Spur Gear Pair Research Articles

RATIONAL SHIFTS OF THE BASIC RACK’S PROFILE FOR WHEELS OF A CYLINDRICAL SPUR GEAR PAIR TO DECREASE TEETH WEAR

Purpose. It is to create a method for determination of such shift coefficients of the basic rack’s profile for driving and driven wheels of a cylindrical spur gear pair, that during gear pair operation time the wear rate of a surface layer on the most wearing sections of the teeth’ involute surfaces will be minimal. Research methods are based on integrated application of the involute gear pair theory and tribological laws. The golden section method is used to minimize the function. Results. We have obtained a dimensionless quantity, which is a function of shift coefficients of the basic rack’s profile for driving and driven wheels of a cylindrical spur gear pair. Values of this function are proportional to the greatest wear rate in the neighborhood of characteristic points on teeth’ involute profiles of these wheels. The method is created for determination of rational shift coefficients of the basic rack’s profile, which minimize the obtained dimensionless function, and which maximize the service life of the spur gear pair until the greatest worn-layer thickness on the teeth’ active surfaces reaches limiting permissible value. Scientific novelty. It is proved, that increasing of the worn-layer thickness during spur gear pair operation time will be occurring most rapidly in the neighborhood of certain characteristic points on teeth’ involute profiles of the wheels. Calculation model is elaborated for determination of the worn-layer thicknesses in the neighborhood of extreme active points on teeth’ profiles of the cylindrical spur gear pair and in the neighborhood of the lowest and the highest bounding points of single-pair contact. This model takes into account the influence of specific slides and numbers of wheels’ teeth as well as hardness values of teeth’ surface layer on the wear rates at characteristic points of teeth’ profiles. In addition, the model takes into account sharing of total force of the wheels interaction between two teeth pairs in double-pair contact and also a change of force transmitted by single pair of teeth when it comes into or go out of engagement. Practical value. Application of the created method for determination of shift coefficients of the basic rack’s profile is expedient in design of cylindrical spur gear pairs for machines and equipment, which operate in conditions when ingress of abrasive particles in the teeth engagement region is possible. An example of application of this method in design calculation of the cylindrical spur gear pair is given.

Solving time-varying mesh stiffness of spur gears based on improved potential energy method

Meshing stiffness is an important factor in gear dynamics analysis. At present, there are many ways to calculate the meshing stiffness of gears, and we usually use the potential energy method to solve and calculate the meshing stiffness of healthy (faulty) gears. In this paper, the potential energy method, the improved potential energy method and the theoretical calculation method are used to solve the time-varying stiffness of the healthy spur gear pair, and the solution results are compared, which verifies the feasibility of the improved potential energy method to solve the gear meshing stiffness.

A comparative experimental research on the diagnosis of tooth root cracks in asymmetric spur gear pairs with a one-dimensional convolutional neural network

Gearboxes transfer rotational motion and handle precision functionalities in many fields, including aviation, wind turbines, and industrial services. Their health management is essential to minimize workforce risks, increase the level of safety, and avoid machine breakdowns. From this standpoint, the present experimental research work developed a convolutional neural network-based method for diagnosing different levels of tooth root cracks (25 %-50 %-75 %-100 %) for symmetric (20°/20°) and asymmetric (20°/30°) profiled gear pairs. A series of vibration experiments were performed on a one-stage spur gearbox to achieve this by using a tri-axial accelerometer under variable working loads. The main purpose of this experimental research study is to explore the influence of the tooth profile on spur gears’ vibration responses and whether utilizing an asymmetric tooth profile would positively impact a deep learning algorithm's classification accuracy to add to the enhancements it provides in terms of fatigue life, mesh stiffness, and impact strength. Experimental results revealed that the overall classification accuracy could be increased by 7.712 % by feeding the proposed deep learning model with vibration data measured using test samples with asymmetric teeth.

Dynamic reduction technique for nonlinear analysis of spur gear pairs

In this study the dynamic response of a spur gear pair is analyzed using a novel nonlinear approach. The actual rolling motion and engagement of the system is simulated using a set of reduced order models obtained in a pre-processing phase using the minimal amount of master degrees of freedom without loss of accuracy or generality. The flexibility of the gear bodies is included by a refined finite element model, and no geometry simplification is introduced while also retaining all nonlinearity sources. To reduce the computational cost the time-varying mesh stiffness is also pre-computed and used depending on the instantaneous loading conditions. Contact loss is also taken into account, and reconnection events are treated as vibro impacts. The results are compared against high quality and demanding experimental results with a computational cost several orders of magnitude lower than models with similar accuracy. Different loading conditions are investigated during the sweep-up and down maneuvers. Mainly, the dynamic transmission error is analyzed, showing remarkable agreement with the test campaign’s results. Different nonlinear phenomena such as hysteretic jumps and sub- and super-harmonic resonances are correctly predicted by the proposed model in terms of both frequency and amplitude. This method allows quick and accurate nonlinear analyses overcoming current limitations and is open to further complications to include other components and effects.

Study on fault mechanism and dynamics of spur gear pair with time-varying backlash

Study on fault mechanism and dynamics of spur gear pair with time-varying backlash

An impact-damping model of collision body with multi-point contact and external load, and its application in gear transmission

The damping term is an important component of the calculation model of the contact force of collision bodies because it simulates the energy dissipation of collision bodies during the contact-impact process. Therefore, an impact-damping model of a collision body with multi-point contact and external load is established by considering the restitution coefficient. The algorithm for solving this model is proposed based on iterative methods. Then, the accuracy of the established model is verified by the consistency between the calculated and tested results. The effect of multi-point contact and external load on the contact-impact process of the collision body is studied. The results show that the hysteresis damping factor increases from small to large when the collision body enters a stable state through repeated impacts. Further, the established damping model is applied to the dynamics model of a spur gear pair with backlash, extended teeth contact, and structure coupling effect of the gear body. The nonlinear dynamic characteristics, multi-state mesh, and teeth impact of the system are studied through the root-mean-square of transmission error, bifurcation diagrams with multi-mapping sections, and contact forces. The calculation results of the gear dynamics model with the impact-damping model are more consistent with the tested results than the calculation results of the gear dynamics model with the traditional damping model. Meanwhile, the connection between system resonance, response coexistence, teeth disengagement, and teeth impact is revealed.

Addendum To: Geometric design, modelling and finite element analysis of spur gear pairs having normal teeth based on the modification of the pressure angle

Addendum To: Geometric design, modelling and finite element analysis of spur gear pairs having normal teeth based on the modification of the pressure angle

Analytical method for time-varying meshing stiffness and dynamic responses of modified spur gears considering pitch deviation and geometric eccentricity

Pitch deviation and geometric eccentricity are almost inevitable in gear transmission systems. Tooth profile modification (TPM) is a common method used to improve gear contact properties. However, the coupling interaction between the tooth profile deviation introduced by TPM and manufacturing errors is complex. There is a lack of an analytical model that can clarify the dynamic behaviors of the modified gear pair affected by multiple manufacturing errors, specifically pitch deviation and geometric eccentricity. To fill this gap, a comprehensive analytical gear mesh model is proposed considering TPM, pitch deviation, and geometric eccentricity. The calculation formulas for the time-varying meshing stiffness (TVMS) and load sharing factor (LSF) are derived considering the iterative change of gear pair meshing parameters with phase under the influence of multiple manufacturing errors. Then, the translation-torsion coupling dynamic model of the spur gear pair based on the lumped mass method is established to investigate the dynamic transmission error (DTE) and vibration characteristics. Results show that TPM is an effective method for improving the comprehensive stiffness of a gear pair within the meshing range containing pitch deviation, but it reduces the meshing phase that includes only geometric eccentricity. Compared with the unmodified gear pair, the suppression effect of changing the modification length on the maximum DTE and vibration acceleration is more significant than that of the modification amount. TPM increases the overall fluctuation threshold range of DTE for gear pairs with only geometric eccentricity. It is expected that the proposed method can provide a theoretical basis for exploring vibration suppression in gear pairs with geometric eccentricity and pitch deviation errors.

Experimental Analysis of Spur Gear Pair with Geometrical and Operating Parameters

Abstract This study presents experimental work to analyze spur gear pair with geometrical and operating parameters. The spur gear pair accommodates the correction in tooth addendum, gear backlash, and linear tip relief profile modification with three levels. As per Taguchi L9 orthogonal array, nine test spur gear pairs are precisely manufactured to analyze the combined influence of these three geometrical parameters on gear dynamics. Other basic gear design parameters and operating parameters were held at a constant level. RMS acceleration in the vertical direction to quantify the dynamic response of test gear pairs. Experimental data are analyzed by using the Taguchi method to investigate the rank of influencing parameters to minimize vibration response. Finally, the confirmation test was performed by using a simulation study to validate the experimental results. The simulation results are similar to the experimental results. Similarly, 20 experiments were conducted with different speed and load combinations to check the effect of operating conditions on gear dynamics. From this experimental study, it is observed that the rank of influencing parameters and optimum level of geometrical are varied with respect to operating conditions. It may be concluded that typically optimized spur gear pair operated at one particular combination of load and speed may not show good performance at other operating conditions. Therefore, this study suggests that the realistic range of operating conditions should be considered while selecting suitable geometrical parameters.

Optimization of polyoxymethylene spur gear pair using meta-heuristic algorithms: A comparative study

Design-optimization of power transmitted elements such as gears is a complicated process to accomplish as the mathematical model acquires to be in compliance with the optimization algorithm used. Utilization of technical standards and handbooks assisted engineers and designers with designing the gears, however the results were inefficient, thus meta-heuristic algorithms were approved for optimization, for example, genetic algorithm. In this paper, optimization of single stage spur gear pair made of polyoxymethylene material was carried out using genetic algorithm (GA), artificial bee colony (ABC), JAYA, grey wolf optimizer (GWO), and whale optimization algorithm (WOA) in MATLAB. Three single functions were used as objectives, those are, weight, power loss, and center distance. Module ( m), face width ( b), number of pinion teeth ( z1), and finally profile shifts ( x1, x2) served as design variables. Unlike steel gears which were constrained by handling root bending stress and flank pressure, more constraints were added to the problem with the increase of complexity such as root and flank temperature, tooth deformation, and wear abrasion. The results showed a significant decrease in all three objectives when optimizing each objective solely with variation of the variables. It is also observed that JAYA has the superiority in decreasing the three objectives in comparison to GA and ABC nevertheless, in agreement with GWO, and WOA.

Linear complementarity formulations for contact analysis and wear prediction in gears

This work presents linear complementarity based formulations for contact analysis and wear prediction in gear trains. Assuming steady motion of gears transmitting constant input torque without transmission errors, three sets of linear complementarity formulations are proposed, first for rigid gear teeth, second for gear teeth with skin compliance, and third for gear teeth with full compliance, the last two being new in linear complementarity literature. The developed formulations are used for contact analysis of two examples of gear trains, a pair of spur gears in mesh, and a planetary gear train. The accuracy of the method is tested by considering various time step sizes and stability of the formulations demonstrated for wide range of time steps. Contact forces between meshing teeth are calculated at a pre-defined set of points on the path of contact, and then wear is calculated on gear tooth flank using Archard’s wear equation. These wear calculations are compared with existing results in literature which demonstrate the applicability of the developed LC formulations. Future work is projected to include more state-of-art friction models and extension to 3D contact analysis in helical gear trains.

Meshing characteristic analysis of spur gear pair with tooth surface wear fault based on improved fractal method

Meshing characteristic analysis of spur gear pair with tooth surface wear fault based on improved fractal method

An improved tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors

The contact quality between geometric features significantly affects the assembly accuracy and dynamic performance of mechanical products. However, accurate and efficient contact problem solving is currently a common challenge in the fields of tolerance analysis and tooth contact analysis (TCA). In this paper, a tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors is proposed. Specifically, tolerance modeling of systematic and random deviations on tooth surfaces is represented by skin model shapes (SMS). Elastic-plastic contact behaviors including contact deformation, status, and stress distribution are modeled and calculated by a modified boundary element method (BEM). The accuracy and efficiency of this BEM algorithm are verified by comparing with the finite element method (FEM), indicating that the BEM can greatly improve the computational efficiency while maintaining high solution accuracy. A case study for a pair of spur gears demonstrates the significant effects of contact behaviors on transmission performance. This research provides an effective method to introduce mechanical contact behaviors in tolerance analysis of gear mechanisms. In addition, the contact simulation results can also be adopted in the analysis of vibration, reliability, and other aspects.

Modeling and contact analysis for spur gear pair with modification and localized crack failure

The modified teeth are deliberately manufactured to mitigate the stress concentration generated by offset components, which makes equipment more susceptible to localized failures. Therefore, this paper presents a model aimed at investigating the meshing characteristic of gear pair with modification and crack failure. In this model, the tooth deformation and force transmission are determined based on slice coupling theory. The torsional deformation of teeth produced by eccentric load distribution is considered and complemented to the calculation method. Finally, the approach is proven through the experimental tests and FE model. The result shows that when the equipment is operated in misalignment condition, eccentric load distribution, and stress concentration at tooth edge are generated, the meshing stiffness will be reduced drastically, which can be suppressed by appropriate modification, and the rapid decline of meshing stiffness is weakened by increasing the effective contact length. Although the contact stiffness would be further reduced by the crack failure and the tooth surface deformation is increased, it seems to alleviate the stress concentration and eccentric load effectively.

A Computational Model for Analysing the Dry Rolling/Sliding Wear Behaviour of Polymer Gears Made of POM.

This study presents a computational model to determine the wear behaviour of polymer gears. Using PrePoMax finite element numerical calculation software, a proposed computational model was built to predict dry rolling/sliding wear behaviour based on Archard's wear model. This allows the calculation of the wear depth in each loading cycle with constant mesh updating using the finite element method. The developed computational model has been evaluated on a spur gear pair, where the pinion made of POM was meshed with a support gear made of steel. The computational results obtained were compared with the analytical results according to the VDI 2736 guidelines. Based on this comparison, it was concluded that the proposed computational model could be used to simulate the wear behaviour of contacting mechanical elements like gears, bearings, etc. The main advantage of the model, if compared to the standardised procedure according to the VDI 2736 guidelines, is the geometry updating after a chosen number of loading cycles, which enables a more accurate prediction of wear behaviour under rolling/sliding loading conditions.

Stochastic uncertain lubrication in gear transmission subjected to tribodynamic loading

A stochastic uncertain tribodynamic model is established for a spur gear pair for the first time. The stochastic uncertainty of pinion rotation speed propagated to lubrication performance is investigated. The probability density function of the minimum lubricant film thickness hmin evolves over time periodically at interval of an engagement process. Correspondence between abrupt increase in meshing force and amplification of hmin uncertainty is found. Robust and reliable lubrication performance can be achieved by suppressing the hmin uncertainty and decreasing the lubrication failure probability. This can be done by increasing lubricant viscosity, and decreasing input torque and uncertainty level of input rotation speed. This work lays a solid foundation for robust and reliability based optimization for tribodynamic gear system.

Calculation of Time-Varying Mesh Stiffness of Internal Gears based on Precise Tooth Profile and Dynamic Analysis of Planetary Systems with Root Cracks

The internal meshing spur gear pair is the research object, and the potential energy method is applied to calculate the time-varying meshing stiffness of the internal gear. The internal gear tooth profile is divided into two parts: involute and transition curves, and the gear tooth stiffness is calculated by numerical integration based on accurate tooth profile, which improves the calculation accuracy. Analyzed the influence of different crack size parameters on the stiffness of internal meshing gears. A coupled dynamic model of a planetary system with internal gear crack faults was established, and the influence of cracks on the dynamic response of the planetary system was studied using Zoom-FFT spectrum and cepstrum analysis methods. The simulation results show that as the crack parameter size increases, the mesh stiffness of the internal gear pair gradually weakens, and the periodic vibration impact of the planetary-internal gear pair is also more severe. Cepstrum analysis can easily capture weak fault characteristic frequency information in the system and discover their variation patterns. The research results can provide a theoretical basis for the state monitoring and fault diagnosis of gear systems with crack defects.

Prediction of mechanical efficiency of spur gear pairs based on tribo-dynamics model of multi-tooth meshing

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.

ALTERNATE SLIDING FRICTION FORMULATIONS OF ELASTO - HYDRODYNAMIC OF SPUR GEARS

In this paper, several sliding friction formulations used in spur gear dynamics are examined and compared in terms of the predictions of interfacial friction forces and off-line-of-action. Computing friction formulations include Coulomb models with time-varying friction coefficients and empirical expressions based on elasto - hydrodynamic and/or boundary lubrication regime principles. Predicted results compare well with friction force measurements and appeared completely conformed with specific objectives of this paper are established as follows: (1) propose an improved MDOF spur gear pair model with time-varying coefficient of' friction, µ(t), given realistic mesh stiffness profiles.(2) comparatively evaluate alternate sliding friction models and predict the interfacial friction forces and motions in the off-line-of-action (OLOA) direction: and (3) validate one particular model (III) by comparing predictions to the benchmark gear friction force measurements

A New Calculation Method for Instantaneous Efficiency and Torque Fluctuation of Spur Gears

As a critical component of the joint gearbox, spur gear pairs play a crucial role in energy conversion, limiting the performance of a collaborative robot. Accurately assessing their instantaneous efficiency and torque fluctuation is essential for developing high-precision robot joint control models. This study proposes a computational model to predict the instantaneous efficiency and torque fluctuation of spur gears under typical operating conditions. The model incorporates a torque balance model, a load distribution model, and a friction model to reflect the relationship between gear meshing position and efficiency. The instantaneous efficiency and torque fluctuation of gear pairs were compared with the Coulomb friction model with an average friction coefficient and the elastohydrodynamic lubrication model with a time-varying friction coefficient. The effect of gear contact ratio on efficiency is analysed, while the instantaneous efficiency and torque fluctuation of gears are studied under varying operating conditions. The results indicate a maximum efficiency difference of 1.86 % between the two friction coefficient models. Under specific operating conditions, the instantaneous efficiency variation of the gear pair can reach 3.34 %, and the torque fluctuation can reach 5.19 Nm. Finally, this study demonstrates the effectiveness and accuracy of the proposed method through comparative analysis.