Investigation on the stiffness of wire web of multi wire sawing machine and its influence on machining accuracy

： Diamond wire sawing has gradually applied as the dominant way of silicon sawing in the photovoltaic and semiconductor industry. In the design and evaluation field of diamond wire sawing machine, little research on the static stiffness are presented. But in the design field of traditional metal cutting machine tools, stiffness is an extremely important performance index. The stiffness of wire web should be mastered to improve the machining accuracy of diamond wire sawing. The special significance of this study is that stiffness of wire web is considered to be the key index of multi wire sawing performance, as a result, wire lag and wire bow due to the loss of the stiffness of wire web could be minimized. Aiming at the problem of the stiffness of wire web, the measurement method, process and evaluation standard were proposed. The influence of the stiffness of wire web on the machining process, wire bow, cutting force, machining accuracy and other factors were studied.


Introduction 
At present, the main processing methods for silicon wafer are diamond wire sawing and slurry wire sawing [ 1 , 2 ]. With the development of technology, diamond wire sawing has made great progress in efficiency and quality, but the reciprocating cutting problems such as wire marks and tension fluctuation are still need to improve [ 3 ]. Generally, the way to improve the processing quality is to use finer diamond wire, improve the synchronicity of moving parts, and increase wire speed and satisfy feed speed to improve tension loss, in order to give full play to the cutting capability of diamond wire to ensure the accuracy [ 4 , 5 ]. In addition, as a special machine tool, the stiffness performance of multi wire sawing machine will also affect the processing quality.
In the design and evaluation of diamond wire sawing machine, little research on stiffness are presented. But in the field of traditional metal cutting machine tools, stiffness is an extremely important performance index. The support stiffness of machine tool should be properly designed for reducing both the ground disturbance vibration and the drive disturbance vibration [ 6 ]. The stiffness of the motion and running parts of the machine tool are considered as the key components that affect the processing efficiency and accuracy. In the research of the spindle stiffness, Sarenac [ 7 ] considered spindle stiffness is the main contribution of the machine tool stiffness. Bearings, inter-distances relations and console length and other constructional parameters were emphasized. Yen [ 8 ] investigated the different approaches on a servo control board and with the ability to change the servo control algorithms on a test machine tool to test the performance of servo stiffness. As for the design and modeling method of stiffness, many scholars have studied it from a long time ago, such as Schenk in 1939 [Error! Bookmark not defined.] proposed model of spindle as elastic beams. Daisuke Kono [Error! Bookmark not defined.] proposed a 3D stiffness model of a machine tool support using contact stiffness which was obtained by multiplying the unit contact stiffness by the real contact area. By using the proposed stiffness model, the natural frequency and vibration mode shape of a machine tool bed was predicted. Li [ 9 ] introduced a novel approach for designing the stiffener layout inside large machine tools by applying the self-optimal growth principle of plant ramifications in nature.
As for the measurement methods of stiffness, several researches have been presented. Majda [ 10 ] adopted a method of measuring machine tool stiffness based on the model of rigid body motion, which enabled measurements to be made in generalized coordinates, including measurements of translational and torsional stiffness. This allowed testing the stiffness of various medium-sized machine tools -lathes, milling machines. Laspas [ 11 ] presented a novel measurement procedure to measure and identify full translational stiffness matrices of 5-axis machining centers using quasi-static circular trajectories. The research expanded the measurement procedure to a calibration procedure for 5-axis machining centers and identify rotational stiffness. Different from the measurement method, CAE method has been used in the research of machine tool stiffness for a long time. Huang [ 12 ] introduced a single module method and a hybrid modeling method for analyzing the stiffness of machine tools by using CAE modelling.
Similar to the traditional machine tool design, in order to improve the machining accuracy of diamond wire sawing, the stiffness design of wire web should be considered as a design item. This is the main purpose of this paper. Because the movement form of DWS is simple as only one movement axis in the feed direction, its dynamic performance is less affected by the movement axis. The elastic diamond wire is different from the rigid tool of traditional machine tool, the influence of vibration on machining is limited by the damping effect of wire. Therefore, unlike the dynamic stiffness used to evaluate the dynamic performance of equipment [ 13 ], the static stiffness is more suitable for the stiffness evaluation of wire web.
As the multi wire sawing machine tool is composed of many parts, each part will deform under the action of load, which will lead to the relative displacement between diamond wire and silicon ingot, the displacement is a comprehensive amount. The stiffness of a machine tool cannot be evaluated by the stiffness of a certain part, but refers to the capability of the whole machine tool to resist deformation under the action of cutting force [ 14 ]. Different from traditional machine tools, multi wire sawing machine tools use abrasive grains to slide silicon material through the relative movement of diamond wire, and the elastic deformation of diamond wire is significantly larger than other parts of the equipment [ 15 ]. In the actual machining, under the action of cutting load, the wire web will deform along the force direction to form a wire bow [ 16 ]. These deviations will bring adverse effects on the machining, such as lose of accuracy and the deterioration of surface roughness. If the stiffness of wire web can be accurately measured, it plays an important role in understanding the processing capability of diamond wire sawing machine tool, and improving the processing accuracy and efficiency. The static stiffness of the machine tool is mainly determined by the stiffness of diamond wire web, the static stiffness of wire web can be used as an important index to evaluate the performance of multi wire sawing machine tool. It will directly affect the machining accuracy, efficiency and surface quality. Therefore, measuring and evaluating the static stiffness of wire web is the basis to establish competitiveness of diamond wire sawing machine tool.

Definition of stiffness of wire web
The static stiffness of diamond wire web refers to the capability of the wire web to resist the relative position change caused by the static force in the specified direction, mainly in the feeding direction of the silicon ingot relative to the wire web. It is an important factor to improve the machining accuracy and stability of multi-wire machines. The static stiffness can be calculated by the ratio between the loading force and the deformation of diamond wire caused by the external force.

Detection and calculation method of static stiffness
The relationship between the magnitude of force and deformation is approximately linear. The average static stiffness can be calculated by the ratio of force increment to displacement increment. The data can also be processed by the successive difference method and the least square method to obtain the static stiffness.
The static stiffness is obtained by Eq.(1). In order to reduce the measurement error, the deformation under the action of multiple incremental forces is usually measured continuously, and then the equation is obtained by linear fitting. The coefficient of the linear equation is the static stiffness value of the wire web.
where, Ff is the loading force applied in the feed direction, Df is the deformation in the feed direction caused by the load. The stiffness measurement method of diamond wire web is shown in Fig. 1. The process is as follows: (1) The loading device moves and contacts the wire web according to the specified number of wires, then record the feed position. (2) Continue to increase the unit feed, record the corresponding force under this feed. (3) Increase the feed position in turn, and record the force under each feed increment until the wire bow of the wire web reaches a certain value and has enough strength when the measurement can be stopped after enough data.

DWS equipment and measuring instruments
(1) Sawing machine tool The sawing equipment used in the experiment is a GC630S Multi wire sawing machine tool added tension adjusting function, as shown in Fig. 2. The silicon ingot is fixed on the force sensor which is set on the feed mechanism during the cutting test, while the press plate is fixed on the force sensor during the stiffness test. (2) Test instrument The loading forces are measured by force measuring system consist of ME NC-3DT60 (K3D60) three component forces sensor and DASP V11 test system. The force is increased by equal spacing increment. The direction of the force action simulates the direction of multi wire sawing, that is, the feeding direction of silicon ingot. The action point of the force is concentrated on the middle position of silicon ingot and diamond wire in each experiment. The displacement test is recorded according to the actual feed rate displayed by the control system of the multi-wire sawing machine tool.

Experimental design (1) Stiffness test of wire web
The stiffness test of wire web is designed by changing the number of diamond wires that make up the wire web. It is repeated that the measurement method mentioned above to measure the static stiffness. In the test, the force sensor is placed on the workpiece feeding mechanism. The feed position along the feeding direction is taken as the reference position when force signal is changed. The feed rate is used to express the displacement. After the end of the first feed position, it stays for about 10 s, and gradually incrementally presses down the wire web. Continue to press the wire web incrementally to realize deformation difference, measure the change of force and record the incremental displacement. After the end of force measurement at each subsequent feed position, the incremental displacement time is about 12 s, and multiple force data corresponding to different positions is measured. The displacements and forces are recorded and taken as the data of static stiffness calculation. The experiment was carried out at ambient temperature of 20 ℃. The diamond wire used in the static stiffness test is 50 core wire with a diameter of 0.066 mm and a preset tension of 6.5 N in the test.
(2) Cutting test The wire web composed of different number of wires is used to cut different lengths to sawing kerfs. The system parameters and cutting forces in the sawing process are recorded, and the wire bows generated during sawing are indirectly measured by the formed kerfs, so as to evaluate the influence of wire web stiffness on sawing accuracy. The process used in the test is shown in the Table 2, and part of the silicon material used in this research is shown in Fig. 3. The feed force measured in the static stiffness test is shown in Fig. 4. The feed force increases with the deformation increases. The relation between the feed force and deformation is linear. The static stiffness curve can be obtained as shown in Fig. 5. The static stiffness of wire web composed of different number of wires could be recognized by the coefficient of linear equation in the figure. In addition, the lateral force and force in the cutting direction corresponding to the feed force also show an increasing trend. The lateral force in Fig. 6a shows a linear positive proportional relationship between the lateral force and deformation. However, the relationship between the force and deformation along the wire speed direction is not completely linear, as shown in Fig. 6b. Fig. 6 The lateral force and force on the wire cutting direction vary with the feeding pressure and the number of wires that forms wire web. (a. lateral force; b. force in the cutting direction) Fig. 7 shows the relationship between the stiffness of wire web and the number of wires forming the wire web. It can be seen that a linear relationship between the stiffness of wire web and the number of wires is existed. In addition, it can be seen from Fig. 7 that the stiffness of a single wire forming the wire web is basically stable between 0.0031-0.0034 N/mm in the incremental range from 5 wires to 40 wires. The stiffness of a single wire can be considered as the unit contact stiffness, that is, the unit contact stiffness of the wire is about 0.0033 N/mm. This law is the same as the stiffness model of machine tool [Error! Bookmark not defined.]. Therefore, the contact stiffness of wire web could be obtained by multiplying the unit contact stiffness by the real contact area which is wire number, as shown in Eq.(2). K = N × Ks (2) where, K is the stiffness of wire web; Ks is the stiffness of single wire; N is the number of wire forming wire web. The stiffness matrix between the average incremental static stiffness of a single diamond wire and the number of wires forming the wire web and the feed position of the feeding mechanism is shown in Fig. 8a. With the increase of wire number, the average incremental static stiffness of a single wire decreases slightly; with the increase of the feed position corresponding to wire bow, the incremental static stiffness of a single wire increases greatly.
The stiffness matrix of the incremental static stiffness of the whole wire web with the number of wires and the feed position is shown in Fig. 8b. With the increase wire number, the stiffness of wire web increases. With the increase of feed position, the tension in the wire increases, and the stiffness of the wire web per unit feed distance increases.
It can be concluded that: (1) Under the condition of the same machine tool, wire and machining technology, the increasing wire number will greatly improve the stiffness of the wire web, and the stiffness of a single wire decreases slightly, which indicates that increasing wire number is helpful to improve the stiffness of the wire web and reduce the loading of a single wire.
(2) With the increase of wire bow, both the stiffness of the whole wire web and the stiffness of a single diamond wire will be greatly increased, and the increasing trend will gradually increase. But this will increase the loading of diamond wire, which is easy to cause wire broken.

Relationship between static stiffness of wire web and wire bow during sawing
The relationship between the wire bow and its corresponding static stiffness of wire web calculated by the actual sawing kerf is shown in Fig. 10. The wire bow can be stable only when the number of wire forming wire web is more than 20. With the increase of the wire number from 5 to 10, the stiffness of wire web increased and the corresponding wire bow decreased significantly. However, when the number of wires increases from 10 to 15 and from 15 to 20, the phenomenon is different at different feed positions with wire bow decreases or increases irregularly. However, as the number of wires continues to increases, the wire bow begins to stabilize. With the stiffness of the wire web continues to increase, the wire bow will not change significantly with the increase of stiffness. It shows that one of the conditions to ensure stable sawing is that the number of wire web should exceed at least 20.
The more the number of wires, the higher the stiffness and the higher the stability of deformation. If the stiffness in Fig.  10 is more than 3 N/mm, the wire bow can be stably predicted for the wire web composed of more than 20 wires. The wire bow corresponding to the number of wire can be seen that the pantograph of 5, 10 and 15 fluctuates greatly. It can also be seen from the proportional coefficient between the kerf length lk and the preset sawing length lp in Fig. 11 that the proportion coefficient lk/lp changes when 5, 10 and 15 wires form a wire web. Only when more than 20 wires form the wire web, the proportion coefficient between the actual sawing length and the preset sawing length can be stable, that is, when the sawing length is set, the stable sawing length can be output, which is helpful for ensuring and predicting the sawing accuracy. The stiffness of wire web can be obtained by calculating the force along the feeding direction of silicon ingot and the preset feed position during cutting, as Fig. 12a shown. It can be seen that the deformation of wire web is also proportional to the feed force during sawing, and the cutting stiffness in Fig. 12b is close to linear with the number of wire that forming wire web.
It can be explained from the static stiffness of the wire web and the cutting stiffness of the wire web. For the multi wire sawing machine, the wire web must contain at least 20 wires to ensure enough stiffness of wire web and can obtain stable cutting accuracy. For single wire and a small number of wire sawing machine, it can realize the rewinding of diamond wire or the arrangement of multi wheel from the aspect machine tool design, which is beneficial to ensure the stiffness and accuracy, reduce the wire bow and machining error.

Finite element model
The static stiffness test is carried out on the finalized equipment of GC630S, but it is inconvenient to change the structure after the equipment is finalized. As the wheel span and wheel diameter affect the stiffness of wire web, the finite element method is used to make qualitative analysis. The variable parameters of FEM are shown in Table 3. The analysis results of wire web deformation on the condition of wheel span 500 mm, wheel diameter 218 mm, wire tension 10 N, contact length 1/4 winding circle are shown in Fig. 13. Under the preset tension and the uniformly distributed load shown in Fig. 13a on the diamond wire, the deformation along the loading direction is generated to form wire bow as shown in Fig. 13b and c. The effects of wheel span, wheel diameter and preset tension on wire stiffness are shown in Fig. 14. With the increase of wheel span, the deformation of wire web increases greatly, and the stiffness of wire web decreases sharply. Among the three selected wheel diameters of 180 mm, 218 mm and 250 mm, the wire deformation of wheel diameter of 218 mm is the smallest and the stiffness of wire web is the best. The larger the preset tension of diamond wire, the smaller the wire deformation and the better the stiffness of wire web.
Therefore, reducing the wheel span and increasing the preset tension on the diamond wire are very helpful to improve the stiffness of wire web. There is an optimal range of wheel diameter for improving the stiffness of wire web. It is not that the larger or smaller the wheel diameter is, the better the stiffness of the wire web is. Wu [ 17 ] proposed three performance indices for stiffness evaluation to investigate the stiffness of 5-DOF machine tool changing along different directions with a give machine configuration. In this paper, the evaluation indices are used to evaluate the stiffness of wire web changing with wheel span, wheel diameter, and preset tension.
(1) C1: the lowest stiffness; (2) C2: the ratio of the highest stiffness to the lowest stiffness; (3) C3: the average stiffness. The minimum deformation of the wire web is 0, so evaluation index of C2 is not considered. Indexes of C1 and C3 are used to evaluate. The results of the evaluation are consistent with the overall distribution trend. The significance of the evaluation indexes is in the stiffness comparison of diamond wire sawing machine tools with different design parameters, such as machine tools produced by different manufacturers, different specifications, different design parameters, and different parts and components.

Conclusion
In this paper, the definition of stiffness of wire web was proposed, and it is considered as one of the indexes that determine the machining accuracy of DWS. The research can draw the following conclusions: (1) The definitions of static stiffness of wire web and cutting stiffness were proposed. The measurement methods of wire web stiffness was provided, including measuring processes and testing instruments.
(2) The static stiffness matrix of the wire web and the single wire were provided by a series of stiffness test. The experimental results show that the number of wires in the wire web cannot be less than 20 in order to ensure enough machining accuracy and make it stable and predictable. Single wire rewinding and adding guide wheel are the measures to improve the stiffness of wire web.
(3) The influences of wire web stiffness on machining process, wire bow, cutting force and machining accuracy were studied. It is found better stiffness of wire web could improve the machining accuracy of diamond wire saw, reduce wire bow and its corresponding forces. Meanwhile, the processing is stable and could be predicted. (4) The finite element method was used to analyze the influence of multiple parameters on the wire web stiffness, including span of cutting wheel, wheel diameter and wire preset tension. Reducing the wheel span and increasing the preset tension of wire could improve the stiffness of wire web. An optimal range of wheel diameter for improving the stiffness of wire web is existed. (5) The lowest stiffness and the average stiffness are used to evaluate the stiffness of wire web of diamond wire machine tools, which compares the performance of DWS machine tools with different design parameters and information.