Analysis and simulation of the grinding process. Part I: Generation of the grinding wheel surface

This paper is in three parts describing the analysis and simulation of the grinding process. This first part is concerned with the generation of the wheel surface by single point diamond dressing. In grinding, the grinding wheel has to be dressed periodically to restore wheel form and cutting efficiency. Understanding the process of generating the grinding wheel surface is important for the control of the grinding process. Generation of the wheel surface is simulated as a single diamond dressing process on a computer generated wheel. The wheel is simulated by grains randomly spaced in the wheel volume. The topography of the wheel cutting surface is generated by simulating the action of an ideal dressing tool as it dresses the wheel. The simulation of the wheel topography takes account of the motion of the dressing tool, grain size, grain spacing, grain fracture and grain break-out. The simulated cutting surface is used for further simulations of grinding. The simulation of grinding using the simulated grinding wheel surface is described in Sections 2 and 3 where a comparison is made of results predicted from simulation with results obtained from experiments. By matching simulated and experimental results, it is possible to explain the relative importance of dressing and grinding parameters.     

Analysis and simulation of the grinding process. Part III: Comparison with experiment

A method of simulating dressing and grinding was described in Parts I and II of this three-part series. In Part III, the effects on grinding performance of varying the dressing conditions are simulated and compared with experimental results. The results show that a coarse dressing condition leads to low grinding force and grinding power but a high workpiece surface roughness. The grinding performance of the wheel in the dwell period for “spark-out” is simulated. Simulated and experimental results both show that grinding power in the dwell period decreases following an exponential decay function, however the reduction of surface roughness does not follow an exponential decay.     

Analysis and simulation of the grinding process. Part IV: Effects of wheel wear

A method of simulating dressing and grinding was described in Parts I and II of this paper. In Part IV, the effects of wheel wear and wheel characteristics on grinding performance are simulated and compared with experimental results. The results show that grinding performance is strongly affected by dressing conditions immediately after dressing. As grinding continues, the grinding power, and also the surface roughness, tends to converge towards similar values for all dressing conditions when the same grinding conditions are employed. Results from the simulation show that the influence of wheel wear is affected by the wheel fracture characteristics. The convergence of the grinding behaviour shown in the simulation and experiments suggests that stable grinding performance in a wheel redress life cycle may be achieved by selecting dressing conditions, taking account of the grinding behaviour.     

Mechanics of self-rotating double-disc grinding process

Unlike most double-disc grinding processes, which use forced workpiece rotation, some double-disc processes rely on workpiece self-rotation driven by non-uniform shear forces resulting from partial wheel-workpiece coverage. This self-rotation is poorly understood, with workpiece angular frequency remaining unknown despite its importance. This paper investigates the kinematics of self-rotation via analytical modelling of the moment-equilibrium conditions, derived from experimentally determined specific-energy values. The model showed that workpiece coverage ratio is the dominant factor governing workpiece angular frequency, allowing for the choice of optimal workpiece coverage ratios that avoid (i) workpiece-stoppage and (ii) excessive frictional heat generation. The predicted velocity was validated with acoustic-emission measurements.     

On the mechanics of the grinding process – Part I. Stochastic nature of the grinding process

Grinding of metals is a complex material removal operation involving cutting, ploughing, and rubbing depending on the extent of interaction between the abrasive grains and the workmaterial under the conditions of grinding. It is also a stochastic process in that a large number of abrasive grains of unknown geometry, whose geometry varies with time, participate in the process and remove material from the workpiece. Also, the number of grains passing through the grinding zone per unit time is extremely large. To address such a complex problem, it is necessary to analyze the mechanics of the grinding process using probability statistics, which is the subject of this investigation. Such an analysis is applicable to both form and finish grinding (FFG), such as surface grinding and stock removal grinding (SRG), such as cut-off operation. In this investigation, various parameters of the process including the number of abrasive grains in actual contact, the number of actual cutting grains per unit area for a given depth of wheel indentation, the minimum diameter of the contacting and cutting grains, and the volume of the chip removed per unit time were determined analytically and compared with the experimental results reported in the literature. Such an analysis enables the use of actual number of contacting and cutting grains in the grinding wheel for thermal and wheel wear analyses. It can also enable comparison of analytical work with the experimental results and contribute towards a better understanding of the grinding process. The analysis is applied to some typical cases of fine grinding and cut-off operations reported in the literature. It is found that out of a large number of grains on the surface of the wheel passing over the workpiece per second (˜million or more per second), only a very small fraction of the grains merely rub or plough into the workmaterial (3.8% for FFG and 18% for SRG) and even a smaller fraction (0.14% for FFG and 1.8% for SRG) of that participate in actual cutting, thus validating Hahn’s rubbing grain hypothesis.     

Modelling and simulation of process: machine interaction in grinding

This article presents an overview of current simulation methods describing the interaction of grinding process and grinding machine structure, e.g., vibrations, deflections, or thermal deformations. Innovative process models which describe the effects of the grinding wheel–workpiece interaction inside the contact zone are shown in detail. Furthermore, simulation models representing the static and dynamic behaviour of a grinding machine and its components are discussed. Machine tool components with a high influence on the process results are modelled more detailed than those with low influence. The key issue of the paper is the coupling of process and machine tool models for predicting the interactions of process and machine. Several coupling methods are introduced and the improvements of the simulation results are documented. On the basis of the presented simulation approaches, grinding processes and machines can be designed more effectively resulting in higher workpiece quality and process stability.     

Mechanics and dynamics of general milling cutters. Part II: inserted cutters

Inserted cutters are widely used in roughing and finishing of parts. The insert geometry and distribution of inserts on the cutter body vary significantly in industry depending on the application. This paper presents a generalized mathematical model of inserted cutters for the purpose of predicting cutting forces, vibrations, dimensional surface finish and stability lobes in milling. In this paper, the edge geometry is defined in the local coordinate system of each insert, and placed and oriented on the cutter body using the cutter's global coordinate system. The cutting edge locations are defined mathematically, and used in predicting the chip thickness distribution along the cutting zone. Each insert may have a different geometry, such as rectangular, convex triangular or a mathematically definable edge. Each insert can be placed on the cutter body mathematically by providing the coordinates of the insert center with respect to the cutter body center. The inserts can be oriented by rotating them around the cutter body, thus each insert may be assigned to have different lead and axial rake angles. By solving the mechanics and dynamics of cutting at each edge point, and integrating them over the contact zone, it is shown that the milling process can be predicted for any inserted cutter. A sample of inserted cutter modeling and analysis examples are provided with experimental verifications.     

On the mechanics of the grinding process, Part II—thermal analysis of fine grinding

Thermal analysis of fine grinding is conducted taking into consideration the stochastic nature of the distribution of abrasive grains and its role under fine grinding (dry) conditions to determine the grinding temperatures and the heat partition at the contacting interface. The analysis considers the grain–workpiece interactions at the local level and the wheel–workpiece interactions at the global level. The workpiece temperature in the grinding zone is taken as the sum of the background temperature due to distributed action of all the previous active grains operating in the grinding zone (global thermal analysis) and the localized temperature spikes experienced at the current abrasive grain tip–workpiece interfaces (local thermal analysis), similar to the work reported in the literature. Since the Peclet number, N_Pe, in the case of fine grinding is very high (a few hundred), the heat flow between the work and the contacting abrasive grains can be considered to be nearly one-dimensional. In this paper, we consider the interaction between an abrasive grain and the workpiece at the contact interface. Consequently, the heat source relative to the grain is stationary and relative to the workpiece is fast moving. The interface heat source on the grain side as well as on the workpiece side is equivalent to an infinitely large plane heat source (with the same heat liberation intensity as the circular disc heat source). However, it will be shown in the paper that the contacting times are different. For example, the abrasive grain contacts the heat source, as it moves over the interface, for a longer period of time (˜milliseconds) whereas the workpiece contacts the heat source for a shorter period of time (˜a few microseconds). The equivalent thermal model developed in the present investigation is simple and represents the process more realistically, especially the heat partition. The analytical results reported here are found to be in good agreement with both the analytical and experimental results reported in the literature by other researchers.     

A dynamic model of the rolling process. Part II: inhomogeneous model

Mathematical models representing the static rolling process have attracted considerable attention in the past, resulting in analytical, numerical, or graphical solutions obtained under various degrees of simplification and constraints. Most models, however, can only be used under steady conditions, and therefore are not suitable for the study of rolling chatter. An enhanced analytical process model that can handle dynamic variations exerted by roll vibrations in multiple directions is proposed in this paper, and its linearized form established for easy analysis. Finally, experiments are presented that verify the accuracy of the proposed dynamic model of the rolling process.     

大尺寸硅片自旋转磨削运动分析及仿真

本文利用数学矩阵方法,建立大尺寸硅片自旋转磨削运动的理论模型,研究了砂轮半径、硅片和砂轮旋转速度、旋转方向等因素的选择及各因素对磨削轨迹的影响,同时还研究了磨粒合成运动速度的变化规律。研究结果表明,随着砂轮半径的增大,磨削轨迹的曲率减小,选用较小直径砂轮将更有利硅片表面质量的提高。当转速比i大于零,随着i值的增大,磨削轨迹的曲率逐渐减小。在转速比i小于零的情形,当转速比i=-2时,磨削轨迹曲率为0,磨削轨迹的形状接近一条直线。磨粒合成运动速度随砂轮转速和硅片转速的增大而增大。     

A local process model for simulation of robotic belt grinding

A local process model to estimate the material removal rate in robotic belt grinding is presented and applied to the process simulation system. It calculate the acting force by incorporating the local geometry information of the workpiece instead of the cutting depth parameter with only one certain value as in a global grinding model. The simulation accuracy can be improved to below 5% even for a non-uniform contact under stable cutting conditions.     

Mechanics of thin ultra-light stainless steel sandwich sheet material: Part II. Resistance to delamination

This study concerns three variants of a novel type of thin sandwich sheet. Details of the core structures, and also the results of an investigation into elastic properties, were presented in the first part of this pair of papers. A study was also made of the tensile properties of single fibres of the type present in the core of these sheets. In this second paper, an investigation is presented of the resistance offered by these materials to delamination of the two faceplates. In one variant of the material, in which the fibres lie approximately normal to the plane of the sheet, delamination occurs predominantly by frictional pull-out of fibres from their sockets in the adhesive. The mode I fracture energy has been measured at about 340 J m⁻². This value is consistent with predictions from a model based on shear-lag theory, with a fibre–adhesive interfacial shear strength of about 5 MPa. It is noted that there should be scope for improving the fracture energy somewhat by raising the strength of the fibre–adhesive bond. For the other two variants studied, in which the fibres are softer (as a result of heat treatment during sintering) and are inclined close to the plane of the sheet, the measured fracture energy is appreciably lower at about 30 J m⁻². In this case, delamination occurs by fracture of the fibres near the mid-plane. Application of a simple model for prediction of the fracture energy in this case leads to the conclusion that some of the fracture was probably of sintered necks between fibres, rather than the fibres themselves, and that this process required considerably less energy.     

磨削加工仿真预报系统的研究现状及发展趋势

磨削加工作为大部分产品成形前的最后一道工序,直接影响产品的加工精度和表面质量。磨削加工的多参数输入、输出和磨削过程中输入、输出参数之间的非线性映射决定了计算机仿真技术在磨削加工中应用的重要性。概述了计算机仿真技术相关理论,基于国内外磨削加工及预报模型,包括磨削运动学、磨削力、磨削温度、磨削液等模型的研究现状,对国内外将计算机仿真技术与磨削加工预报模型理论相结合的磨削加工仿真预报系统的研究现状进行了阐述。着重叙述了国内学者在磨削加工仿真预报系统软件开发方面的研究工作,最后展望磨削加工仿真预报系统的研究发展趋势。     

曲面零件带槛拉深的力学分析

给出了曲面零件带槛拉深时的阻力计算方法，并以此为基础，推导了槛的成形力、最小压边力公式。为了保证压槛时不起外皱并使毛坯完全包槛，初始毛坯的几何参数还应满足一定的限制条件。     

Simulation of precision grinding process, part 1: generation of the grinding wheel surface

This paper is in two parts describing the kinematic simulation of the grinding process. The first part is concerned with the generation of the grinding wheel surface. A numerical procedure for effectively generating the grinding wheel topography is suggested. The procedure is based on the transformation of a random field. The sufficient condition for the transformation is discussed, and two transformations satisfying the condition are introduced. Numerical examples are used to illustrate the viability of the approach. It will be shown that the generated and measured grinding wheel topography share the same probabilistic characteristics.     

Mechanics and dynamics of general milling cutters.: Part I: helical end mills

A variety of helical end mill geometry is used in the industry. Helical cylindrical, helical ball, taper helical ball, bull nosed and special purpose end mills are widely used in aerospace, automotive and die machining industry. While the geometry of each cutter may be different, the mechanics and dynamics of the milling process at each cutting edge point are common. This paper presents a generalized mathematical model of most helical end mills used in the industry. The end mill geometry is modeled by helical flutes wrapped around a parametric envelope. The coordinates of a cutting edge point along the parametric helical flute are mathematically expressed. The chip thickness at each cutting point is evaluated by using the true kinematics of milling including the structural vibrations of both cutter and workpiece. By integrating the process along each cutting edge, which is in contact with the workpiece, the cutting forces, vibrations, dimensional surface finish and chatter stability lobes for an arbitrary end mill can be predicted. The predicted and measured cutting forces, surface roughness and stability lobes for ball, helical tapered ball, and bull nosed end mills are provided to illustrate the viability of the proposed generalized end mill analysis.     

Simulation based optimization of the NC-shape grinding process with toroid grinding wheels

For achieving high material removal rates while grinding free formed surfaces, shape grinding with toroid grinding wheels is favored. The material removal is carried out line by line. The contact area between grinding wheel and workpiece is therefore complex and varying. Without detailed knowledge about the contact area, which is influenced by many factors, the shape grinding process can only be performed sub-optimally. To improve this flexible production process and in order to ensure a suitable process strategy a simulation-tool is being developed. The simulation comprises a geometric-kinematic process simulation and a finite elements simulation. This paper presents basic parts of the investigation, modelling and simulation of the NC-shape grinding process with toroid grinding wheels.     

On the mechanics of the grinding process, Part III—thermal analysis of the abrasive cut-off operation

This is Part III of a 3 part series on the Mechanics of the Grinding Process. Part I deals with the stochastic nature of the grinding process, Part II deals with the thermal analysis of the fine grinding process and this paper (Part III) deals with the thermal analysis of the cut-off operation. Heat generated in the abrasive cut-off operation can affect the life of resin bonded grinding wheels and cause thermal damage to the workpiece. Thermal analysis of the abrasive cut-off operation can, therefore, provide guidelines for proper selection of the grinding conditions and optimization of the process parameters for improved wheel life and minimal thermal damage to the workpiece. In this investigation, a new thermal model of the abrasive cut-off operation is presented based on statistical distribution of the abrasive grains on the surface of the wheel. Both cutting and ploughing/rubbing that take place between the abrasive grains and the work material are considered, depending on the depth of indentation of the abrasives into the work material. In contrast to the previous models, where the apparent contact area between the wheel and the workpiece was taken as the heat source, this model considers the real area of contact, namely, the cumulative area of actual contacting grains present at the interface as the heat source. It may be noted that this is only a small fraction of the total contact area as only a small percentage of the abrasive grains present on the surface of the cut-off wheel are in actual contact with the workpiece at any given time and even a smaller fraction of them are actual cutting grains taking part in the cut-off operation. Since, the Peclet number, N_Pe in the case of cut-off grinding is rather high (a few hundred), the heat flow between the work and the contacting abrasive grains can be considered to be nearly one-dimensional. In this paper, we consider the interaction between an abrasive grain and the workpiece at the contact interface. Consequently, the heat source relative to the grain is stationary and relative to the workpiece is fast moving. The interface heat source on the grain side as well as on the workpiece side is equivalent to an infinitely large plane heat source with the same heat liberation intensity as the circular disc heat source. However, it will be shown in the paper that the contacting times are different. For example, the abrasive grain contacts the heat source, as it moves over the wheel-work interface, for a longer period of time ( milliseconds) whereas the workpiece contacts the heat source for shorter period of time ( a few microseconds). The temperature in the grinding zone is taken as the sum of the background temperature due to the distributed action of the previous active grains operating in the grinding zone (global thermal analysis) and the localized temperature spikes experienced at the current abrasive grain tip-workpiece interfaces (local thermal analysis), similar to the work reported in the literature [Proc Roy Soc (London) A 453 (1997) 1083]. The equivalent thermal model developed in the present investigation is simple and represents the process more realistically, especially the heat partition. The model developed provides a better appreciation of the cut-off operation; a realistic estimation of the heat partition between the wheel, the workpiece, and the chip; thermal gradients in the workpiece due to abrasive cut-off operation, and an insight into the wear of the cut-off wheels.     

Modeling and predicting surface roughness of the grinding process

A new method for predicting the surface roughness of the workpiece for the grinding process is developed in this paper. The conventional method determines the surface roughness based on the model using the mean value of the grain protrusion heights, which leads to a formula. However, the analytical value based on the formula is substantially smaller than measurement. To overcome this problem, the proposed method takes into consideration the random distribution of the grain protrusion heights. As such, there is no formula, and a numerical solution is developed. To solve this problem, first the intersecting points of any two grains with different heights are determined. To determine the final profile considering thousands of the grains, a search method is developed to systematically solve the workpiece profile, starting with the highest protruded grain in a descending order of the grain protrusion heights. Furthermore, a truncated Gaussian distribution model is developed to relate the wheel volume wear to the change in the mean value of the protrusion heights. Simulation shows that the proposed method yields the results that are consistent with measurement, thereby proving the effectiveness of the method.     

Measurement and control of the drill point grinding process

The accuracy of the drill point grinding is a problem of concern to the drill user, the drill manufacturer and the manufacturer of the drill point grinder. In this paper, an algorithm is developed to check and control the drill point grinding operation based upon the mathematical model and is illustrated using the conical, hyperboloidal, and ellipsoidal drills. The discrepancies of the drill point geometry estimated from the designed geometry, due to the errors of the estimation are investigated to evaluate the performance of the grinding process and to indicate where adjustments might be made. The coordinate system for the measurement of the drill flank configuration, the objective function, the initial guess values, the number of observations and the parameter for specifying the radial direction of the drill with the curvilinear cutting edges are discussed.