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So far it is believed that, for every series-parallel planetary gear system (PGS), as a coupled gear, a very harmful phenomenon of power circulation must occur in at least one of its closed circuits. In this paper (Part I) and in the next two (Part II and Part III), it will be shown that it is possible to construct a three-row series-parallel PGS in which this phenomenon can be avoided. For this purpose, in Part I, a detailed analysis of the kinematics and statics of a planetary gear with power circulation inside a closed loop was carried out. The determination of the angular velocities of gears and carriers is carried out using Willis formulas and the graphical-analytical method (for verification), while the torques are determined using free body diagrams. The magnitudes of angular velocities and torques were used to determine the directions of power flows with improved energy balance equations in the reference frame related to the stationary gear body and, additionally, only to verify the energy balance equation in the mobile reference frame related to the carrier hi (i=2,5,8). The improvement of the methods was based on the use of the original concept of distinguishing active torque from reactive torque, as well as active power from reactive power, which made it very easy to determine the directions of the power flow. The determined paths of the power flow, including the power circulation in the analysed PGS, are presented graphically.

The objective of this study is the application of multi-criteria optimization to the two-carrier two-speed planetary gear trains. In order to determine mathematical model of multi-criteria optimization, variables, objective functions and conditions should be determined. The subject of the paper is two-carrier two-speed planetary gears with brakes on single shafts. Apart from the determination of the set of the Pareto optimal solutions, the weighted coefficient method for choosing an optimal solution from this set is also included in the mathematical model.

For planetary gear has the characteristics of small volume, light weight and large transmission ratio, it is widely used in high speed and high power mechanical system. Poor working conditions result in frequent failures of planetary gear. A method is proposed for diagnosing faults in planetary gear based on permutation entropy of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) Adaptive Neuro-fuzzy Inference System (ANFIS) in this paper. The original signal is decomposed into 6 intrinsic mode functions (IMF) and residual components by CEEMDAN. Since the IMF contains the main characteristic information of planetary gear faults, time complexity of IMFs are reflected by permutation entropies to quantify the fault features. The permutation entropies of each IMF component are defined as the input of ANFIS, and its parameters and membership functions are adaptively adjusted according to training samples. Finally, the fuzzy inference rules are determined, and the optimal ANFIS is obtained. The overall recognition rate of the test sample used for ANFIS is 90%, and the recognition rate of gear with one missing tooth is relatively high. The recognition rates of different fault gears based on the method can also achieve better results. Therefore, the proposed method can be applied to planetary gear fault diagnosis effectively.

Given local weak feature information, a novel feature extraction and fault diagnosis method for planetary gears based on variational mode decomposition (VMD), singular value decomposition (SVD), and convolutional neural network (CNN) is proposed. VMD was used to decompose the original vibration signal to mode components. The mode matrix was partitioned into a number of submatrices and local feature information contained in each submatrix was extracted as a singular value vector using SVD. The singular value vector matrix corresponding to the current fault state was constructed according to the location of each submatrix. Finally, by training a CNN using singular value vector matrices as inputs, planetary gear fault state identification and classification was achieved. The experimental results confirm that the proposed method can successfully extract local weak feature information and accurately identify different faults. The singular value vector matrices of different fault states have a distinct difference in element size and waveform. The VMD-based partition extraction method is better than ensemble empirical mode decomposition (EEMD), resulting in a higher CNN total recognition rate of 100% with fewer training times (14 times). Further analysis demonstrated that the method can also be applied to the degradation recognition of planetary gears. Thus, the proposed method is an effective feature extraction and fault diagnosis technique for planetary gears.

The nonlinear dynamic model of the NW (planetary gear structure with internal and external meshing and without planet carrier) planetary gear bearing was established in this study, taking into account factors such as random wind speed, time-varying support stiffness, bearing clearance, transmission error, tooth backlash, flexible ring gear, time-varying meshing stiffness, and tooth surface friction. The system's nonlinear behavior was described using phase trajectory plane, time–frequency analysis, time history, 3D frequency spectrum, FFT spectrum, phase diagram, and Poincaré map, as well as bifurcation diagram. Additionally, the superharmonic resonance characteristics of the system were analyzed using a multi-scale method, and the stability conditions for superharmonic resonance were determined through numerical analysis. Furthermore, the effects of meshing damping, displacement control parameters, and speed control parameters on the amplitude–frequency characteristics of the NW planetary gear-bearing system were examined. The conclusions indicate that the NW planetary gear-bearing system exhibits various nonlinear characteristics, and the system's stability can be improved by increasing damping and selecting appropriate time delay parameters.

Planetary bearings are the key components of the double planetary gear set because of their high loading capacity. As the supporting parts for the planet gears, their behaviors have pronounced influences on the double planetary gear set dynamics. Most previous works only focused on the influence of gear wear on single planetary gear set dynamics. This work creatively proposes a time-varying wear (TVW) model of the planetary bearing for a double planetary gear set, which can be helpful to understand the performance degradation analysis for the double planetary gear set dynamics. The time-varying behavior of relative sliding speeds and contact forces for the planetary bearing contact parts are considered in the TVM model. The bearing and gear excitations are all included in the dynamic model. The TVW model is established by using Archard wear theory. The wear depths of different operating times are analyzed. The influences of the ring wear of planetary bearing on the roller, carrier, planet gear, and cage dynamics are investigated. The results obtained by simulation and experimental methods are evaluated to confirm the accuracy of dynamic model. Moreover, the results show that the ring wear of inner planetary bearing is more severe compared to that of the outer planetary and the outer ring wear is more significant than that of the inner ring.

This paper addresses the challenging issues of transmission quality degradation and difficulty in obtaining dynamic response characteristics caused by the nonlinear behavior of non-circular planetary gear systems (NPG). A dynamic model for NPG was developed, encompassing axial elastic displacement, backlash, tooth surface friction, time-varying meshing stiffness, and viscoelastic damping. Fourier fitting matrices for time-varying mesh stiffness and polynomial models for dynamic backlash in non-circular gears were acquired to enhance model precision. Various analysis techniques including phase trajectory diagrams, bifurcation diagrams, time history diagrams, Poincaré mapping diagrams, and phase amplitude frequency characteristic curves were used to evaluate the nonlinear behavior of NPGs. Research results indicate that increasing the damping ratio benefits frequency response bandwidth, reduces phase lag, and improves system stability. The friction coefficient on the surface of non-circular gears also plays a role in ensuring the stability and phase consistency of NGP, although excessive coefficients can induce chaos. The output solar gear is more sensitive to internal excitation, with higher internal excitation leading to stronger system chaos.

As one of the most vital energy conversation systems, the safe operation of wind turbines is very important; however, weak fault and time-varying speed may challenge the conventional monitoring strategies. Thus, an entropy-aided meshing-order modulation method is proposed for detecting the optimal frequency band, which contains the weak fault-related information. Specifically, the variable rotational frequency trend is first identified and extracted based on the time–frequency representation of the raw signal by constructing a novel scaling-basis local reassigning chirplet transform (SLRCT). A new entropy-aided meshing-order modulation (EMOM) indicator is then constructed to locate the most sensitive modulation frequency area according to the extracted fine speed trend with the help of order tracking technique. Finally, the raw vibration signal is bandpass filtered via the corresponding optimal frequency band with the highest EMOM indicator. The order components resulting from the weak fault can be highlighted to accomplish weak fault detection. The effectiveness of the proposed EMOM analysis-based method has been tested using the experimental data of three different gear fault types of different fault levels from a planetary test rig.

The lever method facilitates a straightforward analysis of the motion dynamics of an automatic transmission under various gear settings, offering a simple and intuitive approach to examining planetary gear transmissions. Specifically, the equivalent lever diagram for a parallel-double planetary gear transmission comprises four component points. From a control perspective, a transmission system with two planetary rows can accommodate up to two reduction gears, one direct gear, one overdrive gear, and one reverse gear. In contrast, the equivalent lever diagram for a parallel three-planetary gear transmission includes five component points, with such systems typically supporting up to three reduction gears, one direct gear, two overdrive gears, and two reverse gears. This methodological approach underscores the lever diagram’s utility in delineating the functional capabilities of complex transmission systems.

Distinguished from parallel-axis gear systems, the revolution of the carrier brings out the pass effect of the planet gear. Compared with the phenomenological descriptions in conventional studies, this paper aims to provide more abundant physical information and the elastodynamics mechanism of the pass effect of the planet gear. To this end, a continuous-discrete model of helical planetary gears is proposed Considering both the in-plane and out-of-plane vibration, a semi-analytical model is established to simulate the elastic ring gear. The elastic and the lumped parameter submodels are synthesized using the moving elastic coupling boundary. The simulated modal and dynamic characteristics are verified using a planetary gear test rig. Based on the proposed dynamic model, the pass effect of the planet gear, the effects of bolt constraint, and the out-of-plane vibration of helical planetary gears are investigated. It is revealed that the “realistic rotation” strategy adopted in the proposed dynamic model can better reflect the physical essence of the pass effect of the planet gear compared with the “pseudo rotation” strategy utilized in the traditional model.