Optimal Tooth Profile Research Articles

Improving the wear resistance of 3D printed spur gears through a free-form tooth flank optimization process

Involute gears have traditionally been the preferred choice for gear transmission systems due to their simplicity and interchangeability. However, there are applications where they do not provide the most durable and efficient solution. While the cost of implementing optimized non-involute gears in most applications often outweighs their comparative advantages, the advent of additive manufacturing has opened up possibilities for designers to explore alternative gear tooth profiles. This is particularly relevant in the realm of plastic gears, where optimized non-involute gears produced through 3D printing can address their primary drawbacks, such as surface durability and wear resistance. In this study, a comprehensive free-form optimization process was conducted to determine the optimal tooth profile that minimizes wear on 3D printed spur gears during operation. The tooth flank geometry was represented using a 4th order B-spline curve, and a genetic algorithm was employed to determine the optimum positions of the control points aiming to minimize wear depth across the tooth flanks. The spur gears were manufactured using Fused Deposition Modeling (FDM) with PLA material. The parameters of the additive manufacturing process were experimentally fine-tuned to achieve the best possible accuracy. To evaluate the performance of the optimized free-form gears, two case studies were implemented, demonstrating that the optimized gears achieved a remarkable reduction of average wear depth by more than 50% and a reduction of maximum wear depth by more than 69% compared to standard involute gears. To further validate the effectiveness of the optimization method, experiments were carried out using an FZG test rig. The profiles of the tooth flanks were measured on a Coordinate Measuring Machine (CMM) before and after the experiments to compare the wear depth against the standard involute gears. The results revealed a significant improvement in the wear resistance of the tooth flanks, with a reduction of wear depth of 44.1%.

Improving the productivity of the cutter-loader

The task of the conducted research is to determine the dependence of the productivity of the cutter-loader on the parameters of the engagement of the teeth of the wheel-rack mover of chainless haulage systems. To solve this problem, the dependence of the influence of the parameters of the toothed gearing of the wheel-rack mover of the chainless haulage system on the performance of the cutter-loader is found using the mathematical apparatus. The equation of the balance of the power of the cutter-loader is made, taking into account the mechanical energy input for the destruction of the formation, loading the destroyed mineral mass onto the face conveyor (relevant for cutter-loaders for excavating thin flat-lying seams) and moving the cutter-loader. The theoretical performance of the cutter-loader depends on the redistribution of the mechanical energy of the drive electric motor, aimed at the destruction of the mineral and the translational movement of the cutter-loader. The correlation of the productivity of the cutter-loader with the parameters of the engagement of the teeth of the wheel-rack mover of chainless haulage system systems is revealed. The obtained dependences of the performance of the cutter-loader allow assessing the degree of influence of the parameters of the toothed gearing of the mover of the chainless haulage system on the performance of the cutter-loader. The possibility of increasing the minute productivity of the cutter-loader by up to 20% when using a wheel-rack mover of a chainless haulage system with an optimal tooth profile is established.

Optimization Design of an Axial Split-Phase Bearingless Flywheel Machine with Magnetic Sleeve and Pole-Shoe Tooth by RSM and DE Algorithm

In order to improve the suspension and torque performance of a bearingless flywheel machine, a new type of axial split-phase bearingless flywheel machine with a magnetic sleeve and pole-shoe tooth is analyzed and optimized as described in this paper. Based on the basic structure and working characteristics of the machine, the response surface methodology (RSM) and differential evolution (DE) algorithm are adopted to further optimize the parameters of the stator teeth of the machine to improve the radial space utilization and motor output performance. Firstly, the Box–Behnken design (BBD) and finite element analysis (FEA) are combined to select the representative optimization parameter combinations to establish the sample data space, and the response surface models of machine torque and suspension force are established using the least square method. Besides this, the DE algorithm is employed to obtain the optimal tooth profile parameter configuration for the multi-objective optimization of machine performance. Finally, the output performances of the machine before and after optimization are compared under initial and optimized winding turns. The results show that, compared with the initial structure, the average torque and suspension force of the optimized machine increase by 36.46 % and 108.22% respectively, which demonstrates the effectiveness of the tooth profile optimization method. At the same time, an experimental prototype is also produced, laying the experimental foundation for further practical exploration.

Optimal tooth profile design of a gear shaper cutter when meshing with internal straight splines

In this paper the possibilities for practical realization of a non-traditional method for internal straight splines cutting by gear shapers, using internal gearing have been analyzed. Based on the theoretical research conducted, the equations of the gearing line have been determined, and later used for defining the parametric equations of shaper teeth profile. Gear shapers with a different number of teeth have been profiled. The effect of the number of shaper teeth over their contour is explored. It is theoretically proved that if the gear-shaper number of teeth is twice as small as that of the splines, the profile of the tool teeth is considerably simplified. In that case, the profile transforms to a part of a circle. It has also been found that when the shaper has only a few teeth, their profile contour is a broken curve. In this regard, a limiting condition for avoiding this curvature and a formula for determining the minimum number of gear shaper teeth has been defined.

On optimal tooth profile modification for reduction of vibration and noise in spur gear pairs

The usefulness of deliberate modification of the tooth profile of a spur gear pair in reducing the level of vibration is studied. The severity of vibration and noise at different operating conditions have been measured by different metrics namely dynamic mesh force, dynamic tooth force, bearing force along line-of-action and off-line-of-action directions. The optimal modification of the tooth profile has been obtained by solving an optimization problem. Two methods, namely, a graphical method using contour plots and a semi-analytical method are presented for the said purpose. It is shown that if properly modified the vibration caused by geometrical errors like profile and pitch error can be reduced considerably.

The Optimal Design Method and Standardized Mathematical Model of Tooth Profile Modification of Spur Gear

The paper reports a tooth profile modification method of spur gear. After establishing a standardized mathematical model for optimized tooth profile and simulating meshing process with ANSYS finite element analysis, we obtained 625 groups of gear models with different modification parameters. The group with minimum transmission errors owns the optimal parameters. Genetic algorithm was adopted in the entire process for the purpose of reducing the variation of transmission errors in meshing process. The arc and parabolic modification were doing the same processing. After comparing the transmission errors fluctuation produced by the meshing process of gear of nonmodification with arc modification and parabolic modification, we found that the best modification effects of arc modification and parabolic modification were both reduced by 90%. The modification method makes the gear drive process more stable and efficient, and it is also promising in general application for gear drive.

Influence of addendum modification on spur gears stability

The aim of the presented research is to develop the new method for studying the influence of main characteristics of spur gears on their stability behaviour. Few different processes and calculations are compiled in algorithm make a method for spur gears stability analysis. In this paper, the main influence factors for spur gears stability are identified and the discussion of appropriate dynamic model of spur gears is given. In order to determine the time-dependent load distribution and mesh stiffness, developed procedure uses verified spur gears model for finite element analysis. The special attention has been devoted to the research and conclusions about the influence of addendum modification on spur gears low-vibration and stability. The discussion is based on comparative study of a series of gear pairs with different addendum modification coefficient in order to select the optimal tooth profile design with gear pair low-vibration and stability as the main aspects.

Nonlinear vibration of the bevel gear with teeth profile modification

The prediction of gear vibration and noise has always been a major concern in gear design. Noise and vibration are inevitable problems that are involved in transmission systems; they have intensified when some nonlinear phenomena such as jump phenomenon, tooth separation and period-doubling bifurcation appear in the system. Tip and/or root modifications are well-known solutions that improve dynamic performance of gears. The present work investigates the complex, nonlinear dynamic behavior of three bevel gear models: (1) model with pure involute profile, (2) model with statically optimized tooth profile, and (3) model with dynamically optimized tooth profile. Tooth profile modification is employed in models by means of genetic algorithm in order to extract the best amount and length of modifications. The dynamic responses obtained from dynamic analyzer were compared qualitatively and quantitatively. By augmenting tooth profile modification, the average value of the dynamic responses is decreased intensely for both statically and dynamically optimized gear pairs. Dynamic load factor is calculated and compared with the involute tooth profile model and the two optimized gear sets. Employing teeth optimization leads to elimination of period- $$2T_{m}$$ in both optimized simulations.

Quadruple-Arc Gear Tooth Profile Parameterize Optimization Based on ANSYS

In this paper, quadruple-arc gear tooth profile parameters are optimized further by finite element analysis software ANSYS, and acquired a group of optimized tooth profile parameters. Which can improve the quadruple-arc gear carrying capacity, and provide a theory for design and calculation of quadruple-arc gear.

Explicit Parametric Method for Optimal Spur Gear Tooth Profile Definition

The gear tooth profile has an immense effect on the main operating parameters of gear pairs (load capacity, working life, efficiency, vibrations, etc). In current engineering research and practice, there is a strong need to develop methods for tooth profile optimization. In this paper a new method for selecting the optimal tooth profile parameters of spur gears is described. This method has been named the Explicit Parametric Method (EPM). The addendum modification coefficient, radius of root curvature, and pressure angle of the basic rack for cylindrical gears, have been identified as the main tooth profile parameters of spur gears. Therefore, the EPM selects the optimal values for these three tooth profile parameters. Special attention has been paid to develop a method of adjustment for the particular working conditions and explicit optimization requirements. The EPM for optimal tooth profile parameters of gears uses contact nonlinear Finite Element Analysis (FEA) for calculation of deformations and stresses of gear pairs, in addition to explicit comparative diagrams for optimal tooth profile parameter selection.

Stress optimization and study of the sensitivity to geometric variations of a spur gear tooth profile

This paper presents an approach for obtaining an optimized geometry for the flank of a tooth by minimizing the equivalent contact stress. The stress calculation method is based on Hertz theory. As the majority of tooth profiles are involute, the geometric variation of the flank of the tooth is achieved variationally relative to the involute profile. The optimum profile is obtained by Monte Carlo simulation. During this optimization, a polynomial expression of the tooth geometry is used. The parameters influencing the simulation are the four characteristic contact points. The approach presented has been applied in a representative case. A study of the geometric sensitivity was conducted on the optimized tooth profile. Two different approaches were considered: by Monte Carlo simulation and analytical propagation. The robust and linear nature of the behavior of the tooth profile was demonstrated when it was subjected to geometric variations.

Parametric Study of Meshing Characteristics With Respect to Different Meshing Rollers of the Antibacklash Double-Roller Enveloping Worm Gear

Roller shapes play a very critical role in the performance of antibacklash double-roller enveloping hourglass worm (ADEHW) gears; however, their influence is seldom reported in the literature. Based on the theories of differential geometry and gear meshing, this paper presents generic models of meshing characteristics for ADEHW gears, including both the contact curve and the tooth profile. We present different meshing functions and their derivatives with respect to each drive type and their associated roller shapes. We also compare the effects of contact curve, tooth profile, tooth undercutting, lubrication angle, induced normal curvature, and autorotation angle in order to design the most optimal tooth profile for the ADEHW gear. Finite element analysis was also conducted in order to minimize the contact stresses as a function of the roller type. Our results show that a spherical roller has the smallest value among the three available meshing roller shapes but is the most difficult design to procure. A cylindrical roller, however, incurs the greatest contact stress.

The Optimization Design for Rack Tooth Profile Curve of Straight Push-Rod Linear Reducer

The straight push-rod linear reducer is a new kind of transmission installation, the rack is the key link of the mechanism. Here the author have discussed the establishment of optimization mathematical model about rack tooth profile curve when the linear reducer are used in different occasion, and also have introduced the adopted optimization methods, finally have analyzed the optimal result of tooth profile curve of two specific conditions. In the end, through the comparison about some corresponding indexes of tooth profile between the optimal result and a few commonly used curve, the optimal tooth profile got by the optimal solution is verified the optimal result.

Mathematical models for manufacturing a novel gear shaper cutter

The design principle and models for novel error-free shaper cutters are discussed to improve the accuracy and service life of gear cutting tools. The modified methods for designing the tooth profile of the shaper cutter are developed by including the inverse envelope method, midpoint-midline method and midpoint-displacement method based on the theory of geometrical reverse problem. The universal mathematical models for manufacturing optimal tooth profile of shaper cutters are presented. The feasibility and reliability of the proposed principle, methods and models are proved by combining the numerical examples with practical application. The design method suggested in this paper is superior to the traditional design scheme, and will be helpful in facilitating the design and manufacture of shaper cutters.

Optimized tooth profile based on identified gear dynamic model

A new method to determine the dynamic model of a practical gear system is proposed by the identification of the relation between the rotational movement of gears and the acoustical noise signal measured on the gear tester. The rotational movement alone the line of action of gears is treated as the input of the dynamic gear system and the acoustical noise signal as the output. The parameters of the model are estimated by the LSM (least square method). By an example that the gear pair works under a certain design load and rotational speed, the identified model is verified and the optimized gear tooth profile modification is derived based on the identified model. The tested results show that the profile modification can greatly reduce gear noise.

Theoretical and Experimental Study on Optimum Design of Gear Profile Modification Curves

Gear profile modification is an important method for reducing system vibration and noise of a gear transmission. In this paper, three kinds of gear profile modification curves are studied, and relevant experiments are conducted. Some useful conclusions based on the theoretical analysis and experimental results have been obtained. Line, parabola and sinusoid modification curves are very useful in improving the dynamic performance of a gear transmission system by reducing system vibration and noise. In order to design the optimal tooth profile modification curve, the curve type must be decided firstly according to the specific working condition, and the curve parameters can be determined by means of optimum programming.

Minimization of transmission error in helical gears

A method of minimizing transmission error in helical gears is presented. Using the concept of optimal modification of the plane of action to generate the new tooth profile, the idea of perfect plane of action of helical gears is introduced. The cubic splines are used to determine the optimal helical tooth profile. The method proposed in this work constitutes a general tooth correction method and makes the design, unlike other modification schemes presented in the literature, less complex and less sensitive to dynamic factors and retains the involute characteristics. Also the proposed method permits easier cutting of the tooth profile because of the simplicity of the process.

Optimal Tooth Modifications for Spur and Helical Gears

A method for the simultaneous calculation of optimal tooth tip relief and tooth crowning for spur and helical gears is presented in this paper. The tooth profile modification is described by a linear function. Two types of crowning are introduced: linear and parabolic. The optimization of the tooth modifications is based on the following conditions: (1) The teeth are entering in mesh smoothly, without interference. (2) The load distribution factor is minimized. A computer program is developed for the calculation of the optimal tooth tip relief and crowning for spur and helical gears. By using this program the influence of type and length of optimal crowning and length of tooth tip relief on load distribution factor is investigated. Also, the influence of gear parameters on optimal tooth profile modification is discussed. On the basis of the obtained results, by regression analysis an equation is derived for the calculation of the optimal tooth tip relief.

The synthesis of tooth profile shapes and helical gears of high hydraulic performance for rotary type pumps

In this work a method is presented for the synthesis of high hydraulic performance of non involute helical gears and tooth profiles. Two gear capacity criteria are used in the synthesis: (1) the capacity based on optimal helical advance equal to the circular pitch for which perturbations of delivery speed and torque can be completely eliminated and (2) the capacity based on optimal tooth profile which consists substantially in circular arcs whose centers fall on the pitch point, and whose connecting part consists of a suitable curve which satisfies the condition for continuous contact without intersection of two mating teeth surfaces. The present paper deals with the theoretical basis and the comparison with test results. The tooth profile recommended as a result of this study has permitted size and cost reduction in a currently manufactured line of pumps with delivery pressure to 20 kg/cm 2.