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.

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 in Planetary Gears With Frequency-Modulated Time-Varying Mesh Stiffness

2014 ◽  
Author(s):  
Xinghui Qiu ◽  
Qinkai Han ◽  
Fulei Chu
Keyword(s):  
Parametric Instability ◽  
Operating Conditions ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Rotational Model ◽  
Planetary Gears ◽  
Gear Mesh ◽  
Stiffness Variation ◽  
Speed Fluctuations ◽  
Gear Mesh Stiffness

A rotational model of planetary gears is developed which incorporates mesh stiffness variation and input speed fluctuations. Gear mesh stiffness is approximated by rectangle wave and different harmonic orders are considered. Because of speed fluctuations, the mesh stiffness is frequency modulated. The parametric instability associated with frequency-modulated time-varying stiffness is numerically investigated. The operating conditions leading to parametric instability are identified using Floquet theory and numerical integration. Whether the general laws derived for steady speed to suppress particular instabilities are applicable for fluctuating speed is verified. The effects of speed fluctuations on parametric instability are examined.

Parametric Instability of Planetary Gear Transmission Mounted on Discrete Support Struts With Minimum Ring Gear Rim Thickness Design

2018 ◽  
Author(s):  
Peng Guan ◽  
Hans DeSmidt
Keyword(s):  
Parametric Instability ◽  
Planetary Gear ◽  
Gear Transmission ◽  
Ring Gear ◽  
Mesh Stiffness ◽  
Gear Mesh ◽  
Stiffness Variation ◽  
Parametric Instabilities ◽  
Gear Mesh Stiffness ◽  
Flexible Ring

This research explores parametric instabilities of the PGT driveline system and a stability-based method for ring gear rim thickness design. Parametric excitation of a planetary gear transmission (PGT) driveline system arises from two sources: 1) gear mesh stiffness variation, 2) Interaction between moving planets, flexible ring gear and boundary struts. Many researchers have studied the parametric instability of planetary gear transmissions due to gear mesh stiffness variation, however, the effect of interaction between moving planets, flexible ring and discrete boundary struts on parametric instabilities has not been fully studied before. Especially, for sufficiently thin ring gears, this kind of effect becomes even more significant. To illustrate the novel PGT rim design proposal, firstly, a structural dynamics model of a complete PGT driveline system with elastic ring gear supported by discrete boundary struts is established. Secondly, by applying Floquet method, the parametric instability behavior due to the second parametric excitation source is fully investigated. Lastly, the design guidelines for planetary gear transmission ring gear rim thickness are proposed based on system stability from a dynamical viewpoint. The analysis and results provide new and important insights into dynamics and design of lightweight planetary gear transmission ring gear rim.

A hybrid analytical-numerical method based on Isogeometric Analysis for determination of time varying gear mesh stiffness

2021 ◽  
Vol 160 ◽  
pp. 104291
Author(s):  
Andreas Beinstingel ◽  
Michael Keller ◽  
Michael Heider ◽  
Burkhard Pinnekamp ◽  
Steffen Marburg
Keyword(s):  
Numerical Method ◽  
Isogeometric Analysis ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Gear Mesh ◽  
Gear Mesh Stiffness

BIDIMENSIONAL FINITE ELEMENT ANALYSIS OF SPUR GEAR: STUDY OF THE MESH STIFFNESS AND STRESS AT THE LEVEL OF THE TOOTH FOOT

2009 ◽  
Vol 33 (2) ◽  
pp. 175-187 ◽  
Author(s):  
Mohamed Nizar Bettaieb ◽  
Mohamed Maatar ◽  
Chafik Karra
Keyword(s):  
Finite Element ◽  
Stress State ◽  
Spur Gear ◽  
Fortran Program ◽  
Mesh Stiffness ◽  
Element Analysis ◽  
Gear Mesh ◽  
Finite Element Software ◽  
Time Calculation ◽  
Finite Element Meshing

The purpose of this work is to determine the spur gear mesh stiffness and the stress state at the level of the tooth foot. This mesh stiffness is derived from the calculation of the normal tooth displacements: local displacement where the load is applied, tooth bending displacement and body displacement [15]. The contribution of this work consists in, basing on previous works, developing optimal finite elements model in time calculation and results precision. This model permits the calculation of time varying mesh stiffness and the evaluation of stress state at the tooth foot. For these reasons a specific Fortran program was developed. It permit firstly, to obtain the gear geometric parameters (base radii, outside diameter,…) and to generate the data base of the finite element meshing of a tooth or a gear. This program is interfaced with the COSMOS/M finite element software to predict the stress and strain state and calculate the mesh stiffness of a gear system. It is noted that the mesh stiffness is periodic and its period is equal to the mesh period.

scholarly journals Experimental Validation Of The Gearbox NVH Parameters

2015 ◽  
Vol 13 (2) ◽  
pp. 16-21 ◽  
Author(s):  
Aleš Prokop ◽  
Kamil Řehák ◽  
Martin Zubík ◽  
Pavel Novotný
Keyword(s):  
Simplified Model ◽  
Mesh Stiffness ◽  
Gear Mesh ◽  
Global Dynamic ◽  
Transmission Area ◽  
Bearing Stiffness ◽  
Noise Vibration ◽  
Main Components ◽  
Gear Mesh Stiffness ◽  
Complex Transmission

Abstract The noise, vibration and harshness (NVH) plays an important role in the transmission area of automotive industry. To understand all the impacts on the gearbox’s global dynamic behavior it is necessary to gain information from a simplified model, create methods and get an appropriate and well correlated results with the experiment. The method itself can be afterwards reused for more complex transmission, which could be supported by other measurements. This paper deals with creation of a gearbox’s simplified model, including essential mechanisms as gear mesh stiffness, backlash, bearing stiffness and modal properties of the main components. Except for the presented model, more models with different difficulty levels are used. Numerical results are compared with data from experiment with good correlation.

Investigation of Power Harvesting via Parametric Excitations

2008 ◽  
Vol 20 (5) ◽  
pp. 545-557 ◽  
Author(s):  
Mohammed F. Daqaq ◽  
Christopher Stabler ◽  
Yousef Qaroush ◽  
Thiago Seuaciuc-Osório
Keyword(s):  
Output Power ◽  
Coupling Coefficient ◽  
Electromechanical Coupling ◽  
Multiple Scales ◽  
Broad Band ◽  
Parametric Instability ◽  
Load Resistance ◽  
Resistive Load ◽  
Parametrically Excited ◽  
Critical Excitation

This article presents an analytical and experimental investigation of energy harvesting via parametrically excited cantilever beams. To that end, we consider a lumped-parameter non-linear model that describes the first-mode dynamics of a parametrically excited cantilever-type harvester. The model accounts for the beam's geometric and inertia non-linearities as well as non-linearities representing air drag. Using the method of multiple scales, we obtain approximate analytical expressions describing the beam response, voltage drop across a purely resistive load, and output power in the vicinity of the first principle parametric resonance. Using these expressions, we study the effect of the electromechanical coupling and load resistance on the output power. We show that these parameters play an imperative role in determining the magnitude of the output power and characterizing the broad-band properties of the harvester. Specifically, we show that the region of parametric instability wherein energy can be harvested shrinks as the coupling coefficient increases. Furthermore, we show that there exists a coupling coefficient beyond which the peak power decreases. We also demonstrate that there is a critical excitation level below which no energy can be harvested. The amplitude of this critical excitation increases with the coupling coefficient and is maximized for a given load resistance. Theoretical findings that were compared to experimental results show good agreement and reflect the general trends.

Dynamic analysis of a helical geared rotor-bearing system with composite rotating shafts

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Ying-Chung Chen ◽  
Xu Feng Cheng ◽  
Siu-Tong Choi
Keyword(s):  
Composite Material ◽  
Dynamic Analysis ◽  
Dynamic Characteristics ◽  
Mesh Stiffness ◽  
Content Type ◽  
Gear Mesh ◽  
Rotor Bearing ◽  
Rotating Shafts ◽  
Gear Mesh Stiffness ◽  
Bearing System

Purpose This study aims to study the dynamic characteristics of a helical geared rotor-bearing system with composite material rotating shafts. Design/methodology/approach A finite element model of a helical geared rotor-bearing system with composite material rotating shafts is developed, in which the rotating shafts of the system are composed of composite material and modeled as Timoshenko beam; a rigid mass is used to represent the gear and their gyroscopic effect is taken into account; bearings are modeled as linear spring-damper; and the equations of motion are obtained by applying Lagrange’s equation. Natural frequencies, mode description, lateral responses, axial responses, lamination angles, lamination numbers, gear mesh stiffness and bearing damping coefficients are investigated. Findings The desired mechanical properties could be constructed using different lamination numbers and fiber included angles by composite rotating shafts. The frequency of the lateral module decreases as the included angle of the fibers and the principal shaft of the composite material rotating shaft increase. Because of the gear mesh stiffness increase, the resonance frequency of the coupling module of the system decreases, the lateral module is not influenced and the steady-state response decreases. The amplitude of the steady-state lateral and axial responses gradually decreases as the bearing damping coefficient increases. Practical implications The model of a helical geared rotor-bearing system with composite material rotating shafts is established in this paper. The dynamic characteristics of a helical geared rotor-bearing system with composite rotating shafts are investigated. The numerical results of this study can be used as a reference for subsequent personnel research. Originality/value The dynamic characteristics of the geared rotor-bearing system had been reported in some literature. However, the dynamic analysis of a helical geared rotor-bearing system with composite material rotating shafts is still rarely investigated. This paper shows some novel results of lateral and axial response results obtained by different lamination angles and different lamination numbers. In the future, it makes valuable contributions for further development of dynamic analysis of a helical geared rotor-bearing system with composite material rotating shafts.

Coupled Bending-Torsion Vibration of Geared Rotors

1995 ◽  
Author(s):  
J. S. Rao ◽  
J. R. Chang ◽  
T. N. Shiau
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Element Model ◽  
Mode Shapes ◽  
Mesh Stiffness ◽  
Present Program ◽  
Gear Mesh ◽  
Pressure Angle ◽  
Torsion Vibration ◽  
Gear Mesh Stiffness

Abstract A general finite element model is presented for determining the coupled bending-torsion natural frequencies and mode shapes of geared rotors. Uncoupled bending and torsion frequencies are obtained for examples available in literature and the present program is verified against these. The effect of the gear box is considered to determine the coupled frequencies. Parameters studied include the pressure angle, gear mesh stiffness, and bearing properties. The gear pressure angle is shown to have no effect on the natural frequencies of rotors supported on isotropic bearing supports. Several case studies with bending-torsion coupling are considered and the results obtained are compared with those available in literature. The results of a general rotor system with 8lodes are also presented.

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