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.

Harmonic Resonance Frequency Factor of Spur Gear System and Oscillation Reduction Approaches

2007 ◽  
Author(s):  
Jianping Wang ◽  
Pengfei Li ◽  
Ziying Wu ◽  
Minghong Zhang
Keyword(s):  
Parametric Excitation ◽  
Linear Time ◽  
Primary Resonance ◽  
Frequency Factor ◽  
Multiple Scale ◽  
Transmission Error ◽  
Spur Gear ◽  
High Order ◽  
Time Varying ◽  
Harmonic Resonance

In this study, a non-linear time-varying dynamic model of a spur gear pair system is used to investigate the dynamic behavior of the system by means of multiple scale approach. Both time-varying stiffness, transmission error and tooth backlash clearance of the system are taken into account in the model. The mesh stiffness fluctuation is developed as high order Fourier series and tooth backlash clearance is fitted by high order polynomial function. The frequency factors of the system are investigated and the frequency-response equations at the case of internal and external excitation, parametric excitation and combined excitation are obtained. The peak value of the amplitude of the primary resonance, super and sub harmonic resonance and combination harmonic under internal, external and parametric excitation are researched. The approaches of vibration reduction are investigated. Finally an example is investigated using the presented process and the results indicate the sensitivity and correctness of the presented analysis approaches.

scholarly journals Dynamic Model of Spur Gear Pair with Modulation Internal Excitation

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Wang ◽  
Lei Zhang ◽  
Yuan-Qing Luo ◽  
Chang-Zheng Chen
Keyword(s):  
Dynamic Model ◽  
Transmission Error ◽  
Spur Gear ◽  
Experimental Result ◽  
Time Varying ◽  
Gear Pair ◽  
Mesh Stiffness ◽  
Gear Mesh ◽  
Internal Excitation ◽  
Modulation Sideband

In the actual measurements, vibration and noise spectrum of gear pair often exhibits sidebands around the gear mesh harmonic orders. In this study, a nonlinear time-varying dynamic model of spur gear pair was established to predict the modulation sidebands caused by the AM-FM modulation internal excitation. Here, backlash, modulation time-varying mesh stiffness, and modulation transmission error are considered. Then the undamped natural mode was studied. Numerical simulation was made to reveal the dynamic characteristic of a spur gear under modulation condition. The internal excitation was shown to exhibit obvious modulation sideband because of the modulation time-varying mesh stiffness and modulation transmission error. The Runge-Kutta method was used to solve the equations for analyzing the dynamic characteristics with the effect of modulation internal excitation. The result revealed that the response under modulation excitation exhibited obvious modulation sideband. The response under nonmodulation condition was also calculated for comparison. In addition, an experiment was done to verify the prediction of the modulation sidebands. The calculated result was consistent with the experimental result.

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.

Influence of Sliding Friction on the Dynamic Characteristics of a Planetary Gear Set With the Improved Time-Varying Mesh Stiffness

2020 ◽  
Vol 142 (7) ◽  
Author(s):  
Wei Luo ◽  
Baijie Qiao ◽  
Zhixian Shen ◽  
Zhibo Yang ◽  
Hongrui Cao ◽  
...  
Keyword(s):  
Dynamic Model ◽  
Dynamic Characteristics ◽  
Sliding Friction ◽  
Planetary Gear ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Friction Forces ◽  
Basic Model ◽  
Internal Excitation ◽  
Improved Model

Abstract Acting as an important internal excitation, sliding friction can cause the vibration and noise of the planetary gear set. In this paper, a dynamic model is developed to study the influence of sliding friction on the dynamic characteristics of the planetary gear set by including the time-varying mesh stiffness (TVMS), sliding friction forces and torques. An improved analytical model is proposed to calculate the TVMS with sliding friction. The explicit analytical expressions of the sliding friction forces and torques are also derived. Three kinds of different models are applied to investigate the influence of sliding friction: (1) the basic model: sliding friction is neglected in the dynamic model; (2) the improved model I: only the sliding friction forces and torques are considered in the dynamic model; and (3) the improved model II: both the influence of sliding friction on the TVMS and the sliding friction forces and torques are introduced into the dynamic model. The planetary gear set with three equally spaced planet gears is applied to analyze the dynamic characteristics under sliding friction. The simulation results show that the dynamic characteristics can be enhanced or disturbed by sliding friction. In the end, the dynamic model is validated by the experiments. Therefore, the influence of sliding friction is non-negligible when investigating the dynamic characteristics of the planetary gear set. The developed dynamic model provides a feasible dynamic research scheme for the planetary gear set with sliding friction.

Strategies for Modeling Friction in Gear Dynamics

2003 ◽  
Vol 125 (2) ◽  
pp. 383-393 ◽  
Author(s):  
Manish Vaishya ◽  
Rajendra Singh
Keyword(s):  
Sliding Friction ◽  
Linear Time ◽  
Operating Conditions ◽  
Time Varying ◽  
System Complexity ◽  
Friction Forces ◽  
Linear Time Invariant ◽  
Balance System ◽  
Time Invariant ◽  
Solution Methodology

Sliding friction between meshing teeth is one of the primary excitations for noise and vibration in geared systems. Yet, there exist very limited studies on this topic. This paper proposes new modeling strategies for incorporating friction in the dynamic analysis of a gear pair. First, some tribological issues are discussed for estimation of the friction forces under different operating conditions. Second, modeling procedures and results are compared for linear time-invariant, linear time-varying and non-linear time-varying formulations. Criteria such as energy balance, system complexity and desired solution methodology are discussed. Finally, sample results from the various analyses along with their benefits and limitations are examined.

Influence of simultaneous time-varying bearing and tooth mesh stiffness fluctuations on spur gear pair vibration

2019 ◽  
Vol 97 (2) ◽  
pp. 1403-1424 ◽  
Author(s):  
Guanghui Liu ◽  
Jun Hong ◽  
Robert G. Parker
Keyword(s):  
Spur Gear ◽  
Time Varying ◽  
Gear Pair ◽  
Mesh Stiffness

Dynamic Analysis of Sliding Friction in a Gear Pair

2003 ◽  
Author(s):  
Rajendra Gunda ◽  
Rajendra Singh
Keyword(s):  
Load Distribution ◽  
Sliding Friction ◽  
Spur Gear ◽  
Time Instant ◽  
Friction Torque ◽  
Speed Ratio ◽  
Gear Pair ◽  
Mesh Stiffness ◽  
Static Mode

Chief objective of this article is to evaluate the role of sliding friction in gear dynamics, and more specifically the effect of the periodic variations in mesh stiffness, load distribution and friction torque during a mesh cycle. A non-unity speed ratio spur gear is considered. Only the torsional degree of freedom of the gear pair, with ideal Coulomb friction law, is analyzed. Previous studies by Vaishya and Singh [1–3] make idealized assumptions about temporal (or spatial) variation of mesh stiffness and load sharing in order to obtain more tractable analytical solutions. In our formulation, an accurate Finite Element/Contact Mechanics analysis code [4] is run in the static mode to compute the mesh stiffness and load distribution at every time instant of the mesh. The computed parametric variation of stiffness is then incorporated into our dynamic formulation that includes frictional torques. Next, we use appropriate numerical techniques to solve for the dynamic response in time domain. This study, though preliminary in nature, examines the effects of pinion speed, coefficient of friction and mean input torque. This, along with work in progress, should yield further insights into the role of friction sources in gear vibro-acoustics.

Low excitation spur gears with variable tip diameter

2021 ◽  
Vol 263 (5) ◽  
pp. 1275-1285
Author(s):  
Joshua Götz ◽  
Sebastian Sepp ◽  
Michael Otto ◽  
Karsten Stahl
Keyword(s):  
Transmission Error ◽  
Spur Gear ◽  
Low Noise ◽  
Spur Gears ◽  
Mesh Stiffness ◽  
Helical Gears ◽  
Axial Forces ◽  
Tooth Width ◽  
Drive Trains ◽  
Static Transmission Error

One important source of noise in drive trains are transmissions. In numerous applications, it is necessary to use helical instead of spur gear stages due to increased noise requirements. Besides a superior excitation behaviour, helical gears also show additional disadvantageous effects (e.g. axial forces and tilting moments), which have to be taken into account in the design process. Thus, a low noise spur gear stage could simplify design and meet the requirements of modern mechanical drive trains. The authors explore the possibility of combining the low noise properties of helical gears with the advantageous mechanical properties of spur gears by using spur gears with variable tip diameter along the tooth width. This allows the adjustment of the total length of active lines of action at the beginning and end of contact and acts as a mesh stiffness modification. For this reason, several spur gear designs are experimentally investigated and compared with regard to their excitation behaviour. The experiments are performed on a back-to-back test rig and include quasi-static transmission error measurements under load as well as dynamic torsional vibration measurements. The results show a significant improvement of the excitation behaviour for spur gears with variable tip diameter.

Sign in / Sign up

Export Citation Format

Share Document