Impact of Tooth Friction and Its Bending Effect on Gear Dynamics

2007 ◽  
Author(s):  
Gang Liu ◽  
Robert G. Parker
Keyword(s):  
Dynamic Response ◽  
Gear Tooth ◽  
Approximate Solutions ◽  
Contact Ratio ◽  
Rotational Model ◽  
Bending Effect ◽  
Modal Damping ◽  
Parametric Instabilities ◽  
Iterative Integration ◽  
The Stability

This work studies the influences of tooth friction on parametric instabilities and dynamic response of a single-mesh gear pair. A mechanism whereby tooth friction causes gear tooth bending is shown to significantly impact the dynamic response. A dynamic translational-rotational model is developed to consider this mechanism together with the other contributions of tooth friction and mesh stiffness fluctuation. An iterative integration method to analyze parametric instabilities is proposed and compared with an established numerical method. Perturbation analysis is conducted to find approximate solutions that predict and explain the numerical parametric instabilities. The effects of time-varying friction moments about the gear centers and friction-induced tooth bending are critical to parametric instabilities and dynamic response. The impacts of friction coefficient, bending effect, contact ratio, and modal damping on the stability boundaries are revealed. Finally, the friction bending effect on the nonlinear dynamic response is examined and validated by finite element results.

scholarly journals On the Existence of Self-Excited Vibration in Thin Spur Gears: A Theoretical Model for the Estimation of Damping by the Energy Method

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 664 ◽  
Author(s):  
Yanrong Wang ◽  
Hang Ye ◽  
Long Yang ◽  
Aimei Tian
Keyword(s):  
Axial Force ◽  
Traveling Wave ◽  
The Self ◽  
Spur Gears ◽  
Contact Ratio ◽  
Symmetric Structure ◽  
Modal Damping ◽  
Excited Vibration ◽  
Cyclic Symmetric ◽  
The Stability

The gear is a cyclic symmetric structure, and each tooth is subjected to a periodic mesh force. These mesh forces have the same phase difference tooth by tooth, which can excite gear vibrations. The mechanism of additional axial force caused by gear bending is shown and examined, which can significantly affect the stability of a self-excited thin spur gears vibration. A mechanical model based on energy balance is then developed to predict the contribution of additional axial force, leading to the proposed numerical integration method for vibration stability analysis. By analyzing the change in the system energy, the occurrence of the self-excited vibration is validated. A numerical simulation is carried out to verify the theoretical analysis. The impacts of modal damping, contact ratio, and the number of nodal diameters on the stability boundaries of the self-excited vibration are revealed. The results prove that the backward traveling wave of the driven gear as well as the forward traveling wave of the driving gear encounter self-excited vibration in the absence of sufficient damping. The model can be used to predict the stability of the gear self-excited vibration.

Influence of the backlash generated by tooth accumulated wear on dynamic behavior of compound planetary gear set

2016 ◽  
Vol 231 (11) ◽  
pp. 2025-2041 ◽  
Author(s):  
Shijing Wu ◽  
Haibo Zhang ◽  
Xiaosun Wang ◽  
Zeming Peng ◽  
Kangkang Yang ◽  
...  
Keyword(s):  
Dynamic Response ◽  
Dynamic Model ◽  
Dynamic Behavior ◽  
Planetary Gear ◽  
Tooth Wear ◽  
Transmission Error ◽  
Gear Transmission ◽  
Tooth Surface ◽  
Contact Ratio ◽  
The Impact

Backlash is a key internal excitation on the dynamic response of planetary gear transmission. After the gear transmission running for a long time under load torque, due to tooth wear accumulation, the backlash between the tooth surface of two mating gears increases, which results in a larger and irregular backlash. However, the increasing backlash generated by tooth accumulated wear is generally neglected in lots of dynamics analysis for epicyclic gear trains. In order to investigate the impact of backlash generated by tooth accumulated wear on dynamic behavior of compound planetary gear set, in this work, first a static tooth surface wear prediction model is incorporated with a dynamic iteration methodology to get the increasing backlash generated by tooth accumulated wear for one pair of mating teeth under the condition that contact ratio equals to one. Then in order to introduce the tooth accumulated wear into dynamic model of compound planetary gear set, the backlash excitation generated by tooth accumulated wear for each meshing pair in compound planetary gear set is given under the condition that contact ratio equals to one and does not equal to one. Last, in order to investigate the impact of the increasing backlash generated by tooth accumulated wear on dynamic response of compound planetary gear set, a nonlinear lumped-parameter dynamic model of compound planetary gear set is employed to describe the dynamic relationships of gear transmission under the internal excitations generated by worn profile, meshing stiffness, transmission error, and backlash. The results indicate that the introduction of the increasing backlash generated by tooth accumulated wear makes a significant influence on the bifurcation and chaotic characteristics, dynamic response in time domain, and load sharing behavior of compound planetary gear set.

scholarly journals Simulation of the Load Evolution of an Anchoring System under a Blasting Impulse Load Using FLAC3D

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Xigui Zheng ◽  
Jinbo Hua ◽  
Nong Zhang ◽  
Xiaowei Feng ◽  
Lei Zhang
Keyword(s):  
Shear Stress ◽  
Dynamic Response ◽  
Axial Force ◽  
Mechanical Characteristics ◽  
Oscillation Amplitude ◽  
Anchoring System ◽  
Impulse Load ◽  
The Stability ◽  
Calculation Software ◽  
Dynamic Perturbation

A limitation in research on bolt anchoring is the unknown relationship between dynamic perturbation and mechanical characteristics. This paper divides dynamic impulse loads into engineering loads and blasting loads and then employs numerical calculation software FLAC3Dto analyze the stability of an anchoring system perturbed by an impulse load. The evolution of the dynamic response of the axial force/shear stress in the anchoring system is thus obtained. It is revealed that the corners and middle of the anchoring system are strongly affected by the dynamic load, and the dynamic response of shear stress is distinctly stronger than that of the axial force in the anchoring system. Additionally, the perturbation of the impulse load reduces stress in the anchored rock mass and induces repeated tension and loosening of the rods in the anchoring system, thus reducing the stability of the anchoring system. The oscillation amplitude of the axial force in the anchored segment is mitigated far more than that in the free segment, demonstrating that extended/full-length anchoring is extremely stable and surpasses simple anchors with free ends.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

Effects of Moving Speeds of Dynamic Loads on the Deflections of Gear Teeth

1981 ◽  
Vol 103 (2) ◽  
pp. 357-363 ◽  
Author(s):  
K. Nagaya ◽  
S. Uematsu
Keyword(s):  
Dynamic Response ◽  
Timoshenko Beam ◽  
Gear Tooth ◽  
Bending Moment ◽  
Dynamic Loads ◽  
Gear Teeth ◽  
Moving Speed

For the dynamic response problems of gear teeth, the dynamic loads which act upon the gear teeth should be considered as a function of both the position and the moving speed. In previous studies, the effects of the moving speed have not been considered. In this paper the effects of the moving speed of dynamic loads on the deflection and the bending moment of the gear tooth are investigated. The results are obtained from the elastodynamic analysis of the tapered Timoshenko beam.

Manufacturing Process for Circular-Arc Curvilinear Cylindrical Gears with Predesigned Fourth Order Transmission Error

2010 ◽  
Vol 37-38 ◽  
pp. 623-627 ◽  
Author(s):  
Jin Zhan Su ◽  
Zong De Fang
Keyword(s):  
Fourth Order ◽  
Gear Tooth ◽  
Transmission Error ◽  
Circular Arc ◽  
Cylindrical Gears ◽  
Polynomial Curve ◽  
Order Polynomial ◽  
Tooth Surfaces ◽  
The Stability ◽  
Fourth Order Polynomial

A fourth order transmission error was employed to improve the stability and tooth strength of circular-arc curvilinear cylindrical gears. The coefficient of fourth order polynomial curve was determined, the imaginary rack cutter which formed by the rotation of a head cutter and the imaginary pinion were introduced to determine the pinion and gear tooth surfaces, respectively. The numerical simulation of meshing shows: 1) the fourth order transmission error can be achieved by the proposed method; 2) the stability transmission can be performed by increasing the angle of the transfer point of the cycle of meshing; 3) the tooth fillet strength can be enhanced.

Solution of third-order Emden–Fowler-type equations using wavelet methods

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Arshad Khan ◽  
Mo Faheem ◽  
Akmal Raza
Keyword(s):  
Nuclear Reactions ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Third Order ◽  
Content Type ◽  
Resolution Level ◽  
The Stability ◽  
Comparison Of The Results ◽  
Wavelet Methods ◽  
Nonlinear Nature

Purpose The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs. Design/methodology/approach Because of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods. Findings This paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased. Originality/value This paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.

Friction Induced Vibrations in Aircraft Disk Brakes

1997 ◽  
Author(s):  
Lisle B. Hagler ◽  
Per G. Reinhall
Keyword(s):  
Friction Coefficient ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Rotational Model ◽  
Stick Slip ◽  
Numeric Simulation ◽  
Constant Friction ◽  
The Stability ◽  
Rotor Stiffness ◽  
Friction Induced Vibration

Abstract This paper presents a detailed analysis of the dynamic behavior of a single rotor/stator brake system. Two separate mathematical models of the brake are considered. First, a non-rotational model is constructed with the purpose of showing that friction induced vibration can occur in the stator without assuming stick-slip behavior and a velocity dependent friction coefficient. Self-induced vibrations are analyzed via the application of the method of multiple scales. The stability boundaries of the primary resonance, as well as the super-harmonics and sub-harmonics are determined. Secondly, rotational effects are investigated by considering a mathematical brake model consisting of a spinning rotor engaging against a flexible stator. Again, a constant friction coefficient is assumed. The stability of steady whirl is determined as a function of the system parameters. We demonstrate that only forward whirl is stable for no-slip motion of the rotor. The interactions between chatter, squeal, and rotor whirl are investigated through numeric simulation. It is shown that rotor whirl can be an important source of the torsional oscillations (squeal) of the stator and that the settling time to no-slip decreases as the ratio of the stator to rotor stiffness is increased.

scholarly journals Dynamic response of Zelazny Most tailings dam to mining induced extreme seismic event occurred in 2016

2019 ◽  
Vol 262 ◽  
pp. 01001
Author(s):  
Aleksandra Korzec ◽  
Waldemar Świdziński
Keyword(s):  
Stability Analysis ◽  
Dynamic Response ◽  
Dynamic Loading ◽  
Time Integration ◽  
Horizontal Displacement ◽  
Material Model ◽  
Implicit Time ◽  
Tailings Dam ◽  
Peak Horizontal Acceleration ◽  
The Stability

The paper deals with the stability analysis of tailings dam subjected to dynamic loading induced by mining shocks which occurred in neighbouring copper mine. The main goal of the paper was to model the dynamic response of the dam during two extreme paraseismic events which occurred in 2016 based on accelerograms recorded at the dam toe. Dynamic response of the tailings dam was calculated using finite element method and the implicit time-integration method implemented in commercial codes. The boundary condition corresponding to dynamic loading was determined by deconvolution procedure. The error analysis showed that most precise signal reproduction is achieved while using target signal with peak value reduced by 40% as a test signal. Both acceleration and displacement time-series were successfully reproduced. Moreover, the stability analysis was conducted for five independent signals with design peak horizontal acceleration and showed that no permanent displacements should occur. The temporary horizontal displacement of the dam crest should not exceed 13 mm, assuming equivalent linear material model.

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

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