Loaded Transmission Error Research Articles

Continuous optimization method of micro geometry for unparallel beveloid gears

To enhance the bearing capacity and minimize the loaded transmission error for unparallel beveloid gears, a continuous optimization method of micro geometry is proposed. Considering that the conjugate characteristic of tooth surfaces is destroyed by misalignments and modification, an unload contact analysis model is first developed. This includes a contact tooth sequence determination model and an unloaded transmission error calculation model for multi-tooth contact. To determine contact due to deformation, a potential contact point matching model of non-contact tooth surfaces is introduced. Further, a numerical loaded contact analysis model based on the influence coefficient method for unparallel beveloid gear is developed. For multi-tooth contact of modified tooth surfaces, the transmission error compatibility condition is introduced. Based on the loaded contact model, a continuous optimization model for the comprehensive performance of contact pressures and loaded transmission error within a meshing cycle is established. To improve optimization efficiency, a calculation strategy for continuous optimization is developed. The feasibility of the proposed method is validated using a numerical example.

Robust modification optimization for helical gear mounting errors based on contact ratio

Mounting errors are a disruptive factor in gear utilization. This paper investigates the contribution of mounting error components in helical gears to the helix slope deviation using the involute equations. A model of the helical gear transmission system has been established, by which the impact of mounting errors on the loaded transmission error and contact ratio of the bevel gear transmission was studied using the Loaded Tooth Contact Analysis method. A bench test was then casted as a verification of this model. A methodology was proposed for optimizing the micro modification of helical gear flanks based on contact ratio control. Robust optimization of the mounting error was achieved by reducing the contact ratio under commonly used torques compared to the contact ratio at the local minimum of the loaded peak-to-peak transmission error. This approach has been demonstrated to enhance the robustness of gear transmission against mounting errors and torque fluctuations.

Analytical model of meshing stiffness, load sharing, and transmission error for internal spur gears with profile modification

The influence of the teeth geometry on the load distribution, transmission error, and meshing stiffness of internal spur gears is very similar to that of external gears. Therefore, the formulation of the models for strength and dynamic calculations for internal and external gears is also similar. However, the internal gears have usually greater values of the contact ratio, frequently above 1.8, even 1.9. In these cases, the lengthening of the contact interval due to the elastic deflections of the teeth may result in an effective contact ratio greater than 2, with the subsequent changes in the load sharing and transmission error. These changes are not properly described by the theoretical models for high contact ratio spur gears. In this paper, a model of load sharing, quasi-static transmission error, and time-varying meshing stiffness for internal spur gears with profile modification is presented, which has been applied to low contact ratio internal gears which, due to the transmitted load, become high contact ratio internal gears. The influence of the variations of the transmitted load on the load capacity is also studied.

Research on noise reduction of drive axle hypoid gear based on tooth surface mismatch modification

In order to reduce the transmission noise of drive axle hypoid gear, a method of tooth surface mismatch modification was proposed. Firstly, on the basis of establishing the grinding mathematical model of hypoid gear with cutter tilt, the theoretical tooth surface equation was derived, and the numerical tooth surface was calculated by dividing the grid points on tooth surface. Secondly, the pinion tooth surface which was full conjugated with gear tooth surface was constructed, and the tooth surface mismatch topography was established and decomposed into the first order and second order mismatch coefficients. According to the modification requirements, the pinion target surface was constructed by modifying the tooth surface mismatch coefficients, and the pinion modified machine setting parameters were acquired by establishing the pinion processing parameters corrected mathematical model. Finally, the loaded tooth contact analysis with finite element simulation and drive axle NVH (Noise, Vibration, Harshness) simulation were carried out for a pair of hypoid gear, therefore the loaded tooth contact areas, transmission errors curves, and noise simulation curves were obtained, and the simulation results show that after modification the tooth contact stress, loaded transmission errors amplitude, and NVH simulation curves value are all reduced. The road test results indicate that the drive axle hypoid gear howling phenomenon under the high speed operation can be eliminated, so the effectiveness of tooth surface modification method and NVH simulation method were verified.

Meshing characteristics and dynamic characteristics of planetary transmission system with new tooth-type flexible ring gear

As one of the important components of the planetary transmission system, dynamic meshing characteristics of ring gear directly affect the performance of planetary transmission system load sharing and dynamic load factor. With the help of the flexibility of ring gear, the planetary gear system with flexible ring gear (FRG) can compensate for the uneven transmission torque of each branch caused by the machining error and installation error and improve the system's load sharing and dynamic load performance. An L-shaped FRG was developed in Italy to improve the traditional ring gear and was used in the latest AW101 helicopter with good feedback. By increasing the ring flexibility, the load distribution between planetary gears is more uniform, and the load-sharing performance of the system is significantly improved. Tooth wear and fracture caused by uneven load distribution can be improved. However, the structure has problems such as unbalanced loading, strength weakening, and stress concentration. In this article, the FRG's meshing and structural characteristics are analyzed in detail by constructing the parametric finite element model. Differences, advantages, and disadvantages of FRG and traditional ring gear are compared. A new type of tooth-type FRG is proposed. It has excellent structural flexibility meshing characteristics of tooth surface and avoids generating stress concentration. It improves the planetary gear train's load-sharing performance and overcomes tooth surface unbalanced loading and large amplitude fluctuation of loaded transmission errors.

Research on Loaded Contact Analysis and Tooth Wear Calculation Method of Cycloid–Pin Gear Reducer

This study establishes the geometric model of cycloid–pin gear meshing transmission based on the multi-tooth meshing characteristics of the cycloid speed reducer. The calculation and analysis of meshing motion parameters of the cycloid speed reducer are carried out. An integrated calculation flow is presented for solving the question of the loaded tooth contact of the cycloid speed reducer by using the elimination clearance method of gradual contact and the quasi-Hertz contact simulation of the tooth surface under loads. The loaded transmission error is obtained, and both the number of pins participating in the meshing and the contact area of tooth surfaces are determined. Using the regression formula of the wear coefficient, the dynamic wear coefficient is quickly solved on the instantaneous contact line of the tooth surface. Thereby, the wear distribution law of two tooth surfaces appears. The results show that there is a singular point in the wear of the pin teeth, with a maximum wear of 100 μm, that seriously affects the meshing accuracy of the tooth surface and thus affects the accuracy and lifespan of the reducer.

An ease-off tooth surface redesign for spiral bevel gears considering misalignment under actual working conditions

To improve the loaded performance of spiral bevel gears, a novel tooth surface redesign method considering misalignment is proposed based on ease-off. First, the digital features of the contact pattern were extracted, and the equivalent misalignment was obtained by an optimal method according to the minimum deviation of the contact path. Second, a pinion target surface whose performance under misalignment was consistent with the original gear in standard position was built, and a pinion surface with good meshing performance under misalignment was redesigned with equivalent misalignment. Third, the flank modification was carried out to cut down on the loaded transmission error of gear under misalignment. Through simulations, it is found the transmission error and the contact path of redesign gear considering misalignment were the same as original gear in standard position. The loaded transmission error amplitude of original gear under misalignment was 43.91% higher than original gear in standard position, and the loaded transmission error amplitude of redesign gear after optimization under misalignment was 44.73% lower than original gear in standard position and 61.60% lower than original gear under misalignment. The tooth surface stress of redesigned gear after optimization under misalignment was also significantly improved. This proposed redesign method, which considers misalignment on the basis of ease-off, can greatly improve the loaded meshing quality of gear under actual working conditions.

Influence of profile modification on the transmission error of spur gears under surface wear

The load distribution, transmission error, and time-varying meshing stiffness of spur gears are all influenced by the teeth dimensions and profiles shape. Any modification on the teeth geometry, as a profile relief frequently used to avoid the mesh-in impact, will affect all these parameters and the entire dynamic behavior of the gearset. In addition, due to the friction between contacting surfaces, wear arises at the teeth flanks. This causes the shape of the meshing profiles to change, which will affect the load distribution and the transmission error for the subsequent meshing cycles. And this new load distribution causes a surfaces wear different form that of the previous meshing cycle. Consequently, the wear depth increase is different at any contact point —because the load and sliding velocity are different— and at any meshing cycle as well —because the load distribution changes with the accumulated wear—. And all the above is also influenced by the profile modification the teeth were manufactured with. In this paper, a simple, analytic model of meshing stiffness, load sharing, and transmission error for spur gears with profile modifications including the influence of surfaces wear is presented. The evolution of the transmission error with the number of wear cycles is studied, and the optimal profile modification for minimizing the dynamic load induced by the transmission error is investigated.

A novel design method of low noise spiral bevel gear tooth surface for polynomial ease-off topological modification

In order to reduce the vibration noise of the spiral bevel gear transmission system (SBGTS), a polynomial ease-off topological modification (PETM) method, which combines the load tooth contact analysis (LTCA) and vibration analysis of the spiral bevel gear, is proposed in this paper to minimize the normal dynamic meshing force and normal relative displacement to reduce the vibration noise. Firstly, a new PETM equation is established based on the complete conjugation principle, which is applied to the initial clearance calculation of LTCA and an explicit LTCA calculation method with PETM without requiring inverse processing parameters is constructed. The calculation accuracy is better than that of a general algorithm. Secondly, the dynamic response of an eight-degree-of-freedom dynamic system of spiral bevel gear is calculated according to the loaded transmission error and time-varying meshing stiffness calculated by the explicit LTCA of PETM. Finally, a PETM low-noise tooth surface design method is constructed with the goal of minimizing the normal meshing force and the normal relative displacement and the PETM equation coefficient as the optimization variable. The numerical results show that the new PETM low-noise tooth surface design method can improve the meshing performance and dynamic performance of spiral bevel gears to a greater extent than those before modification and after the second-order topological modification, thus greatly reducing the vibration noise of SBGTS.

Exploration of trade-offs between NVH and efficiency in bevel gear design

Minimizing NVH and friction-induced power losses is becoming paramount in the design of geared transmissions. The aim of this paper is to present an automatic methodology to explore Pareto-optimal designs of bevel gears when minimization of noise and frictional losses is essential. In the first part, a semi-empirical model to estimate frictional power losses under elasto-hydrodynamic lubrication is described. The model has been validated against experimental data available in the literature in previous works by the authors. The efficiency calculation is coupled with a state-of-the-art loaded tooth contact analysis (LTCA) tool to obtain accurate predictions of the instantaneous load shared by the mating tooth pairs during the meshing cycle. In the second part, an automatic framework based on multi-objective optimization (MOO) is presented where the tooth micro-geometry is systematically designed. The design variables are represented by few coefficients of a polynomial basis that embodies the tooth flank ease-off topography. To ensure manufacturability, the polynomial modifications are projected onto the feasible set of the machine-tool envelopes. This step is achieved through a state-of-the-art identification algorithm that the authors have developed in previous work. Frictional losses are estimated with the aforementioned model, whereas the NVH level is measured by the loaded transmission error (LTE), directly available from the simulation tool. The maximum contact pressures are limited by the material properties, thus proper nonlinear constraints are prescribed. Application to a test case involving the design of a spiral bevel gearset reveals that the methodology presented allows the designer to obtain Pareto-optimal solutions in a systematic and automatic manner.

Research on Active Precontrol Strategy for Shape and Performance of Helicopter Spiral Bevel Gears

Gear vibration becomes unmanageable under extreme conditions, which shortens the time between overhauls of the helicopter reducer and affects the service efficiency of helicopters. For this purpose, an active precontrol strategy for shape and performance was developed for spiral bevel gear (SBG) systems, thus prolonging the reliable service period of helicopters. Firstly, the perfect pinion surface was built based on the preset contact path (CP) and transmission error (TE). Secondly, through a combination of tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA), the loaded transmission error (LTE), meshing stiffness (MS), and meshing impact (MI) of the SBG transmission were obtained. Thirdly, an eight-degrees-of-freedom (8-DOF) dynamic model was built, and a simplified dynamic model was generated based on this. As regards the dynamic response, the root mean square of normal vibration acceleration (RMA) of the SBG transmission was obtained by solving the differential equation of motion. The RMA of the SBG transmission was also analysed based on the basic theory of vector structural mechanics. Finally, an optimization model aimed at reducing the RMA of the SBG transmission was established using ease-off, and a redesigned SBG transmission with good dynamic performance was obtained through optimization. Simulation analysis showed that the LTE and MS amplitudes of the optimized gear were smaller than those of the original gear. The impact force and impact velocity of the optimized gear were 27.02% and 25.81%, respectively, lower than those of the original gear. The RMA value of the optimized gear was much lower than that of the original gear. The normal vibration velocity and displacement of the optimized gear transmission were also cut down. Therefore, the design approach can effectively increase the performance of gear transmission.

Acoustical behavior of periodic flank modifications under dynamic operating conditions

Noise emissions belong to the crucial design characteristics in modern transmissions. The most efficient way to achieve beneficial noise behavior is to reduce the excitation in the tooth contact, since the tooth contact represents the main source of noise in gearboxes. Special flank modifications may optimize the noise behavior, but the common design approach uses static assumptions, whereas operating conditions include dynamic operation.The application of periodic flank modifications as a special type of flank form provides a possibility to avoid the typical compromise between the flank design objectives load carrying capacity and low excitation behavior which results from the exclusive usage of standard modifications such as crownings or tip- and root relieves. The principle of periodic modifications is based on the direct compensation of the inconstant part of the elastic deformations in the tooth mesh. For this reason, the static loaded transmission error (LTE) is typically used as specific value to evaluate the effect of these flank forms on the excitation. In theoretical and experimental studies, the effectiveness of periodic modifications to optimize the excitation behavior could be verified. However, the analysis of these modifications under dynamic operating conditions is still scarcely documented. In this paper, the mesh excitation of periodic and standard modifications under various dynamic operating conditions in the subcritical, resonance and supercritical operating range is simulated and compared to each other. In addition, experimental investigations at the dynamic test rig of the Gear Research Center (FZG) were performed. Due to the measurement of the torsional acceleration level during continuous speed runups, the simulated results can be validated by experimental data.

Nonlinear dynamics of non-orthogonal offset face gear-bearing transmission system

A nonlinear dynamic model of a non-orthogonal offset face gear-bearing transmission system is established, in which nonlinear factors such as input load fluctuation, gear clearance, bearing clearance, and transmission error are considered. The dynamic response of the system is analyzed by solving the differential equation with the numerical method. The subharmonic resonance and primary resonance characteristics of the torsional vibration displacement are analyzed by the multi-scale method. The results show that the change of meshing damping will make the system present rich nonlinear characteristics, and the equivalent static load and meshing damping ratio will affect the resonance characteristics of the system. Choosing the appropriate system parameters can make the system in a stable state and suppress the amplitude of primary resonance and subharmonic resonance.

Nonlinear excitation and mesh characteristics model for spiral bevel gears

Accurate and rapid calculation of nonlinear excitation (NE) and loaded mesh characteristics (LMC) of gear pairs is a key to achieving rapid iterative optimization of gear system designs. Therefore, an analytical model (AM) was proposed for the accurate and numerically efficient calculation of NE and LMC for application to spiral bevel gears (SBG). First, a tooth contact analysis of an SBG was completed using Coons surface technology and mesh theory to determine the contact path and unloaded transmission error (UTE). Thus, an AM for the principal and relative normal curvatures of the SBG was derived using the Euler's formula. Based on the traditional calculation method for a Hertz contact ellipse, the relation between the relative curvatures was introduced to derive the formulas for the major and minor axes using the Muller's method, which could improve the programmability and avoid numerical instability of the program. Then, analytical models for the contact ellipse and pressure distribution on the contact tooth surfaces were derived using the Hertz model and geometric theory. Subsequently, based on a single tooth LMC, infinitesimal method, elasticity, and series stiffness model, a single mesh stiffness (SMS) model for SBGs was developed, which incorporates transverse tooth stiffness, transverse gear foundation stiffness, axial tooth stiffness, axial gear foundation stiffness, and Hertz nonlinear contact stiffness. Based on the SMS and UTE, compliance and load matrices were established, and the time-varying mesh stiffness, loaded transmission error, mesh force, and multi-tooth LMC models of the SBG were determined. Finally, the proposed method was validated against nonlinear FE simulations using examples. Compared to FE simulations, the proposed method requires significantly less computational effort and can be further extended to optimization or system analysis problems.

Computerized design, simulation of meshing, and stress analysis of curvilinear cylindrical gears based on the active design of the meshing line function

The computerized design of curvilinear cylindrical gears based on the active design of the meshing line function is presented. The parametric equations of their meshing line functions are based on a parabola to control the curvilinear shape of the teeth along the face width of the gears. Different numbers of control points and different combinations of curves, including parabolic curves, circular arcs, and involutes, smoothly connected with each other at the control points are chosen to form the active tooth profile. The meshing performance, mechanical behavior, and influence of the parameters of combined curves are studied in terms of the contact patterns, variation of the maximum stresses and loaded functions of transmission errors and compared to two types of curvilinear cylindrical gear drives generated by face-milling cutters as a reference. The investigated curvilinear gear drives with five control points show lower maximum contact and bending stresses as well as lower peak-to-peak level of loaded transmission errors compared to other cases of curvilinear gear drives.

Transform of parameter and teeth surface modification for non-standard planetary spur gear train with X-gears

In view of the lack of ready formulae for modification calculation of non-standard planetary spur gear trains, the paper is focused on new modification methods of non-standard planetary spur gear trains with X-gears. Firstly, a novelty transform method from the parameters on the pitch circle to the parameters on the reference circle for a non-standard planetary spur gear train with X-gears is proposed in order to study the modification method. Secondly, a scheme of teeth surface modification for non-standard planetary spur gear train with X-gears is put forward. Thirdly, the calculation method of loaded transmission error and flash temperature on the tooth surface under mixed elastohydrodynamic lubrication for external meshing spur gear pairs is proposed using tooth contact analysis as well as loaded tooth contact analysis. Then, a new modification optimization method for non-standard planetary spur gear train with X-gears is proposed, and formulae of the maximal modification length along the tooth height are derived. Finally, the effectiveness of the modification optimization method is confirmed by an example, and the transform formulae of the optimized modification parameters for sun gear are derived.

Study on nonlinear dynamic characteristics of power six-branch herringbone gear transmission system with 3D modification

The power six-branch herringbone gear transmission system has the advantages of large power transmission and transmission ratio. Because of its multi-way transmission and over-constraint structure, to prevent loaded tooth interference, there are large backlashes between teeth. The system shows complex nonlinear dynamic characteristics under the influence of backlashes, which seriously affects meshing performance. The modification technology can effectively improve gear meshing performance. Hence, on the basis of optimizing meshing performance of the active pair of the six-branch herringbone gear transmission system, this article will combine 3D modification with system nonlinear vibration characteristics and propose an analysis method of the system nonlinear vibration characteristics under 3D modification to reduce vibration and noise. First, the 3D modified tooth surface equation is determined by forming grinding, the loaded transmission error (LTE) of the system's active pair is obtained by tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) technology, and the optimal modification parameters are obtained by Ant Lion Optimizer (ALO) with the minimum error amplitude as optimization objective. Then, the time-varying meshing stiffness of system under the optimal modification is calculated, and the pure torsional nonlinear dynamic model with backlashes, meshing stiffness, and static transmission error (STE) is established. Finally, the global vibration characteristics in parameter field are studied through time domain and bifurcation diagram of system. Results show that the 3D modification can eliminate edge contact of tooth and improve tooth contact performance. With the increase of input power, the root mean square (RMS) values of acceleration increase and the RMS values and jump decrease after 3D modification. With the increase of input speed, the RMS curves appear multiple resonance peaks and jumps at low input speed. After 3D modification, the RMS, jump, and resonance peak values decrease. Compared with level І, the backlashes and STEs of level II and phase difference ϛsII have great influence on system dynamic characteristics and easily make system in chaotic motion, while the 3D modification reduces their influence and makes system motion periodic.

Calculation of gear mesh stiffness and loaded tooth contact analysis based on ease-off surface topology

Based on the theory of rack-gear common-tangential conjugate generating, the constructing of ease-off surface has been done by means of the topology modification of polynomial surface. By analyzing topological structure characteristics of ease-off surface, the meshing characteristics such as the contact area, the contact line, and transmission error are determined. By using the geometric analysis of ease-off surface and the potential energy method, the calculation methods of gear tooth stiffness and loaded transmission error are given, and a series of results for the tooth meshing stiffness, loaded transmission error, and load distribution map under varied loads are obtained. In order to solve the stress problem of tooth surface edge contact, the calculation method of quasi-Hertz contact is proposed by subdividing finite contact elements. Taking the second-order parabolic topological modification as an example, the time-varying history characteristics of tooth surface meshing are analyzed. The simulation results of the maximum contact stress and contact area are in good agreement with those calculated by MASTA software. Based on the integration of the geometric analysis of ease-off surface, the tooth stiffness calculation of potential energy method and the quasi-Hertz contact, it provides a convenient method for loaded tooth contact analysis.

Design and Load Distribution Analysis of the Mismatched Cycloid-Pin Gear Pair in RV Speed Reducers

The current loaded tooth contact analysis of cycloid drives based on the assumption of theoretical positions of ring pins ignores the deviations caused by manufacturing errors and elastic deformations, which are not in agreement with reality. To fill this gap, an improved load distribution model of the mismatched cycloid-pin gear pair with ring pin position deviations is presented for a component-level analysis. Firstly, with the cycloid gear tooth profile geometry defined, the unloaded tooth contact analysis is applied as a pre-processor to determine the potential contact points, the gear backlash, and the rotation angle of the cycloid gear. Secondly, due to the statically indeterminate structure of the multi-tooth contact, a varying nonlinear contact stiffness is introduced to establish the relation between force and deformation. Then, the force and moment equilibrium equations with compatibility conditions are solved by using an iterative approach. With this, detailed parametric case studies are presented to verify the correctness of the proposed model by comparing it with those predicted by the current model and to demonstrate the influences of ring pin position deviations on the distributed load, contact stress, loaded transmission error, and instantaneous gear ratio of the mismatched cycloid-pin gear pair. This study provides a deeper investigation into the load distribution characteristics of the cycloid drive and therefore can be employed to assist in gear design.

Dynamic Modeling, Optimization and Experiment for a High-Speed Spur Gear Set

This study proposed a novel methodology for reducing the vibration of a high-speed spur gear pair (up to 30,000 rpm) by performing multi-objective optimization of the peak-to-peak loaded transmission error (PPLTE) under three assembly conditions. The optimum tip relief parameters for the high-speed spur gear set were studied by considering the practical manufacturing tolerances on the gear shaft and bearing bores. First, the PPLTE under three assembly conditions was determined by loaded tooth contact analysis. Multi-objective optimization with a genetic algorithm was used to determine the optimum linear tip relief parameters, minimizing the PPLTE under three assembly conditions by 70%. Moreover, the dynamic characteristics of the original and optimum designs were simulated and compared under ideal conditions as well as axial misalignments. The optimum design exhibited a >50% reduction in the peak RMS of acceleration at the natural frequency. Finally, dynamic experiments were performed and the RMS values of the acceleration at various speeds were computed for comparison. The results from both dynamic simulation and experiment indicated that the optimum design exhibits superior dynamic characteristics to the original design.