machine design, Vol.3(2011) No.3, ISSN pp

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1 machine deign ol3() No3 ISSN 8-59 pp 5-56 Original cienific paper MATHEMATICAL MODELING OF THE DYNAMIC PROCESSES OF A HIGH ELOCITY FORGING MACHINE alenin ABADJIE * - Emilia ABADJIEA - Dochka PETROA 3 3 Ine of Mechanic Blgarian Academy of Science Sofia Blgaria Received (73); Revied (6); Acceped (69) Abrac: The applicaion of high-velociy hammer deigned o realize differen echnological procee (a a cloeddie forging of eel deail pile amping of oil by pile fondaion ec) ha are conneced wih ing of rocke engine for propelling of he impac device (hammer ram) i a new modern field in indry The paper rea he creaed mahemaical model decribing he dynamic behavior of a rocke impac device in he preparaory phae which one enre an opimal impac inflence on providing of he echnological proce The baic kinemaic and dynamic characeriic of a ypical preparaory phae being before an arbirary echnological proce are defined A concree evalaed experimen i given Key word: high-velociy hammer die forging pile amping mahemaical modeling INTRODUCTION In Blgaria he e of indrial rocke engine for propelling of mechanical hammer ha daed ince 8 of h cenry [4 5] A ha momen he fir Blgarian paen of a machine for a cloed-die forging wih hammer ram rocke propelling wa creaed [6] The die made dring hi period how [] ha he eel piece obained by a high-velociy die forging (abo 6-3m/) have larger drabiliy han analogical one prodced by a mechanical reamen or by forging wih velociie from 7 m/ o 9 m/ I i alo eablihed ha a he forging velociy increae a he lo of fricion beween he inrmen and he mechanical reamen decreae and herefore he reorce of he ed forging inrmen increae In he end of 9 of he la cenry in Blgaria an indrial rocke engine i deigned and manfacred ed for a heoreical and experimenal dy of a high-energeic cloed-die forging [7] A he beginning of he new cenry i i aring an idea o apply one Blgarian paen [8] repreening an eqipmen for pile amping in differen ype of oil and of rronding medim driven by indrial rocke engine The developmen of he high-energeic and high-velociy proce of cloed-die forging and pile amping reqire dying he dynamic behavior of he impac device of indrial rocke hammer The reearch aim a pporing he proce of inrodcing a high-velociy impac in indrial pracice by applying he poibiliy of hammer ram permanen moion by rocke engine wih a racion conrolled dring he whole period of acion The complex heoreical olion of he problem p in he defined reearch reqire working o a e of mahemaical model decribing he dynamic procee enring in an adeqae way concree echnologie ch a: pile amping in oil wih differen phyical characeriic; obaining of forging par wih complex configraion in condiion of maximm accracy of he form and of he dimenion and maximm efficiency of he procee ec The preen paper deal wih he mahemaical model decribing he moion of an impac rocke device verically p by a reacive racion PRINCIPLES ENSURING THE TECHNOLOGICAL PROCESS I i hown (Fig) a cheme of a high-velociy hammer wih an impac device ha cold be p in moion in wo direcion aonomoly by a yem of indrial rocke Fig Doble-acing high-velociy forging machine wih a rocke impac device : - anvil block; -die holder; 3- impac device; 4- indrial rocke engine lifing he impac device 3; 5 hammer body; 6- indrial rocke engine dropping he impac device; 7 hammer ram The working cycle of he rocke impac device coni of wo phae: A preparaory one: for lifing of he impac device p o a given heigh where he velociy of moion become eqal o zero; A working one: for moving he impac device down and faening becae of rocke racion *Correpondence Ahor Addre: Ine of Mechanic Blgarian Academy of Science Acad G Bonchev Sr block 4 3 Sofia Blgaria abadjiev@imbmbabg

2 alenin Abadjiev Emilia Abadjieva Dochka Perova: Mahemaical Modeling of he Dynamic Procee of a High elociy Forging Machine; Machine Deign ol3() No3 ISSN 8-59; pp MATHEMATCAL MODELLING OF THE DYNAMIC PROCESS IN THE PREPARATORY PHASE 3 Fncion of he raigh line moion velociy The aim of he preen reearch i o define mahemaically he raigh-line moion of he rocke impac device in he preparaory phae in rronding medim of waer when he high-velociy forging machine amp pill ino he ea floor In Fig one can ee he dynamic cheme when he rocke device move verically p if n indrial rocke engine work imlaneoly The dy of he dynamic behavior of he rocke device i performed afer he following exploiaion characeriic have been defined [ 3 ]: q nqi conan - fel air mixre conmpion rae; q i con an - fel air mixre conmpion rae for he indrial rocke engine; m m q - remaining ma of he device a an arbirary momen of he died proce; m - oal iniial ma inclding he ma of he body of he rocke device and he ma of he fel-air mixre neceary for he rocke device moion; - velociy of he rocke impac device a he momen ; U - reacive velociy of he reacive ga flow ream (ie a reacive velociy wr o he impac device body); e + U - abole velociy of he reacive ga ream; G - weigh of he rocke device; S - rronding medim reiance force; A - Archimede force; U e G S A - magnide of he correponding vecor Applying he principle of imple and momenm beween ime and ime + for he mechanical yem (hown in Fig ) one can wrie m + ( G + A + S ) ( m m )( + ) + m Afer he vecor eqaion () i projeced on he axi h (ee Fig ) and remodeling he differenial eqaion of moion of he rocke impac device i obained a follow: d A G + S qu + () d m q m q m q The preen reearch die he cae when m q m conan (3) and he reiance force magnide S depend on he qare of he raigh-line velociy magnide ie S k k k ρf (4) where k i a coefficien relaed o he form of reamlined body ρ i he rronding medim deniy; F i he area of he reamlined body projecion on a plane perpendiclar o he velociy Replacing in () eqaliie (4) and q U e () k T + A G T β (5) m k one arrive a d d ( β ) (6) I i amed T n T i con an where T i i he reacive force creaed by one indrial reacive engine; G i he weigh magnide The olion of (6) i of he form β + ln β β + C (7) Fig Dynamic of a rocke impac device of a highvelociy forging machine A i i illraed in Fig a he momen he remaining ma of he rocke device i m and he rocke velociy i Dring an inerval a fel-air mixre m q wih a ma phe o of he combion n n chamber of each indrial rocke engine which enre he moion of he rocke device wih a relaive velociy U where C i a conan of inegraion calclaed ing he iniial condiion: h a Cae < β ( β > ) Eqaliy (7) reolved wr o i β ae β (8) ae + β β + β where a > 5

3 alenin Abadjiev Emilia Abadjieva Dochka Perova: Mahemaical Modeling of he Dynamic Procee of a High elociy Forging Machine; Machine Deign ol3() No3 ISSN 8-59; pp 5-56 When i follow ha a and herefore (8) can be expreed in he form β h( β) (9) A i i een from (9) when increae from o infiniy hen he velociy magnide increae from o β ( > Cae > β β ) Then from (7) i deermined a follow β e + a β () β e a When ie a eqaliy () i of he form β ch( β) () Analyzing () when [ + ) he velociy magnide decreae from o β and herefore > β a every momen ( > Cae β β ) In hi cae β 3 Eqaion of moion Cae < β The olion of eqaion (7) wr can be preened in he form h Arch β + β β () Then inegraing () he eqaion of moion of rocke device i chβ + Arch h ln β (3) ch Arch β where h i he verical diplacemen of he rocke device according o Fig Cae > β In hi cae he velociy magnide and he eqaion of moion of he rocke device are ch Arcch β β + (4) β hβ + Arcch h ln β h Arcch β (5) Eqaliie (3) and (5) are obained relevan o he iniial condiion: h a 3 Eqaion of he velociy magnide and eqaion of moion of rocke device afer he brning in he rocke engine ha been opped Le rewrie he differenial eqaion () aking ino accon he condiion (3) and U ie d G A k (6) d m m Solving (6) in condiion ha a he momen one obain λ anarcan λ (7) λ G A where λ k Le repreen (7) in he form dh λ anarcan λ (8) d λ Solving (8) in accordance wih he iniial condiion: h a he eqaion of moion of he rocke device i coarcan λ λ h ln (9) co arcan λ 4 DYNAMICS OF THE ROCKET IMPACT DEICE FOR A REALIZATION OF THE TEHNOLOGICAL PREPERATORY MODE Here i reaed he cae where he preparaory phae inclde he following wo age: Fir age: The rocke impac device move verically p wih working rocke engine The proce i characerized wih an iniial velociy and he reacive ream ha a nominal velociy U A he end of hi age defined by a ime and a velociy he reacive ream velociy i U Second age: The impac device conine moving verically p wih iniial velociy dring a ime When he device reache a preliminary defined diance H wr he iniial poiion i re immovable Uing he obained olion of he differenial eqaion ha decribe he dynamic of he rocke impac device we can wrie he following yem of eqaion 53

4 alenin Abadjiev Emilia Abadjieva Dochka Perova: Mahemaical Modeling of he Dynamic Procee of a High elociy Forging Machine; Machine Deign ol3() No3 ISSN 8-59; pp 5-56 β β e e β β β e + e β e + e h ln β () h ln H h + h + λ The olion of () define he baic dynamic characeriic of he concree echnological proce of he preparaory phae namely The magnide of he velociy of he impac device moion a he end of he fir age H e λβ () H λ e + β The velociy magnide repreened by () i he iniial velociy magnide for he moion of he impac device when he econd age ar The heigh he rocke impac device lif p o o ha velociy become eqal o H λ e + β ln λ + β h The ime needed o cover he diance h H e Arch H e λ β λ + β () (3) The heigh (he diance) which he rocke impac device goe wih no working indrial rocke engine ( U ) h + H h ln (4) λ The ime needed o cover he diance h H λ e arcan H g e β λ + β (5) The ime Σ needed he rocke impac device o reach a heigh H where he velociy of device eqal o Σ (6) + 5 COMPUTER EXPERIMENT The comper experimen i worked o for he cae when he rocke impac device ha hree indrial reacive engine he wo of hem realize he moion of he impac device verically p (a preparaory phae) (ee Fig Poiion No ) and he hird one move he device verically down (a working phae) (ee Fig Poiion No 6) The conidered cae i characerized wih he following baic parameer: Ma of he impac device ( m + m ) + m 35kg m 3 (7) where e g h m e 5kg i he ma of he reacive engine [7]; m g 9kg i he ma of he fel-air mixre in one reacive engine which enre a conino work dring [7]; m h kg i he ma of hammer ram Magnide S of he reiance force S of he rronding medim A i ha been menioned he rronding medim reiance force vale depend on he following conan: k 64 - a reian coefficien of he rronding medim when he reamlined body ha a qare form; 3 ρ 5 kg m - i he ea waer deniy; F 3 m i he rface of projecion of reamlined body on a plane perpendiclar o he velociy Then he coefficien k (ee (4)) i k k ρ F 968 (8) Tracion of he impac rocke device The preparaory phae i realized if n ie T nt i N (9) Deerminaion of he reacive force creaed by a ingle reacive engine T i i realized according o he fir principle of he hermodynamic [9] Weigh magnide of he impac device G mg 3453 N (3) where g 98 m i he graviy acceleraion Archimede force magnide A 64 N The magnide of Archimede force i deermined when he capaciy of he rocke impac device which i 3 conied of hree indrial rocke i 3 m Dynamic conan k T + A G 559 β 755 (3) m k elociy fncion In condiion of working indrial rocke engine βh(β) (3) 54

5 alenin Abadjiev Emilia Abadjieva Dochka Perova: Mahemaical Modeling of he Dynamic Procee of a High elociy Forging Machine; Machine Deign ol3() No3 ISSN 8-59; pp 5-56 In condiion of no- working indrial rocke engine λ anarcan λ (33) λ where i he magnide of he velociy of he impac device a momen when he brning proce in he combing chamber of he indrial rocke engine i opped Table No Parameer Symbo l ale Eqaion of moion In condiion of working indrial rocke engine h ln[ ch( β)] (34) In condiion of no-working indrial rocke engine arcan λ h ln λ (35) coarcan λ A magnide of he moion velociy of he impac device in condiion of working indrial rocke engine 686 m/ A verical diplacemen of he rocke impac device in condiion of working rocke engine h 9556 m Fig3 Fncion of he velociy of he impac rocke device moion in he cae of working indrial rocke engine 3 A ime for a realizaion of he verical diplacemen of he impac device in cae of working rocke engine A verical diplacemen of he rocke impac device in condiion of noworking rocke engine A ime for a realizaion of he diplacemen of he impac device in condiion of noworking rocke engine h 446 m 44 Fig4 Fncion of he velociy of he impac rocke device moion in he cae of no-working indrial rocke engine 6 A maximm diplacemen of he impac device H m 7 A ime for a realizaion of he maximm diplacemen of he rocke impac device Σ 6775 Fig5 Graphic cheme of he verical diplacemen of he impac rocke device in he cae of working indrial rocke engine 55

6 alenin Abadjiev Emilia Abadjieva Dochka Perova: Mahemaical Modeling of he Dynamic Procee of a High elociy Forging Machine; Machine Deign ol3() No3 ISSN 8-59; pp 5-56 The concree olion from () o (6) of he e of eqaion () are illraed in Table They give informaion abo he parameer of he preparaory phae of a high-velociy hammer wih a rocke drive of he impac device ACKNOWLEDGMENT The financial ppor of he Naional Science Fnd a he Miniry of Edcaion Yoh and Science of Blgaria nder Gran No DO -6/88 i graeflly acknowledged REFERENCES Fig6 Graphic cheme of he verical diplacemen of he impac rocke device in he cae of no-working indrial rocke engine Baed on he characeriic defined here and on he comper program he fncion (3) - (35) are vialized (ee Fig 3 - Fig6) Thee fncion are baic one for defining he concree dynamic proce which enre given echnological parameer in preparaory phae realizaion 6 CONCLUSION The preened reearch inclde he heoreical bai for defining differen echnological condiion needed o realize he preparaory phae of he proce repreening impac by he rocke device of a high-velociy hammer Wha are died are he poible cae of an impac rocke device moion when he high-velociy hammer work in preparaory phae and in waer medim The preened analyical relaion ake ino conideraion all exernal force ha are applied o he impac device in i moion verically p The worked-o algorihm i a bae of a comper program for calclaing he dynamic characeriic of he rocke impac device of a high-velociy hammer The calclaion made in Paragraph 5 are baed on he echnical daa of he indrial rocke device (objec of [7]) which are placed a or dipoal [] ABADJIE ABADJIEA E Reearch of he Pile Fondaion Proce Dynamic Mahemaical Modelling of he Pile Tamping Dynamic Compe Rend De L Academie Blgare De Science Sofia ol 63 No pp [] ABADJIE ABADJIEA E Reearch of he Pile Fondaion Proce Dynamic Simlaion Rel Compe Rend De L Academie Blgare De Science Sofia ol 63 No pp [3] BEER F JOHNSTON R E ecor Mechanic For Engineer Fifh ed McGraw-Hill Pblihing Company New York p [4] BODURO P Abo he Poibiliie of Applying Rocke Engine ino High-velociy Hammer Jornal Mahinoroene No pp6-6 (in Blgarian) [5] BODURO P GENO J Engineering Srcre for Deforming he Meal by Mean of Rocke Engine Proceeding of Naional Scienific- Technical Conference: A Modern Eqipmen Mechanizaion Aomaizaion in Samping of Meal Pleven 976 pp8-9 [6] BODURO P A High-elociy Hammer A Paen of Blgaria No [7] BODURO P PENCHE T Indrial Rocke Engine and I Applicaion for Propelling of Forging Hammer Jornal of Maerial Proceing Technologyol6 No 3 5 pp54-58 [8] BODURO P A Device for Pile Tamping A Paen of Blgaria No [9] KRASTE J Thermodynami Sofia p (in Blgarian) [] LOJTZJANSKY L LUR E A Theoreical Mechanic Ed Tehnika ol 96 6 p (in Blgarian) [] SOGRISHIN JU GRISHIN L OROBJE Forging of High-elociy Hammer ed Mahinoroenie Mocow p (in Rian) 56