Nonlinear Dynamics of Planetary Gears With Equal Planet Spacing

2007 ◽  
Author(s):  
Cheon-Jae Bahk ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Response ◽  
Harmonic Balance ◽  
Gear Tooth ◽  
Harmonic Balance Method ◽  
Planetary Gear ◽  
Nonlinear Effects ◽  
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. The resulting vibration causes tooth separation leading to nonlinear effects such as classical jump phenomena and sub- and superharmonic resonance. The nonlinear dynamics of the planetary gear is examined by both 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. The harmonic balance method with arclength continuation confirms the perturbation solutions. The accuracy of the analytical and harmonic balance solutions is validated by parallel finite element and numerical integration simulations.

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.

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 Study on Planetary Gear Dynamics With Tooth Profile Modification

2011 ◽  
Author(s):  
Cheon-Jae Bahk ◽  
Robert G. Parker
Keyword(s):  
Dynamic Response ◽  
Gear Tooth ◽  
Planetary Gear ◽  
Transmission Error ◽  
Tooth Profile ◽  
Peak Amplitude ◽  
Profile Modification ◽  
Tooth Profile Modification ◽  
Static Transmission Error ◽  
The Impact

This study investigates the impact of tooth profile modification on planetary gear dynamic response. Micro-scale geometric deviations from an involute gear tooth profile add an additional excitation source, potentially reducing gear vibration. In order to take account of the excitation, tooth profile modification is included in an analytical planetary gear model. Nonlinearity due to tooth contact loss is considered. Time-varying mesh stiffness and both rotational and translational gear motions are modeled. The accuracy of the proposed model for dynamic analysis is correlated against a benchmark finite element analysis. Perturbation analysis is employed to obtain a closed-form approximation of planetary gear dynamic response with tooth profile modification. Mathematical expressions from the perturbation solution allow one to easily estimate the peak amplitude of resonant response using known parameters. Variation of the peak amplitude with the amount and the length of profile modification illustrates the effect of tooth profile modification on planetary gear dynamic response. For a given external load, the tooth profile modification parameters for minimal response are readily obtained. Static transmission error and dynamic response are minimized at different amounts of profile modification, which contradicts common practical thinking regarding strong correlation between static transmission error and dynamic response. Contrary to the expectation of further reduced vibration, the combination of the optimum sun-planet and ring-planet mesh tooth profile modifications that minimizes response when applied individually increases dynamic response.

The Homotopy Analysis for the Nonlinear Dynamics of Planetary Gear Trains

2018 ◽  
Author(s):  
Chao Xun ◽  
Sujuan Jiao ◽  
Yong Chen ◽  
Xinhua Long
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Response ◽  
Nonlinear Problems ◽  
Planetary Gear ◽  
Displacement Function ◽  
Strongly Nonlinear ◽  
Planetary Gear Trains ◽  
Homotopy Analysis ◽  
Gear Trains ◽  
The Impact

In this paper, the homotopy analysis method (HAM) is proposed to study the nonlinear oscillators of planetary gear trains, in which the periodically time-varying mesh stiffness and gear backlash are included through a nonlinear displacement function. In contrast to the traditional perturbation methods, the HAM does not require a small parameter in the equation under study, and then can be applied to both of the weakly and strongly nonlinear problems. In this article, firstly the closed-form approximations for the dynamic response of planetary gear trains are obtained by HAM. The analytical solutions give insight into the nonlinear dynamics and the impact of system parameters on dynamic response. The accuracy of HAM solutions is evaluated by numerical integration simulations. Results indicate that with large tooth separation times, the amplitude-frequency curves obtained by HAM agree better with the results obtained by NI than those obtained by the MMS.

Influence of the backlash generated by tooth accumulated wear on dynamic behavior of compound planetary gear set

2016 ◽  
Vol 231 (11) ◽  
pp. 2025-2041 ◽  
Author(s):  
Shijing Wu ◽  
Haibo Zhang ◽  
Xiaosun Wang ◽  
Zeming Peng ◽  
Kangkang Yang ◽  
...  
Keyword(s):  
Dynamic Response ◽  
Dynamic Model ◽  
Dynamic Behavior ◽  
Planetary Gear ◽  
Tooth Wear ◽  
Transmission Error ◽  
Gear Transmission ◽  
Tooth Surface ◽  
Contact Ratio ◽  
The Impact

Backlash is a key internal excitation on the dynamic response of planetary gear transmission. After the gear transmission running for a long time under load torque, due to tooth wear accumulation, the backlash between the tooth surface of two mating gears increases, which results in a larger and irregular backlash. However, the increasing backlash generated by tooth accumulated wear is generally neglected in lots of dynamics analysis for epicyclic gear trains. In order to investigate the impact of backlash generated by tooth accumulated wear on dynamic behavior of compound planetary gear set, in this work, first a static tooth surface wear prediction model is incorporated with a dynamic iteration methodology to get the increasing backlash generated by tooth accumulated wear for one pair of mating teeth under the condition that contact ratio equals to one. Then in order to introduce the tooth accumulated wear into dynamic model of compound planetary gear set, the backlash excitation generated by tooth accumulated wear for each meshing pair in compound planetary gear set is given under the condition that contact ratio equals to one and does not equal to one. Last, in order to investigate the impact of the increasing backlash generated by tooth accumulated wear on dynamic response of compound planetary gear set, a nonlinear lumped-parameter dynamic model of compound planetary gear set is employed to describe the dynamic relationships of gear transmission under the internal excitations generated by worn profile, meshing stiffness, transmission error, and backlash. The results indicate that the introduction of the increasing backlash generated by tooth accumulated wear makes a significant influence on the bifurcation and chaotic characteristics, dynamic response in time domain, and load sharing behavior of compound planetary gear set.

Dynamics of the Gear Train Set in Wind Turbine

2011 ◽  
Vol 697-698 ◽  
pp. 701-705
Author(s):  
D.D. Ji ◽  
Y.M. Song ◽  
J. Zhang
Keyword(s):  
Wind Turbine ◽  
Dynamic Model ◽  
Harmonic Balance Method ◽  
Planetary Gear ◽  
Gear Train ◽  
Dynamic Responses ◽  
Multi Stage ◽  
Lumped Parameter ◽  
Cartesian Coordinate ◽  
The Impact

A lumped-parameter dynamic model for gear train set in wind turbine is proposed to investigate the dynamics of the speed-increasing gear box. The proposed model is developed in a universal Cartesian coordinate, which includes transversal and torsional deflections of each component, time-varying mesh stiffness, gear profile errors and external excitations. By solving the dynamic model, a modal analysis is performed. The results indicate that the modal properties of the multi-stage gear train in wind turbine are similar to those of a single-stage planetary gear set. A harmonic balance method (HBM) is used to obtain the dynamic responses of the gearing system. The responses give insight into the impact of excitations on the vibrations.

Finding Periodic Solutions of Forced Systems With Local Nonlinearities: A Mixed Shooting Harmonic Balance Method

2015 ◽  
Author(s):  
Frederic Schreyer ◽  
Remco Leine
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Shooting Method ◽  
Mechanical Systems ◽  
Harmonic Balance Method ◽  
Nonlinear Effects ◽  
Balance Method ◽  
Pros And Cons ◽  
Numerical Approaches

Several numerical approaches have been developed to capture nonlinear effects of dynamical systems. In this paper we present a mixed shooting-harmonic balance method to solve large mechanical systems with local nonlinearities efficiently. The Harmonic Balance Method as well as the shooting method have both their pros and cons. The proposed mixed shooting-HBM approach combines the efficiency of HBM and the accuracy of the shooting method and has therefore advantages of both.

Nonlinear Effects of Surface Texturing on the Performance of Journal Bearings in Flexible Rotordynamic Systems

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Jocelyn Rebufa ◽  
Fabrice Thouverez ◽  
Erick Le Guyadec ◽  
Denis Mazuyer
Keyword(s):  
Harmonic Balance ◽  
Vibration Amplitude ◽  
Surface Texturing ◽  
Harmonic Balance Method ◽  
Nonlinear Effects ◽  
Coupled System ◽  
Journal Bearings ◽  
Internal Damping ◽  
Rotating Shaft ◽  
Mean Pressure

A dynamic model of a rotating shaft on two textured hydrodynamic journal bearings is presented. The hydrodynamic mean pressure is computed using multiscale periodic homogenization and is projected on a flexible shaft with internal damping. Harmonic balance method (HBM) is used to study the limit cycles of unbalance response of the coupled system discretized by finite element method (FEM). Stability is analyzed with Floquet multipliers computation. An example of an isotropic texturing pattern representing laser dimples on a lightweight rotor is analyzed. Vibration amplitude and stability zone are compared with plain bearing lubrication. It is shown in an example that full surface texturing leads to relatively higher vibration amplitude compared to plain bearings.

On the Development of a Synchronized Harmonic Balance Method for Multiple Frequencies and its Application to LPT Flows

2020 ◽  
Author(s):  
Javier Crespo ◽  
Jesús Contreras
Keyword(s):  
Boundary Conditions ◽  
Time Domain ◽  
Harmonic Balance ◽  
Fourier Transforms ◽  
Harmonic Balance Method ◽  
Single Passage ◽  
Computational Performance ◽  
Runge Phenomenon ◽  
Numerical Instabilities ◽  
The Impact

Abstract The aim of this paper is to describe the development and application of a multi-frequency harmonic balance solver for GPUs, particularly suitable for the simulation of periodic unsteadiness in nonlinear turbomachinery flows comprised of a few dominant frequencies, with an unsteady multistage coupling that bolsters the flow continuity across the rotor/stator interface. The formulation is addressed with the time-domain reinterpretation, where several non-equidistant time instants conveniently selected are solved simultaneously. The set of required frequencies in each row is driven into the governing equations with the help of almost-periodic Fourier transforms for time derivatives and time shifted boundary conditions. The spatial repetitiveness inside each row can be exploited to perform single-passage simulations and the relative circumferential positioning of the rotors or stators and the different blade or vane counts is tackled by means of adding fictitious frequencies referring to non-adjacent rows therefore taking into account clocking and indexing effects. Existing multistage row coupling techniques of harmonic methods rely on the use of non-reflecting boundary conditions, based on linearizations, or time interpolation, which may lead to Runge phenomenon with the resulting numerical instabilities and non-preserving flux exchange. Different sets of time instants might be selected in each row but the interpolation in space and time across their interfaces gives rise to robustness issues due to this phenomenon. The so-called synchronized approach, developed in this work, consist of having the same time instances among the whole ensemble of rows, ensuring that flux transfer at sliding planes is applied more robustly. The combination of a set of shared non-equidistant time instances plus the use of unequal frequencies (real and fictitious) may spoil the Fourier transforms conditioning but this can be dramatically improved with the help of oversampling and instants selection optimization. The resulting multistage coupling naturally addresses typical numerical issues such as flow that might reverse locally across the row interfaces by means of not using boundary conditions but a local flux conservation scheme in the sliding planes. Some examples will be given to illustrate the ability of this new approach to preserve accuracy and robustness while resolving them. A brief analysis of results for a fan stage and a LPT multi-row case is presented to demonstrate the correctness of the method, assessing the impact in the modeling accuracy of the present approach compared with a time-domain conventional analysis. Regarding the computational performance, the speedup compared to a full annulus time-domain unsteady simulation is a factor of order 30 combining the use of single-passage rows and time spectral accuracy.

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