Analytical Study of Helical Gear Dynamics With Sliding Friction Using Floquet Theory

2007 ◽  
Author(s):  
Song He ◽  
Rajendra Singh
Keyword(s):  
Floquet Theory ◽  
Sliding Friction ◽  
Initial Conditions ◽  
Transmission Error ◽  
Helical Gear ◽  
Degree Of Freedom ◽  
Analytical Models ◽  
Mesh Stiffness ◽  
Forcing Function ◽  
Dynamic Transmission Error

Analytical models of a helical gear pair are developed in order to examine the effect of sliding friction on the dynamic transmission error. Simplified 6 degree-of-freedom and single degree-of-freedom analytical models are developed. These models characterize the contact plane dynamics and capture the velocity reversal at the pitch line due to sliding friction. By assuming a constant mesh stiffness density along the contact lines, a linear time-varying model (with parametric excitation) is obtained. The effect of sliding friction is quantified by an effective mesh stiffness term. Floquet theory is then used to obtain closed-form solutions to the dynamic transmission error given periodic piece-wise linear tooth stiffness function. Responses to both initial conditions and forcing function under a nominal torque are derived. Analytical models are validated by comparing predictions with numerical simulations. Finally, parametrically-induced instability issues are briefly mentioned.

Dynamic Transmission Error Prediction of Helical Gear Pair Under Sliding Friction Using Floquet Theory

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Song He ◽  
Rajendra Singh
Keyword(s):  
Floquet Theory ◽  
Sliding Friction ◽  
Initial Conditions ◽  
Transmission Error ◽  
Helical Gear ◽  
Marginal Effect ◽  
Gear Pair ◽  
Mesh Stiffness ◽  
Dynamic Transmission Error ◽  
Helical Gear Pair

An analytical solution to the dynamic transmission error of a helical gear pair is developed by using a single-degree-of-freedom model with piecewise stiffness functions that characterize the contact plane dynamics and capture the velocity reversal at the pitch line. By assuming a constant mesh stiffness density along the contact lines, a linear time-varying model (with parametric excitations) is obtained, where the effect of sliding friction is quantified by an effective mesh stiffness term. The Floquet theory is then used to obtain closed-form solutions to the dynamic transmission error, and responses are derived to both initial conditions and the forced periodic function under a nominal preload. Analytical models are validated by comparing predictions with numerical simulations, and the effect of viscous damping is examined. Stability analysis is also briefly conducted by using the state transition matrix. Overall, the sliding friction has a marginal effect on the dynamic transmission error of helical gears, as compared with spur gears, in the context of the torsional model.

The Effect of Tooth Wear on the Dynamic Transmission Error of Helical Gears with Smaller Number of Pinion Teeth

2014 ◽  
Vol 657 ◽  
pp. 649-653 ◽  
Author(s):  
Virgil Atanasiu ◽  
Cezar Oprişan ◽  
Dumitru Leohchi
Keyword(s):  
Dynamic Contact ◽  
Tooth Wear ◽  
Transmission Error ◽  
Analytical Investigation ◽  
Helical Gear ◽  
Wear Depth ◽  
Contact Ratio ◽  
Mesh Stiffness ◽  
Helical Gears ◽  
Dynamic Transmission Error

The paper presents an analytical investigation of the effect of the tooth wear on the dynamic transmission error of helical gear pairs with small number of pinion teeth. Firstly, the dynamic analysis is conducted to investigate only the effect of the time-varying mesh stiffness on the variation of dynamic transmission error along the line of action. Then, the tooth wear effect on the dynamics of helical gear with small number of pinion teeth is being researched. In the analysis, instantaneous dynamic contact analysis is used in wear depth calculations. A comparative study was performed to investigate the relation between total contact ratio, mesh stiffness and dynamic transmission error of helical gear pairs with small number of teeth.

Construction of Semianalytical Solutions to Spur Gear Dynamics Given Periodic Mesh Stiffness and Sliding Friction Functions

2008 ◽  
Vol 130 (12) ◽  
Author(s):  
Song He ◽  
Todd Rook ◽  
Rajendra Singh
Keyword(s):  
Dynamic Models ◽  
Sliding Friction ◽  
Linear Time ◽  
Transmission Error ◽  
Spur Gear ◽  
Degree Of Freedom ◽  
System Model ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Friction Forces

Gear dynamic models with time-varying mesh stiffness, viscous mesh damping, and sliding friction forces and moments lead to complex periodic differential equations. For example, the multiplicative effect generates higher mesh harmonics. In prior studies, time-domain integration and fast Fourier transform analysis have been utilized, but these methods are computationally sensitive. Therefore, semianalytical single- and multiterm harmonic balance methods are developed for an efficient construction of the frequency responses. First, an analytical single-degree-of-freedom, linear time-varying system model is developed for a spur gear pair in terms of the dynamic transmission error. Harmonic solutions are then derived and validated by comparing with numerical integration results. Next, harmonic solutions are extended to a six-degree-of-freedom system model for the prediction of (normal) mesh loads, friction forces, and pinion/gear displacements (in both line-of-action and off-line-of-action directions). Semianalytical predictions compare well with numerical simulations under nonresonant conditions and provide insights into the interaction between sliding friction and mesh stiffness.

scholarly journals Geometry Modification of Helical Gear for Reduction of Static Transmission Error

2018 ◽  
Vol 167 ◽  
pp. 02013
Author(s):  
Jeonghyun Park ◽  
Changjun Seo ◽  
Kwangsuck Boo ◽  
Heungseob Kim
Keyword(s):  
Transmission Error ◽  
Helical Gear ◽  
Necessary Condition ◽  
Mesh Stiffness ◽  
Profile Modification ◽  
Gear Mesh ◽  
Static Transmission Error ◽  
Excitation Mechanisms ◽  
Gear Systems ◽  
Gear Mesh Stiffness

Gear systems are extensively employed in mechanical systems since they allow the transfer of power with a variety of gear ratios. So gears cause the inherent deflections and deformations due to the high pressure which occurs between the meshing teeth when transmit power and results in the transmission error. It is usually assumed that the transmission error and variation of the gear mesh stiffness are the dominant excitation mechanisms. Predicting the static transmission error is a necessary condition to reduce noise radiated from the gear systems. This paper aims to investigate the effect of tooth profile modifications on the transmission error of helical gear. The contact stress analysis was implemented for different roll positions to find out the most critical roll angle in view point of root flank stress. The PPTE (peak-to-peak of transmission error) is estimated at the roll angles by different loading conditions with two dimensional FEM. The optimal profile modification from the root to the tip is proposed.

Nonlinear Dynamic Analysis of a New Antibacklash Gear Mechanism Design for Reducing Dynamic Transmission Error

2015 ◽  
Vol 137 (5) ◽  
Author(s):  
S. R. Besharati ◽  
V. Dabbagh ◽  
H. Amini ◽  
Ahmed A. D. Sarhan ◽  
J. Akbari ◽  
...  
Keyword(s):  
Mechanism Design ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic Analysis ◽  
Transmission Error ◽  
Mesh Stiffness ◽  
Gear Teeth ◽  
Dynamic Transmission Error ◽  
Gear Mechanism ◽  
Lower Transmission ◽  
Input Torque

In this study, a new antibacklash gear mechanism design comprising three pinions and a rack is introduced. This mechanism offers several advantages compared to conventional antibacklash mechanisms, such as lower transmission error as well as lower required preload. Nonlinear dynamic modeling of this mechanism is developed to acquire insight into its dynamic behavior. It is observed that the amount of preload required to diminish the backlash depends on the applied input torque and nature of periodic mesh stiffness. Then, an attempt is made to obtain an approximate relation to find the minimum requiring preload to preserve the system’s antibacklash property and reduce friction and wear on the gear teeth. The mesh stiffness of the mated gears, rack, and pinion is achieved via finite element method. Assuming that all teeth are rigid and static transmission error is negligible, dynamic transmission error (DTE) would be zero for every input torque, which is a unique trait, not yet proposed in previous research.

A parameterized numerical model for the evaluation of gear mesh stiffness variation of a helical gear pair

2008 ◽  
Vol 222 (7) ◽  
pp. 1321-1327 ◽  
Author(s):  
J Hedlund ◽  
A Lehtovaara
Keyword(s):  
Numerical Model ◽  
Transmission Error ◽  
Helical Gear ◽  
Hertzian Contact ◽  
Mesh Stiffness ◽  
Fe Method ◽  
Gear Design ◽  
Gear Mesh ◽  
Stiffness Variation ◽  
Gear Mesh Stiffness

One of the most common challenges in gear drive design is to determine the best combination of gear geometry parameters. These parameters should be capable of being varied effectively and related to gear mesh stiffness variation in advanced excitation and vibration analysis. Accurate prediction of gear mesh stiffness and transmission error requires an efficient numerical method. The parameterized numerical model was developed for the evaluation of excitation induced by mesh stiffness variation for helical gear design purposes. The model uses linear finite-element (FE) method to calculate tooth deflections, including tooth foundation flexibility. The model combines Hertzian contact analysis with structural analysis to avoid large FE meshes. Thus, mesh stiffness variation was obtained in the time and frequency domains, which gives flexibility if comparison is made with measured spectrums. Calculations showed that a fairly low number of elements suffice for the estimation of mesh stiffness variation. A reasonable compromise was achieved between design trends and calculation time.

Static and Dynamic Transmission Error Measurements of Helical Gear Pairs With Various Tooth Modifications

2019 ◽  
Vol 141 (10) ◽  
Author(s):  
M. Benatar ◽  
M. Handschuh ◽  
A. Kahraman ◽  
D. Talbot
Keyword(s):  
Dynamic Models ◽  
Nonlinear Behavior ◽  
Transmission Error ◽  
Helical Gear ◽  
Forced Response ◽  
Test Machine ◽  
Response Curves ◽  
Gear Mesh ◽  
Dynamic Transmission Error ◽  
Static Transmission Error

This paper presents a set of motion transmission error data for a family of helical gears having different profile and lead modifications operated under both low-speed (quasi-static) and dynamic conditions. A power circulatory test machine is used along with encoder and accelerometer-based transmission error measurement systems to quantify motion transmission behavior within wide ranges of torque and speed. Results of these experiments indicate that the tooth modifications impact the resultant static and dynamic transmission error amplitudes significantly. A design load is shown to exist for each gear pair of different modifications where static transmission error amplitude is minimum. Forced response curves and waterfall plots are presented to demonstrate that the helical gear pairs tested act linearly with no signs of nonlinear behavior such as tooth contact separations. Furthermore, static and dynamic transmission error amplitudes are observed to be nearly proportional, suggesting that static transmission error can be employed in helical gear dynamic models as the main gear mesh excitation. The data presented here is intended to fill a void in the literature by providing means for validation of load distribution and dynamic models of helical gear pairs.

A Comparative Study of Selected Gear Mesh Interface Dynamic Models

1992 ◽  
Author(s):  
G. Wesley Blankenship ◽  
Rajendra Singh
Keyword(s):  
Dynamic Models ◽  
Transmission Error ◽  
Analytical Models ◽  
Primary Sources ◽  
Mesh Stiffness ◽  
Mathematical Basis ◽  
Gear Mesh ◽  
Interface Dynamic ◽  
Gyroscopic Effects ◽  
Gear Drives

Abstract There is a consensus among modern gear researchers that variation in gear mesh stiffness and transmission error are the primary sources of vibratory excitation in most moderately to heavily loaded gear drives. However, several schools of thought exist in the literature on how to incorporate these mesh stiffness and transmission error concepts into a dynamic model. In this paper, a formal expression for an elastic gear mesh force vector is developed and selected gear mesh interface models, which exemplify most of the common modeling approaches in use today, are compared on a common mathematical basis. The various modeling strategies, their inherent philosophies and assumptions are made clear. All of the models examined employ a common spatial domain analysis methodology which pervades the field of modem gear dynamics. The focus of this study is limited to the quasi-steady state, non-resonant dynamic analysis of a single involute gear pair operating below critical shaft speeds such that shaft whirling and gyroscopic effects are negligible, and under loading conditions sufficiently high to prevent loss of tooth contact due to gear backlash phenomenon. The need for extended analytical models, which consider multi-dimensional excitation and better describe force transmissibility via the gear mesh interface, is identified. This forms the basis of an on-going comprehensive investigation which expects to clarify several unresolved issues in gear dynamic modeling; future publications will report such studies.

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