The Generation Principle and Mathematical Models of Double-Enveloping Internal Gear Pairs

In this paper, the planar double-enveloping method is presented for the generation of tooth profiles of the internal gear pair for various applications, such as gerotors and gear reducers. The main characteristic of this method is the existence of double contact between one tooth pair such that the sealing property, the load capacity and the transmission precision can be significantly improved as compared to the conventional configuration by the single-enveloping theory. Firstly, the generation principle of the planar double-enveloping method is introduced. Based on the coordinate transformation and the envelope theory, the general mathematical model of the double-enveloping internal gear pair is presented. By using this model, users can directly design different geometrical shape profiles to obtain a double-enveloping internal gear pair with better meshing characteristics. Secondly, to validate the effectiveness of the proposed model, specific mathematical formulations of three double-enveloping internal gear pairs which apply circular, parabolic and elliptical curves as the generating curves are given. The equations of tooth profiles and meshing are derived and the composition of tooth profiles is analyzed. Finally, numerical examples are provided for an illustration.

Dynamic Characteristics of Double Helical Planetary Gear Train With Tooth Friction

A new nonlinear dynamics model of the double helical planetary gear train with 44 degrees of freedom is developed, and the coupling effects of the sliding friction, time-varying meshing stiffness, gear backlashes, axial stagger as well as gear mesh errors, are taken into consideration. The solution of the differential governing equation of motion is solved by variable step-size Runge-Kutta numerical integration method. The influence of tooth friction on the periodic vibration and nonlinear vibration are investigated. The results show that tooth friction makes the system motion become stable by the effects of the periodic attractor under the specific meshing frequency and leads to the frequency delay for the bifurcation behavior and jump phenomenon in the system.

A Three-Dimensional Load Sharing Model of Planetary Gear Sets Having Manufacturing Errors

Planet-to-planet load sharing is a major design and manufacturing tolerancing issue in planetary gear sets. Planetary gear sets are advantageous over their countershaft alternatives in many aspects, provided that each planet branch carries a reasonable, preferably equal, share of the torque transmitted. In practice, the load shared among the planets is typically not equal due to the presence of various manufacturing errors. This study aims at enhancing the models for planet load sharing through a three-dimensional formulation of N-planet helical planetary gear sets. Apart from previous models, the proposed model employs a gear mesh load distribution model to capture load and time dependency of the gear meshes iteratively. It includes all three types of manufacturing errors, namely constant errors such as planet pinhole position errors and pinhole diameter errors, constant but assembly dependent errors such as nominal planet tooth thickness errors, planet bore diameter errors, and rotation and assembly dependent errors such as gear eccentricities and run-outs. At the end, the model is used to show combined influence of these errors on planet load sharing to aid designers on how to account for manufacturing tolerances in the design of the gears of a planetary gear set.

On the Correlation Between Dynamic Transmission Error and Dynamic Tooth Loads in Three-Dimensional Gear Systems

A three-dimensional dynamic model is presented to simulate the dynamic behavior of single stage gears by using a combination of classic shaft, lumped parameter and specific 2-node gear elements. The mesh excitation formulation is based on transmission errors whose mathematical grounding is briefly described. The validity of the proposed methodology is assessed by comparison with experimental evidence from a test rig. The model is then employed to analyze the relationship between dynamic transmission errors and dynamic tooth loads or root stresses. It is shown that a linear dependency can be observed between the time variations of dynamic transmission error and tooth loading as long as the system can be assimilated to a torsional system but that this linear relationship tends to disappear when the influence of bending cannot be neglected.

Method of Predicting Differential Case Bolt Behavior

This paper proposes a CAE-based method to predict the maximum load at which bolts start to loosen. The qualitative validity of this method was confirmed using the fastening bolts between the differential case and ring gear. In general, the differential case and ring gear are fastened with bolts. Therefore, it is essential to estimate the maximum load of bolt loosening when designing these bolts. Moreover, prototypes found that tightening bolts are more likely to loosen as the thickness of the differential case or ring gear decreases and becomes easier to deform. This indicates that the deformation of the differential case and ring gear must be considered in bolt design. However, predicting the maximum load is relatively difficult because the behavior of both the screw and the contact surface between the differential case and ring gear is complicated. In contrast, recent transmissions require further weight reduction without sacrificing reliability. Consequently, a method of predicting the maximum load of bolt loosening is required. First, this paper describes a CAE model for estimating the pressure and friction generated at the contact surface between the differential case and ring gear, as well as at the screw surface and bolt seating surface. Furthermore, a method for determining bolt loosening is described that incorporates the pressure and friction on the bolt seating surface into friction circle theory. This method was used to derive the maximum load of bolt loosening. In addition, it was also confirmed that the results qualitatively agree with actual cases of bolt loosening. Second, this paper identifies the relative sliding of the screw surface and contact surface when the load increases. In this case, it was verified that the sliding on the contact surface between the differential case and ring gear induces relative sliding of the screw, followed by sliding of the bolt bearing surface and loosening of the bolt. Finally, this paper refers to design guidelines for reducing the weight of the differential case using an experimental design method. Certain ribs cause non-uniform bolt bearing surface pressure distribution, which likely affects bolt loosening. Through this research, the validity of the method was confirmed and the bolt behavior was clarified when a differential case and ring gear are loaded in one direction. Based on these results, it should be possible to apply this method to more complicated load cases in the future.

Kinematics Model for Tooth-Skipped Gear Honing

As gears that have been honed offer excellent wear characteristics and are extremely quiet, gear honing has become an essential part in the production of high-speed transmissions. In this paper, a novel gear honing technology based on toothskipped gearing theory is proposed. So far, this theory has been used only in the gear dynamic integrated error measuring machine, which was invented by a Chinese engineer. It is the first time that the theory has been introduced into gear manufacturing. Firstly, the whole tooth-skipped honing process is analyzed by taking the rack and pinion as an example. Secondly, the kinematics model of relationship between the rack and the gear is established, which takes the rack addendum meshing process, the gear addendum meshing process and the involute meshing process into consideration. Finally, the boundary points of each process are analyzed. The velocity curve of the gear to be manufactured is obtained. Tooth-skipped gear honing as a gear fine finishing approach has high processing efficiency and can increase the gear quality.

Loaded Tooth Contact Analysis of Power-Split Gear Drives Considering Shaft Deformation and Assembly Errors

The power-split gear mechanisms is widely applied in power transmission because of the advantages of compact design, lighter weight and high power density. The load sharing and the load distribution are the important performance issues while designing the power split mechanisms. The paper propose a computerized approach based on the influence coefficient method for loaded tooth contact analysis of such the gear transmission. Not only the load sharing of the multiple contact tooth pairs and the loaded transmission errors, but also the distributed contact stresses and the corresponding contact patterns on all the engaged tooth flanks can be calculated by using the proposed LTCA approach. Some analysis results are also discussed with a study case of the first planetary stage of a compound cycloid planetary gear drive.

Harmonic Balance Method for Nonlinear Vibration of Compound Planetary Gear Sets

In this paper, a new nonlinear time-varying dynamic model for compound planetary gear sets, which incorporates the time-varying meshing stiffness, transmission errors and gear backlash, has been presented. The harmonic balance method (HBM), which is an analytical approach widely used for nonlinear oscillators, is employed to investigate the dynamic characteristics of the gear sets. The matrix form iteration algebraic equations has been established and solved by HBM and single rank inverse Broyden method to reveal the effect of transmission error and gear backlash on the frequency response characteristic of the system. Sub-harmonic resonant, super-harmonic resonant and jump phenomenon have been illustrated by several examples.

Topology Optimization of Gearbox to Reduce Radiated Noise

In this paper, topology optimization of gearbox to reduce the radiated noise is studied based on the analysis of modal acoustic contribution and panel acoustic contribution. Firstly, the bearing dynamic loads are obtained by solving the dynamic equations of gear system. Secondly, the vibration of gearbox is calculated using FEM and the radiated noise is simulated using BEM by taking these bearing dynamic loads as excitations. Thirdly, the panel having larger contribution to the sound pressure level (SPL) at a specific field point is found by panel acoustic contribution analysis (PACA), and this panel is taken as design domain. The mode order with larger contribution is determined by modal acoustic contribution analysis (MACA), and making corresponding natural frequency becomes far away from excited frequency is taken as a constraint. Finally, the topology optimization of gearbox is completed using SIMP method, and the ribs are arranged according to the optimization results. The results show that the equivalent sound pressure level at objective field point can be reduced obviously by using this method.

Design of a Clutchless Hybrid Transmission for a High-Performance Vehicle

There exists the potential for major simplifications to current hybrid transmission architectures, which can lead to advances in powertrain performance. This paper assesses the technical merits of various hybrid powertrains in the context of high-performance vehicles and introduces a new transmission concept targeted at high performance hybrid applications. While many hybrid transmission configurations have been developed and implemented in mainstream and even luxury vehicles, ultra high performance sports cars have only recently begun to hybridize. The unique performance requirements of such vehicles place novel constraints on their transmissions designs. The goals become less about improved efficiency and smoothness and more centered on weight reduction, complexity reduction, and performance improvement. To identify the most critical aspects of a high performance transmission, a wide range of existing technologies is studied in concert with basic physical performance analysis of electrical motors and an internal combustion engine. The new transmission concepts presented here emphasize a reduction in inertial, frictional, and mechanical losses. A series of conceptual powertrain designs are evaluated against the goals of reducing mechanical complexity and maintaining functionality. The major innovation in these concepts is the elimination of a friction clutch to engage and disengage gears. Instead, the design proposes that the inclusion of a large electric motor enables the gears to be speed-matched and torque-zeroed without the inherent losses associated with a friction clutch. Additionally, these transmission concepts explore the merits of multiple electric motors and their placement as well as the reduction in synchronization interfaces. Ultimately, two strategies for speed-matched gear sets are considered, and a speed-matching prototype of the chosen methodology is presented to validate the feasibility of the proposed concept. The power flow and operational modes of both transmission architectures are studied to ensure required functionality and identify further areas of optimization. While there are still many unanswered questions about this concept, this paper introduces the base analysis and proof of concept for a technology that has great potential to advance hybrid vehicles at all levels.