Discontinuous Simulation Techniques for Worm Drive Mechanical Systems Dynamics

Transcription

1 Discotiuous Simultio Techiques for Worm Drive Mechicl Systems Dymics Rostyslv Stolyrchuk Stte Scietific d Reserch Istitute of Iformtio Ifrstructure Ntiol Acdemy of Scieces of Ukrie PO Box 5446, Lviv-3, 793, Ukrie Phoe/Fx: E-mil: rostyks@ddiiilvivu Astrct I this pper we cosider pplictio of Two-Sided Wedge Mechism (TSWM) with clerce d frictio cotct s Desig Dymic Model of Worm Drives Mchies The Mthemticl model formultio sed o prtitioig co-cordite method for idetify kiemtic d force costrits together with dymic equtios i closed loop form The residul equtio of motio holds Discotiuities d switchig coditios for simultio cotct trsitios d Force Trsfer Fuctio ccordig to system stte The simultio results re sed o switchig lgorithm which relised step y step itegrtio scheme d cotrol procedure which implemeted i Mtl ODE progrms Itroductio The discotiuities hve presece i costried Multiody System Dymics whe cotct forces reltios etwee some movig mchie prts my e istteously chges t time depedig o systems stte d simultio prmeters Cotctcses modifictio result i discotiuities of correspodig costrits equtios udergoig structure-vrit modellig is give i literture [,,3,4] The mthemticl formultio of such Multiody System (MBS) i residul form hve ee preseted y ODEs with Discotiuities i the right-hd side The systems itegrtio scheme sed o cocept of switchig coditios i order to moitored d otis time poits whe discotiuities re preset i itegrtio step Differet spects d pproches of this umericl prolem re well discussed i [5,6] The force costrits prolem udergoig discotiuity lysis is very ctully for Worm Drives Mechicl System, which use s oject of simultio i these pper It is well kow tht Worm gerig hs wo wide cceptce for idustril drives ecuse of its my dvtges of cojugte tooth ctio, compctess, high-speed reductio rtios d lod-crryig cpcity Sice meshig ctio etwee worm thred d the teeth of the drive worm ger is predomitly slidig where slidig frictio forces i vrile closed loop cotct-cses depeds from vlue d directio of slidig velocity tht hve gret ifluece o result efficiecy of force trsmissio By presece of clerce, which hs usully mufcturig d costructive ture for provides the reltive moility of gerset prts we c expli the trsitios etwee differet cotct-cses idicted y discotiuities i the iterl force cotour This situtio poses the prolem of how to tke dvtges of Multiody Dymics d discotiuous pproch to develop effective Desig d Mthemticl Model for simultio mechicl efficiecy of the Worm Drives Uit Due to the worm geometry tht is of screw thred cofigurtio d orthogol cross-xis of ssemly motio we propose to cosider the Two- Sided Wedge Mechism (fig) s Desig Dymic Model of Worm Drives Mchie All stges of Mthemticl Modellig re presets i follows: Idetifictios d ssumptios prolems for TSWM; Mthemticl formultio of costrits d dymic equtios suitle for computig; Costructio of Switchig Fuctio d Switchig Algorithm; Simultio exmples Model Idetifictio d Iitil Assumptios Cofigurtio of TSWM (fig) is sed o orthogol complemets i terms of prtitio coordites [,] d uder rigid-ody pproch

2 3 F m R ρ m x F S R S Fig Desig Dymic Model We strt y mkig the followig ssumptios: TSWM is represet the Multiody System (MBS) with movig i orthogol complemet d itercoected rigid d iertil Wedge segmets (odies d ) y iclitio surfce; TSWM relized two differet cotct cses o iclitio surfce (-) or (-) with holoomic costrits due to existig clerce; We ssumed tht clerces size i TSWM is so smll, tht llows to ccept shock-free d istteous (zero i time) trsitios from oe cotct cse to other Those trsitios evets chrcterise discotiuities o sttiory rch of model motio; The slidig frictio forces durig ssemly motio of TSWM ssumed oly etwee Wedge segmets o iclitio surfce correspod to frictiol worm kiemtics pir; Cotcts etwee orthogol fixed guides 3 d movig wedge segmets (odies) d re relised idel type of joits where frictio forces re eglected Prmeter Idetifictio: x, x - joit co-ordites (time-depeded ) tht ttched to movig odies d i orthogol complemets They ritrry seprted o oe idepedet (mi) d depedet co-ordites tht eeded to tret virtul ssemly displcemets of movig odies; F, F - exterl ctuted forces pplied to odies d log the co-ordite directios x, x which ssumed s kow systems prmeters; S, - iterl reduced rectio forces s the projectios of rectios R, R which pplied to odies d log the co-ordite directios x, x d ssumed s ukow prmeters They formed i ssemly vrile iterl force x cotour for ech closed-loop cotct-cses with correspodig Force Trsfer Fuctio (FTF); m,m - reduced msses of odies, which re prtitioed y cotct They idicte the ierti elemets of Drives System tht coected with the worm shft d the shft of worm wheel respectively; - iclitio gle of cotct surfce, which idetify led gle of worm; ρ - frictiol gle idictes the vlue of slidig frictio forces, which depedet olierly o the velocity of slidig V S The lyticl descriptio of ρ = f ( V s ) my e preseted y o-lier fuctio ρ ( x& ) = / ( ( x& / cos ) + c), [7], where V S = x& / cos ;,, c - coefficiets, otied from pproximtio of experimetl dt upo ccoutig mechicl properties of cotct mterils Kiemtic costits formultio The kiemtic costrits equtios of TSWM hs geerted from geometricl iterprettio of cotct collisio etwee ssemly movig odies i orthogol directios d writte s x = or x xtg x tg = () The velocity d ccelertio costrit equtios we otied fter doule time derivtio of Eqs ssumig cotct s holoomic costrits x& = x& tg (), & = tg (3) x & The costt vlue mrk s u w = / tg mes stile kiemtic rtio etwee TSWMs movig prts Force costrits d Discotiuity Kerel Formultio We cosider here two force pls represeted o fig d fig3 which hve descrie the reltioships etwee iterl reduced forces S d correspod to ech cotct cses sed o kietosttic lysis d equilirium system stte

3 ρ R +ρ S X N 3 N 3 R Fig Force pl for ctive (-) cotct surfce ρ ρ R X N 3 S R r N 3 Fig3 Force pl for ctive (-) cotct surfce Accordig to those pls we oti force costrits writte s : S ψ = = tg( + ρ), (4) S S ψ = = tg( ρ), (5) S where ψ - the Force Trsfer Fuctio (FTF) mrk y idex for ctive cotct-cse (-) which idetify the type of System structure with direct force/eergy strem d udergoig coditios S >, S < ; ψ - the Force Trsfer Fuctio (FTF) mrk y idex for ctive cotct-cse (-) which idetify the type of System structure with iverse force/eergy strem d udergoig coditios S <, > Ech FTF descrie lyticlly the level of effectiveess of iterl force trsformtio for ech ctive cotct-cses The set of FTFs my e presets y oe prmeter ψ = { ψ j, j =,} d med the Discotiuity Kerel (DK) of Force costrits with correspodig compoets (5) d (6) The DK is piecewise cotiuous fuctio which idicte ve X X jump discotiuities, eg cotct d force trsitios durig dymic process I geerl form the Force costrit equtio hve ee writte s S = ψ, (6) where = {, j =,} ψ - preseted the ψ j Discotiuity Kerel of System which formulted y pir compoets - FTFs For estimtio the level of icresig force strem o systems output we c use followig coefficiets for every cotct-cses: K = / ψ d K = / ψ Mthemticl model d switchig fuctio formultio The dymic equtios of motio re writte i Newtos formultio together with costrits equtios (3),(6) s DAEs : m&& x = F S, m&& x = F S, (7) & = &tg, S = Sψ, ( ψ = { ψ (4), ψ (5) }) These DAEs we re le to reduce to miiml set, ie oe equtio per system degree of freedom, y lgeric elimitig of ukow iterl costrit forces S,S d depedet ccelertio coordite selected s & from dymic equtios The residul equtio of motio with respect to idepedet co-ordites x, x, d DK ψ i the right hd side we otied i the form F Fψ =,( ψ = { ψ j, j =,}), (8) m m ψtg with dded iitil systems coditios x ( ) x, x& () = x& = (8) The umericl tretmet of prolem (8),(8) is complicted y the presece of DK tht eed to itroduce some switchig fuctio d switchig coditios i cojuctio with stdrd step-ystep itegrtio methods [5] Thus we preset the specilly desiged coditios for moitored systems DK y switchig fuctio ψ for ϕ( >, ψ = (9) ψ for ϕ( < where ϕ( t ) = Φ( t, x, x& ) - time d stte depeded Switchig Fuctio (SF) The coditio ϕ ( > descrie system structure d cotct cse type

4 withψ = ψ ; the coditio ϕ ( < - respectively withψ = ψ The coditios uder which the discotiuity occurs o the time step [t i, t i+ ] of curret process c e formulted s ϕ( τ ) = Φ( τ, x ( τ ), x& ( τ )) =, () S ( τ ) = S( τ ) =, () where τ = τ k,( k =,,3), τ [ t i, ti+ ] deotes the Switchig Poits (SPs) if ϕ ( chges its sig o the time scle The qulity () hs stisfied istteous trsitio d criticl stte of system whe cotct-cses chges d iterl forces S, re dectivted tht provide jump chges i the DK Sustituted () i (7) we c oti the Switchig Fuctio of System s lgeric differece etwee ow odies ccelertios ϕ( = ( (, () tg where ( = F ( / m - deotes ow ccelertio of ody ; ( = F ( / m - deotes ow ccelertio of ody I geerl the developed Mthemticl Model with Switchig Fuctio must e solved itertively y usig cotiuous represettio of the umericl solutios x (, x& ( of systems motio ccordig to Switchig Algorithm show elow Switchig Algorithm For simultio discotiuities o itegrtio time step y stdrd computer progrm we costruct the Switchig Algorithm, which is sed o ssumptio of djustig solutio t the ed of the step i which discotiuity ws occurs d fix time τ = t i+ s Discotiuity Poit This ssumed to e implicitly defied the time τ s root of SF o step legth I geerl, the switchig lgorithm works o oe time step i followig optios : Iitiliztio: i=, t i =t, y i =y, ϕ(t i )=ϕ(t ) Check sig ( ϕ ( ti )) d defie DK ψ j ( t i ) 3 Itegrtio of oe step t i t i+ usig f ( t, y, ψ j ) 4 Check sig ( ϕ( t i+ )) o the ed of step 5 If o sig chge, cotiue the itegrtio otherwise loclize the discotiuity t τ t if ϕ ( τ ) < ε = i+ 6 Chge to the ew DK s ψ j ± ( ti+ ) i the right hd side d restrtig the itegrtio o oe step 7 Check coditio t i+ < Ted Τ We put here stte vector s y = ( x, x& ) These Algorithm d Mthemticl Model re implemeted to Mtl ODE 5s solver i Multiody code d cofirm y the simultio results Simultio exmple d results For simultio exmple we use followig prmeters of TSWM which correspod to lift mchie (3 kg lod cpcity) with worm uit Costt prmeters: m =586 kg, m =75 kg, F = 4774 N, =57, tg=, for worm gerig R =,3 m, R =,8 m d kiemtic rtio u=4 Vrile prmeters: electric motor mechicl chrcteristic s exterl force F ( x& ) = Fk ( W + / ), W Vc = 3,35 m/sec, V k =,7 m/sec, F k =4375 N; Frictiol gle ρ ( x& ) = / ( x& / cos ) + c, =,74; =,6; c =, The simultio results re preseted o fig4,5,6,7 d demostrte the ehviour of iterl forces d FTFs usig solutios of systems stte vriles x (, x& (, ( d ψ (, where S = m F ( x& ), = S /ψ The results re for two differet lod cses: ) F >; ) F < Iterl forces (S,S), N ( ) 4 S Time(secs) Fig4 Simultio results for S, i test cse with oe switchig poit with sttes trsitio ST ST

5 Force trsfer fuctio, ψ 5 5 ψ ψ 5 τ 5 5 Time(secs) Fig5Simultio results for ψ i test - cse, Iterl forces (S,S), N τ =,486sec - switchig poit S Time(sec) Fig6 Simultio results for S, i test-cse without switchig poits Refereses Che S, Hse JM, Tortorelli DA Ucoditiolly eergy stle implicit itegrtio: pplictio to muitiody system lysis d desig Itertiol Jourl for Numericl methods i Egieerig ; 48: 79-8 Nikrvesh PE Computer-Aided Alysis of Mechicl Systems Pretice-Hll, Eglewood Clifs NJ, Pfeiffer F Uilterl prolems of dymics Archive of Applied Mechics,69 (999) pp Klisch T, Cotct Mechics i Multiody Systems Multiody System Dymics : , Eich Soeller E, Fuhrer C, Numericl Methods i Multiody Dymics Lud, Corrected reprit of Teuer Stuttgrt, HyJL, Crossie RE, Chpli RI Itegrtio routies for systems with discotiuities The Computer Jourl, Vol7, No3, pp Stolyrchuk R Approximtig of the osttiory chrcteristic of slidig frictio i the worm gerig // News of Teropil Stte Techicl Uiversity Teropil:, Vol5, No3, pp 9-34 (i Ukrii) 4 Force trsfer fuctio, ψ ψ 5 5 Time(sec) Fig7 Simultio results for ψ i test-cse Coclusios The developed Mthemticl Model with switchig coditios d discotiuous simultio techique for lysis iterl force cotour i TSWM with clerce presece show good results d my e used ot oly for Worm Drives Mchies ut lso useful for Mchies with screw joits type