Modelling Clamping Force Deection in Dexel-Based Material Removal Simulations

Machining simulations of material removal that predict workpiece quality are a key factor in gain-ing an understanding of the possible causes of manufacturing defects. Particularly in the case of thin-walled workpieces, as are frequently produced in the aerospace industry, the workpiece stiffness is of utmost impor-tance. Form deviations on the ﬁnal workpiece can result due to the the process force or the clamping situation. This article presents a method for modelling the deformation due to the clamping force in dexel-based material removal simulations. To prevent distortion of the dexel model, triangulated surface meshes are generated separately for the start and end points of a dexel ﬁeld by means of a Delaunay triangulation for the ﬁnal contour. With the help of an FE simulation of the near contour state, the resulting displacements for the corner points of the triangles are determined and then inversely displaced. Subsequently, the new start and end points of the machined dexels are determined through a 2D interpolation. The method is validated for ﬂatness and roundness deviations using two specimen workpieces. It shows that the prediction can be signiﬁcantly improved, especially for thin-walled components. was written in the context of the research project ”Quality Predictive CAM Simulation for Machining (QUAPS)”. The IGF project 19828 N of the Forschungsvere- inigung Programmiersprachen f¨ur Fertigungseinrichtungen e. V. (FVP) via the Industrial Research (AiF) as part of the program to promote joint industrial research (IGF) by the of Eco- nomics and on the basis of a resolution of the German Bundestag.


Introduction
Clamping systems are used in production for positioning and fixing workpieces. This can be done for both production and quality control by means of a friction and/or a form fit. In addition to accessibility to the component, it is important to ensure that the clamping force is large enough to prevent any movement due to external forces, e.g. process forces. At the same time, it must be guaranteed that the deformations occuring as a result of the clamping force do not exceed the permissible tolerances. [1] In the course of increasing digitalisation, process planning with the help of CAM systems has become an industrial standard. In addition to path planning, they usually include both a material removal simulation and a collision detection based on geometric-kinematic machine models. The clamping situation can also be taken into account. By integrating the control behaviour, production times can also be predicited. All this offers potential for process optimisation even before the beginning of production. [2][3][4] While analytical/continuous approaches, such as constructive solid geometry representations (CSG) [5], are mainly used for process planning, e.g. for the simulation of material removal, discrete approaches are preferred for process modelling. Especially the modelling by means of contour lines, dexels and voxels should be mentioned here. [6] Possible areas of application of process modelling are quality-predictive CAM simulations [7] or the monitoring of processes [8][9][10][11]. Here, different effects such as thermal and mechanical influences can be mapped [10,[12][13][14]. Mechanical influences can lead to deformations on the tool side as well as on the workpiece side and cause shape deviations on the component. Especially with thin-walled work-  [15] and due to the clamping force [16,17]. The present article presents a method for modeling the deformation due to the clamping force in dexel-based material removal simulations. Since dexels can only be lengthened or shortened and not distorted, a direct threedimensional deformation of the workpiece is not possible. The method presented in this article circumvents this problem by transforming the dexel model into a boundary point cloud (BPC) model and then transforming it back.

Related Work
Modelling of influences on the accuracy of machining processes as a result of the clamping situation of workpieces is subject of many research papers. In addition to effects arising due to incorrect adjustment of the component, the focus is primarily on work that uses FE simulations to determine the deformation of the part and its resulting stiffness due to the clamping [1,16,18,19]. In many cases, iterative methods are used to minimise the clamping force/deformation or to realise a uniform deformation [20,21]. Huang and Yoshi compensate the deformation caused by process heat by specifically modifying the clamping force [22]. More recent work deals with an adjustment of the cutting parameters or the tool path at locations with low stiffness or high deformation of the workpiece [17,23].
In [24], Knape et. al. present a method for modelling the workpiece deformation due to the clamping force in a dexel-based material removal simulation. With the help of an FE simulation, the dexels are initially lengthened/shortened at the beginning of the process and shortened/lengthened at the end of the machining process. In a first step, the prediction accuracy regarding the flatness is improved. In a later work, this methodology is validated for further tolerances [25]. The disadvantage of this method, however, is that it is only suitable for modelling the effect in the near-net-shape state, or for machining operations in which the deformation of the workpiece does not change significantly during the process. This article therefore presents a method making it possible to model the effect of workpiece deformation as a result of the clamping force independently of the raw state of the workpiece.

Principal Modelling Approach
The present work uses the existing machining simulation from [10]. Machine-internal data is read out parallel to the process from the NC control to determine the material removal. The machine behaviour can be modelled with the help of the available data. In addition, data can be used to execute a process force estimation [26]. Together with background information such as the machine and tool stiffness, the displacement at the tool centre point (TCP) can be determined. In contrast to [24], the simulation is initially executed without taking the clamping situation into account. After machining is completed, the Dexel model is converted into a BPC model. Separate point clouds are generated for the start and end points for each Dexel direction. Furthermore, the individual point clouds can be further segmented/clustered. This can be done featureor position-related and is necessary, for example, if a dexel has several start and end points. Possible positionrelated segmentations can be done with manually, defined, equal or geometric interval or also using natural breaks (Jenks) [27,28]. Subsequently, the individual point clouds are meshed by a Delaunay triangulation. The triangulation is done in the respective dexel plane so that a triangular net is created. For the individual corner points of the triangles, the displacements due to the clamping forces are now determined using a scatterd interpolant [29] and the FE simulation. By moving the corner points in the negative displacement direction, the component distortion due to the clamping forces is reproduced. If no further production step follows after declamping, the displaced point clouds can be used directly for quality evaluation, e.g. according to ISO 1101 [30]. If further production steps follow, the point clouds must be transformed back into a Dexel model. By means of a 2D interpolation and the corresponding triangular net, the new start and end points of the processed Dexel are determined. Finally, a Dexel model in a declamped state results that can be used for further processing.

Machining Simulation
Machining Process

Conversion to BPC-Model
Inverse Shifting

Conversion to Dexel-Model
Quality Evaluation

Experimental Validation
To validate the approach, two different specimen workpieces were machined. Here, attention was paid to the fact that the workpiece distortion has a relevant influence on the workpiece quality. On the first part the clamping deformation was validated based on the flatness of the part. For the second part, the roundness was used. Both parts were manufactured from a (100 mm x 100 mm x 40 mm) C45 steel block, as seen in Fig.  4, and with a clamping force of 20 and 40 kN. Production was done on a DMG Mori DMU65 Monoblock, a 5-axis machining centre. In order to reduce thermal effects in the cutting zone and to keep the influence of the process force as low as possible, all processes were executed with cooling lubricant and a finishing process with low depth of cut. In addition, to parameterise the material removal simulation, the machine was measured before the first and after the last machining operation to determine the axis alignment errors using a double-ball bar (DBB), as these errors are particularly relevant for roundness [31]. A Zeiss Contura coordinate measuring machine (CMM) was used to measure the parts in the declamped state. It was shown that the simulation could be significantly improved with the help of the developed method.  For validation, similar to [32], the point clouds from the material removal simulation and the CMM measurement are compared using a point cloud registration and flatness deviation. First, the point clouds are mapped using the 3D Normal Distribution Transform (3D-NDT) algorithm [33]. This is particularly suitable for point clouds with different resolutions and is robust against outliers [33,34]. The first occurs due to the fact that the dexel density of the simulation is usually significantly higher than the resolution of the measurement using CMM. After the point clouds are mapped to each other, the flatness and the resulting deviations in the Z-direction are determined for the congruent area.     For the simulation results, in addition to the flatness of the declamped part in the congruent area, the flatness of the entire surface, neglecting the centring hole, and the flatness in the congruent area of the part without taking the clamping conditions into account are also shown.
Since the flatness is only represented by the difference between the upper and lower deviation from the reference plane and does not give any conclusions about the form similarity of both surfaces, this approach is only sufficient for validation. One possibility to compare both surfaces is the resulting chord error. Since outliers, as seen in Fig. 7 at part no. 1 and 12, can also strongly influence the evaluation here, boxplots of the resulting chord error between CMM measurement and simulation are generated for the individual parts. Due the convexity in the present application is much lower compared to the remaining part dimensions, it is assumed via small-angle feeding that the resulting chord error is equal to the difference of the Z-values.
Comparing the resulting cord errors between a considered (Fig. 8) and a neglected clamping situation (Fig.  9), it becomes apparent that the prediction quality of the simulation is improved significantly. Therefore, the interquartile range and the whiskers show that the deviations of the (absolute) chord error are significantly reduced.

Roundness Error
While mainly the displacements in Z-direction influence the workpiece quality for flatnesses, the displacements in the X-and Y-direction are relevant for the roundness. Here, in addition to the displacement of the circle edge, there is also a displacement of the centre point. To validate the approach, the blank was facemilled to a height of 20 mm first, as in the case of the flatness validation. Then, a hole with a diameter of 60 mm was milled in the centre of the part using a Ø16 mm end mill.Twelve parts were produced with a clamping force of 20 kN (parts no. 1-6) and 40 kN (parts no. 7-12), respectively. After production, the roundness of the parts was measured on the CMM at three different heights. The three measurements are aggregated to a total point cloud. Afterwards, this is aligned to the relaxed BPC model using the 3D-NDT algorithm so that the relevant points of the simulation can be identified. For each measurement, a separate point cloud from the simulation is selected for both the clamped and declamped states. With the help of a circle fitting according to [35], the centres of the respective circles are determined so that the resulting radius can be derived over the angle. By comparing the deviations between the radii of the measured and the simulated parts, here the interquartile range and the whiskers also show that the deviations are significantly reduced. Fig. 11 and 12 illustrate this for all measurements.

Conclusion
In the machining process, various effects can compromise components that influence the quality of the workpiece. Especially parts with a low stiffness in the final state are deformed by clamping forces. In this article, a method was presented that makes it possible to reproduce this effect in dexel-based material removal simulations. The presented approach has the advantage that it works independently of the raw part and the virtual workpiece can be used for successive process simulations. In addition, three-dimensional deformations can be modelled. Furthermore, it offers the possibility of a more precise error breakdown in process monitoring with the help of machine-internal data. A disadvantage of this approach is that only the start and end points can be changed while new/old start and end points cannot be added/removed. Nevertheless, it was shown that the presented approach can improve dexel-based material removal simulations for parts with a relevant clamping force deflection. The approach was validated on the basis of two specimen workpieces, so that the effect could be reproduced in all three directions.      Resulting Radius Error without De ection -Roundness Evaluation