The analysis of brushing tool characteristics

Transcription

1 ARCHIVES OF CIVIL AND MECHANICAL ENGINEERING Vol. IV 004 No. 4 The analysis of rushing tool harateristis Kiele University of Tehnology, al. Tysiąleia P. P. 7, 5-34 Kiele In this paper, an analytial proedure is developed in order to evaluate the filament loading of a irular rush. Filament deformation is omputed ased on the mehani analysis in onjuntion with kinemati onstraints for a rigid flat surfae with frition taken into aount. Numerial results whih reveal the relationship etween rotation angle and fore distriution are reported. Keywords: flexile eletrode, filament, interation fores. Introdution So far the mahining proess using filamentary metal rushes in the shape of disks has een used in surfae mahining to remove orroded layers, to prepare metal surfaes to e galvanized, and to produe surfaes of high adhesion to e oated with paint, glue, et. Reently the proess has een developed to inlude operations suh as removing sharp edges and urrs [,0], flashes and osses from mahine parts made of alloys of non-ferrous metals, as well as leaning welds. Using rushes with densely paked filaments made of hard steel roadens the range of uses to inlude the miromilling of ordinary onstrutional steels of low hardness, whih are mahined with the tips of the filaments. To summarize, the typial uses of metal rush tools are limited to mahining materials whose hardness is lower than that of the material the filaments of the rush are made of. Using the filaments made of arasive-grain-filled polymers allows the rushes to e used to mahine the surfaes of materials of high hardness. On analysis of the advantages of using rush tools the author suggests a new mahining operation that omines mehanial, eletrohemial, and eletro-erosive proesses ating on the mahined item [3 6]. Due to the synergisti effet this type of hyrid mahining makes the metal removal proess more ost-effetive. Soft-mahining parameters allow not only the removal of the exess material from large items of low stiffness ut also the highly effiient volumetri mahining of metals, alloys, and ondutor-ased omposites. The numerous uses of rush eletrodes result from suh their harateristis as: flexiility of individual filaments, type, shape, and paking density of the filaments, possiility of operating at various settings of eletrode defletion,

2 8 large ontat zone of the tool and the mahined item, possiility of using the eletrode until it is worn out, large working area of the hot eletrode allowing the mahining of oth flat and omplex-shape items. Beause of their onstrution rush eletrodes are haraterized y: uniform distriution of the filaments, whih helps to form a disrete struture suitale for maintaining stale onditions in the mahining zone, radial, axial, and tangential flexiility, whih makes the filaments fit easily omplex geometry surfaes, thus permitting a uniform removal of surfae layers without hanging signifiantly the geometry of the mahined part, easy disposal of the erosion y produts from the disharge zone, suitaility for automated operations. The use of rushing tools in an automation environment will neessitate a lear understanding of an important harateristis of rush performane suh as fores. An understanding of suh harateristis is important, as surfae preparation proesses require a detailed knowledge of interrelationships etween produtivity of mahining and rush operating onditions [3, 7]. For example, it is reognized that eletrial disharges generated during eletroerosion-mehanial proesses are losely related to the mehanial harateristis of the filament [5].. Statis and kinematis of a single filament Sine the elements of a disk rush tend to deform easily, the use of the rush in erosion mehanial mahining hanges the harater of mehanial interations with the mahined surfae in ontrast to deformation-resistant eletrodes. An inrease in the value of the pressure fore at the filament tip as a funtion of displaement along the surfae inevitaly leads to a reak in the anodi film and initiates disharges whose frequeny an e determined, among others, y the virations of individual filaments of the eletrode. The mehanis of the movement and the interations etween the filament wire and the mahined surfae are very omplex. The wire eomes deformed in a way that is diffiult to analyse. This is aused y onfounded oundary onditions whih allow only an approximate solution to the equation of its motion. Only a tentative analysis of the interations etween the rush elements and the surfae has een presented. Let us onsider a tentative analysis of a filament load. The asi assumptions are: inertial fores are negleted, the filament tip moves along a rigid surfae. Additionally, due to low paking density, interations etween individual wires are ignored. It is assumed that the filaments are plaed radially from the hu entre and are restrained at the hu outside radius and oey Hooke s law [8].

3 The analysis of rushing tool harateristis 9 The filaments are straight efore they ome into ontat with the mahined surfae. They are defleted perpendiularly to the axis of rotation, with the radial run-out of the disks eing ignored. Filament defletion is examined in a moile referene system K ξ η (Figure ), where η = η (ξ) is its elasti defletion assuming that there is no influene of non-dilatational strain. ω0 r a l K O η η ξ S α α 0 h f x ξ O y η K P y P η P x P ξ x ξ y Fig.. Geometry of a partiular filament deformation The differential equation of the ending line is: η EI = F ( ξ ) [ η( ) η( ξ )], 3 / η + Fξ () ( + η ) EI filament flexural rigidity, η ( ξ ) = d η / dξ,, η ( ξ ) = d η / dξ 0 ξ. The geometry of the prolem examined produes the following relationships: h = f / sinα ;

4 30 a + r = r + hosα = d + hosα, sinα a + r d = r. sinα We assume that: F y = µ F x, Fξ = F x, Fη = F x, = sinα µ osα, = osα + µ sinα, α = ω 0 t, µ oeffiient of frition etween the filament tip and the mahined surfae, ω 0 angular veloity of the rush. The funtion η(ξ ) should also satisfy the following ondition: l 0 + [ η ( ξ )] dξ = 0. () It is very diffiult to otain numerial solutions for Equation () with onstraint (). Analytial solutions an e otained if the values of η(ξ ) are small enough to enale the linearization of the left-hand side of Equation (). The details of the solution of Equation () with initial onditions: η ( 0) = 0 and η '(0) = 0 and with the assumption that the wire tip (for ξ = ) has point ontat with the surfae (then η"() = 0)) have een presented elow. We will examine a ase of a single filament load under tentative onditions presented aove. The defletion of the part is desried in a moile referene system K ξ η (Figure ). In suh a ase, η = η(ξ ) is its elasti defletion with the assumption that there is no influene of non-dilatational strain.

5 The analysis of rushing tool harateristis 3 An approximate analytial solution an e otained if we assume that the values of η '( ξ ) are small enough to enale the linearization of the left-hand side of Equation (), whih is the ase l a l <<. Then we assume that: η '( ξ ) <<. Consequently, in plae of Equations (), () we an have EI 3 η ( ξ ) η ( ξ ) = [ C ( ξ ) + C ( f η ) F x, ] (a) [ η ( ξ) ] d = 0, l ξ 0 (a) f = η(). As a result of a susequent approximation the aove equations are replaed y: [ ( ξ ) + ( f η ] Fx EI η = ) (3a) 0 [ η ( ξ )] d ξ = l. (3) Furthermore, we will onsider solutions for Equations (3) valid only if l a is small. The first equation is as follows: Fx [ d + ( tanα + ) f ξ η ] EI η = and we assign: F ω = x, EI

6 3 F x EI thus ω =, η + ω = eause [ d + ( tanα + ) f ξ η] ω ξ, tanα + = / sinα, so η + ω η = D ω ξ, (4) ω ω D = ( d + h) = onst (does not depend on ξ). The solution to Equation (4) with initial onditions: η ( 0) = 0 and η '(0) = 0 and with the wire tip (ξ = ) having point ontat with the surfae (then η"() = 0) is: ( α) η ( ξ ) = [( osωξ ) tanω ( ωξ + sinωξ )], (5) ( α) ω ω = ( α) F EI x. The prolem is intratale eause: ω is unknown (dependent on the unknown F x ) and is unknown (dependent on h or f ).

7 The analysis of rushing tool harateristis 33 In (4), we should require for ξ = to e η () = f = h sinα, then we otain: ( α) hsinα = = ( α) ω [( osωξ ) tanω ( ωξ + sinωξ )] η( ). Thus, after employing geometri relationships, we otain the following equation: d(tanω ω ) h =. (6) ω osα tanω Condition (4) will e satisfied after employing (5), so the equation is rewritten as: osω( ξ ) η ( ξ ) =, osω then tanω ω [ η ( ξ )] dξ = Employing os ω a + r = r + hosα = d + hosα sinα we otain: a + r tanω + r hosα = 3. sin + os α ω ω (7) Sustituting (6) for (7) we otain a transendental equation with the unknowns ω. Only the lowest roots of the equation alulated as a funtion of α are physially feasile. These roots enale the value of ω to e alulated. It applies to all the omponents of the interation fore, that is: F x EIω =, F y = µ F x,

8 34 F = F x ξ, F η = F, x as well as to the ending line for η(ξ ). Fig.. Changes in the parameter ω as a funtion of the rotation angle and filament flexural rigidity Figure shows the hanges in the parameter ω as a funtion of α = ω 0 t and filament flexural rigidity EI at the following parameters: l = 0.05 m, a = 0.04 m, r = 0.0 m, µ = 0.5. Fig. 3. The values of fore omponent F x, interation with the surfae as a funtion of rotation angle α and filament flexural rigidity EI

9 The analysis of rushing tool harateristis 35 Fig. 4. The values of fore omponent F y, interation with the surfae as a funtion of rotation angle α and filament flexural rigidity EI Graphs of the hanges in the values of fore omponents F x, F y as a funtion of the rotation angle and filament flexural rigidity EI have een presented in Figures 3 4. Graphs of the hanges in the values of fore omponents Fξ, Fη as a funtion of the rotation angle and filament flexural rigidity EI have een presented in Figures 5 6. Fig. 5. The values of fore omponent F ξ, interation with the surfae as a funtion of rotation angle α and filament flexural rigidity EI

10 36 Fig. 6. The values of fore omponent F η, interation with the surfae as a funtion of rotation angle α and filament flexural rigidity EI The hanges in the relationship F η as a funtion of the angle α = α(t) have to e pointed out. At α 6 o the sign of the fore is reversed. Consequently, the fore auses the filament to straighten when it loses ontat with the mahined surfae. 3. Dynamis of a single filament of a irular filamentary rush The equation of the motion of a filament with its mass taken into aount an e shown y oordinates K ξ η as an equation [] desriing relative motion. We assume that η(ξ, t) << and omit Coriolis inertial fores whih are negligily small in this ase in order to otain: 4 η η η EI + Fξ ρ η δ ξ ρ ξ) & α + A = F ( ) + A( r +. (8) 4 ξ ξ t It an e shown that when inertial fores are negleted the equation an e rewritten as (3a). Fores F ξ and F η are marked F ξ and F η, respetively, eause these are not the same fores as in the previous expressions. Solutions have to e looked for with oundary onditions eing:

11 The analysis of rushing tool harateristis 37 η for ξ ( 0) = 0 we have η ( 0, t) = 0 and ξ = 0= 0, ξ η for ξ = (t) we have ( ) = 0 ξ = t ξ and initial onditions: η η ( ξ,0) = 0, t= 0 = 0. t 3 η η and [ EI + F ] 3 ξ ξ ( t) = Fη ξ ξ =, (9) In addition, the geometri onditions mentioned earlier have to e met as well as ondition (). This oundary-initial prolem annot e solved y onventional methods. It is very hard to otain even approximate numerial solutions for the equation. If we assume that the filaments maintain ontat with the mahined surfae, the following ondition is satisfied: 0 [ η ( ξ) ] dξ = l. (0) It is very hard to otain even approximate solutions for the equation. This oundary-initial prolem annot e solved y onventional methods. Based on the solutions presented aove, the Galerkin approximation was used. At α = ω 0 t the last omponent of Equation (8) disappears. Let us assume that the first approximation is η ( ξ, t) Y ( ξ ) S( t), () the value S(t) desries the shift of the filament tip towards the axis η when is multiplied y Y(), the funtion of Y(ξ ) has een hosen aritrarily; it satisfies the onditions Y(0) = 0 and Y'(0)=0 and will e integrated using the variale limits of 0 (t). As a result we otain an ordinary differential equation ontaining variale oeffiients eause = ( α ) = ( t), m r ( ) S & () t + k ( ) S() t = F Y ( ), & () r η

12 38 m r ) = ρ A Y ( ξ dξ, 0 k r = EI IV II Y ξ ) 0 0 ( ξ ) Y ( ξ ) dξ + F Y ( ξ ) Y ( ξ dξ, whose solution should satisfy ondition (0). Consequently, we otain a system of two equations with two unknowns S(t) and (t). Figure 7 shows a numerial solution for the equation desriing filament tip displaement along the mahined surfae as a funtion of the angle α (eing simultaneously a funtion of time α = ω 0 t) for ρ A = kg/m, that is: r + a y = ( r + )osα 0 S( t) Y ( ) / sin α, (3a) tanα r + a α 0 = arsin. (3) r + l Fig. 7. Dynami displaement of the filament tip along the line of ontat with the surfae as a funtion of the rotation angle of the disk α(t) Equation (3a) an e rewritten as:

13 The analysis of rushing tool harateristis 39 r y = R osα0 S( t) Y ( ) / sinα, tanα R the disk outside radius, the filament radial defletion value applied. Figure 7 shows that the movement of the filament along the mahined surfae is not monotoni. It demonstrates that the influene of the filament dynamis on its load an e quite onsiderale. The paper offers only a rief outline of the prolem whih requires further researh. 4. Conlusions Lower paking densities of filament wires diminish the effet of mutual filament support, thus making the rush more deformation-prone. It makes it possile to adjust the defletion ( ) parameter within a wider range of settings, with the disk retaining its original size. The pressure fore the filament tip exerts on the surfae along the displaement path inreases in a non-linear manner, with its value suddenly dropping towards the end of the displaement path. Upon analysis of the differential equation of a single filament displaement path it an e stated that: hanges of the fore of the filament interations with the surfae are diretly proportional to the hanges of the filament stiffness, thus a solution for (EI ) is also appliale to (EI ), the shapes of the ending line are idential if for a given position of a workpart (speified y the angle α) the values of interation fores (F(α)i) are proportional to the orresponding stiffness values of the elements (EI )i. Referenes [] Duwell E.J., Bloeher U.: Deurring and Surfae Conditioning with Brushes Made with Arasive Loaded Nylon Fier, Soiety of Manufaturing Engineers, Tehnial Paper MR83-684, Dearorn, MI [] Gutowski R., Swietliki W. A.: Dynamika i drgania układów mehaniznyh, PWN, Warszawa, 986. [3] Nowiki B., Spadło S.: Smoothing the surfae y rush eletrodisharge mehanial mahining BEDMM, Central European Exhange Program for University Studies Projet PL-- CEEPUS, Siene Report, Kiele, 998, pp [4] Spadło S.: Complex Shape Surfae Finishing Proess, Patent PL 7559, 997.

14 40 [5] Nowiki B., Pierzynowski R., Spadło S.: New Possiilities of Mahining and Eletrodisharge Alloying of Free-Form Surfaes, Journal of Materials Proessing Tehnology, 00, Vol. 09, No. 3, pp [6] Nowiki B., Pierzynowski R., Spadło S.: The Superfiial Layer of Parts Mahined y Brush Eletrodisharge Mehanial Mahining (BEDMM), Proeedings of II International Conferene on Advanes in Prodution Engineering. Part II, Warsaw, June, 00, pp [7] Spadło S.: Experimental Investigations of the Brush Eletrodisharge Mehanial Mahining Proess BEDMM, Advanes in Manufaturing Siene and Tehnology, Quarterly of the Polish Aademy of Sienes, 00, Vol. 5, No. 3, pp [8] Osieki J., Spadło S.: A model of mehanial interations of a rush eletrode with a flat surfae, Pro. of 9th Int. Si. Conf. on Prodution Engineering, Computer Integrated Manufaturing and High Speed Mahining, Croatian Assoiation of Prodution Engineering, Lumarda, 003, pp. IV066 IV07. [9] Timoshenko S.P., Gere J.M.: Theory of Elasti Staility, seond edition, MGraw-Hill, In., London, 96, pp [0] Wik C., Veilleux R.F.: Mehanial and Arasive Deurring and Finishing, SME Tool and Manufaturing Engineers Handook, Chapter 6, Vol. 3, 985. Analiza harakterystyk narzędzi szzotkowyh Przedstawiono analityzne rozwiązanie zagadnienia sił, z jakimi oddziaływują pojedynze włókna szzotki orotowej z powierzhnią. Przeprowadzono analizę deformaji pojedynzyh druików szzotki, uwzględniają występująe więzy kinematyzne dla przypadku powierzhni płaskiej niepodatnej z występowaniem taria. Przedstawiono wyniki symulaji komputerowyh w postai zależnośi sił oddziaływań druików z powierzhnią w funkji kąta orotu szzotki.