Nonlinear Vibration of Gear Pairs With Tooth Surface Modifications at Primary Resonance Using a Perturbation Method

2011 ◽  
Author(s):  
Tugan Eritenel ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Vibration ◽  
Half Space ◽  
Multiple Scales ◽  
Gear Tooth ◽  
Primary Resonance ◽  
Surface Modifications ◽  
Element Model ◽  
Elastic Behavior ◽  
Tooth Surface ◽  
Contact Vibrations

An analytical solution for the nonlinear vibration of gear pairs that exhibit partial and total contact loss is found. The gear teeth can have arbitrary tooth surface modifications. Such modifications and dynamic displacements separate parts of gear tooth surface otherwise designed to be in contact. This is partial contact loss. The excitation and the nonlinearity are not specified but are found from the force-deflection function of the gear pair, which comes from independent analysis, such as a finite element model. Fourier and Taylor series expansions of the force-deflection function capture the flexibility, nonlinearity, and the excitation in a few coefficients. The gear elastic behavior includes Hertz contact, bending, and shear. The nonlinearity arises chiefly from tooth surface modifications due to the dependence of contact upon the instantaneous dynamic mesh force. Although this work focuses on gear pairs with tooth surface modifications, the physical system from which the force-deflection function comes is not limited to gear pairs. Sphere/half-space contact vibrations are also analyzed. The dynamic frequency-amplitude relation at the steady-state is found using the method of multiple scales. Comparisons with experiments from the literature on gear vibrations and sphere/half-space contact vibrations verify the method.

scholarly journals Nonlinear Vibration of Gears With Tooth Surface Modifications

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Tugan Eritenel ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Gear Tooth ◽  
A Priori ◽  
Surface Modifications ◽  
Element Model ◽  
Transmission Error ◽  
Tooth Profile ◽  
Tooth Surface ◽  
Partial Contact

This work provides an analytical solution for the nonlinear vibration of gear pairs that exhibit partial and total contact loss. Partial contact loss is where parts of contact lines lose contact although other parts remain in contact. The gear tooth surface modifications admit an arbitrary combination of profile and lead modifications. Modifications are a source of partial contact loss. The analysis also applies for total contact loss. Unlike models in the literature that are excited by static transmission error or time-varying mesh stiffness, the excitation and the nonlinearity are not a priori specified. Instead, the force-deflection function of the gear pair is provided by an independent source, such as a finite element model or Hertz contact formula. The manipulation of the single-degree-of-freedom oscillator equation of motion yields the excitation and the nonlinearity that arise from Fourier and Taylor series expansions of the force-deflection function. These expansions capture the essential contact behavior that includes tooth profile and lead modifications as well as the bending and shear flexibility of the gear teeth and gear blanks. The method of multiple scales gives the steady-state dynamic response in terms of a frequency-amplitude relation. Comparisons with gear vibration experiments and simulations from the literature that include spur and helical gears with tooth profile and lead modifications verify the method.

Robust Optimization of Cylindrical Gear Tooth Surface Modifications Within Ranges of Torque and Misalignments

2013 ◽  
Author(s):  
Alessio Artoni ◽  
Massimo Guiggiani ◽  
Ahmet Kahraman ◽  
Jonny Harianto
Keyword(s):  
Performance Metrics ◽  
Gear Tooth ◽  
Surface Modifications ◽  
Transmission Error ◽  
Robust Design Optimization ◽  
Tooth Surface ◽  
Global Optimality ◽  
Cylindrical Gear ◽  
Geometry Design ◽  
Design Variables

Tooth surface modifications are small, micron-level intentional deviations from perfect involute geometries of spur and helical gears. Such modifications are aimed at improving contact pressure distribution, while minimizing the motion transmission error to reduce noise excitations. In actual practice, optimal modification requirements vary with the operating torque level, misalignments, and manufacturing variance. However, most gear literature has been concerned with determining optimal flank form modifications at a single design point, represented by fixed, single load and misalignment values. A new approach to the design of tooth surface modifications is proposed to handle such conditions. The problem is formulated as a robust design optimization problem, and it is solved, in conjunction with an efficient gear contact solver (LDP), by a direct search, global optimization algorithm aimed at guaranteeing global optimality of the obtained micro-geometry solutions. Several tooth surface modifications can be used as micro-geometry design variables, including profile, lead, and bias modifications. Depending on the contact solver capabilities, multiple performance metrics can be considered. The proposed method includes the capability of simultaneously and robustly handling several conflicting design objectives. In the present paper, peak contact stress and loaded transmission error amplitude are used as objective functions (to be minimized). At the end, two example optimizations are presented to demonstrate the effectiveness of the proposed method.

Robust Optimization of Cylindrical Gear Tooth Surface Modifications Within Ranges of Torque and Misalignments

2013 ◽  
Vol 135 (12) ◽  
Author(s):  
Alessio Artoni ◽  
Massimo Guiggiani ◽  
Ahmet Kahraman ◽  
Jonny Harianto
Keyword(s):  
Performance Metrics ◽  
Gear Tooth ◽  
Surface Modifications ◽  
Transmission Error ◽  
Robust Design Optimization ◽  
Tooth Surface ◽  
Global Optimality ◽  
Cylindrical Gear ◽  
Design Variables ◽  
Gear Contact

Tooth surface modifications are small, micron-level intentional deviations from perfect involute geometries of spur and helical gears. Such modifications are aimed at improving contact pressure distribution, while minimizing the motion transmission error to reduce noise excitations. In actual practice, optimal modification requirements vary with the operating torque level, misalignments, and manufacturing variance. However, most gear literature has been concerned with determining optimal flank form modifications at a single design point, represented by fixed, single load and misalignment values. A new approach to the design of tooth surface modifications is proposed to handle such conditions. The problem is formulated as a robust design optimization problem, and it is solved, in conjunction with an efficient gear contact solver (Load Distribution Program (LDP)), by a direct search, global optimization algorithm aimed at guaranteeing global optimality of the obtained microgeometry solutions. Several tooth surface modifications can be used as microgeometry design variables, including profile, lead, and bias modifications. Depending on the contact solver capabilities, multiple performance metrics can be considered. The proposed method includes the capability of simultaneously and robustly handling several conflicting design objectives. In the present paper, peak contact stress and loaded transmission error amplitude are used as objective functions (to be minimized). At the end, two example optimizations are presented to demonstrate the effectiveness of the proposed method.

scholarly journals Nonlinear Vibration Analysis of Curved Piezoelectric-Layered Nanotube Resonator

Energies ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 8031
Author(s):  
Zia Saadatnia
Keyword(s):  
Nonlinear Vibration ◽  
Piezoelectric Layer ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Point Load ◽  
Vibration Response ◽  
Structural Vibration ◽  
Small Scale ◽  
Sensors And Actuators ◽  
Harmonic Voltage

Piezoelectric-based nano resonators are smart structures that can be used for mechanical sensors and actuators in miniature systems. In this study, the nonlinear vibration behavior of a curved piezoelectric-layered nanotube resonator was investigated. The curved structure comprises a core nanotube and a slender layer of piezoelectric material covering the inner nanotube where a harmonic voltage is applied to the piezoelectric layer. Applying the energy method and Hamiltonian principle in association with non-local theories, the governing equations of motion of the targeted system are obtained. Then, the problem is solved using the Galerkin and multiple scales methods, and the system responses under external excitation and parametric load are found. Various resonance conditions are investigated including primary and parametric resonances, and the frequency responses are obtained considering steady state motions. The effects of different parameters such as applied voltage, piezoelectric thickness, and structural curvature on the system responses are investigated. It is shown that the applied harmonic voltage to the piezoelectric layer can cause a parametric resonance in the structural vibration, and the applied harmonic point load to the structure can cause a primary resonance in the vibration response. Considering two structural curvatures including quadratic and cubic curves, it is also found that the waviness and curve shape parameters can tune the nonlinear hardening and softening behaviors of the system and at specific curve shapes, the vibration response of the targeted structure acts similar to that of a linear system. This study can be targeted toward the design of curved piezoelectric nano-resonators in small-scale sensing and actuation systems.

USING ENERGY-PHASE METHOD TO STUDY GLOBAL BIFURCATIONS AND SHILNIKOV TYPE MULTIPULSE CHAOTIC DYNAMICS FOR A NONLINEAR VIBRATION ABSORBER

2012 ◽  
Vol 22 (01) ◽  
pp. 1250001 ◽  
Author(s):  
S. B. LI ◽  
W. ZHANG ◽  
M. H. YAO
Keyword(s):  
Nonlinear Vibration ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Standard Form ◽  
Primary Resonance ◽  
Phase Method ◽  
Vibration Absorber ◽  
Global Bifurcations ◽  
Modulation Equation ◽  
Nonlinear Vibration Absorber

Global bifurcations and Shilnikov type multipulse chaotic dynamics for a nonlinear vibration absorber are investigated by using the energy-phase method for the first time. A two-degree-of-freedom model of a nonlinear vibration absorber is considered. After the nonlinear nonautonomous equations of this model are given, the method of multiple scales is used to derive four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of the two interacting modes in the presence of 1:1 internal resonance and primary resonance. Using several coordinate transformations to transform the modulation equation into a standard form, we can apply the energy-phase method to show the existence of the multipulse chaotic dynamics by identifying Shilnikov-type multipulse orbits in the perturbed phase space. We are able to obtain the explicit restriction on the damping, forcing excitation and the detuning parameters, under which the multipulse chaotic dynamics is expected. These multipulse orbits represent the repeated departure from purely vertical oscillations for the nonlinear vibration absorber. Numerical simulations also indicate that there exist different forms of the multipulse chaotic responses and jumping phenomena for the nonlinear vibration absorber.

scholarly journals Times Three Dimensional Spur Gear Static Contact Investigations Using Finite Element Method

2016 ◽  
Vol 10 (5) ◽  
pp. 145 ◽  
Author(s):  
Ahmed Mohammed Abdelrhman ◽  
Haidar F. AL-Qrimli ◽  
Husam M. Hadi. ◽  
Roaad K. Mohammed ◽  
Hakim S. Sultan
Keyword(s):  
Finite Element ◽  
Contact Stress ◽  
Gear Tooth ◽  
Three Dimensional ◽  
Element Model ◽  
Spur Gear ◽  
Industrial Applications ◽  
Tooth Surface ◽  
Surface Strength ◽  
Static Contact

<p>A gear is a critical component and can be found in many industrial applications. This investigation develops a three dimensional finite element spur gear model to calculate the contact stress on the gear tooth surfaces. Contact stress is one of the main factors that is used to decide the gears tooth surface strength. In addition there are other important factors such as frictional forces and micro-pits that influence the gear tooth surface. Different analytical techniques have been used to calculate the contact stress of the gear surfaces namely; Hertzian theory and AGMA standards. The analytical results have been compared to the numerical analysis to verify the spur gear finite element model.</p>

Nonlinear vibration control of nano-beam based on capacitive acoustic sensors

2020 ◽  
Vol 64 (1-4) ◽  
pp. 421-429
Author(s):  
Lei Wan ◽  
Canchang Liu ◽  
Weixu Kong ◽  
Yingchao Zhou ◽  
Chicheng Ma
Keyword(s):  
Vibration Control ◽  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Graphene Film ◽  
Primary Resonance ◽  
High Sensitivity ◽  
Vibration Signal ◽  
System Stability ◽  
Acoustic Sensors ◽  
Nonlinear Vibration Control

The nonlinear vibration control of the Euler–Bernoulli beam is studied based on capacitive micro-mechanical acoustic sensors The graphene film has the characteristics of high sensitivity and high accuracy which can be applied to sense the vibration signal. Capacitive micro-mechanical acoustic sensors can be used to detect the acoustic signal of the vibration of the nano-beam. The nonlinear vibration control equation of nano-beam can be established with the displacement and velocity voltage feedback controller based on capacitive micro-mechanical acoustic sensors. The amplitude-frequency response equation of the primary resonance of nano-beam can be gotten by using the multiple scales method. The relationship between the nonlinear vibration of nano-beam and system parameters is investigated. The influencing factors of how to weak system nonlinearity and enhance system stability are analyzed. The static bifurcation behavior of the system is discussed. The numerical results show that the nonlinearity of vibration can be reduced and the stability of the system can be improved by selecting the appropriate control gains and appropriately reducing the amplitude of DC and AC excitation voltages.

scholarly journals Accuracy Improvement of the Method of Multiple Scales for Nonlinear Vibration Analyses of Continuous Systems with Quadratic and Cubic Nonlinearities

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Akira Abe
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Continuous Systems ◽  
Driving Frequency ◽  
Accuracy Improvement ◽  
Definition Of ◽  
Quadratic And Cubic Nonlinearities

This paper proposes an accuracy improvement of the method of multiple scales (MMSs) for nonlinear vibration analyses of continuous systems with quadratic and cubic nonlinearities. As an example, we treat a shallow suspended cable subjected to a harmonic excitation, and investigate the primary resonance of the th in-plane mode () in which and are the driving and natural frequencies, respectively. The application of Galerkin's procedure to the equation of motion yields nonlinear ordinary differential equations with quadratic and cubic nonlinear terms. The steady-state responses are obtained by using the discretization approach of the MMS in which the definition of the detuning parameter, expressing the relationship between the natural frequency and the driving frequency, is changed in an attempt to improve the accuracy of the solutions. The validity of the solutions is discussed by comparing them with solutions of the direct approach of the MMS and the finite difference method.

Modeling of surface roughness in a novel magnetorheological gear profile finishing process

2020 ◽  
pp. 095440622097826
Author(s):  
Ravi Datt Yadav ◽  
Anant Kumar Singh ◽  
Kunal Arora
Keyword(s):  
Surface Roughness ◽  
Gear Tooth ◽  
Surface Characteristics ◽  
Spur Gear ◽  
Tooth Profile ◽  
Tooth Surface ◽  
Magnetic Flux Density ◽  
Element Analysis ◽  
Finishing Process ◽  
Gear Profile

Fine finishing of spur gears reduces the vibrations and noise and upsurges the service life of two mating gears. A new magnetorheological gear profile finishing (MRGPF) process is utilized for the fine finishing of spur gear teeth profile surfaces. In the present study, the development of a theoretical mathematical model for the prediction of change in surface roughness during the MRGPF process is done. The present MRGPF is a controllable process with the magnitude of the magnetic field, therefore, the effect of magnetic flux density (MFD) on the gear tooth profile has been analyzed using an analytical approach. Theoretically calculated MFD is validated experimentally and with the finite element analysis. To understand the finishing process mechanism, the different forces acting on the gear surface has been investigated. For the validation of the present roughness model, three sets of finishing cycle experimentations have been performed on the spur gear profile by the MRGPF process. The surface roughness of the spur gear tooth surface after experimentation was measured using Mitutoyo SJ-400 surftest and is equated with the values of theoretically calculated surface roughness. The results show the close agreement which ranges from −7.69% to 2.85% for the same number of finishing cycles. To study the surface characteristics of the finished spur gear tooth profile surface, scanning electron microscopy is used. The present developed theoretical model for surface roughness during the MRGPF process predicts the finishing performance with cycle time, improvement in the surface quality, and functional application of the gears.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

2017 ◽  
Vol 9 (6) ◽  
pp. 1485-1505
Author(s):  
Lingchang Meng ◽  
Fengming Li
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Composite Laminated Plate ◽  
Vibration Localization ◽  
Composite Laminated Plates ◽  
Localization Phenomenon ◽  
Harmonic Resonance

AbstractThe nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

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