The influence of the carriage speed speed on the compliance of the tool-holder Kals, H.J.J.; Hoogenboom, A.J.

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The ifluece of the carriage speed speed o the compliace of the tool-holder Kals, HJJ; Hoogeboom, AJ Published: 111969 Documet Versio Publisher s PDF, also kow as Versio of Record (icludes fial page,

1 The ifluece of the carriage speed speed o the compliace of the tool-holder Kals, HJJ; Hoogeboom, AJ Published: Documet Versio Publisher s PDF, also kow as Versio of Record (icludes fial page, issue ad volume umbers) Please check the documet versio of this publicatio: A submitted mauscript is the author's versio of the article upo submissio ad before peer-review There ca be importat differeces betwee the submitted versio ad the official published versio of record People iterested i the research are advised to cotact the author for the fial versio of the publicatio, or visit the DO to the publisher's website The fial author versio ad the galley proof are versios of the publicatio after peer review The fial published versio features the fial layout of the paper icludig the volume, issue ad page umbers Lik to publicatio Citatio for published versio (APA): Kals, H J J, & Hoogeboom, A J (1969) The ifluece of the carriage speed speed o the compliace of the tool-holder (TH Eidhove Afd Werktuigbouwkude, Laboratorium voor mechaische techologie e werkplaatstechiek : WT rapporte; Vol WT227) Eidhove: Techische Hogeschool Eidhove Geeral rights Copyright ad moral rights for the publicatios made accessible i the public portal are retaied by the authors ador other copyright owers ad it is a coditio of accessig publicatios that users recogise ad abide by the legal requiremets associated with these rights Users may dowload ad prit oe copy of ay publicatio from the public portal for the purpose of private study or research You may ot further distribute the material or use it for ay profit-makig activity or commercial gai You may freely distribute the URL idetifyig the publicatio i the public portal? Take dow policy f you believe that this documet breaches copyright please cotact us providig details, ad we will remove access to the work immediately ad ivestigate your claim Dowload date: 19 Ja 219

2 techische hogeschool eidhove laboratorium voor mec:haisc:he tec:hojogie e werkplaatstec:hiek rapport va de sectie: Verspaigstechologie titel: The ifluece of the carriage speed o the compliace of the tool-holder biz 1 va2 biz rapport r 227 coderig: P7c auteur{s): sec:tieleider: ir HJJ Kala ir AJ Hoogeboom drir ACH va der Wolf trefwoord: Stabiliteit Gereedschapswerktuige hoogleraar: samevattig prof dr FC Veestra Summary The effect of the motio of the carriage at the compliace of the Peter - Vaherck test rig has bee measured for several coditios A qualitative eplaatio of the pheomeo is give Actually this eplaatio is based o the reslts computer eperimets _H of aalogue progose datum:,,, aatal biz 2 gesc:hikt voor publ ic:atie i: ote to be preseted to ClRP Group Ma

3 rapport r" 227 biz 2 va2 blzl - troductio 1-15 r- Displacemet measuremet of the testrig ad of the carriage as well are carried out durig harmoic ecitatio of the test-rig With the aid of the results of these measuremets some coclusios are made These coclusios lead to a simple mathematical model, which has bee ivestigated o aalogue computer The computer results are compared with the origial eperimetal data H werkplaatstechfek techische hogeschool eidhove

4 rapport r 227 biz 3 va2 blzl o 1 Eperimets 5 11 Program of Measuremets 1Q The followig measuremets are carried out See Fig 1 a) The modulus of the trasferfuctio ;y (strai gauges measuremet) at v a ad v 2 ms for three values of the dampig ratio 1: the tes trig viz Case, Case ad Case See Figs 2, 3,ad 4 b) The amplitude ratio l measured at atural frequecy a of the testrig (155 Hz) as a fucyio of the carriage speed also for the three cases metioed See Fig 5 c) The absolute amplitude l measured at atural frequecy for v - a ad v = 2 ms See Table also for the cases,, ad d) The absolute amplitude yl measured at atural frequecy for v - a ad v 2 ms See Table also for the three cases 5 werkplaatstechjek tchisch hogschool eidhove

5 rapport t 227 bll 4 va2 blzl 1-12 Remarks lq - 15 r- a) A compariso of absolute motio l ad the relative motio -y shows a same tedecy with respect to decrease of the amplitude See Table ad the Figs 2, 3, ad 4 b) The frequecy at which the maimum amplitude of Xpl occurs (about atural frequecy) does ot chage with a varyig carriage speed See Figs 2, 3, ad 4 c) The low frequecy compliace ( 4 Hz) is ot iflueced by a chage i the carriage speed See Figs 2, 3, ad 4 d) A icrease of carriage speed V results i a decrease of the testrig motio ad at the same time a icrease of the carriage motio y See Table 3-13 otes a) The i Table metioed y data for v = are measured i a absolute way As a matter of fact, there is o relative motio betwee carriage ad frame i this case b) However, at v - 2 ms the frame movemet is close to zero so that the absolute movemet y equas the relative displacemet betwee bed ad carriage we,kplaatltechlek techische hogeschool eidhove

6 rapport r 227 biz 5 va2 blzl - 14 Coclusios The movemet of the carriage iflueces the couplig of carriage ad frame - v = O The harmoic forces betwee carriage ad frame are 1 r ' r- 4 r- ot large eough to eceed Coulomb-frictio forces i order to cause a relative displacemet y betwee carriage ad frame The couplig betwee the carriage ad frame is rigid See ote 13a - v = large ow, we have to deal with a relative velocity betwee carriage ad frame The Coulomb-frictio trasformes ito a viscous frictio The couplig betwee carriage ad frame is viscous ad almost idepeded o the carriage speed See ote 13b - v = small The velocity amplitude of the carriage is equal or larger tha the omial speed V this case the carriage will periodically stad still The couplig betwee carriage ad frame is periodically rigid ad viscous, ad depeds upo the carriage speed So we ca divide the velocity rage i two parts See Fig 5 the followig the situatio of a very loose ad a rigid couplig of carriage ad frame is simulated by varyig the quatity p This quatity represets the viscous frictio y betwee carriage ad frame z) Z) The viscous couplig betweecarriage ad frame is also idepeded 45 - o the preload That is the reso why the dyamic behaviour will ot be iflueced by lockig the carriage at a certai speed 5 -- werkplaatltec:hlek techische hogeschool eidhove

7 5Otrapport r 227 bll 6 va 2 bill 2 Mathematical Model 21 Differetial equatios lo f Of- the followig part a very simplified model of the combiatio test rig-carriage-frame is formuhted There are two simultaeous differetial equatios which describe the displacemets ad y The solutio of these equatios is formed with the aid of a aalogue computer m + p (-y) + c (-y) - r P (-y) + c (-y) - m y+ p y y y 25- where m mass of the testrig 3- m y P mass of the carriage dampig cos tat:b the ::ttis;t rig P y clampig costari "betwee cardage ad bed 351- c stiffess of the testrig See Fig 1 Substitute of c 2 2S c 2S c y W - - P m W Py w ad m _l:;,; m y k m where werkplaatatechlek techische hogeschool eidhove

8 rapport r 227 biz 7 va 2 blzl - S dampig ratio of the test rig, reduced dampig ratio of the carriage, gives 1Q r- ] 26 p C') + (-y) ad y -- til C til X X (-y) + (-y) y : Y til til til 2k m lsi- 3Ot bte- werkplaatstechfek techlsche hogeschool eidhove

9 rapportr 227 biz 8 va 2 bll 1 22 Dimesioless Computer Eguatios f lq - The quatities,, i X, y, Y ad yare related to their maimum values, m ' m ' m Ym' Y m ad Y m respectively case of a harmoic ecitatio, with the maimum circular frequecy W t the followig relatio betwee these maimum ma values will eist: - W m ma m 2 W m ma m 2 - Let us assume m - y m ow the eqs modify ito: 2 W W P c W W [( ) - ( )] () ==- ( ) -' { 213 ( ) m m,ma ma m Y m 2 + (_:_)' [( >_(}-)]} ma m m ad The aalogue computer circuit is give i Fig werkplaattec:hlek techische hog school eidhove

10 r , SOlrapport r 227 biz 9 vall2 biz - 23 Aalogue Eperimet 1Q - Most of the parameters of both eqs of B y W We assume --- 5, k -,16, S - 1 w ma m y are kow ecept the value ad a - 3 y Prosram of Measuremets The followig measuremets are carried out: 2S - The modulus of the trasferfuctio -Y p for B 1 ad By - 3 for three y values of B viz Case, Case,ad Case 3 - See ad compare Fig 7, 8, ad 9 with Fig 2, 3, ad 4 respectively, ote t ca be proved that the observed effects eist is a wide rage of values Bad k Y m werkplaatltec:h lek techische hogeschool eidhove

11 rapport r 227 biz 1 a va 2 blzl - 3 Coclusios 1Q r After comparig the eperimetal ad'computed data, the mathematical model seems to be fairly good The ifluece of the carriage speed o the trasferfuctio may be decreased by addig mass to the carriage, adjustig backlash betwee carrige viscosity ad frame or usig a lubricate with a high We have to otice that the system is oliear Therefore the dyamic behaviour will deped o the amplitude of the ecitig force Usig the test rig characteristics for computig the stability limit-values we must keep i mid either this oliearity as the velocity ifluece 5 - werkplaattechlek techische hogeschool eidhove

12 rapport r 227 biz 11 va2 b,zl o 5 1Q p testrig 1----""" X m v m y y 3 35 frame Fig 1 Testrig mouted o the carriage 4S 5 werkplaatltechlek techische hogeschool eidhove

13 L- :E O,4m "C D : ;- : Ct t tj: '! t:= c;: ;; Ct Ct a "C 1 :: : '" 4- f; y l _ ::' ::: ft' ::' ::' f) ::' 9 -, ::: a : «> ::: 2 Hz Fig 2 Trasfer fuctio, case, -- -L- < DO :: o 5!:,

14 4 ll: '111:" "Q Q a f :s - '111:" ; y -o Col Ct 3 "Q 1 :s "'-J it :: ' CO t 2- :i* a -' 1 :J Fig_) Trasferfuctio, \ V = ' J f, ",---, p" V 2"msec\ case! " 2 Hz S!:: f'" - - w <» re S!:: f'"

15 :!t ljlu 71:" 4 " -ir D - ;;r t t t to: :;: 1:1 '" 1 1 T a "!:: V = ; \ - Ct :: r' ;- 71:" "-J 4 \ - \ f \' J ; Y f ;- : ::a : C: : :f? ; V - \\ pi! v = 21lmsec: \ ' \!" e -----!2: C: : C: 1:'1 :i CL :>c : 1» 2 Hz ::: <, ::a ) Fig 4 Trasferfuctio, case!2: 1:'1 - --

16 !C :1:" "' D D - ft :: ;- :1:" ;- ::s =- i' =- =- CQ Ct : 2- cd ::s to r X - X f-, P f this part: r----- V ' case r------r y:> v case case '" 16, ' J 1 1 Fig 5 Amplitude ratio X;X at the atural frequecy versus carriage-speed 32 llmsec a "' - "'-J S!:, -"", < := Er Q ::s- "4- S!:,

17 rapport r 227 biz 16 vao2 blzl case 1 The sigals are to small for accurate measuremets 1 r- case 15 r- V " V = 2llmsec 45 mv 36 mv 2 - 'y 1,5 mv 3,2 mv 25 - case V V 2tJmsec 3 r- - 9 mv 66 mv ely 2,2 mv 3,7 mv 35 r- y, 44 llm Table 1 Eperimet data Or- 5Orwrkplaatstechlek techische hogeschool eidhove

18 " Q a & ':T : i" - + C> - X- Y m Y m - to) C> 1ft Y m D Q " " :::, a:- ::s iii" et rv C et t 2- et 5r + - m Fig6 Aalogue computer circuit Q 6 < et ::s 2 7 m Potetiometers til k 'l m til ma 26 til 82- til ma 1 til + til :: ma o 5!: J' k m - m,

19 : "Q '" ) iii"" D - if, : - '" J - co a "Q! : S' :T :s ;;" : :T ca to :T 2- :f Q : < to roo ----'" _-'- y 1 :s 6 = 1 y L -'-- B = 3 Fig 7 Aalogue computer trasferfuctio, case ---,!,;;: 2 Hz ----!2:!" -C1:> < D> ::!2:!" ---'

20 o, 1' :E a 't 't " D! D :::: '" : Go :::: i" Jt'" s -y p ey= 1 V-,," ' \ V" y : :s :,, : -- CJ ", -- ; ' a = 3 " -, i -- : - 2- S!:!'" -\D Q : < > 1 2 Hz :::s :s Fig 8 Aalogue computer trasferfuctio, case S!:!'" ----l

21 o -- - a J " D! a - " - :: \ 1\ i" 1 S' :' :s fi' -y P )," ' a y = \ i,"" i lay = 3,,! j, \ \ ", :' ",", :', cc til --- -"- :' - V,, ' g, 5!: t" :' < > 1, 2 Hz ;:s :s Fig 9 Aalogue computer trasferfuctio, case 5!: t" --l! i! i " ",

On a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G.

On a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G. O a problm of J. d Graaf coctd with algbras of uboudd oprators d Bruij, N.G. Publishd: 01/01/1984 Documt Vrsio Publishr s PDF, also kow as Vrsio of Rcord (icluds fial pag, issu ad volum umbrs) Plas chck