing force is analyzed. Based on the new dynamic model and new calculation method of dynamic meshing force, the influence of different factors on dynamic response and dynamic meshing force are further discussed. The results show that, compared with the traditional calculation method, this new method can be used to effectively calculate the dynamic meshing force and the load sharing ratio of each pair of teeth with different meshing characteristics. The presented method for the calculation of the dynamic meshing force and the load sharing ratio provides an important reference for analyzing and predicting the dynamic behavior or the load capacity of spur gears, especially the high contact ratio (HCR) gears with contact ratio more than two.

An improved analytical model for gear mesh stiffness calculation

The tiltrotor drive system is a complex and coupled system with different torsional dynamics compared to conventional rotorcraft. It is designed to guarantee safety and reliability and tested for the effects of its dynamics and loads on component sizing. The epicyclic gearing system, the final reduction stage in most aircraft gearboxes, is most impacted by such dynamics. The components of this subsystem must withstand all peak loads due to novel control strategies and torsional dynamics. In this paper, a gear contact analysis model is coupled with a validated torsional dynamics model of a tiltrotor interconnected drive system. The coupled model predicts component-level design loads in the interconnected drive system originating from its torsional dynamic behavior. An accurate gear macro and micro geometry is generated in the gear contact model which is used to derive the time varying mesh stiffnesses at the sub-component interfaces. The variable mesh stiffnesses are applied in the gear dynamic model to simulate the dynamic responses of the gear system under load. Finally, a dynamic overload factor relating the peak dynamic and static loads is analytically assessed, and it is shown to exceed conventional predictions in the case of an asymmetrically loaded interconnected drive system. This dynamic factor can be implemented to inform design decisions during the development of future multi-rotor VTOL aircrafts to mitigate component damage and, in turn, increase the safety and extend the lifespan of critical components.