Compound Planetary Gear Train Research Articles

Influence of eccentricity error on the orbit of a two-stage double-helical compound planetary gear train with different mesh phasing configurations

The eccentricity error is an unavoidable error during the manufacturing and mounting process and has a significant influence on the response of gear systems, especially for planetary gear train. First, based on the most basic quasi-static model, pressure angle, contact ratio and meshing stiffness are considered to be the time-varying parameters due to the eccentricity error. Then, a new quasi-static model of a two-stage compound planetary gear train with eccentricity error is established, and the accuracy of the established model is verified by analyzing the orbit result of healthy gear. Furthermore, the influence of eccentricity error on the response in the time-domain (orbit) under various error combinations is studied and discussed. Finally, it is found that the orbit of the component with eccentricity error is very different from that without eccentricity error, and the formation law of orbit for gear with eccentricity error is revealed in this paper. The results of the quasi-static model could provide a fundamental understanding to the dynamic model for the studied compound planetary gear train.

Study on nonlinear dynamics characteristics of dual speed dual clutch transmission system based on bond graph

The variable speed transmission system employs compound planetary gear trains and clutches. Compared to traditional aviation transmission systems, the added control parts in the variable transmission system result in a stronger coupling between components and more complex dynamic response characteristics. Modeling using the central mass method is complex and abstract. This study proposes a bond graph-based method for modeling and analyzing the dynamic behavior of the variable transmission system. The transmission system is divided into multiple component-level real-time dynamic models, and obtain a nonlinear multi-energy domain shift dynamic models based on energy flow paths coupling. The mathematical model is numerically solved using the Runge-Kutta method. Simulation results show that there is a delay in the speed response during the shifting process. Further analysis reveals that this “give and take” phenomenon is related to the mechanical structure and passive working principle of the one-way clutch. Subsequently, the time-domain response of the output speed under the hydraulic loading characteristic curves of linear, exponential, S-shaped, and composite functions, as well as the clutch torque, were studied. The impact of different hydraulic loading durations based on the exponential curve on component speed and torque was also analyzed. The simulation results provide a theoretical basis for designing low-shift impact, high-reliability, and high-performance variable transmission systems.

Key characteristics of generic bond graphs for planetary gears

This paper deals with generalized power flow models of compound planetary gear trains based on standard planetary gears. In this context, the Bond Graph approach provides an optimal tool for a large variety of structures. Starting with compound gears based on two standard planetary gears, two kinds of parameter definitions are discussed and two specifications for power ratio and efficiency are compared. Next, the influence of elastic shaft connections is clarified. Essential methods for these findings are the use of symmetry and the definition of sub-systems and, if applicable, of scalar or vectorial models. The second part deals with the effect of the number of constraints on constructions based on three standard planetary gears. Typical consequences, such as the number of free running elements, of complete Willis equations or of self-locking impacts, are systemized and illustrated by means of reference instances. It is shown by example that constructions featuring five external shafts would avoid such effects. Corresponding outcomes for the equivalent dynamic system equations, the parameter definitions and the Bond Graph modelling are given.

Bond Graph-Based Approach to Modeling Variable-Speed Gearboxes with Multi-Type Clutches

The modeling and simulation of gearboxes is important for analyzing the dynamic characteristics and designing control strategies of transmission systems. Variable-speed gearboxes include compound planetary gear trains and clutches, which complicates dynamic modeling. Here, a procedural bond graph-based modeling method that considers many uncertainties is proposed. The proposed method yields a constant system–structure model. First, bond graph models of the two most common planetary gears were summarized, and were used as sub-models of a compound planetary gear train. Then, the Karnopp friction sub-model of the friction clutch and a relative angular displacement sub-model of the one-way clutch were constructed. Based on the dynamic coupling between the sub-models, the modeling steps of the gearbox, including the compound planetary gear train friction clutch one-way clutch coupling system, are described in detail. Next, the main sources of uncertainties of gearbox were analyzed and the simulation methods were given. Finally, the novel uncertain bond graph model was used to simulate the double planetary gearbox; the transmission ratio before and after the shift was 2.42 and 1.72, compared with the design values of 2.41 and 1.71, respectively; the deviation is within 5.8%; The average rotating speeds of the output shaft fluctuated by 6 and 2.5% respectively, which was within a reasonable range, so the effectiveness of the method is verified.

Analysis of nine-speed planetary change-gears through the torque method

An alternative method for kinematic and power analysis of compound gear train is presented, which uses the torques and lever analogy of a given gear train. The method combines the accuracy of the Willis analytical method with the clarity of the graphic method of Kutzbach. Unlike these methods, the torque method allows determining not only the gear ratio, but the magnitude and direction of power flows, and hence the determination of efficiency of a given compound planetary gear train. The application of the method is illustrated with example of novel arrangement of complex nine-speed change-gear. Structural, kinematic, and power (efficiency) analysis are made. A software for these analysis facilitation is developed. A numerical example is presented and discussed.

Dynamically Balanced Pointing System for CubeSats: Study and 3D Printing Manufacturing

The increasing presence of additive manufacturing (AM) in the space sector prompted us to investigate the feasibility of a single degree of freedom (DoF) pointing system (PS) made by means of a compound planetary gear train system (C-PGTS) integrating a dynamic balancing system (DBS) and entirely realized in AM. We analyzed in detail the dynamics of the system dealing with the design and the realization of the prototype. Of fundamental importance for this paper is the careful selection of materials for AM suitable for the prohibitive conditions of space. The results, deriving from the comparison between the experimental part and the simulations, underline the correct dimensioning of the PS and the fundamental importance of DBS in maintaining the satellite attitude. The results also confirm the capabilities of AM in the production of complex mechanical systems, allowing high precision, combined with interesting mechanical properties and low weight.This suggests the potential of AM in the space domain, both for structural parts and active components, such as those listed in this work.

A Novel Wolfrom-Based Gearbox for Robotic Actuators

In many robotic applications, high torque density and highly efficient actuators are required to handle the high torque motion without hampering mobility through excessive weight and footprint. Traditionally, engineers designing actuators for these types of applications use high-speed electrical motors in combination with high-ratio speed reducers such as harmonic and cycloid drives or lever arms in order to achieve the required torques. However, these systems are not ideal since they suffer from one or more of the following limitations: high cost, load hysteresis, nonlinear behavior, and torque ripple. An alternative to achieving the required torque increase is using an interesting type of compound planetary gear trains (C-PGTs). In this article, a novel type of such a C-PGT, based on the Wolfrom topology, is presented. An extra degree of freedom is created, which can be used to increase the performance of the overall gearbox, i.e., increase its efficiency, reduce its inertia, increase its torque capability, etc. This modified Wolfrom is discussed in detail by deriving a general framework for the design of such a transmission system. This framework consists of listing the specific fitting conditions for such a Wolfrom planetary gear train, the derivation of an efficiency model based on the Rolling Power concept, and a load analysis to determine the weakest gearing element. Finally, a prototype is built to demonstrate the design framework and the feasibility of the Wolfrom variant. Additionally, the trend of the efficiency model is confirmed.

Effects of Vibration Isolator on Compound Planetary Gear Train with Marine Twin-Layer Gearbox Case: a Dynamic Load Analysis

This study aims to take the compound planetary gear train as an example to investigate how the vibration isolator affects the dynamic characteristics of the gear system. The dynamic parameters are obtained from the finite element model of the gearbox case with the vibration isolators by the substructure method. The lumped mass method is used to construct the systematic coupled dynamic model of the compound planetary gear train with the marine twin-layer gearbox case. The dynamic response of the coupled system is calculated by the numerical calculation method of the Fourier series. The number of vibration isolators negatively affects the bearing forces of the input shaft, output shaft, and planets. The influence of the number of vibration isolators on LSCs is different in the differential stage and the encased stage. When the vibration isolators stiffness is less than 200 N/μm, the stiffness of the vibration isolator has a significant effect on the bearing forces of the input shaft, output shaft, and planets. The increase in the stiffness of the vibration isolator would increase the LSCs of the differential stage and decrease the LSCs of the encased stage. The number and stiffness of the vibration isolator have influences on the bearing forces and the LSCs. Compared with the LSCs in the differential stage, the LSCs in the encased stage are more sensitive to the number and stiffness of vibration isolators.

Robust optimisation of dynamic and NVH characteristics for compound power-split hybrid transmission

In this paper, vibro-acoustic characteristics of a power-split hybrid transmission including a compound planetary gear set are investigated by numerical procedure and refined system dynamics modelling. For validation of the numerical predictions, bench tests are performed for dynamic and acoustic responses of the hybrid transmission, contribution rates of acoustic radiation power induced due to the planetary gears, support bearings, transmission shafts and the gearbox housing are estimated. In improving the noise, vibration and harshness (NVH) performance of the transmission during hybrid vehicle acceleration, traction torques of the motors against the planetary gear parametric resonance are formulated and an optimal control strategy is proposed. By real road NVH test results acquired on board of the midsize hybrid car, it is demonstrated that a significant reduction of the planetary gear whine noise is achieved. As a result, numerical approaches applied to establish relationships between torques of the two traction motors and parametric excitations of the compound planetary gear train are experimentally validated.

Design of Compound Planetary Gear Train

Design of Compound Planetary Gear Train

Configuration synthesis of parallel hybrid transmissions

This paper presents a systematic design approach to synthesize feasible configurations of parallel transmissions for a hybrid electric vehicle using single electric motor/generator. Based on the studies of existing parallel hybrid systems, four main types of parallel hybrid transmissions are classified and studied. The mechanisms, which were studied in present research, are two degrees of freedom compound planetary gear trains consisting of two simple planetary gear trains. By applying the proposed design approach, twelve feasible mechanisms for parallel hybrid transmissions are generated. Then, four potential mechanisms, which are used in existing designs, are employed to systematically synthesize 37 corresponding feasible parallel hybrid transmissions, including some existing parallel systems. To demonstrate the feasibility of the synthesized systems, a new parallel hybrid transmission is taken arbitrarily as a numerical example to illustrate its working principle with operation modes, and kinematics and power flow analysis.

EMBODIMENT DESIGN OF NOVEL 5-SPEED REAR DRIVE HUBS FOR BICYCLES

This paper presents embodiment design of 5-speed rear drive hubs for bicycles. A 7-link, 2-degrees of freedom (DOF) compound planetary gear train as the main body of a rear drive hub is introduced. The relationship between the number of coaxial links of a planetary gear train and the number of gear stages that a drive hub can provide with is discussed. By means of kinematic analysis, four speed ratios of the planetary gear train are derived, which represents four forward gears of the rear drive hub. By adding a direct-drive gear, five forward gears can be provided and two feasible clutching sequence tables are synthesized. Manual translational-type gear-shifting mechanisms are further designed to incorporate with the planetary gear train for appropriately controlling the gear stage. The power-flow path at each gear stage is checked to verify the feasibility of the proposed design. Finally, two novel 5-speed bicycle rear drive hubs are presented.

Design of a six-speed transmission hub with an integrated brushless permanent-magnet motor used for electric bicycles

Purpose – The electric motor and multi-speed transmission hub are essential components for an electric bicycle. Traditionally, these two devices have been designed and manufactured independently. The purpose of this study is to propose a novel electromechanical device that artfully integrates an exterior-rotor brushless permanent-magnet (BLPM) motor into a transmission hub to become a compact structural assembly. Design/methodology/approach – A novel design that integrates a three-phase, 20-pole/18-slot exterior-rotor BLPM motor with a multi-speed transmission hub composed of a six-link, two-degrees-of-freedom (2-DOF) compound planetary gear train (PGT) is presented in this study to overcome inherent drawbacks of existing designs. An analytical approach, based on fundamental circuits, is developed to synthesize the clutching sequences and numbers of teeth of all gears of the PGT. Findings – The integrated device provides six forward speeds, including two underdrives, two direct drives, and two overdrives, as well as two drive modes: the motor-drive mode and the human-drive mode, for electric bicycles. The main feature of the proposed design is the spur gear teeth merged with the pole shoes of the stator to dramatically minimize the detent torque, which is an oscillatory torque that always induces vibration and acoustic noise of the BLPM motor. Originality/value – The gear teeth on the pole shoes of the stator provide functions not only for transmission, but also act as dummy slots for adjusting the magnetostatic field of the BLPM motor to effectively reduce the detent torque. The peak value of the detent torque of the integrated design is only 9 percent of the original BLPM motor with identical magnet properties and motor dimensions. Such a feature is contributive in suppressing the vibration and acoustic noise of the electric bicycle's BLPM motor. A BLPM motor rated at 310 W and 250 r/min for the integrated device is presented and analyzed by using the commercial finite-element package Ansoft/Maxwell 2D Field Simulator.

Nonlinear Dynamic Characteristics of Compound Planetary Gear Train Sets Based on Harmonic Balance Method

A purely rotational model of Ravigneaux compound planetary gear train sets including time-varying mesh stiffness,synthetic mesh errors and gear backlashes is developed to show the nonlinear dynamic behavior of the system with the action of multi-clearances.The gap function is expressed as describing function and harmonic balance method(HBM) is used to convert the differential equations to nonlinear algebraic equations,which is solved iteratively by single rank inverse Broyden method.The steady state response of fundamental frequency is obtained.The influences of gear backlashes,time-varying mesh stiffness and synthetic mesh errors are analyzed by changing the value of the parameter.It is showed from the research that multiple value and amplitude jump discontinuities are presented on the dynamic curves,there the impact phenomenon is reflected.Meanwhile the nonlinearity degree of the system is increased by the coupling of stiffness fluctuation,mesh errors and backlashes.The HBM based on describing function can be used for more complicated model to obtain the steady state response of fundamental frequency,which provides a method for deeply researching the dynamic behavior of compound planetary gear train sets.

Dynamics analysis of the Minuteman cover drive

Recursive matrix relations in kinematics and dynamics analysis of a 1-DOF compound planetary gear train are established in the paper. The mechanism of the Minuteman cover drive is a system with four moving links and three gear pairs controlled by one electric motor. Knowing the rotation motion of the output link, the inverse dynamic problem is solved using an approach based on the principle of virtual work, but it has been verified the results in the framework of the Lagrange equations . Finally, some simulation graphs for the input and output angles of rotation , the torque and the power of the actuator are obtained.

Non-linear dynamic behaviour of compound planetary gear trains: Model formulation and semi-analytical solution

A discrete, non-linear, time-varying, torsional dynamic model of a multi-stage planetary train that is formed by any number of simple planetary stages is proposed in this study. Each planetary stage has a distinct fundamental mesh frequency and any number of planets spaced in any angular positions. The model allows the analysis of the gear train in all possible power flow configurations suitable for various gear drive ratios. It includes periodic variation of gear mesh stiffnesses as well as clearance (backlash) non-linearities that allow tooth separations. Equations of motion for the general case are formulated and solved semi-analytically using a hybrid harmonic balance method (HBM) in conjugate with inverse Fourier transform. Relative mesh displacements along lines of action of individual gear pairs were used as the continuation parameters to pass singular points and ill-conditioned equations in their proximity. At the end, a case study of a two-stage planetary train is used to demonstrate the effectiveness of the model and solution methods. The HBM solutions are compared to those obtained by a direct numerical integration method to assess their accuracy.

A new structure of combined gear trains with high transmission ratios

The aim of our research has been to perform the synthesis of a new structure of combined gear trains that would feature a very compact construction and a relatively inexpensive manufacturing process as well as very high transmission ratios and good efficiencies. After the analysis of existing structures, the new structure that represents the combination of a worm gearing and a compound planetary gear train connected in a specific way has been suggested. The synthesis procedure, the description and the characteristics of the suggested structure, the program for computer dimensioning, the verification through the three-dimensional model and the test of the physical model of the new worm-planetary gear train are presented in this paper.

Free vibration characteristics of compound planetary gear train sets

In this study, a plane model of a compound planetary gear train (CPGT) is proposed in order to study its free vibration characteristics. Two translations and one rotation are considered for each element (sun, carrier, ring-1, ring-2, and planets) of the CPGT. The contact between the teeth in mesh is modelled by linear springs with stiffness called gearmesh stiffness. The phasing between these stiffness is taken into account. Using Lagrange formulation, the equation of motion is derived. Starting from the eigenvalue problem of the system, the influence of the planets’ number and position, and the effect of the gyroscopic phenomena on the free vibration of the CPGT are studied. The natural modes are classified into three groups: translational (the carrier, ring 1, and sun have pure modal translational deflection with no rotation), rotational (the carrier, ring 1, and sun have pure modal rotational deflection with no translation), and planets’ modes (only the planets have a modal deflection). It was found that the change of the planets’ angular position does not affect this classification. On the other hand, the study of the influence of the carrier rotation speed show that the gyroscopic effect separates the repeated translational modes into distinct ones. The obtained results are much important in the design process in order to avoid critical operating conditions which lead to undesirable vibrations.

Duality between the kinematics of gear trains and the statics of beam systems

This paper provides geometric insight into the duality between the first-order kinematics of gear trains and the statics of beam systems. The two devices have inherent geometrical relationships that will allow the angular velocities of the gears in a gear train to be investigated from a knowledge of the forces acting on the beams of the dual beam system, and vice versa. The primary contribution of the paper is the application of this duality to obtain the dual beam system for a given compound planetary gear train, and vice versa. The paper develops a systematic procedure to transform between the first-order kinematics of a gear train and the statics of the dual beam system. This procedure provides a simple and intuitive approach to study the speed ratios of a planetary gear train and the force ratios of the dual beam system. The speed ratios are expressed in terms of kinematic coefficients, which are a function of the position of the input gear and provide insight into the geometry of the gear train. Two numerical examples of simple and compound planetary gear trains are presented to demonstrate the simplicity of the proposed approach. A third example is included to illustrate the systematic procedure to transform from a given beam system to a planetary gear train. The examples take advantage of the principle that the speed ratios for gearing are dual to the force ratios for beam systems, and vice versa.

Extension of Graph Theory to the Duality Between Static Systems and Mechanisms

This paper is a study of the duality between the statics of a variety of structures and the kinematics of mechanisms. To provide insight into this duality, two new graph representations are introduced; namely, the flow line graph representation and the potential line graph representation. The paper also discusses the duality that exists between these two representations. Then the duality between a static pillar system and a planar linkage is investigated by using the flow line graph representation for the pillar system and the potential line graph representation for the linkage. A compound planetary gear train is shown to be dual to the special case of a statically determinate beam and the duality between a serial robot and a platform-type robot, such as the Stewart platform, is explained. To show that the approach presented here can also be applied to more general robotic manipulators, the paper includes a two-platform robot and the dual spatial linkage. The dual transformation is then used to check the stability of a static system and the stationary, or locked, positions of a linkage. The paper shows that two novel platform systems, comprised of concentric spherical platforms inter-connected by rigid rods, are dual to a spherical six-bar linkage. The dual transformation, as presented in this paper, does not require the formulation and solution of the governing equations of the system under investigation. This is an original contribution to the literature and provides an alternative technique to the synthesis of structures and mechanisms. To simplify the design process, the synthesis problem can be transformed from the given system to the dual system in a straightforward manner.