Zeszyty Naukowe Państwowej Wyższej Szkoły Zawodowej im. W itelona w Legnicy ISSN 1896-8333, e-ISSN 2449-9013 nr 29(4)/2018

Martyna WiciakPoznan U niversity o f Technology e-mail: m artyna.r.w iciak@ doctorate.put.poznan.pl

Agata FelusiakPoznan U niversity o f Technology e-mail: agata.z.felusiak@ doctorate.put.poznan.pl

Piotr KierujPoznan U niversity o f Technology e-mail: piotr.a.kieruj@ doctorate.put.poznan.pl

Piotr SzablewskiThe President Stanisław W ojciechowski State U niversity o f Applied Sciences in K alisz e-mail: piotr.szablewski@ pwk.com .pl

Stiffness analysis of clamping the workpiece in the turning process

SUMMARY

The achievement of the highest accuracy is an important factor in machining. The dynam­ics and statics of the system, ultimately, is affecting the process precision1. The stiffness of MGFT system (machine tool - grip - fixture - tool) is significant in process of ma­chining. Dynamics of the machining system is related to its some parts like machine dynamics, tool dynamics or clamping workpiece2. The rigidity of wokrpiece - toolholder subsystem has an enormous influence of machining3. The paper provides stiffness analy­sis of workpiece made of Waspaloy, nickel-base superalloy material. The workpiece was clamped into hydraulic three-jaw chuck. The wokrpiece - turning turret subsystem was strained in the radial direction of the shaft. These measurements were repeated on vari­ous phases of turbine shaft turning. This component was burdened and disburdened in terms of 0-2000 N. Based on this research the deflection characteristics in relation to load yx = f(F) were determined. The purpose of the study was to identify at what stage of the turning process subsystem stiffness will be forfeited. Additionally, impact of number of machined passes on loss of stiffness was determined.

Key words: turning, static stiffness, three-jaw chuck clamping.

1 N. Seghedin, M. Horodincâ, D. Chitariu, R. Drosescu, Dynamic Behavior o f Long Turned Work­pieces, „Applied Mechanics and Materials” 2014, vol. 657, p. 78-82.

2 Ibidem.3 Z. Murcinkova, K. Vasilko, The real shape o f the workpiece after turning and milling, „Journal

of Production Engineering” 2013, vol. 16, p. 33-36.

310 Martyna Wiciak, Agata Felusiak, Piotr Kieruj, Piotr Szablewski

Introduction

The developm ent o f new m achining technologies like H igh Speed C utting (HSC), dry m achin ing (w ithout use o f cu tting fluid) o r hard m achin ing is rela ted to the app lica tion o f m ore pow erfu l and cu tting sp e e d 4. It is im portan t to se lected op tim al m achining co n ­ditions in o rder to m in im al surface ro u g h n ess5. These requirem ents shall apply to tu rn ­ing d ifficu lt-to -cu t m ateria ls like W aspaloy. N ow adays, the n ickel-base superalloy such W aspaloy is very o ften used in m achining. Turning o f th is m ateria l is difficult, so there are som e new techniques fo r exam ple L aser A ssisted M ach in ing (L A M )6. This type o f hybrid m ethod allow ed increasing the m achinability o f m aterial. The cu tting forces o r cen trifugal fo rce arises during m achining. As a result, the h igh rad ia l and axial forces o r bend ing m om ents are c re a te d 7. This fo rces and m om ents are carried fo rm w ork ­piece on the spindle. This is done by clam ping device along w ith the suitable g ripping force. D efin ition o f the requ ired g ripp ing force is necessary. It ensures safety clam ping o f the w orkpiece, especially during H igh Speed M achining. In tu rn ing process, the three jaw -ch u ck clam ping is the p referred option, because o f its cen tering ca p ab ilitie s8. D ur­ing the last years, papers about dynam ic stability o f w orkpiece - too l subsystem have b ee n developed. F o r exam ple, various studies show analysis o f deform able w orkpieces like th in -w alled co m p o n en ts9. M ost research bu ilds on defining acceleration , velocity o r v ib ra tion am plitude fo r various parts o f system . There are som e m onitoring system s b u ilt on analysis o f v ib ra tio n signals capable the deflection o f w o rk p iec e10 . The determ i­nation o f natural frequency fo r d iffe ren t elem ents o f m achin ing system m ay be relevant. This w ill allow to avoid the instab ility caused by resonance. It is im portan t to deter­m ine w h ich com ponent has the b iggest influence on the stability o f M G FT sy s tem 11. D uring the m achining, the d im ensional and geom etric inaccuracies are created. This m eans a d ifferen t stiffness along the entire leng th o f w orkpiece. The cu tting forces have an im pact on these varia tions. A dditionally , these forces have influence on the elastic

4 P. F. Feng, Z. J. Wu, D. W. Yu, E. Uhlmann, An improved computation model fo r critical bending force o f three-jaw chucks, „Journal of Materials Processing Technology” 2008, vol. 208, p. 124-129.

5 D. Przestacki, T. Chwalczuk, The analysis o f surface topography during turning o f Waspaloy with the application o f response surface method, „MATEC Web of Conference” 2017, vol. 136,p. 02006.

6 A. Bartkowska, M. Kukliński, P. Kieruj, The influence o f laser heat treatment on the geometric structure o f the surface and condition ofthe surface layer and selected properties o f Waspaloy, „MATEC Web of Conference” 2017, vol. 121, p. 03006.

7 P. F. Feng, Z. J. Wu, D. W. Yu, E. Uhlmann, An improved computation model...8 E. Uhlmann, P. F. Feng, S. Doll, Dynamic clamping force fo r jaw-chucks fo r high speed cutting,

„Production Engineering” 2004, 11 (1), p. 149-152.9 K. Kolluru, D. Axinte, Coupled interaction ofdynamic responses o f tool and workpiece in thin wall

milling, „Journal of Materials Processing Technology” 2013, vol. 213, p. 565-1574.10 P. Twardowski, M. Tabaszewski, S. Wojciechowski, Turning process monitoring o f internal

combustion engine p iston ’s cylindrical surface, „MATEC Web of Conference” 2017, vol. 112, p. 10002.

11 A. Otto, F.A. Khasawneh, G. Radons, Position-dependent stability analysis o f turning with tool and workpiece compliance, „The International Journal of Advanced Manufacturing Technology” 2015, vol. 79, p. 1453-1463.

311

deflection o f w o rk p iec e12. In th is regard, the m ain purpose is to ob ta in the d im ensional to lerance, w h ich is req u ire d 13.

D uring the m achining, w orkpiece is burdened by cutting force F. In turning, the m ain cutting force F has three com ponents: cutting force F c, feed force F f and radial force Fp. R adial force F p has the m ost significant im pact on the deflection o f w orkpiece. F urther­m ore, this constituent o f force affects the m achine and too l stiffness. To know about value o f radial force F p provides opportunity o f estim ate am ount o f s tiffness14.

B ased on investigations, the clam ping device stiffness is h igher than the ta ilstock stiff­ness. H owever, it is not a rule, it depends on chuck ty p e 15. F or exam ple, in 16 studies, the circularity o f w orkpiece clam ped in three jaw -chuck w as m easured. The m easurem ents w ere carried out applying m inim al gripping force Fu min. A dditionally, the offset o f the axle shaft w as determ ined. F igure 1 depicts the profile o f deform ed shaft. The triangular shape seen at the picture, can be correlated w ith places, w here the jaw s w ere closed.

Stiffness analysis o f clamping workpiece in turning process

Fig. 1. Profile of deformed shaft fixed into three-jaw chuck (own elaboration based on:Z. Murcinkova, K. Vasilko, The real shape o f the workpiece after turning and milling, „Journal

of Production Engineering” 2013, vol. 16, p. 33-36.)

12 M.A. Salgado, L. N. López de Lacalle, A. Lamikiz, J. Muñoa, J.A. Sánchez, Evaluation o f the stiff­ness chain on the deflection o f end-mills under cutting forces, „International Journal of Machine Tools & Manufacture” 2005, 45, p. 727-739.

13 Z. Murcinková, K. Vasilko, op. cit.14 Ibidem.15 M. Estrems, M. Arizmendi, W. E. Cumbicus, A. López, Measurement o f clampingforces in a 3 jaw

chuck through an instrumented Aluminium ring, Procedia Engineering, The Manufacturing Engineering Society International Conference, MESIC 2015, vol. 132, p. 456-463.

16 Z. Murcinková, K. Vasilko, op. cit.

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In pap e r17 the dynam ics o f the system while turning o f long and susceptible workpiece, w as presented. In this study, the natural frequency o f com ponent w as determined. The inves­tigation states that w ith the increase o f workpiece diameter, increased the natural frequency and stiffness o f com ponent. Then, the high precision o f m achining can be obtained. The axial force w as applied along the tailstock, the increase o f force resulted in decrease o f natural frequency, due to excessive burden o f shaft.

The problem occurring mainly during H igh Speed Cutting, w hen workpiece is clam ped into three-jaw chuck. In turning, the great axial force that has an im pact on the fixture appears18. Influence o f axial force m ight lead to issues in HSC. Insufficient gripping force could result in to eject a workpiece from fixture. It may lead to very serious injuries o r dam ­ages. In order to decrease a gripping tim e, the automatic clam ping is often u se d 19. There are a lo t o f types o f grippers in CNC (Com puter N um erical Control) machines like hydraulic, pneum atic or electrom agnetic clamping. Nowadays, three-jaw chucks are upgraded fo r exam ­ple: light jaw s made o f alum inium or m aterials w ith glass fiber. The m ain purpose o f these changes is reduction o f dynam ics clam ping force losses. This allows fo r m axim um spindle speed20. Testing o f the three-jaw chuck stiffness and its influence on dynam ics clam ping forces can provide safety H SC process during turning. In p aper21 the relations between cen­trifugal force and clam p stiffness are presented. The low er the workpiece stiffness and higher three-jaw chuck rigidity, then the losses o f dynam ics clam ping force is m uch lower. This leads to efficient H SC process during turning.

Martyna Wiciak, Agata Felusiak, Piotr Kieruj, Piotr Szablewski

Experimental details

The objective o f research involved the m easurem ent o f static stiffness o f workpiece made o f Waspaloy. The wokrpiece - turning turret subsystem w as strained in the radial direction o f the shaft. These m easurem ents o f static stiffness were repeated on various phases o f tur­bine shaft turning. This com ponent was burdened and disburdened in term s o f 0-2000 N. F igure 2 depicts the scheme o f m easurem ent o f subsystem static stiffness. Three m easure­m ent points w ere selected, w hich was show n on figure. Extrem e points were chosen in order to stiffness changing along the entire length o f shaft. The A position is the m ost integral point, w hich is the nearest to tailstock. This is why, the follow ing section o f paper increas­ingly is based on loss o f subsystem stiffness in A position. The displacem ent o f shaft was

17 N. Seghedin, M. Horodinca, D. Chitariu, R. Drosescu, op. cit.18 P. F. Feng, D.W. Yu, Z. J. Wu, E. Uhlmann, Jaw-chuck stiffness and its influence on dynamic

clamping force during high-speed turning, „International Journal of Machine Tools & Manufacture” 2008, vol. 48, p. 1268-1275.

19 T. Alquraan, Yu. Kuznetsov, T. Tsvyd, High-speed clamping mechanism o f the CNC lathe with compensation o f centrifugal forces, International Conference on Industrial Engineering, ICIE 2016, vol. 150, p. 689-695.

20 P. F. Feng, Z. J. Wu, D. W. Yu, E. Uhlmann, An improved computation model...; P. F. Feng, D.W. Yu, Z. J. Wu, E. Uhlmann, Jaw-chuck stiffness and its influence...; J. Sharana Basavaraja, Sayad M. Shanawaz Mujawar, Modeling, simulation and analysis o f gripping force loss in high speed power chuck, Procedia Materials Science, International Conference on Advances in Manufacturing and Materials Engineering 2014, AMME 2014, vol. 5, p. 1417-1423.

21 P. F. Feng, D.W. Yu, Z. J. Wu, E. Uhlmann, Jaw-chuck stiffness and its influence...

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m onitored by H eidenhain increm ental length gauge M T 12B. The dim ensions o f clam ped shaft: diam eter D = 82,5 mm, length L = 317,5 mm. Semi-finished product was made from nickel-base superalloy m aterial Waspaloy. Density o f m aterial p = 8138 kg-m-3.

Stiffness analysis o f clamping workpiece in turning process

Fig. 2. Scheme of measurement of the wokrpiece - turning turret subsystem static stiffness(own elaboration)

W orkpiece w as clam ped into hydraulic three-jaw chuck in CNC machine CTX 310 E C O ­LINE. Figure 3 shows the set-up o f experim ental apparatus: force sensor K1505-5 kN con­nected w ith indicator fo r m easuring strain gauge force sensors and increm ental length gauge M T 12B connected w ith digital display H eidenhain D-83301.

Force sensorK1505 5kN

indicator tormeas urine strain

e a use :orce

sensors

IncrementalDigital display length gauge MT

Heidentum12E

D-Ej j OI

Fig. 3. The scheme of experimental apparatus (own elaboration)

314 Martyna Wiciak, Agata Felusiak, Piotr Kieruj, Piotr Szablewski

The m easurem ents based on the follow ing procedure:• fo r force F = 0 N, the displacem ent yx0 = 0 m m w as assumed,• next, the system w as strained by force F = 100 N and a displacem ent yx1 w as saved,• then, the force F was increased every 100 N as far as the value o f m axim um force

F = 2000 N and displacem ent yx w as w ritten at a time,• this procedure has been repeated during relieve o f workpiece from 2000 N to 0 N,• all m easurem ents o f displacem ent yx were saved.

B ased on investigations, the deflection characteristics in relation to load yx = f(F) were determined.

Results and discussion

Research results show the m easured displacem ent yx during burdening o f workpiece. O n the basis o f investigations, the linear function y = ax+b in the form o f y x = aF+b was determined. In this equation the ctefficient a is a value o f static susceptibility. I f friction forces w ould not exist in system, then a coefficient b = 0. The static rigidity is a characteristic o f element, m oreover this is a ratio o f relief gain to the displacem ent gain (which is m easured in certain direction). B ased on coefficient a (from linear function), the static stiffness j was indicated. Because the static rigidity is a opposite o f susceptibility. F igure 4 depicts change o f workpiece m ovem ent yx in function o f load F (during burdening and disburdening in term s o f 0 - 2000 N) in the three position (A,B,C).

a)Force/ [N]

SO_ 70I 60• Î 50

1 JO I 30

1Q 10

0

stiffnessj _ B = 34 N-jinr1

0 500 1000 1500 2000Force / [X]

b)

Stiffness analysis o f clamping workpiece in turning process 315

C)Fig. 4. Change of workpiece movementyx in function of load F: a) in A position, b) in B position, c)

in C position (own elaboration)

The value o f rigidity j in every place o f shaft is indicated by diagram s (fig. 4). N everthe­less in this w ork the position w ith the low est stiffness o f workpiece w as inspected. The p ri­mary objective o f turning large-size shafts is to determ ine the changes in stiffness. F igure 5 shows the com parative analysis o f rigidity in every extreme point. Every m easurem ent posi­tion w as placed in different distance lx from three-jaw chuck.

Fig. 5. Comparative analysis of rigidity in every measurement point (own elaboration)

Research results show the highest static stiffness in C position. This point is m easured in the nearest position o f three-jaw chuck. It has been linked to relevant gripping force. H ow ­ever, shaft loses its stiffness near the tailstock (in A position). This is a drop o f 48% in com ­parison w ith C position. This affect in particular on m achining thoroughness. A long the whole shaft length, the surface roughness is also changing (w ith constants turning parameters).

316

In this conjunction it is good to rem em ber w hen low surface roughness is required. M ost experim ents have been found that the larger the stiffness, the low er roughness o f workpiece. This result from the fact that apart from spindle, the rigidity is higher than beside tailstock.

A fter the conducted tests, the stiffness o f subsystem after turning w as determined. The d is­placem ent w as m easured in the same positions A, B, C as before. A fter ten m achined cuts, m easurem ent procedure o f deflection w as repeated. F igure 6 depicts change o f workpiece deflection under the action o f load F in A and B position.

O n the basis o f the presented results, one can observe that these relations are sim ilar to analysis in Figure 3. It can be noted that the bigger the distance o f spindle, the loss o f stiff­ness is lower. The m easurem ent procedure w as repeated again, after fifteen m achined cuts. The com parison o f rigidity in A position (on the next point o f tailstock) w as the last step of researchers.

Martyna Wiciak, Agata Felusiak, Piotr Kieruj, Piotr Szablewski

Fig. 6. Change of workpiece displacement under the action of load F: a) A position, b) B position (own elaboration)

The stiffness o f wokrpiece - turning turret subsystem in point as far away from spindle changed only slightly. The next m achined cuts is not influenced by system rigidity in A posi­tion. Figure 7 depicts change o f shaft displacem ent after fifteen m achined passes.

Stiffness analysis o f clamping workpiece in turning process 317

Fig. 7. Change of workpiece displacement after fifteen machined passes under the action of load Fin A position (own elaboration)

Additionally, during turning the radial deflection o f workpiece w as m easured using pho­toelectric length sensor optoNCDT ILD1700-10 LL M icro-Epsilon. The m easurem ent range o f sensor - 10 mm, having a m easurem ent accuracy o f 0,5 pm. Type o f ILD sensor has num er­ous value o f sampling. Therefore, it is perfect fo r displacem ent evaluation in dynam ics proc­esses. M easurem ent range must be adapted accordingly before experiments.

A t this stage, the displacem ent was obtained in different rotational speed n: 450, 600, 800, 1000 and 1200 rpm. Figure 8 depicts influence o f rotational speed n on radial displacem ent yx.

O n the basis o f the presented graph, one can observe that displacem ent increases w ith an increasing o f rotational speed. Therefore, the doubling rotational speed n (from 600 to 1200 rpm) leads to m ore than 50% o f radial displacem ent growth. This leads to reduction o f static stiffness in H igh Speed Cutting.

Fig. 8. Relations between rotational speed n and radial displacement y. (own elaboration)

318

Finally, the stiffness o f sem i-finished shaft and workpiece after turning was compared. F igure 9 show rigidity o f system in A position in different turning phases.

Martyna Wiciak, Agata Felusiak, Piotr Kieruj, Piotr Szablewski

Fig. 9. Comparison of system rigidity for different turning phases [own elaboration]

The difference betw een various system rigidity is not numerous. The prelim inary m achin­ing has not influence o f subsystem stiffness on the point o f the m ost workpiece deflection, value o f rigidity changes only slightly. This is due to the ratio o f static stiffness to workpiece diameter. Change o f diam eter in short range o f values affects the change o f stiffness is unno­ticeable.

Conclusions and summary

B ased on this research, the follow ing conclusions were drawn:• The subsystem has the m inim um stiffness in the nearest position o f tailstock. Despite

the workpiece w as supported by tailstock, in this point the subsystem lost its rigidity mainly. I f we com pare a result o f shaft rigidity in C position (near the spindle) w ith stiff­ness in A position (near the tailstock), one can observe losses o f A point rigidity by alm ost 48%. O n the basis o f the provide exam ple it can be seen that the longer workpiece is, the place near tailstock support is more burden. It is im portant in turning o f large-size shafts o f high density like W aspaloy (p = 8138 kg-m-3).

• Research shows that the prelim inary m achining has not influence o f subsystem stiffness on the next point o f tailstock. Value o f rigidity in the m ost burdened part o f shaft changes only slightly. Rigidity o f system in A position in different turning phases oscillates around 20 N -gm -1.

• O n the basis o f the presented investigation, one can observe that displacem ent increases w ith an increasing o f rotational speed n. This is lead to reduction o f static stiffness in H igh Speed Cutting.

• F in ish m achining o f long shafts is highly problem atic. W hen the workpiece stiffness is changing along the w hole length, the surface roughness is also changing. In this con­junction it is good to rem em ber w hen low surface roughness is required.

Stiffness analysis o f clamping workpiece in turning process 319

AcknowledgmentsThis scientific research w ork is supported by National Centre fo r Research and D evelop­

m ent (NCBR) o f Poland grant A pplied Research Program No. PBS 244445.

Bibliography

A lquraan T., Kuznetsov Yu., Tsvyd T., H igh-speed clamping mechanism o f the C N C lathe with compensation o f centrifugal forces, International Conference on Industrial E ngi­neering, ICIE 2016, vol. 150, p. 6 8 9 -695 .

Bartkow ska A., Kukliński M., Kieruj P., The influence o f laser heat treatm ent on the geo­m etric structure o f the surface and condition o f the surface layer and selected properties o f Waspaloy, „M ATEC Web o f Conference” 2017, vol. 121, p. 03006.

Estrem s M., A rizm endi M., Cum bicus W. E., López A., M easurem ent o f clamping fo rces in a 3 ja w chuck through an instrum ented A lum inium ring, Procedia Engineering, The M anufacturing Engineering Society International Conference, M ESIC 2015, vol. 132, p. 4 5 6 -463 .

Feng P. F., W u Z.J., Yu D. W., U hlm ann E., A n im proved computation m odel fo r critical bending force o f three-jaw chucks, „Journal o f M aterials Processing Technology” 2008, vol. 208, p. 124-129.

Feng P. F., Yu D.W., W u Z. J., U hlm ann E., Jaw -chuck stiffness and its influence on dynamic clamping force during high-speed turning, „International Journal o f M achine Tools & M anufacture” 2008, vol. 48, p. 1268-1275.

K olluru K., Axinte D., Coupled interaction o f dynam ic responses o f tool and workpiece in thin w all milling, „Journal o f M aterials Processing Technology” 2013, vol. 213, p. 565-1574.

M urcinková Z., Vasilko K., The real shape o f the workpiece after turning and milling, „Journal o f P roduction Engineering” 2013, vol. 16, p. 3 3 -3 6 .

Otto A., K hasaw neh F. A., Radons G., Position-dependent stability analysis o f turning with tool and workpiece compliance, „The International Journal o f A dvanced M anufacturing Technology” 2015, vol. 79, p. 1453-1463.

Przestacki D., Chwalczuk T., The analysis o f surface topography during turning o f Wasp- aloy with the application o f response surface m ethod , MATEC Web o f Conference 2017, vol. 136, p. 02006.

Salgado M. A., López de Lacalle L. N., Lam ikiz A., M uñoa J., Sánchez J. A., Evaluation o f the stiffness chain on the deflection o f end-m ills under cutting forces, „International Journal o f M achine Tools & M anufacture” 2005, 45, p. 7 2 7 -739 .

Seghedin N., H orodinca M., Chitariu D., D rosescu R., 2014, Dynam ic B ehavior o f Long Turned W orkpieces, „A pplied M echanics and M aterials” , vol. 657, p. 7 8 -8 2 .

Sharana Basavaraja J., Sayad M Shanawaz M ujawar, M odeling, simulation and analysis o f gripping fo rce loss in high speed pow er chuck , P rocedia M aterials Science, Inter­national Conference on Advances in M anufacturing and M aterials Engineering 2014, A M M E 2014, vol. 5, p. 1417-1423.

320

Twardowski P., Tabaszewski M., W ojciechowski S., Turning process m onitoring o f internal combustion engine p is to n ’s cylindrical surface, „M ATEC Web o f Conference” 2017, vol. 112, p. 10002.

U hlm ann E., Feng P.F., D oll S., D ynam ic clamping fo rce fo r jaw -chucks fo r high speed cutting, „Production Engineering” 2004, 11 (1), p. 149-152.

Martyna Wiciak, Agata Felusiak, Piotr Kieruj, Piotr Szablewski

STRESZCZENIE

M artyna Wiciak, A gata Felusiak, P iotr K ieruj, P iotr Szablewski

Analiza sztywności zamocowania półfabrykatu na różnych etapach procesu toczenia

Ważnym czynnikiem przy obróbce skrawaniem jest osiągnięcie możliwie największej dokładności. Na dokładność procesu wpływa przede wszystkim statyka i dynamika systemu22. Sztywność układu OUPN (obrabiarka - uchwyt - przedmiot obrabiany - narzędzie) ma decydujące znaczenie w procesie skrawania. Dynamika systemu ob­róbkowego związana jest z jego poszczególnymi elementami, tj. dynamiką maszyny, narzędzia oraz mocowania przedmiotu obrabianego. W procesie skrawania ważnym aspektem jest sztywność subsystemu: przedmiot obrabiany - imak narzędziowy23. W pracy przeprowadzono analizę sztywności zamocowania przedmiotu obrabianego wykonanego ze stopu Waspaloy. Wał zamocowano w hydraulicznym uchwycie trój- szczękowym. Pomiar sztywności statycznej był przeprowadzany poprzez odkształcanie układu przedmiot - głowica rewolwerowa w kierunku promieniowym wału (wzdłuż osi X), na różnych etapach procesu toczenia. Na podstawie obciążania i odciążania układu w zakresie 0 - 2000 N możliwe było wyznaczenie charakterystyki ugięcia w funkcji obciążenia yx = f(F). Celem badań było wskazanie, na jakim etapie obróbki układ znacząco traci swoją sztywność. Dodatkowo określono wpływ ilości przepro­wadzonych przejść, po których element obrabiany utraci sztywność w wyniku zużycia ostrza skrawającego.

Słowa kluczowe: toczenie, sztywność statyczna, mocowanie w uchwycie trójszczękowym.

D ata w pływ u artykułu: 15.03.2018 r. D ata akceptacji artykułu: 15.05.2018 r.

22 N. Seghedin, M. Horodinca, D. Chitariu, R. Drosescu, op. cit.23 Z. Murcinkova, K. Vasilko, op. cit.