On the Identification of Machine Settings for Gear Surface Topography Corrections

2011 ◽  
Author(s):  
Marco Gabiccini ◽  
Alessio Artoni ◽  
Massimo Guiggiani
Keyword(s):  
Iterative Methods ◽  
Gear Tooth ◽  
Nonlinear Least Squares ◽  
Tooth Surface ◽  
Generation Process ◽  
Sensitivity Matrix ◽  
One Step ◽  
Reliability And Robustness ◽  
Step Control

In this paper we set out to investigate the performances of some of the algorithms proposed in the gear literature for identifying the machine-settings required to obtain predesigned gear tooth surface topographies, or needed to compensate for flank form deviations of real teeth. For the ease of comparison, the problem is formulated as a nonlinear least-squares minimization, and the most widely employed algorithms are derived as particular cases. The algorithms included in the analysis are: (i) one-step methods; (ii) iterative methods; (iii) iterative methods with step control. The performance index is devised in their ability of returning practical solutions in the presence of: (i) strong model nonlinearities, (ii) ill-conditioning of the sensitivity matrix, (iii) demanding topographic shapes purposely selected. Instrumental here is an original classification of topographic modifications as either “simple” or “complex”, based on the SVD analysis of the sensitivity matrix. On the basis of the numerical tests documented, iterative techniques with step control seem the most convenient, due to reliability and robustness of the solutions produced. The generation process here considered is face-milling of hypoid gears, even though the methodology is general enough to cope with any gear cutting method requiring only some minor technical changes.

On the Identification of Machine Settings for Gear Surface Topography Corrections (DETC2011-47727)

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Marco Gabiccini ◽  
Alessio Artoni ◽  
Massimo Guiggiani
Keyword(s):  
Iterative Methods ◽  
Gear Tooth ◽  
Nonlinear Least Squares ◽  
Tooth Surface ◽  
Generation Process ◽  
Sensitivity Matrix ◽  
Special Cases ◽  
Reliability And Robustness ◽  
Value Decomposition ◽  
Step Control

In this paper, we set out to investigate the performances of some algorithms proposed in the gear literature for identifying the machine-tool settings required to obtain predesigned gear tooth surface topographies, or needed to compensate for flank form deviations of real teeth. For ease of comparison, the problem is formulated as a nonlinear least squares problem, and the most widely employed algorithms are derived as special cases. The algorithms included in the analysis are (i) one-step methods, (ii) iterative methods, and (iii) iterative methods with step control. The performance index is devised in their ability of returning practical solutions in the presence of (i) strong model nonlinearities, (ii) ill-conditioning of the sensitivity matrix, and (iii) demanding topographic shapes. Instrumental here is an original classification of topographic modifications as either “simple” or “complex,” based on the singular value decomposition (SVD) analysis of the sensitivity matrix. Some selected numerical examples demonstrate that iterative techniques with step control are the most convenient in terms of reliability and robustness of the obtained solutions. The generation process considered here is face-milling of hypoid gears, although the methodology is general enough to cope with any gear cutting/grinding method.

Revisiting Plane-Generated Gear Tooth Surfaces: A Novel Design Perspective

2015 ◽  
Author(s):  
Alessio Artoni ◽  
Massimo Guiggiani
Keyword(s):  
Gear Tooth ◽  
Transmission Function ◽  
Tooth Surface ◽  
Generation Process ◽  
Noise Emission ◽  
Helical Gears ◽  
Beveloid Gears ◽  
Tooth Surfaces ◽  
Involute Gears ◽  
The One

The teeth of ordinary spur and helical gears are generated by a (virtual) rack provided with planar generating surfaces. The resulting tooth surface shapes are a circle-involute cylinder in the case of spur gears, and a circle-involute helicoid for helical gears. Advantages associated with involute geometry are well known: in particular, the motion transmission function is insensitive to center distance variations, and contact lines (or points, when a corrective surface mismatch is applied) evolve along a fixed plane of action, thereby reducing vibrations and noise emission. As a result, involute gears are easier to manufacture and assemble than non-involute gears, and silent to operate. A peculiarity of their generation process is that the motion of the generating planar surface, seen from the fixed space, is a rectilinear translation (while the gear blank is rotated about a fixed axis): the component of such translation that is orthogonal to the generating plane is the one that ultimately dictates the shape of the generated, envelope surface. Starting from this basic fact, we set out to investigate this type of generation-by-envelope process and to profitably use it to explore new potential design layouts. In particular, with some similarity to the basic principles underlying conical involute (or Beveloid) gears, but within a broader scope, we propose a generalization of these concepts to the case of involute surfaces for motion transmission between skew axes (and intersecting axes as a special case). Analytical derivations demonstrate the theoretical possibility of involute profiles transmitting motion between skew axes through line contact and, perihaps more importantly, they lead to apparently novel geometric designs featuring insensitivity of transmission ratio to all misalignments within relatively large limits. The theoretical developments are confirmed by various numerical examples.

Simulation Analysis of Spiral Bevel Gear Tooth Surface Generation Based on VERICUT

2013 ◽  
Vol 273 ◽  
pp. 175-179
Author(s):  
Zhao Bin Hong ◽  
Zhao Jun Yang ◽  
Bai Chao Wang ◽  
Xue Cheng Zhang
Keyword(s):  
Simulation Analysis ◽  
Gear Tooth ◽  
Bevel Gear ◽  
Surface Generation ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Generation Process ◽  
Gear Cutting ◽  
Nc Program ◽  
Spiral Bevel

Based on the deeply study on VERICUT, the generating process of spiral bevel gear surface is simulated in this paper. A certain surface generation process of spiral bevel gear is analyzed as an example. First the 3D solid models of gear cutting machine, cutter and gear blank are built. Then the NC program is compiled based on the generating principle of tooth surface. Finally the surface generating movement of spiral bevel gear is realized through selecting control system and importing the NC program. It can be indicated that the simulation analysis of surface generation can optimize the cutting movement and provide theoretical basis for the movement analysis of gear cutting.

Geometry of Tooth Profile and Fillet of Face-Hobbed Spiral Bevel Gears

2007 ◽  
Author(s):  
Yi Zhang ◽  
Zhi Wu
Keyword(s):  
Gear Tooth ◽  
Mathematical Formulation ◽  
Tooth Profile ◽  
Tooth Surface ◽  
Generation Process ◽  
Tooth Contact Analysis ◽  
Spiral Bevel Gears ◽  
Bevel Gears ◽  
Spiral Bevel

The determination of the geometry for the whole tooth profile, including the meshing profile and the tooth fillet, is important for tooth contact analysis and the mesh generation for FEM analysis of gear pairs. This paper presents a systematic approach for the determination of the complete tooth geometry of face-hobbed hypoid and spiral bevel gears. The detailed mathematical formulation for the generation of gear tooth surface and the equations for the tooth surface coordinates are provided in the paper. The surface coordinates and normal vectors are calculated at grid points selected based on the gear blank dimension. Using the machine tool settings as input, the computer model simulating the gear generation process precisely calculates the tooth geometry parameters on the selected grid. A numerical example is included in the paper to illustrate the presented approach.

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.

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

scholarly journals Parallel One-Step Control of Parametrised Boolean Networks

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 560 ◽  
Author(s):  
Luboš Brim ◽  
Samuel Pastva ◽  
David Šafránek ◽  
Eva Šmijáková
Keyword(s):  
Control Problem ◽  
Boolean Networks ◽  
Actual State ◽  
Worst Case ◽  
Symbolic Data ◽  
One Step ◽  
The One ◽  
Step Control ◽  
Symbolic Algorithm ◽  
Naive Approach

Boolean network (BN) is a simple model widely used to study complex dynamic behaviour of biological systems. Nonetheless, it might be difficult to gather enough data to precisely capture the behavior of a biological system into a set of Boolean functions. These issues can be dealt with to some extent using parametrised Boolean networks (ParBNs), as this model allows leaving some update functions unspecified. In our work, we attack the control problem for ParBNs with asynchronous semantics. While there is an extensive work on controlling BNs without parameters, the problem of control for ParBNs has not been in fact addressed yet. The goal of control is to ensure the stabilisation of a system in a given state using as few interventions as possible. There are many ways to control BN dynamics. Here, we consider the one-step approach in which the system is instantaneously perturbed out of its actual state. A naïve approach to handle control of ParBNs is using parameter scan and solve the control problem for each parameter valuation separately using known techniques for non-parametrised BNs. This approach is however highly inefficient as the parameter space of ParBNs grows doubly exponentially in the worst case. We propose a novel semi-symbolic algorithm for the one-step control problem of ParBNs, that builds on symbolic data structures to avoid scanning individual parameters. We evaluate the performance of our approach on real biological models.

Investigation of Noise Features of Planetary Gear Trains Based on Human Aural Characteristic

2017 ◽  
Author(s):  
Masao Nakagawa ◽  
Dai Nishida ◽  
Deepak Sah ◽  
Toshiki Hirogaki ◽  
Eiichi Aoyama
Keyword(s):  
Gear Tooth ◽  
Structural Complexity ◽  
Planetary Gear ◽  
Transmission Error ◽  
Sound Level ◽  
Tooth Surface ◽  
Surface Error ◽  
Planetary Gear Trains ◽  
Tooth Stiffness ◽  
Gear Trains

Planetary gear trains (PGTs) are widely used in various machines owing to their many advantages. However, they suffer from problems of noise and vibration due to the structural complexity and giving rise to substantial noise, vibration, and harshness with respect to both structures and human users. In this report, the sound level from PGTs is measured in an anechoic chamber based on human aural characteristic, and basic features of sound are investigated. Gear noise is generated by the vibration force due to varying gear tooth stiffness and the vibration force due to tooth surface error, or transmission error (TE). Dynamic TE is considered to be increased because of internal and external meshing. The vibration force due to tooth surface error can be ignored owing to almost perfect tooth surface. A vibration force due to varying tooth stiffness could be a major factor.

New Geometry of Generating Spiral Bevel Gear

2007 ◽  
Author(s):  
Kaihong Zhou ◽  
Jinyuan Tang ◽  
Tao Zeng
Keyword(s):  
Gear Tooth ◽  
Previous Method ◽  
Bevel Gear ◽  
Tooth Surface ◽  
Spiral Bevel Gear ◽  
Surface Geometry ◽  
Numerical Examples ◽  
Cone Surface ◽  
Spiral Bevel ◽  
Involute Curve

New geometry of generating spiral bevel gear is proposed. The key idea of the new proposed geometry is that the gear tooth surface geometry can be investigated in a developed curved surface based on the planar engagement principle. It is proved that the profile curve on the back of generating cone surface is a conical involute curve. The equations of generated gear tooth surface are achieved by the conical involute curve sweeping along the tooth trace of gear. The obtained equations are explicit and independent of the machine-tool settings. This differs from previous studies. The developed theory is illustrated with numerical examples to compare with the previous method, the comparison approves that the method is possible in this way. The new method indicates that there are new solutions to the design the production of spiral bevel gear.

Revisiting Generation and Meshing Properties of Beveloid Gears

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Alessio Artoni ◽  
Massimo Guiggiani
Keyword(s):  
Tooth Surface ◽  
Generation Process ◽  
Spur Gears ◽  
Helical Gears ◽  
Additional Degree ◽  
Beveloid Gears ◽  
Fixed Space ◽  
Fixed Axis ◽  
The One ◽  
Special Case

The teeth of ordinary spur and helical gears are generated by a (virtual) rack provided with planar generating surfaces. The resulting tooth surface shapes are a circle-involute cylinder in the case of spur gears, and a circle-involute helicoid for helical gears. Advantages associated with involute geometry are well known. Beveloid gears are often regarded as a generalization of involute cylindrical gears involving one additional degree-of-freedom, in that the midplane of their (virtual) generating rack is inclined with respect to the axis of the gear being generated. A peculiarity of their generation process is that the motion of the generating planar surface, seen from the fixed space, is a rectilinear translation (while the gear blank is rotated about a fixed axis); the component of such translation that is orthogonal to the generating plane is the one that ultimately dictates the shape of the generated, envelope surface. Starting from this basic fact, we set out to revisit this type of generation-by-envelope process and to profitably use it to explore peculiar design layouts, in particular for the case of motion transmission between skew axes (and intersecting axes as a special case). Analytical derivations demonstrate the possibility of involute helicoid profiles (beveloids) transmitting motion between skew axes through line contact and, perhaps more importantly, they lead to the derivation of designs featuring insensitivity of the transmission ratio to all misalignments within relatively large limits. The theoretical developments are confirmed by various numerical examples.

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