Deformation Calculating of Electromagnetic Launcher s Rail Subjected to Logarithmic Magnetic Pressure

Transcription

1 Deformaon Calculang of Elecromagnec auncher s al Subjeced o ogarhmc Magnec Pressure Wen u (Correspondng auhor) & Mn School of Scences, Yanshan Unversy, Qnhuangdao 664, Chna Xangzhong Ba School of Cvl Engneerng & Mechancs, Yanshan Unversy, Qnhuangdao 664, Chna E-mal: luwen1961@homal.com ele Wang School of Scences, Yanshan Unversy, Qnhuangdao 664, Chna Ths projec s suppored by Naonal Naural Scence Foundaon of Chna (Gran NO ). The Excellen Gong Abroad Expers' Tranng Program n Hebe Provnce. Absrac To solve he accurae calculaon of force-deformaon of he elecromagnec launcher s ral, hs s helpful o exend he ral lfe and mprove he frng accuracy. Therefore, he elecromagnec launcher s ral can be modeled as a beam on elasc foundaon wh smply suppored beam by movng load. In hs paper Euler beam heory s appled o buld he Mechancal model and he analycal soluon of he equaon subjeced o logarhmc magnec pressure s derved n deal, whch has successfully avoded he errors whch are caused by usng he unform pressure o approxmaely replace he varable force. The numercal analyss brngs from he elasc coeffcen, he dampng coeffcen, he mass of ral and he load s velocy have nfluence on he deformaon of beam by he MATAB sofware. The consequence shows ha he elasc coeffcen and he load s velocy have que obvous affec on he deformaon of he beam whle he dampng coeffcen and he mass of ral have no obvous affec on he deformaon of he beam. I lad he foundaon for solve he elecromagnec launcher s ral subjeced o magnec pressure of arbrary funcon and promoe he praccaly of he elecromagnec guns. Keywords: Elecromagnec gun, auncher s ral, Elasc foundaon beam, Dampng force, Mechancal model, agrange equaon 1. Inroducon The elecromagnec gun s a new concep weapon, he echnology of s has nesmable applcaon poenal no only n he mlary feld, bu also n avaon, aerospace, ransporaon, ndusral producon, scenfc research and oher felds. Snce he 8s, especally n he recen en years, wh he developmen of new echnology and new maeral, he research of launcher, launchng wegh, projecle velocy and hgh effcency power source n he elecromagnec ralgun have reached a seres of achevemen. The Su ense.vermore Naon aboraory and The owes.alamos Naon aboraory, once have cooperaed o acceleraed a projecle weghed.g o a supervelocy of 1km/s. Flud Physcs Insue of he Chnese Engneerng Academy had bul he frs elecromagnec ral launcher, whch can accelerae he projecle weghed.34g o 16.8km/s. Whle he velocy of he convenonal cannon s only km/s, whch s so closed o he lmaon of physcs ha he range s no possble o be farher. On he conrary, he hrus of he elecromagnec ralgun s en mes bgger han ha of he radonal launcher. The projecle can be acceleraed o several klomeers or even o dozens of klomeers n one second, for possesses huge knec energy whch grealy enhance he range and power of he weapon. T.Tzeng used he elasc foundaon beam o buld mechancal model of he elecromagnec ralgun and deduced he solve process of governng equaon. HU Yuwe analyzed heorecal bul model and smulaon analyss for he work of he process of elecromagnec ralgun. WANG Sheng adoped he Fourer ransform o sudy dsplacemen feld due o a movng load on Euler beam resng on an elasc half-space. However, he dampng force o he response of beam s gnored n he above researches. There s no doub ha Publshed by Canadan Cener of Scence and Educaon 17

2 he calculaon of hs knd of suaon has he defecs of analyss and calculaon of mechancal. As a hgh-ech and hgh-precson elecromagnec launcher, accurae heorecal analyss and calculaon n engneerng are requre. Bu unl now, no researcher has gven any exac analycal soluon, hus furher analyc soluon of he equaon subjeced o varable pressure s of grea sgnfcance. In fac, s of heorecal value o research he heorecal analyc soluon of varous dscplnes, ncludng he analycal soluon of praccal engneerng problems. On he one hand, s mechancal pcure can be compleely saed, on he oher hand can be used as a sandard soluon, o wdely produce varey of numercal soluon. In hs paper, regardng he ral as smply suppored beam on he elasc foundaon and consderng he dampng force, a mechanc model whch s under he effec of movng load s proposed. Moreover, makng use of varable mehod and he agrange equaon whch consderng he dampng force, he analycal soluon (u Wen and San u,9)of he governng equaon subjeced o nonlnear funcon pressures s derved and he nfluences brough from he elasc coeffcen, he dampng coeffcen, he mass of ral and he load s velocy on he response of beam s analyzed.. Mechancal Model Fg.1 shows a schemac of an elecromagnec ralgun composed of power source, ral, armaure and projecle. When he elecrc curren of armaure goes hrough he ral, forms a srong magnec feld n he area of her encrclemen. Wh he reacon by he magnec feld and he elecrc curren, emerges powerful elecromagnec force, whch pushes he armaure and projecle o do he accelerang moon along he ral ll he projecle be launched ou of he ral. Fg. s he mechancal model of he ralgun smply suppored beam parally subjecs o nonlnear load n a me-varyng regon sng on he elasc foundaon. Consderng he effec of he beam by he dampng force and basng on he Euler beam heory, we oban he governng equaon of elasc foundaon beam by movng load whch s a ransen fourh order dfferenal equaon as follows (S.Tmoshenko,1965 and YU Yanl,): 4 w w w =, 4 m EI kw c p( x ) (1) x Where w s he deflecon, m = ρbhs he mass per un lengh, ρ s he densy of ral maeral, B and h are respecvely he wdh and hckness of he ral, EI s he bendng sffness of beam, k s he elasc consan, c s he dampng coeffcen. The funcon p( x, ) = q 1 H( x v) n(1), represens he magnec pressure fron ravelng along he ral wh velocy v represened by a Heavsde sep funcon H ( x v) ( Jerome T. Tzeng,5), and q = qlog a, a > Soluon of he Homogeneous Equaon The homogeneous equaon s a fourh order paral dfferenal equaon, n order o change no he ordnary dfferenal equaon, we solve by he mehod of varable separaon. The soluon of he homogeneous equaon of (1)can be expressed as follows: w( x, ) = φ( ) θ () Subsung()no he homogeneous equaon of(1): Tha can be expressed as follows: from equaon(4), le: And φ ( 4) m θ + EIφθ + kφθ + cθ = φ ( 4) EI θ k = + (4) cφ mφ cm θ cm φ (3) = λ (5) cφ mφ 18 ISSN E-ISSN

3 ( 4) EI θ k λ cm θ + cm = (6) Tha s 4 k cm Where, β = λ cm EI soluon of equaon(5)can be expressed as follows: Where, = c 4cm λ > Based on he boundary condon of he smple beam, ( ) θ = θ = x x= φ ( 4) 4 θ β θ = m m () Ae Be (7) = + (8) and ( ) θ = θ = x x= soluon of equaon(7)can be expressed as follows( ZHU Shjan,6): θ = Accordngly: nπ w (, ) ( ) ( ) sn m m x = θ x φ = x Ae + Be m nπ sn x m In erms of he orhogonaly of θ (ZHANG Xangng,6), we oban:, θθ jdx = 1, a j a = j Hence, deformaon w( x, of ) he beam can be expressed by he lnear combnaon of θ. nπ w( x, ) ( ) ( ) sn m m = θ x φ = x Ae + Be m Where consans A, B are deermned by he nal condons. 4. Analycal Soluon of Governng Equaon The analycal soluon of (1) can be obaned by he agrange equaon ncludng he dampng force. WhereT s he knec energy of he beam, U s he oal san energy, G s he dsspaon funcon(zhang Xangng,). d T T U G + = Q d φ The knec energy of he beam T can be expressed as follows (OU Png,3): 1 T = M Where M = mθ dx represens he general mass of he beam. The oal sran energy of he beamu s conssed by he sran energyu of he beam, and he sran energyu of b f Publshed by Canadan Cener of Scence and Educaon 19

4 he foundaon. ( ), θ U EI dx EI θ dx M 1 w x 1 j 1 EI 4 b = j = φ φ = β φ x j x x m 1 k k k f = = θ ϕ = φφ j θθ j = φ j m U kw dx dx dx M The oal sran energyu s obaned as: 1 EI 4 k 1 EI 4 k 1 U = Ub + U f = β Mφ + M φ = Mφ β + = cmφ λ m m m m The dsspaon funcon G can be expressed as: 1 w 1 j c G = c dx c jdx M = θθ = j m The vrual work done by he magnec pressure p( x, ) = q 1 H( x v) n a vrual dsplacemen δϕ can be expressed as follows: W= p x, δwdx= φ p x, θ x dx = Q Where we defne ( ) ( ) ( ) Q as he generalzed force = ( ) θ ( ) = ( ) θ ( ) = v v nπ x Q p x, x dx q x x dx q log a sn dx m q m nπ v = ln( v) 1 cos + C1e nπ ln ( a) Subsung TUGQno,,, he agrange equaon ncludng he dampng force, we oban an ordnary dfferenal equaon: φ c Q ( ) + + cλφ = = F() (9) m M Where: v v nπ x F() = p( x, ) θ dx = q θ dx = qlog asn dx m q m nπ v = ln( v) 1 cos + C1e M nπ ln ( a) The general soluon of equaon(9)s: m ( ) c+ m ( ) m φ () = + φ ( ) + + φ ( ) m m c m e e c m ( ξ) ( ξ) F m m + ( ξ) e e d So he general soluon of(1)can be expressed as follows: nπ m ( ) c+ w( x, ) = θ φ( ) = sn x + φ( ) e m m The nal condons are as: So ( ) m + + φ ( ) e m m ( ξ) ( ξ) F m m + ( ξ) e e d ξ ( ) () ξ c m φ = φ = = m (1) 13 ISSN E-ISSN

5 m vξ ( ξ) ( ξ) φ m m () = F( ξ) e e dξ (11) Accordng, subsung he soluon of (11) no he equaon (1), so we can ge he soluon w( x, of ) (1). The momen and he shear force of he beam n he ral can be furher derved from w( x,, ) whch provdes bass for he overall nvesgaon of he dynamc behavor of he elecromagnec ralgun. 5. Numercal analyss Snce here are dfferences among maerals of elecromagnec ral launcher, he dampng force and he rae of movng load wll possbly brng nfluence o he response of he ral(jerome T. Tzeng 5, 41: 46-5.). Thus, s necessary o consder he elasc coeffcen, he dampng coeffcen, he mass of ral and he load s velocy o compare he response of he ral. A known Maeral s modulus of ral maeral 1 E = 1GPa,he elasc consan k =.53 1 N/m, he densy of 3 ral maeral ρ = 87kg/m, he wdh of ral B = 3 1 m,he hckness of ral h = 1 1 m,he lengh of ral = m,he magnec load collecon degree q ( x ) =11snx MPa (JF,6). Fg.3 shows he deformaon of he beam by he elasc coeffcen. Along wh he elasc coeffcen ( k ) ncreasng, he curve of me-deformaon s a decreasng rend. Under he calculang condons gven by hs 1 paper, for he ral of whch k equals o.53 1 N/m, he deformaon ( w) of he beam s.9 1 m,when he 1 3 armaure moves o he momen = 1 1. Whle for he ral of whch k equals o N/m, he 3 deformaon ( w) of he beam s. 1 m a he same momen, we can see ha he former s 78% smaller han he laer. Fg.4 shows he deformaon of he beam by he dampng coeffcen. Along wh he dampng coeffcen ( c) ncreasng, he curve of me-deformaon s a slowly decreasng rend. Fg.5 shows he deformaon of he beam by he mass of ral. Compared wh copper and alumnum ral, he curve of me-deformaon has no sgnfcan changes. Fg.6 shows he deformaon of he beam by he load s velocy. Along wh he load s velocy ( v) ncreasng, he curve of me-deformaon s a ncreasng rend. Under he calculang condons gven by hs paper, for he ral of whch v equals o1m/s, he deformaon ( w) of he beam s1.7 1 m when he armaure moves o he momen = s. Whle for he ral of whch v equals o1m/s, he deformaon ( w) of he beam s m a he same momen, we can see ha he former s 8% smaller han he laer. 6. Conclusons (1)Takng he ral as a smply suppor beam on he elasc foundaon and consderng he dampng force,a mechancal model for he elecromagnec ralgun s bul. "We don have general soluon o nonlnear problems and some parcular soluons are as few as reasures n hsory."( Zheng Zhemn,1994) In hs paper, makng use of varable mehod and he agrange equaon ncludng he dampng force, he general soluon of he homogeneous par and he analycal soluon of he governng equaon subjeced o logarhmc pressures s derved whch enrched and developed he heory of elasc mechancs wh he hope o lay he foundaon for solvng he dffculy problem of elecromagnec ral subjeced o arbrary dsrbuon funcon pressures. ()The deformaon of beam whch s nfluenced by he elasc coeffcen, he dampng coeffcen, he mass of ral and he load s velocy are analyzed by he MATAB sofware. When he elasc coeffcen s larger, he deformaon of beam s smaller; when he load s velocy s larger, he deformaon of beam s larger, he dampng coeffcen and he mass of ral have no obvous affec on he deformaon of he beam. eferences Anhony J. Johnson1 and Francs C.Moon. (6). Elasc Waves and Sold Armaure Conac Pressure n Elecromagnec aunchers Transacons on Magnecs 6.3, HU Yuwe. (7). Modelng and smulaon of elecromagnec ral gun sysem[d]: [Dsseraon for he Maser Degree]. Harbn Insue of Technology, 7. HU Yuwe. (7). Modelng and smulaon of elecromagnec ralgun sysem[d]: [Dsseraon for he Maser Degree]. Harbn Insue of Technology, 7. Jerome T. Tzeng and We Sun. (7). Dynamc esponse of Canlevered al Guns Arbued o Projecle/Gun Ineracon Theory. Transacons on Magnecs, 7, 43: Publshed by Canadan Cener of Scence and Educaon 131

6 Jerome T. Tzeng. (5). Dynamc esponse of Elecromagnc algun Due o Projecle Movemen Transacons on Magnecs, 5, 41: Jerome T. Tzeng. (5). Srucural Mechancs for Elecromagnec alguns Transacons on Magnecs, 5,1, JIN Shangnan, MA Yongl. (). Theorecal Mechancs. Hgher Educaon Press,, 11. u Wen, San u. Mahemac Model and Analyc Soluon for Cylnder Subjec o Exponenal Funcon. Chnese Journal of Mechancal Engneerng, 9,(4) : u Wen. Mahemac Model and Analyc Soluon for Cylnder Subjec o Uneven Pressures. Chnese Journal of Mechancal Engneerng, 6,19(4) OU Png,ZENG Qngyuan. (3). Fne elemen analyss of nfnely long beam resng on connuous vscoelasc foundaon subjeced o movng loads. Journal of Traffc and Tran sporaon Engneerng 3, 3: 1. S.Tmoshenko. (1965). Mechancs of maeral. Scence press 1965,1-15. Wang sheng. (7). Dsplacemen feld due o a movng load on Euler beam resng on an elasc half-space [D]: [Dsseraon for he Maser Degree]. Harbn Insue of Technology, 7. WANG Yng, XIAO Feng. (1994). Prncple of elecrcgun. Naonal Defense Indusry Press, YU Yanl. (). The research of he dynamc response of he ral sysem and vaduc by movng load [D]: [Dsseraon for he Maser Degree]. Wuhan Unversy of Technology,. ZHANG Xangng, WANG Zhpe. (6). Srucure Vbraon Mechancs. TongJ Unvesy Press 6, Zheng Zhemn. ZhengZheMn corpus. (1994). Bejng: scence press, P86hp:// ZHU Shjan, OU lngjun. (6). Vbraon Theory and Vbraon Isolaon. Naonal Defense Indusry Press 6,366. Fgure 1. The general dagram of he ralgun Fgure. The ral s modeled as a beam on elasc foundaon 13 ISSN E-ISSN

7 4 x 1-3 K=.53*1 1 N/m K=5.64*1 1 N/m x 1-3 Fgure 3. Deformaon curve by dfferences elasc coeffcen 4 x 1-3 c=1.3*1 5 Ns/m c=1.3*1 6 Ns/m x 1-3 Fgure 4. Deformaon curve by dfferences dampng coeffcen 4 x 1-3 m=.67kg m=.81kg x 1-3 Fgure 5. Deformaon curve by dfferences mass of ral 4 x 1-3 v=1m/s v=1m/s x 1-3 Fgure 6. Deformaon curve by dfferences velocy of he load Publshed by Canadan Cener of Scence and Educaon 133