2. Materials and methods  Basic material for  research was hardened stainless  steel wire X10CrNi18-8 (steel 1.4310). Drawing technology was selected in such a way that the wire  featured  high  strength, the same  as  is required from wire used for spring production. Such selection of the  material  was  determined by its usability (a lot  of ways of application of narrow  and thin  strips made  of this steel for production of elastic elements for precision goods industry) [1,2]. Tested material chemical composition is given in Table 1. Table 1. Chemical composition od steel 1.4310 C Mn Si P S  Cr  Ni 0.08 0.91  0.68 0.028  0.001 17.96  8.42 Flattening performed  in  ball rolling mill (fig.1) in which cylindrical rollers  have been  replaced with  balls  of ~ 15mm diameter. The balls are supported by driven resistance rollers. The device is used for flattening of wire with diameter below 0,5 mm [8-10]. Proper  realisation of flattening  process is dependent on numerous parameters, including( among other things diameter and condition of roller (ball) surface and relation d/D (where d means wire diameter and D – roller diameter), hardening condition and structure of wire to be flattened, draft, pull and counter-pull and the  conditions of friction in the area of contact with  rollers [11-16].  The  listed above parameters may  be of great influence on metal flow during plastic strain, and therefore they may influence geometrical  characteristics  of obtained strips.  That  is  why it is purposeful  to elaborate mathematical relations  for the process of strip  flattening  in  ball rolling  mill and thus  to  determine quantitative  relations  between spread and strip shape as well as dimensions accuracy  indicators, and technological process parameters. Fig.2 presents rolling unit of ball rolling mill.
2.  Materials and methods  Application of optimization techniques enables to analyse the importance of interaction between technological  parameters and product characteristics. They enable to determine mathematical relations that are the basis for design and control of manufacturing process. In the performed tests three-level planning was used [17]. The adoption of fractional experiment plan of 35-1 type enabled to analyse the importance of interaction of five  initial  variables (selected process parameters) on  resulting characteristics (spread coefficient    and indicators of accuracy  of strip shape  and  dimensions – edge camber indicator ww, parallelism indicator ww ,deflection from  flatness indicator wp. Resulting variables were: ball roughness,  relation of wire diameter d  to  roller diameter D, relative draft,  relation of front  tension stress  !N to wire  yield stress !p, relation of back tension stress !N to wire yield stress !p. Realisation of fractional experiment plan  enabled to reduce the  number of  tests  to  81. Performance of complete  and full experiment  would  require  realisation of 243  tests. For static calculations significance level p(") < 0.05 was used.  After  81 tests  (flattening attempts), strips were measured  by means of computer-aided method of vector analysis  of digital image [18]. It consisted in measurement carried out on the photo, taken by means of a camera connected to the microscope, of strip cross-section metallographic specimen, and then  image computer analysis with application of Corel-Draw graphics package.  Statistic  calculations were made  by  means  of Statistica programme.  4 types of models were  used to evaluate the significance  of interaction of respective factors in relation to obtained results:   model with linear-square effects together  with bifactor analysis    model with main linear effects with bifactor analysis   model with linear-square effects   model with linear effects.  Calculations  enabled to  adopt optimal models for 81 performed tests. 3. Results  On the ground of tests  and statistic calculations  it was determined that  mathematical  optimal  models  of flattening process are as follows:   initial variable - spread coefficient !"model with linear-square effects together with bifactor analysis:!=-3,235d/D+0,679Ra+3,246(Ra)2+1,295#h-0,002$Prz/$p--0,015#h%$Prz/$p+0,944,            (1)   initial variable - deflection from flatness indicator wpmodel with linear effects: wp=-0,051#h+0,04.               (2) It results from  the  calculations  and analysis  of obtained models, that spread  of wire made  of 1.4310 steel is negatively influenced by the relation of wire diameter to ball diameter, which means  that  spread  increases  when wire  diameter  decreases. Spread is to increase with deterioration of ball surface quality and when the  process is carried out with  bigger  relative  draft.  Spread  is to  decrease  when back-tension stress increases. The model has also shown  mutual interaction (negative) between relative draft and the relation of back  tension stress to wire yield stress. No significant influence of tension stress to strip spread has been observed. Deflection from  flatness indicator depends only  on draft, which means that increase of relative draft will result in decrease of strip flatness. When analysing  the influence  of initial  variables, no significance of their influence on edge camber  indicators ww and parallelism wr was found for tested wire. The size  of those indicators  rather results  from  the  way of flattening process realisation. Mathematical  models  of flattening  process,  that  have been obtained, are not universal, they have to be adapted separately for respective tested material and adopted level of initial variables. Mathematical models have then  been  successfully verified. Performed  flattening operations proved the  accordance of measurement  results  with calculations arising from  adopted models. Both, impact direction and  impact strength of respective variables on spread coefficient and deflection from  flatness ratio have been confirmed.  Determined indicators  of shape and dimensions  accuracy enabled to establish the degree to which dimensions of flattened strip cross-sections differ from strips featuring ideally parallel flat surfaces. It has been adopted that in such  a case all those indicators would  equal 0. It has  been  assumed that indicator of edge camber  (expressed in %) of  tested strip should  not  exceed 10%, indicator of parallelism should not exceed 5% and indicator of deflection from  flatness should not exceed  1%. Then  it is possible to assume that the strip is characterised by required surface parallelism  as  well as uniform  thickness throughout  its width, therefore - the required flatness. It has to be added that due to the fact that there are no normative regulations here, the size of the indicators  was  adopted  conventionally.  Performed measurements  proved  that all tested strips  feature edge  camber indicator smaller than 10%. Majority of strips featured parallelism indicator smaller than  5%. 60% of all tested strips featured indicator of deflection from flatness smaller than 1%.  Measurement  proved that ball rolling mill enables flattening of narrow  and thin  strip  characterised by uniform  thickness throughout their width and required parallelism of flat surfaces. If the  same range of flatness is to  be obtained on flat rollers,  it requires proper preparation of the rollers  and their  frequent recovery (polishing). Those operations  make  production more complicated and increase its costs. As far  as  ball rolling mill is concerned, the tool here are bearing balls, that are produced on a large scale.  Due to extremely low cost of ball purchase  their frequent replacement  is  not a  problem,  which enables  strip flattening with  narrow dimension  tolerance. Low price  of balls influences economical indexes of strip production process.  Observation of metallographic specimen under microscope proved  that  edge quality  was  correct for the majority  of strips.  No  fractures were detected,  even when  flattening  of strips with large draft.  Visual inspection shown  that  strips feature high quality of surface, that is sufficient even for medical purposes. No cracks or scratches that could impair the operation of elastic elements have been detected.  In addition, roughness measurement  was  made  on profile measurement gauge MITUTOYO SJ-20. It was found that strips flattened with balls that featured roughness of Ra = 0,01 &m have roughness of 0,1'0,2 &m. Strips that were  flattened by means of balls  with the  worst surface quality (Ra =  25  &m), feature roughness of Ra = 0,23'0,3 &m. 4. Conclusions  The process used to manufacture narrow and  thin strips, which are characterised by parallelism of flat surfaces, high requirements concerning dimensional tolerances  and appropriate quality of edges is flattening, which was till now realised with flat rollers.  The  new possibility of flattening appeared with  ball rolling machine. In this device working element is in the form of ball-shaped  tools with  15 mm diameter, placed  on two  driven supporting rolls. The results of research indicate that ball roller is a device  that  allows obtaining narrow  and thin  steel strips, 4.   Conclusions3.   Results characterised by uniform thickness over the width, high quality of  surface and softly rounded side edges.  The use  of computer-aided  methodology of image  analysis allowed precise  determination of  geometric features of flattened strips. The  time of measurements with a vectorial measurements is  considerably  shorter and precision of their workmanship is incomparably better  than measurements  with classical  methods (slide caliper, micrometer, toolroom microscope). The accuracy of measurement is determined by finishing  of the used digital camera and its resolution capacity. The  use of popular software allows realisation of measurements on any computer. This innovative measuring  method can  be  used in many other applications. An analysis of statistic  calculations  of fractional experiment of type 35-1 enabled to prove  significance degree of influence of initial variables on spread coefficient and the indicators of strips shape and dimensions accuracy. Obtained results made it possible to establish mathematical models of flattening. Static calculations showed that draft, ratio of wire diameter to the diameter of balls, roughness of balls  and ratio of back tension to yield stress have significant influence on spread coefficient. Flatness of all examined strips is dependent on used drafts.  Uniformity  of  thickness over  strip width, parallelism  of flat surfaces, small dimensional tolerance,  high  surface quality and naturally  rounded  side edges allow to assume that flattening method with the use of round tool will become an sass, which are used among other things to make  elastic elements and elements for medicine [19,20]. 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