Nonlinear dynamics of lumped-parameter planetary gears with general mesh phasing

2022 ◽  
pp. 116682
Author(s):  
Chenxin Wang ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Dynamics ◽  
Planetary Gears ◽  
Lumped Parameter ◽  
Mesh Phasing

Nonlinear Dynamics of Planetary Gears Using Analytical and Finite Element Models

2007 ◽  
Author(s):  
Robert G. Parker ◽  
Vijaya Kumar Ambarisha
Keyword(s):  
Finite Element ◽  
Period Doubling ◽  
Element Model ◽  
Lumped Parameter Model ◽  
Finite Element Models ◽  
Two Dimensional ◽  
Tooth Contact ◽  
Planetary Gears ◽  
Lumped Parameter ◽  
Mesh Phasing

Vibration induced gear noise and dynamic loads remain key concerns in many transmission applications that use planetary gears. Tooth separations at large vibrations introduce nonlinearity in geared systems. The present work examines the complex, nonlinear dynamic behavior of spur planetary gears using two models: (i) a lumped-parameter model, and (ii) a finite element model. The two-dimensional lumped-parameter model represents the gears as lumped inertias, the gear meshes as nonlinear springs with tooth contact loss and periodically varying stiffness due to changing tooth contact conditions, and the supports as linear springs. The two-dimensional finite element model is developed from a unique finite elementcontact analysis solver specialized for gear dynamics. Mesh stiffness variation excitation, corner contact, and gear tooth contact loss are all intrinsically considered in the finite element analysis. The dynamics of planetary gears show a rich spectrum of nonlinear phenomena. Nonlinear jumps, chaotic motions, and period-doubling bifurcations occur when the mesh frequency or any of its higher harmonics are near a natural frequency of the system. Responses from the dynamic analysis using analytical and finite element models are successfully compared qualitatively and quantitatively. These comparisons validate the effectiveness of the lumped-parameter model to simulate the dynamics of planetary gears. Mesh phasing rules to suppress rotational and translational vibrations in planetary gears are valid even when nonlinearity from tooth contact loss occurs. These mesh phasing conclusions, however, are not valid in the chaotic and period-doubling regions.

DYNAMIC MODELING AND ANALYSIS OF MULTI-STAGE PLANETARY GEARS COUPLED WITH CENTRAL BEARINGS AND HOUSING

2016 ◽  
Vol 40 (5) ◽  
pp. 1007-1018 ◽  
Author(s):  
Zheng-Ming Xiao ◽  
Xing Wu ◽  
Qing Chen
Keyword(s):  
Degrees Of Freedom ◽  
Lagrange Equation ◽  
Central Component ◽  
Parameter Method ◽  
Planetary Gears ◽  
Modeling And Analysis ◽  
Multi Stage ◽  
Lumped Parameter ◽  
Set Up ◽  
Mesh Phasing

This paper proposes a coupling dynamic model for multi-stage planetary gears train (PGT) based on the gearing theory and Lagrange equation, which is developed by analyzing the displacement relationships of gearing system and the influence of floating components. The time-varying mesh stiffness, bearings in housing, and the torsional support stiffness of ring gears are also considered. This model has 26 degrees-of-freedom (DOF) and adopts three planar DOFs for each central component, mesh phasing relations among planetary gears, and the rotational DOF for the planets of each stage. The modified transverse-torsional model is established in the rotating Cartesian coordinates by using the lumped-parameter method; thus this model is more accurate than the purely torsional model for describing the physical dynamics. The acceleration data is tested by using the scheme of a back-to-back power circulation set-up, and the vibration properties are also studied.

Dynamic Modeling of Multi-Stage Planetary Gears Coupled with Bearings in Housing

2011 ◽  
Vol 86 ◽  
pp. 30-34
Author(s):  
Zheng Ming Xiao ◽  
Da Tong Qin
Keyword(s):  
Planetary Gear ◽  
Degree Of Freedom ◽  
Design Parameters ◽  
Parameter Method ◽  
Rotational Degree ◽  
Planetary Gears ◽  
Multi Stage ◽  
Lumped Parameter ◽  
Shield Tunnelling ◽  
Gyroscopic Effects

This work develops an analytical model of multi-stages planetary gear transmission (PGT) coupled with bearings in housing based on analyzing the displacement relationships of gearing system. The model adopts three planar degree-of-freedom for each of the central components, and the rotational degree-of-freedom for the planets of each stage. Considering the gyroscopic effects, the modified transverse-torsional model is established in the rotating Cartesian coordinates by lumped-parameter method, which is more accurate and may match with the physical model better than the purely torsional model. According to the design parameters of the 3-stage planetary gears of main reducer of shield tunnelling machine, the natural frequencies and vibration modes are investigated by using this transverse-torsional model.

Analytical Solution for the Nonlinear Dynamics of Planetary Gears

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Cheon-Jae Bahk ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Response ◽  
Harmonic Balance ◽  
Gear Tooth ◽  
Harmonic Balance Method ◽  
Nonlinear Effects ◽  
Mesh Stiffness ◽  
Planetary Gears ◽  
Parallel Finite Element ◽  
The Impact

Planetary gears are parametrically excited by the time-varying mesh stiffness that fluctuates as the number of gear tooth pairs in contact changes during gear rotation. At resonance, the resulting vibration causes tooth separation leading to nonlinear effects such as jump phenomena and subharmonic resonance. This work examines the nonlinear dynamics of planetary gears by numerical and analytical methods over the meaningful mesh frequency ranges. Concise, closed-form approximations for the dynamic response are obtained by perturbation analysis. The analytical solutions give insight into the nonlinear dynamics and the impact of system parameters on dynamic response. Correlation between the amplitude of response and external torque demonstrates that tooth separation occurs even under large torque. The harmonic balance method with arclength continuation confirms the perturbation solutions. The accuracy of the analytical and harmonic balance solutions is evaluated by parallel finite element and numerical integration simulations.

Dynamic Analysis and Multi-Object Optimization of the Forced Torsional Vibration for Vehicular Multi-Stage Planetary Gears

2011 ◽  
Vol 86 ◽  
pp. 263-267 ◽  
Author(s):  
Hui Liu ◽  
Zhong Chang Cai ◽  
Chang Le Xiang ◽  
Ming Zheng Wang
Keyword(s):  
Steady State ◽  
Time Domain ◽  
Torsional Vibration ◽  
Vibration Response ◽  
Resonance Vibration ◽  
Lumped Parameter Model ◽  
Lagrange Method ◽  
Planetary Gears ◽  
Multi Stage ◽  
Lumped Parameter

On the basis of lumped parameter model and the Lagrange method, the model of powertrain was built. Resonance vibration response and non-resonance vibration response were calculated respectively in time domain and frequency domain, characteristics of forced torsional vibration in steady–state were concluded. Comparability and difference of response of parts in different stage were explained. Multi-object optimization was applied to reduce vibration.

Effects of Bearing Radial Internal Clearance on Dynamic Behavior and Bifurcations in Planetary Gears

2011 ◽  
Author(s):  
Yi Guo ◽  
Robert G. Parker
Keyword(s):  
Dynamic Response ◽  
Harmonic Balance Method ◽  
Nonlinear Behavior ◽  
Nonlinear Effects ◽  
Lumped Parameter Model ◽  
Solution Stability ◽  
Planetary Gears ◽  
Gear Mesh ◽  
Bearing Clearance ◽  
Lumped Parameter

This study investigates the dynamics of planetary gears where nonlinearity is induced by bearing clearance. Lumped-parameter and finite element models of planetary gears with bearing clearance, tooth separation, and gear mesh stiffness variation are developed. The harmonic balance method with arc-length continuation is used to obtain the dynamic response of the lumped-parameter model. Solution stability is analyzed using Floquet theory. Rich nonlinear behavior is exhibited in the dynamic response, consisting of nonlinear jumps and a hardening effect induced by the transition from no bearing contact to contact. The bearings of the central members (sun, ring, and carrier) impact against the bearing races near resonance, which leads to coexisting solutions in wide speed ranges, grazing bifurcation, and chaos. Secondary Hopf bifurcation is the route to chaos. Input torque can significantly suppress the nonlinear effects caused by bearing clearance.

A Lumped Parameter Model to Analyse the Dynamic Load Sharing in Planetary Gears with Planet Errors

2011 ◽  
Vol 86 ◽  
pp. 374-379 ◽  
Author(s):  
Xiao Yu Gu ◽  
Philippe Velex
Keyword(s):  
High Speed ◽  
Load Sharing ◽  
Lumped Parameter Model ◽  
Mesh Stiffness ◽  
Planetary Gears ◽  
Position Errors ◽  
Load Distributions ◽  
Lumped Parameter ◽  
Non Linear ◽  
Dynamic Load Sharing

A non-linear dynamic model of planetary gears is presented which accounts for planet position errors, time-varying non-linear mesh stiffness along with the interactions between deflections and instantaneous meshing conditions. The quasi-static load distributions agree well with the experimental results in the literature thus validating the contact simulation. Extensions towards high-speed behaviour are presented which show how dynamic effects may impact the instantaneous load sharing amongst the planets.

Vibration Modes of Helical Planetary Gears

2009 ◽  
Author(s):  
Tugan Eritenel ◽  
Robert G. Parker
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Three Dimensions ◽  
Lumped Parameter Model ◽  
Vibration Modes ◽  
Planetary Gears ◽  
Gear Mesh ◽  
Modal Properties ◽  
Lumped Parameter

This paper examines the vibration modes of single stage helical planetary gears in three dimensions with equally spaced planets. A lumped-parameter model is formulated to obtain the equations of motion. The gears and shafts are modeled as rigid bodies with compliant bearings at arbitrary axial locations on the shafts. A translational and a tilting stiffness account for the force and moment transmission at the gear mesh interface. The modal properties generalize those of two-dimensional spur planetary gears; there are twice as many degrees of freedom and natural frequencies due to the added tilting and axial motion. All vibration modes are categorized as planet, rotational-axial, and translational-tilting modes. The modal properties are shown to hold even for configurations that are not symmetric about the gear plane, due to, for example, shaft bearings not being equidistant from the gear plane. Computational modal analysis are performed to numerically verify the findings.

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