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.

Dynamic Analysis of Planetary Gears With Bearing Clearance

2012 ◽  
Vol 7 (4) ◽  
Author(s):  
Yi Guo ◽  
Robert G. Parker
Keyword(s):  
Harmonic Balance Method ◽  
Period Doubling ◽  
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 with bearing clearance, tooth separation, and gear mesh stiffness variation are developed. The harmonic balance method with arc length continuation is applied to the lumped-parameter model to obtain the dynamic response. Solution stability is analyzed using Floquet theory. Rich nonlinear behavior is exhibited, consisting of nonlinear jumps, a hardening effect induced by the transition from no bearing contact to contact, and softening induced by tooth separation. Bearings of the central members (sun, carrier, and ring) impact against the bearing races near resonances, which leads to coexisting solutions in wide speed ranges, grazing bifurcation, and chaos. Secondary Hopf and period-doubling bifurcations are the routes to chaos. Input torque can suppress some of the nonlinear effects caused by bearing clearance.

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.

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.

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.

scholarly journals Friction Damper Optimisation: Simulation of Rainbow Tests

1999 ◽  
Author(s):  
Kenan Y. Sanliturk ◽  
David J. Ewins ◽  
Robert Elliott ◽  
Jeff S. Green
Keyword(s):  
Harmonic Balance Method ◽  
Lumped Parameter Model ◽  
Friction Damper ◽  
Blade Vibration ◽  
Friction Dampers ◽  
Lumped Parameter ◽  
Engine Order ◽  
Resonance Response ◽  
Theoretical Predictions ◽  
Different Characteristics

Friction dampers have been used to reduce turbine blade vibration levels for a considerable period of time. However, optimal design of these dampers has been quite difficult due both to a lack of adequate theoretical predictions and to difficulties in conducting reliable experiments. One of the difficulties of damper weight optimisation via the experimental route has been the inevitable effects of mistuning. Also, conducting separate experiments for different damper weights involves excessive cost. Therefore, current practice in the turbomachinery industry has been to conduct so-called ‘rainbow tests’ where friction dampers with different weights are placed between blades with a predefined configuration. However, it has been observed that some rainbow test results have been difficult to interpret and have been inconclusive for determining the optimum damper weight for a given bladed-disc assembly. A new method of analysis — a combination of Harmonic Balance Method and structural modification approaches — is presented in this paper for the analysis of structures with friction interfaces and the method is applied to search for qualitative answers about the so-called ‘rainbow tests’ in turbomachinery applications. A simple lumped-parameter model of a bladed-disc model was used and different damper weights were modelled using friction elements with different characteristics. Resonance response levels were obtained for bladed discs with various numbers of blades under various engine-order excitations. It was found that rainbow tests, where friction dampers with different weights are used on the same bladed-disc assembly, can be used to find the optimum damper weight if the mode of vibration concerned has weak blade-to-blade coupling (the case where the disc is almost rigid and blades vibrate almost independently from each other). Otherwise, it is very difficult to draw any reliable conclusion from such expensive experiments.

A Lumped Parameter Electromechanical Model for Describing the Nonlinear Behavior of Piezoelectric Actuators

1997 ◽  
Vol 119 (3) ◽  
pp. 478-485 ◽  
Author(s):  
M. Goldfarb ◽  
N. Celanovic
Keyword(s):  
System Analysis ◽  
Nonlinear Behavior ◽  
Piezoelectric Actuators ◽  
Bond Graph ◽  
Lumped Parameter Model ◽  
Input Output ◽  
Mechanical Stiffness ◽  
Analysis And Design ◽  
Lumped Parameter ◽  
Proposed Model

A lumped-parameter model of a piezoelectric stack actuator has been developed to describe actuator behavior for purposes of control system analysis and design, and in particular for control applications requiring accurate position tracking performance. In addition to describing the input-output dynamic behavior, the proposed model explains aspects of nonintuitive behavioral phenomena evinced by piezoelectric actuators, such as the input-output rate-independent hysteresis and the change in mechanical stiffness that results from altering electrical load. Bond graph terminology is incorporated to facilitate the energy-based formulation of the actuator model. The authors propose a new bond graph element, the generalized Maxwell resistive capacitor, as a lumped-parameter causal representation of rate-independent hysteresis. Model formulation is validated by comparing results of numerical simulations to experimental data.

Tuning the hygro-mechanical response of paper-based systems using glycerol

2020 ◽  
pp. 1045389X2094716
Author(s):  
Isaias Cueva-Perez ◽  
Roque Alfredo Osornio-Rios ◽  
Aurelio Dominguez-Gonzalez ◽  
Ion Stiharu ◽  
Angel Perez-Cruz
Keyword(s):  
Dynamic Response ◽  
Resonance Frequency ◽  
Mechanical Response ◽  
Water Solution ◽  
Low Cost ◽  
Mechanical Systems ◽  
Lumped Parameter Model ◽  
Glycerol Water ◽  
Energy Harvesters ◽  
Lumped Parameter

In recent years, the need for portable, low-cost, and eco-friendly devices for testing and monitoring has arisen. Paper-based devices have emerged as a response to these needs due to the properties induced by capillarity, flexibility, disposability, and biodegradability. In this work, the authors explored the possibility of tuning the hygro-mechanical response of paper-based cantilever beams using glycerol. A lumped-parameter model with non-linear stiffness is used to describe the dynamic response of the beams using three parameters. An experimental method based on resonance frequency tests is used to study the influence of glycerol on the dynamic response of four different beam configurations. The obtained results demonstrate that the resonance frequency of paper-based mechanical systems can be easily tuned by the imbibition of a glycerol–water solution. This study could lead to the development of tunable paper-based mechanical systems for specific applications such as energy harvesters and hygro-mechanical-based sensors.

Dynamic Modeling and Analysis of a Planetary Gear Involving Tooth Wedging and Bearing Clearance Nonlinearity

2009 ◽  
Author(s):  
Yi Guo ◽  
Robert G. Parker
Keyword(s):  
Gear Tooth ◽  
Planetary Gear ◽  
Lumped Parameter Model ◽  
Back Side ◽  
Modeling And Analysis ◽  
Bearing Clearance ◽  
Gravity Forces ◽  
Lumped Parameter ◽  
Bearing Failures ◽  
Wide Range

Tooth wedging occurs when a gear tooth comes into contact on the drive-side and back-side simultaneously. Tooth wedging risks bearing failures from elevated forces. This work studies the nonlinear tooth wedging behavior and its correlation with planet bearing forces by analyzing the dynamic response of an example planetary gear based on a real application of a wind turbine geartrain. The two-dimensional lumped-parameter model [1] is extended to include tooth separation, back-side contact, tooth wedging, and bearing clearances. The simulation results show significant impact of tooth wedging on planet bearing forces for a wide range of operating speeds. To develop a physical understanding of the tooth wedging mechanism, connections between planet bearing forces and tooth forces are studied by investigating physical forces and displacements acting throughout the planetary gear. A method to predict tooth wedging based on geometric interactions is developed and verified. The major causes of tooth wedging relate directly to translational vibrations caused by gravity forces and the presence of clearance-type nonlinearities in the form of backlash and bearing clearance.

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.

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.

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