Perturbation analysis of the two-to-one internal resonance of planetary gear trains

2021 ◽  
pp. 095440622110095
Author(s):  
Chao Xun ◽  
He Dai ◽  
Xinhua Long ◽  
Jie Bian
Keyword(s):  
Internal Resonance ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Planetary Gear ◽  
Resonance Phenomenon ◽  
Mesh Stiffness ◽  
Rotational Model ◽  
Planetary Gear Trains ◽  
Gear Trains

In this study, the two-to-one internal resonance between the first two rotational modes of planetary gear trains (PGTs) is investigated. A purely rotational model is applied considering mesh stiffness variations, tooth separations, and tooth profile modifications (TPMs). Semi-analytical solutions for the internal resonance case are obtained using the method of multiple scales (MMS). The solution equations indicate that the mesh stiffness variations and tooth separations are the main factors causing internal resonance. A validation of the MMS was performed by numerical integration (NI). The results from an example analysis indicate that there exists an internal resonance phenomenon in the case of ωN+2 ≈ ω2, where ω2 and ωN+2 are the natural frequencies associated with the rotational modes, and N is the number of planet gears. Internal resonance in PGTs causes chaos, and part of the energy is transmitted from the ring gear to the sun gear through shocks. Proper TPMs that eliminate the tooth separations could suppress the internal resonance. The internal resonance, in turn, affects the optimal areas of the TPM magnitudes.

Dynamic Modeling of the Torsional Vibration of the Slewing Mechanism of a Hydraulic Excavator

2012 ◽  
Vol 253-255 ◽  
pp. 2102-2106 ◽  
Author(s):  
Xu Juan Yang ◽  
Zong Hua Wu ◽  
Zhao Jun Li ◽  
Gan Wei Cai
Keyword(s):  
Torsional Vibration ◽  
Natural Frequencies ◽  
Planetary Gear ◽  
Moment Of Inertia ◽  
Hydraulic Excavator ◽  
Planetary Gear Trains ◽  
Vibration Model ◽  
Gear Trains ◽  
The Moment ◽  
Relationship Of

A torsional vibration model of the slewing mechanism of a hydraulic excavator is developed to predict its free vibration characteristics with consideration of many fundamental factors, such as the mesh stiffness of gear pairs, the coupling relationship of a two stage planetary gear trains and the variety of moment of inertia of the input end caused by the motion of work equipment. The natural frequencies are solved using the corresponding eigenvalue problem. Taking the moment of inertia of the input end for example to illustrate the relationship between the natural frequencies of the slewing mechanism and its parameters, based on the simulation results, just the first order frequency varies significantly with the moment of inertia of the input end of the slewing mechanism.

Multi-Frequency Multi-Modal Excitation of a Heated Annular Plate

2006 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Nonlinear Dynamics ◽  
Internal Resonance ◽  
Radial Direction ◽  
Natural Frequencies ◽  
Annular Plate ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Plate Theory ◽  
Frequency Excitation ◽  
The Difference

The forced nonlinear dynamics of a pre-buckled thermally loaded annular plate are investigated. The plate is modeled using the von Ka´rma´n plate theory and the heat equation. The heat, which is generated by the difference between the uniformly distributed temperatures at the inner and outer boundaries, is assumed to symmetrically flow in the radial direction. The amount of heat affects the natural frequencies, which may give rise to different internal resonance conditions. The method of multiple scales is used to examine the system axisymmetric responses when it is driven by an external multi-frequency excitation. The plate responses could be very complex exhibiting Hopf and cyclic-fold bifurcations, quasi-periodicity, chaos, and multiplicity of attractors.

Vibration of Planetary Gears With Elastically Deformable Ring Gears Parametrically Excited by Mesh Stiffness Fluctuations

2009 ◽  
Author(s):  
Robert G. Parker
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Parametric Instability ◽  
Invariant System ◽  
Mesh Stiffness ◽  
Elastic Continuum ◽  
Planetary Gears ◽  
Time Invariant ◽  
Time Invariant System

The parametric instability of planetary gears having elastic continuum ring gears is analytically investigated based on a hybrid continuous-discrete model. Mesh stiffness variations of the sun-planet and ring-planet meshes caused by the changing number of teeth in contact are the source of parametric instability. The natural frequencies of the time invariant system are either distinct or degenerate with multiplicity two, which indicates three types of combination instabilities: distinct-distinct, distinct-degenerate and degenerate-degenerate instabilities. By using the structured modal properties of planetary gears and the method of multiple scales, the instability boundaries are obtained as simple expressions in terms of mesh parameters. Instability existence rules for in-phase planet meshes are given. The instability boundaries are validated numerically.

Self-Adaptive Underactuated Hybrid Rolling/Walking Locomotion

2012 ◽  
Author(s):  
Carl A. Nelson
Keyword(s):  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Planetary Gear ◽  
Legged Locomotion ◽  
Robotic Locomotion ◽  
Uneven Terrain ◽  
Planetary Gear Trains ◽  
Obstacle Clearance ◽  
Gear Trains ◽  
And Control

In planetary exploration and other similar robotic applications, it is possible to encounter obstacles on multiple scales, making it difficult to design wheeled locomotion that works well for all terrain types. Legged locomotion tends to be less efficient and slower, but allows better obstacle clearance. This paper describes a novel method of achieving robotic locomotion over uneven terrain using a passive underactuation technique. Using planetary gear trains with one input degree of freedom and two output degrees of freedom, the natural obstacle-based locking of select outputs can cause the transition of power through the alternate outputs. By designing the primary outputs as wheels and the secondary outputs as legs with more ground clearance, a naturally adaptive hybrid gait incorporating both rolling and walking can be generated without the need for sophisticated sensing and control. Derivation and simulation validation are presented.

Optimization of Compound Planetary Gear Train by Improved Mesh Stiffness Approach

2017 ◽  
Author(s):  
Jianwu Zhang ◽  
Han Guo ◽  
Liang Zou ◽  
Haisheng Yu
Keyword(s):  
Planetary Gear ◽  
Coupling Model ◽  
Gear Train ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Planetary Gear Train ◽  
Planetary Gear Trains ◽  
Hybrid Transmission ◽  
Gear Trains ◽  
Potential Energy Method

An improved mesh stiffness approach is presented for optimization of vibration and noise performance of the planetary gear trains in a full power split hybrid transmission, in which mesh stiffness time-variability and biaxial gear stiffness couplings in gear pairs are taken into account. For improving accuracy of the mesh stiffness in double teeth-meshing region for spur gear pairs, a simplified solution to the loading gear deformations counting for time-varying mesh stiffness of the helical gear pairs is proposed, based on the integral potential energy method and FEM simulation. By the new biaxial coupling model, effects of gear body and tooth coupled stiffnesses on gear pair vibro-acoustic responses are also investigated and approved to be considerable. Numerical examples with optimal analyses of the specified planetary gear trains for the full hybrid transmission are provided. Numerical solutions of eigen frequencies and vibration modes for the gear pairs with a variety of time-varying mesh stiffnesses are constructed by the biaxial coupling model and Fourier Series. The dynamic parameters optimization of the compound planetary gear train is then conducted. The optimized planetary gear system is applied in the full hybrid transmission and bench tests for its vibro-acoustic performance are also undertaken. Computational predictions and experimental results are shown to be in fairly good agreement.

Investigation of Noise Features of Planetary Gear Trains Based on Human Aural Characteristic

2017 ◽  
Author(s):  
Masao Nakagawa ◽  
Dai Nishida ◽  
Deepak Sah ◽  
Toshiki Hirogaki ◽  
Eiichi Aoyama
Keyword(s):  
Gear Tooth ◽  
Structural Complexity ◽  
Planetary Gear ◽  
Transmission Error ◽  
Sound Level ◽  
Tooth Surface ◽  
Surface Error ◽  
Planetary Gear Trains ◽  
Tooth Stiffness ◽  
Gear Trains

Planetary gear trains (PGTs) are widely used in various machines owing to their many advantages. However, they suffer from problems of noise and vibration due to the structural complexity and giving rise to substantial noise, vibration, and harshness with respect to both structures and human users. In this report, the sound level from PGTs is measured in an anechoic chamber based on human aural characteristic, and basic features of sound are investigated. Gear noise is generated by the vibration force due to varying gear tooth stiffness and the vibration force due to tooth surface error, or transmission error (TE). Dynamic TE is considered to be increased because of internal and external meshing. The vibration force due to tooth surface error can be ignored owing to almost perfect tooth surface. A vibration force due to varying tooth stiffness could be a major factor.

A New Technique Based on Loops to Investigate Displacement Isomorphism in Planetary Gear Trains

2002 ◽  
Vol 124 (4) ◽  
pp. 662-675 ◽  
Author(s):  
V. V. N. R. Prasad Raju Pathapati ◽  
A. C. Rao
Keyword(s):  
Graph Isomorphism ◽  
Planetary Gear ◽  
Degree Of Freedom ◽  
Gear Train ◽  
Graph Representation ◽  
Kinematic Structure ◽  
Graph Representations ◽  
Planetary Gear Trains ◽  
Gear Trains ◽  
A New Technique

The most important step in the structural synthesis of planetary gear trains (PGTs) requires the identification of isomorphism (rotational as well as displacement) between the graphs which represent the kinematic structure of planetary gear train. Previously used methods for identifying graph isomorphism yielded incorrect results. Literature review in this area shows there is inconsistency in results from six link, one degree-of-freedom onwards. The purpose of this paper is to present an efficient methodology through the use of Loop concept and Hamming number concept to detect displacement and rotational isomorphism in PGTs in an unambiguous way. New invariants for rotational graphs and displacement graphs called geared chain hamming strings and geared chain loop hamming strings are developed respectively to identify rotational and displacement isomorphism. This paper also presents a procedure to redraw conventional graph representation that not only clarifies the kinematic structure of a PGT but also averts the problem of pseudo isomorphism. Finally a thorough analysis of existing methods is carried out using the proposed technique and the results in the category of six links one degree-of-freedom are established and an Atlas comprises of graph representations in conventional form as well as in new form is presented.

scholarly journals The multi-attribute topological graph method and its application on power flow analysis in closed planetary gear trains

2018 ◽  
Vol 10 (8) ◽  
pp. 168781401879410 ◽  
Author(s):  
Yahui Cui ◽  
Jing Gao ◽  
Xiaomin Ji ◽  
Xintao Zhou ◽  
Haitao Yan
Keyword(s):  
Power Flow ◽  
Flow Analysis ◽  
Planetary Gear ◽  
Topological Graph ◽  
Template Model ◽  
Power Flow Analysis ◽  
Analysis Process ◽  
Planetary Gear Trains ◽  
Graphical View ◽  
Gear Trains

The concept of multi-attribute topological graph is proposed in this article to represent the characteristics of both structure and state for typical one-degree-of-freedom planar spur closed planetary gear trains. This method is well applied in power flow analysis and provides a graphical view for the types, values, directions, and transmission relationship of power flow, especially for the recirculation power representation. Furthermore, a template model of multi-attribute topological graph for closed planetary gear trains is also presented, which would be helpful to the multi-attribute topological graph generation for some certain types of closed planetary gear trains just by changing symbols in the template model. A corresponding software is also developed to make the analysis process more convenient. By inputting different parameters, the different visual results can be obtained automatically, thus benefiting engineers in conceptual design.

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