Simulation of dressing process for continuous generating gear grinding

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12th 2018, 12thCIRP CIRPConference ConferenceononIntelligent IntelligentComputation ComputationininManufacturing ManufacturingEngineering, Engineering,18-20 CIRPJuly ICME '18 12th CIRP Conference on Intelligent Computation in Manufacturing Engineering, CIRP ICME '18 Gulf of Naples, Italy 28th CIRP Design Conference, May 2018, Nantes, France gear grinding Simulation of dressing process for continuous generating

Simulation of dressing process for continuous generating gear grinding a, a Jonas Böttger *, Simon Kimme , Welf-Guntram Drossela,barchitecture of A new methodology to analyze the functional and physical a, a Jonas Böttger *, Simon Kimme , Welf-Guntram Drossela,b Institute for Machine Tools and Production IWP, Chemnitz University of Technology, Reichenhainer Str. 70, 09126 Chemnitz, Germany existing products for anProcesses assembly oriented product family identification a

Fraunhofer Institute for Machine Tools and Forming Technology IWU, Reichenhainer Str. 88, 09126 Chemnitz, Institute for Machine Tools and Production Processes IWP, Chemnitz University of Technology, Reichenhainer Str. 70, 09126Germany Chemnitz, Germany b Fraunhofer Institute for Machine Tools and Forming Technology IWU, Reichenhainer Str. 88, 09126 Chemnitz, Germany * Corresponding author. Tel.: +49-371-531-32029; fax: +49-371-531-8-32029. E-mail address: [email protected] * Corresponding author. Tel.: +49-371-531-32029; fax: +49-371-531-8-32029. E-mail address: [email protected] École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France a

b

Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat

Abstract

*Abstract Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]

Requirements for process accuracy in generating gear grinding are particularly high when it comes down to low noise emission of gears. Coincidentally, dressing crucialininfluence on gear the grinding and the high flankwhen topography gear wheels, whereas the number of Requirements for processhas accuracy generating grinding process are particularly it comesof down to low noise emission of gears. investigations thecrucial dressing process on is fairly small. Inprocess order toand examine the topography impact of dressing on whereas the flankthe surface of the Coincidentally,focusing dressingonhas influence the grinding the flank of gearerrors wheels, number of Abstract grinding worm,focusing this workondeals with the development of a kinematic cuttingtomodel for dressing using truing rolls. Apart required simulation investigations the dressing process is fairly small. In order examine the impact of dressing errors onfrom the flank surface of the accuracy, the influence of deals selected errors andofcorresponding process parameters is discussed. grinding worm, this work withdressing the development a kinematic cutting model for dressing using truing rolls. Apart from required simulation In today’s business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of © 2018 The by dressing Elsevier errors B.V. and corresponding process parameters is discussed. accuracy, theAuthors. influencePublished of selected agile and reconfigurable production systems emerged to cope with various products and product families. To design and optimize production Peer-review under responsibility ofElsevier the scientific © 2019 2018 The The Authors. Authors. Published by by Elsevier B.V. committee of the 12th CIRP Conference on Intelligent Computation in Manufacturing © Published B.V. systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to Engineering. Peer-review under underresponsibility responsibilityofofthe thescientific scientific committee the 12th CIRP Conference on Intelligent Computation in Manufacturing Peer-review committee of of the 12th CIRP Conference Intelligent Computation in Manufacturing analyze a product or one product family on the physical level. Different product families,onhowever, may differ largely in terms of theEngineering. number and Engineering. nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production Keywords: Dressing; Grinding; Gear; Simulation system. A new methodology is proposed to analyze existing products in view of their functional and physical architecture. The aim is to cluster Keywords: Dressing; Grinding; Gear; Simulation these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and a1. functional analysis is performed. Moreover, a hybrid functional and physical architecture (HyFPAG) is the output which depicts and the Introduction and state of the art kinematics [4,5].graph Türich took Looman`s analytic approach similarity between product families by providing design support to both, production system planners and product designers. An illustrative 1. Introduction and state of the art kinematics [4,5]. Türich to took Looman`s analytic approach and extended it by a function describe profile corrections [2]. The example of a nail-clipper is used to explain the proposed methodology. An industrial case study on two product families of steering columns of extended it by a function toto describe profile corrections [2]. The Generating gear grinding is a widespread finishing process mathematic fundamentals get the geometry of the dressing thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. Generating gear grindingby is Elsevier aconstraints widespread finishing process mathematic fundamentals get the geometry of the dressing gear wheels. Especially roll will be explained here to in short. ©of2017 The Authors. Publishedhigh B.V.in series production of gear wheels. Especially high constraints in series production roll will be explained here in short. in the automotive industry are reasons for a closer look at There are numerous possibilities to simulate cutting Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018.

in the automotive industry are reasons a closerare look at influences on machining quality. Dressingfor processes often

influences quality. Dressing processessurface are often assumed Assembly; toonbemachining ideal with regard to identification the generated of Keywords: Design method; Family

assumed tools to beand ideal with topublications the generated surface of grinding there are regard very few on it [1]. This grinding tools and there very few to publications This investigation shows anareapproach simulate on theit [1]. dressing investigation shows rolls. an approach the dressing process using truing Therefore to thesimulate exact geometry has to 1.process using truing rolls. the exact geometry hasthe to beIntroduction determined. Since thisTherefore is a macroscopic analysis of be determined. Since is a macroscopic analysis of the generated surface, all this microscopic interactions of utilized Due to the fast development insurface the properties domain of generated surface, alltheir microscopic interactions of utilized dressing diamonds and influence on will communication and an ongoing trend of digitization and dressing diamonds and their influence on surface properties will be neglected. digitalization, manufacturing be The neglected. dressing process usingenterprises a truing rollare is,facing strictlyimportant speaking, challenges in today’s market environments: a dressing continuing The dressing process usingofa gears truing where roll is, the strictly speaking, similar to profile grinding roll tendency towards ofofproduct development times and similar to profilereduction grindingwheel. gears theworm dressing rolla equals the profile grinding Thewhere grinding equals shortened In addition, there isworm ana increasing equals theproduct profile grinding wheel. Theteeth grinding equals high helixa gear wheel with alifecycles. small number of 𝑧𝑧0 and demand customization, being atofthe same a global and approach ainhigh helix gear with a small[3]number teeth 𝑧𝑧0 time anglewheel 𝛽𝛽of[2]. Looman showed an analytic to competition with competitors all over the world. This trend, angle 𝛽𝛽 [2]. showed an Later analytic approach to determine the Looman geometry[3] of the dresser. investigations of which is the development macro micro determine the geometry of on the numerical dresser.from Later investigations of Weck andinducing Escher focused methods totocalculate markets, results in focused diminished lot asizes duecutting to augmenting Weck and Escher on with numerical methods to operation calculate the searched profile geometry reverse product varieties (high-volume to low-volume production) [1]. thethe searched geometry with a reverse cutting of tool byprofile simulating the workpiece geometry andoperation process To cope with this augmenting variety as well as to be able to of the tool by simulating the workpiece geometry and process identify possible optimization potentials in the existing 2212-8271 ©system, 2017 TheitAuthors. Published Elsevier B.V. knowledge production is important tobyhave a precise

There are numerous possibilities cutting processes which are lucidly presentedto insimulate [6]. Tool and processes which are lucidly presented in [6]. and workpiece can be described by voxel [7], dexel [8], Tool polyhedra workpiece be[9]. described voxel [7], dexel solid [8], polyhedra and contourcan lines The so by called constructive geometry and contour lines [9]. The sothe called constructive solid geometry (CSG) method [10] pursues approach to use mathematically (CSG) method [10] pursues the approach to use mathematically describable bodies to represent the constant shape of a of the product range Itto and characteristics manufactured describable bodies theforconstant shapeand/or ofanda respective element. isrepresent often used tool geometries assembled in this system. In this context, the main challenge in respective element. It is often used for tool Ingeometries might reduce required computing resources. this case, and the modelling and analysis is now not only to cope with single might reduce required computing Inorder this case, the dressing geometry will be describedresources. with second surfaces. products, a limited product range existing product families, dressing geometry will be described second order surfaces. This method, in combination withora with dexel based description of but also to be able to analyze and to acompare products to define This method, in combination with dexel based description of the worm geometry, is shown using algorithms and coordinate new product families. can observed that classical existing the worm geometry, isItshown using algorithms andapplications coordinate transformations from 3D be Computer Graphic product regrouped inisfunction of clients or features. transformations from 3D Computer applications [11,12].families Before are the simulation carried Graphic out, it is important to However, assembly oriented product families are hardly toissues. find. [11,12]. Before the simulation carried out, it is important to consider the attainable accuracyisregarding discretization On thethe product family level, products differ mainly two consider attainable accuracy regarding discretization issues. Eventually, the simulation results with different processin errors main characteristics: (i) the results numberwith of components and (ii) the Eventually, the simulation different process errors will be demonstrated. type of components (e.g. mechanical, electrical, electronical). will be demonstrated. Nomenclature Classical methodologies considering mainly single products Nomenclature or solitary, already existing product families analyze the Center distance between worm and dressing rollwhich 𝐴𝐴 product structure on a physical level (components level) Center distance between worm and dressing roll 𝐴𝐴 causes difficulties regarding an efficient definition and comparison of different product families. Addressing this

Peer-review the scientific committee 2212-8271 ©under 2017responsibility The Authors. of Published by Elsevier B.V.of the 11th CIRP Conference on Intelligent Computation in Manufacturing Engineering. Peer-review under responsibility of the scientific committee of the 11th CIRP Conference on Intelligent Computation in Manufacturing Engineering. 2212-8271©©2017 2019The The Authors. Published by Elsevier 2212-8271 Authors. Published by Elsevier B.V. B.V. Peer-reviewunder underresponsibility responsibility scientific committee of the CIRP Conference on 2018. Intelligent Computation in Manufacturing Engineering. Peer-review of of thethe scientific committee of the 28th12th CIRP Design Conference 10.1016/j.procir.2019.02.067



𝐴𝐴 𝑎𝑎𝑆𝑆 ⃗⃗⃗⃗ 𝑎𝑎, 𝑏𝑏 … 𝑙𝑙 𝑏𝑏 𝐷𝐷𝑎𝑎 𝑑𝑑𝑎𝑎 𝑑𝑑𝑏𝑏 𝑑𝑑𝑓𝑓 𝑑𝑑0 𝑒𝑒0 , ⃗⃗⃗ ⃗⃗⃗ 𝑒𝑒1 𝐸𝐸⃗ 𝑓𝑓 𝑓𝑓𝑀𝑀 𝑖𝑖𝐵𝐵 𝑚𝑚𝑛𝑛0 𝑛𝑛⃗ 𝑛𝑛𝑆𝑆 ⃗⃗⃗⃗ 𝑛𝑛𝑑𝑑 𝑛𝑛0 𝑃𝑃1…3 𝑞𝑞𝑛𝑛 𝑄𝑄 𝑅𝑅 𝑅𝑅⃗ 𝑟𝑟 𝑟𝑟𝑎𝑎 𝑟𝑟𝑏𝑏 𝑟𝑟𝑓𝑓 𝑡𝑡 𝑣𝑣𝑏𝑏0 𝑣𝑣𝑑𝑑 𝑣𝑣𝑦𝑦0 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 𝑌𝑌𝐴𝐴 𝑧𝑧0 𝛼𝛼𝑛𝑛0 𝛽𝛽 𝛽𝛽𝑏𝑏 γ0 Δ Δ𝑠𝑠 𝜀𝜀 𝜉𝜉, 𝜏𝜏, 𝜁𝜁 𝜑𝜑 ψ0

Jonas Böttger et al. / Procedia CIRP 79 (2019) 280–285 J. Böttger et al./ Procedia CIRP 00 (2018) 000–000

Amplitude of vibration Vector along axis of profiling roll Coefficients of quadric function Width of gear Outer diameter of dressing roll Head diameter Base circle diameter Foot diameter Pitch diameter of worm Position and unit vector of a dexel Vector to each point of gear flank surface Frequency of vibration Correction function of generated gear surface Increment of B-axis rotation Normal module of worm Normal vector on gear surface Normal vector of axis section of profiling roll Rotational speed of dresser Rotational speed of worm Coefficients of a quadratic function Speed ratio of dresser and worm Solution set of second order surface Radius of dresser Ray equation Radius of worm Head radius Radius of gear base circle Foot radius Vector length of allowance Circumferential speed of worm Circumferential speed of dresser Linear movement of y-axis (worm) Spatial coordinates Center distance component in Y-direction Number of teeth of the worm Normal pressure angle of worm Helix angle of gear at pitch circle Helix angle of gear at base circle Pitch angle of worm Deviation from ideal geometry Absolute deviation from ideal geometry Tilting angle of position- and axial runout error Coordinates to describe gear flank surface Inclination angle of dresser towards worm Evaluation angle on worm surface

2. Determination of dressing roll geometry In contradiction to a spur gearing the contact path between roller profile and the gear tooth is a three-dimensional bended line which does not run along a specific section [3]. An example of this contact line is represented in Fig. 1. If the gear wheel is located with the center distance 𝐴𝐴, equation (2), in y-direction towards the origin of coordinates and with its rotational axis

281

Fig. 1. Contact path between profiling roll and gear flank [3]

parallel to the z-axis, the equation of the worm can be written as follows: 𝑓𝑓𝑀𝑀 (𝜉𝜉) ) ∙ cos 𝜁𝜁 𝑟𝑟𝑏𝑏 𝑓𝑓𝑀𝑀 (𝜉𝜉) 𝐸𝐸⃗ (𝜉𝜉, 𝜏𝜏) = 𝑟𝑟𝑏𝑏 ∙ cos 𝜁𝜁 + (𝜉𝜉 + ) ∙ sin 𝜁𝜁 − 𝑌𝑌𝐴𝐴 𝑟𝑟𝑏𝑏 [ 𝜏𝜏 ∙ cot 𝛽𝛽𝑏𝑏 ] sin 𝜁𝜁 − (𝜉𝜉 +

𝐴𝐴 =

𝐷𝐷𝑎𝑎 + 𝑑𝑑𝑓𝑓 = 𝑟𝑟𝑏𝑏 ∙ 𝑌𝑌𝐴𝐴 2

(1)

(2)

Equation (1) describes the vector from the origin to each point of the surface of the worm, if the surface parameters 𝜉𝜉 and 𝜏𝜏 are chosen in the interval of the inequations (3) and (4). 𝜁𝜁 describes the angle to the starting point of the involute on the base circle 𝑑𝑑𝑏𝑏 , which is half the tooth space angle. √( −

2

𝑑𝑑𝑓𝑓 𝑑𝑑𝑎𝑎 2 ) − 1 ≤ 𝜉𝜉 ≤ √( ) − 1 𝑑𝑑𝑏𝑏 𝑑𝑑𝑏𝑏

𝑏𝑏 ∙ tan 𝛽𝛽𝑏𝑏 𝑏𝑏 ∙ tan 𝛽𝛽𝑏𝑏 ≤ 𝜏𝜏 ≤ 𝑑𝑑𝑏𝑏 𝑑𝑑𝑏𝑏

(3)

(4)

The normal vector 𝑛𝑛⃗ of every point on the flank surface can be calculated by the cross product of the partial derivatives of the surface equation (1) along two directions 𝜉𝜉 and 𝜏𝜏. 𝑛𝑛⃗ =

𝜕𝜕𝐸𝐸⃗ (𝜉𝜉, 𝜏𝜏) 𝜕𝜕𝐸𝐸⃗ (𝜉𝜉, 𝜏𝜏) × 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

(5)

The dressing roll is inclined towards the worm with the angle 𝜑𝜑, which is commonly chosen as the helix angle at the pitch circle (𝜑𝜑 = 𝛽𝛽). A contact point is defined at each 𝜉𝜉 and 𝜏𝜏 where 𝑎𝑎𝑆𝑆 the surface vector 𝐸𝐸⃗ , the normal vector 𝑛𝑛⃗ and the axis vector ⃗⃗⃗⃗ of the profile roll describe a common surface (see Fig. 1). This mathematical context can be expressed by:

Jonas Böttger et al. / Procedia CIRP 79 (2019) 280–285 J. Böttger et al./ Procedia CIRP 00 (2018) 000–000

282

−cos 𝜑𝜑 𝑎𝑎𝑆𝑆 = ( 0 ) ⃗⃗⃗⃗ − sin 𝜑𝜑

(6) (7)

𝑛𝑛𝑆𝑆 = 𝑛𝑛⃗ × ⃗⃗⃗⃗ ⃗⃗⃗⃗ 𝑎𝑎𝑆𝑆

(8)

0 = 𝐸𝐸⃗ (𝜉𝜉, 𝜏𝜏) ∙ ⃗⃗⃗⃗ 𝑛𝑛𝑆𝑆

Equation (8) written out in full can be solved numerically within the respective borders of 𝜉𝜉 and 𝜏𝜏. The calculated line of contact can be transformed and projected into the axial section of the dressing roll and therefore provides the basis for further calculations. 3. Development of simulation The principal elements of the simulation model are described in the following. The chosen data for worm and dresser in Table 1 comply with a typical automotive gearing. Table 1. Basic data of worm and dresser Data name

symbol / unit

Value

Number of teeth

𝑧𝑧0 / −

5

𝛼𝛼𝑛𝑛0 / 𝑑𝑑𝑑𝑑𝑑𝑑

16

𝐷𝐷𝑎𝑎 / 𝑚𝑚𝑚𝑚

120

Normal module Normal pressure angle Pitch diameter Head diameter of dresser

3.1. Modeling of dresser

𝑚𝑚𝑛𝑛0 / 𝑚𝑚𝑚𝑚

2

𝑑𝑑0 / 𝑚𝑚𝑚𝑚

200

𝑄𝑄 =

2

2

+ 𝑏𝑏𝑦𝑦 + 𝑐𝑐𝑧𝑧 + 𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑒𝑒𝑒𝑒𝑒𝑒 + 𝑓𝑓𝑓𝑓𝑓𝑓 + 𝑔𝑔𝑔𝑔 + ℎ𝑦𝑦 + 𝑘𝑘𝑘𝑘 + 𝑙𝑙 = 0 | (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) ∈ ℝ3 }

refers to the example in Fig. 2.a. Generally can be said, that with a stronger bending of the contact line and thus the dressing profile, the deviation increases and the representation of dresser with a quadric function becomes less accurate. 3.2. Modeling of worm and cutting operation

Since the surface of the dresser and its shape are considered constant and ideal, it could be appropriate to describe it with a mathematic function rather than using extensive numerical models. Second order surfaces (quadrics) are taken into account which are generally given by the equation: {𝑎𝑎𝑥𝑥 2

Fig. 2. Deviation between ideal geometry and quadric over radius 𝑅𝑅 of dressing roll (a), absolute deviation ∆𝑠𝑠 with varying 𝑑𝑑0 , 𝑧𝑧0 , 𝑚𝑚𝑛𝑛0 , 𝐷𝐷𝑎𝑎 , 𝛼𝛼𝑛𝑛0 (b-f)

(9)

Due to the restrictions that the dressing roll has rotational symmetric geometry with z as the rotational axis, the parameters 𝑑𝑑, 𝑒𝑒, 𝑓𝑓, 𝑔𝑔 and ℎ equal 0 and 𝑎𝑎 = 𝑏𝑏 . The axial section of this quadric describes a hyperbolic function which points should ideally correspond with the calculated points of the dressing roll. For the reason that not the whole point set lies on the curve, the function parameters can be calculated with an optimization algorithm. It has to be ensured that deviations from the ideal geometry can be neglected. Fig. 2a shows the deviation of the calculated quadric function from the ideal geometry of the dressing roll in the axial section from the calculated inner to outer radius of the dresser 𝑅𝑅. In this case, the sum of minimum and maximum deviation ∆𝑠𝑠 is not higher than 0.004𝜇𝜇𝜇𝜇 which is sufficiently accurate. The geometry of the contact line between dressing roll and worm and therefore the deviation between ideal and quadric geometry depends on the input parameters of the gearing which are 𝑑𝑑0 , 𝑧𝑧0 , 𝑚𝑚𝑛𝑛0 , 𝐷𝐷𝑎𝑎 , 𝛼𝛼𝑛𝑛0 . The influence of each of these parameters on the accuracy are shown in Fig. 2.b-f., where the orange point

The geometry of the worm can be described with equation (1) but there are also other forms [13] which will not be explained here any further. The surface has a discrete number of points. For later evaluation it is reasonable to describe the worm flank with points in direction of the axis section and with points in peripheral direction. Each dexel is represented by the position vector on the surface 𝑒𝑒⃗⃗⃗0 and a unit vector 𝑒𝑒⃗⃗⃗1 from the reference geometry in normal direction with the length 𝑡𝑡 of the initial allowance. 𝑅𝑅⃗ = ⃗⃗⃗ 𝑒𝑒0 + 𝑡𝑡 ∙ ⃗⃗⃗ 𝑒𝑒1

(10)

𝑃𝑃1 𝑡𝑡 2 + 𝑃𝑃2 𝑡𝑡 + 𝑃𝑃3 = 0

(11)

The substitution of the coordinates of 𝑄𝑄 (9) with the components of 𝑅𝑅⃗ (10) results in a quadratic equation (11) which can be efficiently solved with an algorithm taken from [12].

The residual length of the dexel describes the deviation on the surface of the worm. In dependence of the error pattern on the generated surface, it may be sufficient to simulate a small section of a thread in order to save simulation time. Conclusions to the remaining geometry on the thread can be drawn accordingly. 3.3. Kinematics The simulation of the process has to cover the significant movement of the machine axes which are illustrated in Fig. 3. The kinematics of the dressing process can be compared with a thread cutting process [2]. The B-axis as the master with a circumferential speed 𝑣𝑣𝑏𝑏0 has to be coupled with the Y-axis (𝑣𝑣y0) as the slave with a certain relation to generate the pitch of



Jonas Böttger et al. / Procedia CIRP 79 (2019) 280–285 J. Böttger et al./ Procedia CIRP 00 (2018) 000–000

283

Fig. 4. Simulation discretization of geometry and process

Fig. 3. Kinematics of dressing process [2]

the worm. The worm is placed on the y-slide which is inclined with an angle 𝜑𝜑 towards the dresser axis. For this, typically the center pitch angle γ0 is chosen. The dresser has a specified circumferential speed 𝑣𝑣𝑑𝑑 which has to match with the speed of the worm for technological reasons, primarily to maintain a certain effective surface roughness on the grinding wheel. However, this axis has no direct contribution for the generation of the gear and therefore no coupling relation with other axes of the machine. The kinematic chain of each position of the worm towards dressing roll can be realized with a transformation of homogeneous coordinates [11]. For the reason that the shape of the dressing roll is not represented with a set of separate transformable points but with a mathematic function, the cutting operation is not calculated in a global coordinate system, but in the local coordinate system fixed in the dressing roll. Thus, the speed ratio of dresser and worm 𝑞𝑞𝑛𝑛 (12) is relevant for the subsequent examination of the axial runout error of the dresser. 𝑞𝑞𝑛𝑛 =

𝑛𝑛𝑑𝑑 𝑛𝑛0

(12)

3.4. Discretization The discretization of the simulation has a substantial influence on the achievable evaluation accuracy. However, a higher resolution of the process can lead to an unacceptable high calculation time. The simulation can only calculate the removal at discrete points in time, whereas the dressing process has a continuous removal rate. The shape of the generated surface is the result of the envelope of discrete cutting operations. Therefore it contains quasi-feeding marks. Fig. 4 shows one tooth of the worm with a certain allowance, the generated cutting contour and the calculated line of contact where the dressing roll cuts at full depth into the surface. A convergence analysis (Fig. 5b) shows the declining deviations from the quasi feeding marks (Fig. 5a) in the simulation with decreasing rotation steps of the B-axis. In order to achieve a waviness of less than 0.05𝜇𝜇𝜇𝜇 an increment of 𝑖𝑖𝐵𝐵 = 0.1𝑑𝑑𝑑𝑑𝑑𝑑 is sufficiently accurate. The given rotational speed of the worm is

Fig. 5. Waviness on surface with 𝑖𝑖𝐵𝐵 = 0.4𝑑𝑑𝑑𝑑𝑑𝑑 (a), attainable accuracy with varying 𝑖𝑖𝐵𝐵 (b)

𝑛𝑛0 = 50 𝑈𝑈/𝑚𝑚𝑚𝑚𝑚𝑚. Both, the increment of the B-axis and the surface mesh, which is split into points in circumferential direction and in radial direction, have to be chosen according to the simulated error pattern. In order to perform a vibrational movement (𝑓𝑓 = 300𝐻𝐻𝐻𝐻) properly, with ten times sampling rate, a maximum time step of 0.3𝑚𝑚𝑚𝑚 and therefore an increment of 𝑖𝑖𝐵𝐵 = 0.14𝑑𝑑𝑑𝑑𝑑𝑑 is necessary. The number of circumferential and radial points has to be accordingly high to show the pronounced error patterns on the surface. 4. Simulation results There are different causes which lead to errors in the dressing process. Here, a certain variety of errors and its influence on the generated surface is shown, such as the static misalignment of worm and dresser, vibrations in x and y direction and an axial runout of the dresser axle. 4.1. Position error The angular error of the dresser towards the worm results in differently high material removal of left and right of a tooth. In Fig. 6, the simulated worm surface of both flanks is shown from 0 to 5 𝑑𝑑𝑑𝑑𝑑𝑑 and from indicated head radius (in the middle) to foot radius (outwards). The angle of the misalignment is 𝜀𝜀 = 4.67 ∙ 10−3 𝑑𝑑𝑑𝑑𝑑𝑑. There is a positive deviation on the left and a negative deviation on the right flank. Moreover, a parallel deviation from foot to head on both flanks can be seen in the sectional view below. In circumferential direction, the deviation stays the same.

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Fig. 6. Surface worm deviation (left and right flank) caused by a tilting error between worm and dresser

4.2. Vibration The results of a simulated vibration in x- and in y-direction with a frequency of 𝑓𝑓 = 300 𝐻𝐻𝐻𝐻 and an amplitude of 𝐴𝐴 = 0.5𝜇𝜇𝜇𝜇 are shown in Fig. 7 and Fig. 8. within a range 𝜓𝜓0 of 0 to 10𝑑𝑑𝑑𝑑𝑑𝑑 on the worm circumference and from foot radius 𝑟𝑟𝑓𝑓 to head radius 𝑟𝑟𝑎𝑎 on the left and the right flank. The generated waves are lined up one after another according to the contact path on the surface of the worm. In contradiction to the y-axis vibration, the x-axis vibration shows a more distinctive contour of the applied sine wave on the surface and has a smaller maximum and minimum of the deviation (0.14𝜇𝜇𝜇𝜇) than the applied amplitude. The reason is the pressure angle of the worm. Furthermore, the waves on right and left flank are cophasal. Due to the fact that a movement towards the ydirection has a more direct impact on the flank geometry, the wave depth is almost equivalent to the applied amplitude. Therefore, it applies for the y-swing: The more material that is

Fig. 7. Surface worm deviation (left and right flank) caused by x-vibration

Fig. 8. Surface worm deviation (left and right flank) caused by y-vibration

removed from the left flank the less material is removed from the right flank and vice versa. The sections drawn on the left and the right of the diagram also show sharper peaks of the generated surface. This sharp-edged structure with a decreasing peak height becomes more pronounced with higher frequencies of the applied vibration. Fig. 9 shows the structure on the flank for x- and y-vibrations in the range of 150𝐻𝐻𝐻𝐻 to 900𝐻𝐻𝐻𝐻.

4.3. Axial runout

An axial runout is defined as a deflection of the actual axis of rotation with a certain angle towards the rotation axis of the geometry. This can occur due to clamping errors of the dressing roll. The chosen deflection angle is 𝜀𝜀 = 4.67 ∙ 10−3 𝑑𝑑𝑑𝑑𝑑𝑑 which leads to a displacement at the head of the dresser of 0.5𝜇𝜇𝜇𝜇. The results with a common speed ratio of 𝑞𝑞𝑛𝑛 = 100 with 𝑛𝑛0 = 50 𝑈𝑈/𝑚𝑚𝑚𝑚𝑚𝑚 are shown in Fig. 10.

Fig. 9. Wave-structure on the surface with applied vibration (𝐴𝐴 = 0.5𝜇𝜇𝜇𝜇) in x-direction (left) and y-direction (right) with varying frequencies



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[2] [3] [4]

[5] [6] [7] [8] [9] [10] Fig. 10. Flank deviation caused by axial runout of dresser on left (a) and right (b) flank

The surface of the worm shows nearly the same structure as with the above-mentioned vibrations. The period length on the surface of the worm varies according to the speed ratio 𝑞𝑞𝑛𝑛 . Due to the movement of the deflected dressing roll, the difference between crest and trough is slightly higher at foot circle radius than it is at the head radius (see section below). 5. Conclusion and outlook This paper shows the development and the implementation of a simulation of the dressing process for generating gear grinding. The mathematical framework in the beginning shows a way to describe the exact geometry of the dressing roll. The representation of this geometry with second order surfaces provides sufficient accuracy. Based on this, the geometric and kinematic modeling of the process as well as the cutting operation is shown. An adequate observation of the discretization assures the proper evaluation of simulation results. Various selected errors of the dressing process covering misalignment, vibration and axial runout show plausibly pronounced patterns on the generated surface of the worm. Future investigations are necessary to focus on one hand on the comparison of the results with real generated and measured grinding wheel surfaces and on the other hand on the influence of error-prone dressing processes on manufactured gear wheel topographies. Acknowledgements The „Saxon Alliance for Material- and Resource-Efficient Technologies“ (AMARETO) is funded by the European Union (European Regional Development Fund) and the Free State of Saxony. References [1]

Wegener K, Hoffmeister H-W, Karpuschewski B, Kuster F, Hahmann W-C, Rabiey M. Conditioning and monitoring of grinding wheels. CIRP Annals 2011;60(2):757–77.

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