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.

Parametric Instability of Planetary Gears Having Elastic Continuum Ring Gears

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Robert G. Parker ◽  
Xionghua Wu
Keyword(s):  
Multiple Scales ◽  
Parametric Instability ◽  
Mesh Deformation ◽  
The Sun ◽  
Mesh Stiffness ◽  
Elastic Continuum ◽  
Planetary Gears ◽  
Gear Mesh ◽  
Number Of Teeth ◽  
Time Invariant

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 and sequentially phased planet meshes are also discovered. For in-phase planet meshes, instability existence depends only on the type of gear mesh deformation. For sequentially phased planet meshes, the number of teeth on the sun (or the ring) and the type of gear mesh deformation govern the instability existence. The instability boundaries are validated numerically.

Frequency Response Characteristics of Parametrically Excited System

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Qinkai Han ◽  
Jianjun Wang ◽  
Qihan Li
Keyword(s):  
Frequency Response ◽  
Natural Frequency ◽  
Invariant System ◽  
Mesh Stiffness ◽  
Parametrically Excited ◽  
Response Characteristics ◽  
Excited System ◽  
Frequency Response Characteristics ◽  
Time Invariant ◽  
Time Invariant System

The frequency response characteristic of a general time-invariant system has been extensively analyzed in literature. However, it has not gained sufficient attentions in the parametrically excited system. In fact, due to the parametric excitation, the frequency response of time-periodic system differs distinctly from that of the time-invariant system. Utilizing Sylvester’s theorem and Fourier series expansion method, commonly used in the spectral decomposition for matrix, the frequency response functions (FRFs) of a single-degree-of-freedom (SDOF) parametrically excited system are derived briefly in the paper. The external resonant condition for the system is obtained by analyzing the specific expressions of FRFs. Then, a spur-gear-pair with periodically time-varying mesh stiffness is selected as an example to simulate the frequency response characteristics of parametric system. The effects of parametric stability, periodic mesh stiffness parameters (mesh frequency and contact ratio), and damping are considered in the simulation. It is shown from both theoretical and simulation results that the frequency response of parametric system has the following properties: there are multiple FRFs even for a SDOF periodic system as the forced response contains many frequency components and each FRF is corresponding to a certain response spectrum; the system has multiple external resonances. Besides the resonance caused by the external driving frequency equals to the natural frequency, the system will also be external resonant if external frequency meets the combination of natural frequency and parametric frequency. When the system is in external resonant state, the dominant frequency component in the response is the natural frequency; damping makes the peak values of FRFs drop evidently while it has almost no impact on the FRFs in nonresonant regions.

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.

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