Optimization design of structures subjected to transient loads using first and second derivatives of dynamic displacement and stress


 Philip Ross Hall
 1 years ago
 Views:
Transcription
1 Shock and Vibration 9 (202) DOI /SAV IOS Prss Optimization dsign of structurs subjctd to transint loads using first and scond drivativs of dynamic displacmnt and strss Qimao Liu a,, Jing Zhang c andliubinyan b a Dpartmnt of Civil and Structural Enginring, Aalto Univrsity, Espoo, Finland b Collg of Civil and Architctur Enginring, Guangxi Univrsity, Nanning, China c Dpartmnt of Mchanical Enginring, Indiana UnivrsityPurdu Univrsity Indianapolis, Indianapolis, IN, USA Rcivd 6 Sptmbr 200 Rvisd 9 July 20 Abstract. This papr dvlopd an ffctiv optimization mthod, i.., gradinthssian matrixbasd mthod or scond ordr mthod, of fram structurs subjctd to th transint loads. An algorithm of first and scond drivativs of dynamic displacmnt and strss with rspct to dsign variabls is formulatd basd on th Nwmark mthod. Th inquality timdpndnt constraint problm is convrtd into a squnc of appropriatly formd timindpndnt unconstraind problms using th intgral intrior point pnalty function mthod. Th gradint and Hssian matrixs of th intgral intrior point pnalty functions ar also computd. Thn th Marquardt s mthod is mployd to solv unconstraind problms. Th numrical rsults show that th optimal dsign mthod proposd in this papr can obtain th local optimum dsign of fram structurs and somtims is mor fficint than th augmntd Lagrang multiplir mthod. Kywords: Dynamic rspons optimization, gradint, Hssian matrix, timdpndnt constraint. Introduction Dynamic rspons analysis of structurs subjctd to transint loads oftn rquirs much mor dmanding computational cost than static analysis. Thrfor th fficincy of th optimization mthod bcoms critical to dynamic rspons optimization problms. Mathmatically, thr ar thr typs of dynamic rspons optimization mthods: zro ordr mthods (nongradintbasd algorithms), first ordr mthods (gradintbasd algorithms) and scond ordr mthods (gradinthssian matrixbasd algorithms). Gnrally, th first ordr mthods ar mor fficint than th zro ordr mthods, and th scond ordr mthods ar mor fficint than th first ordr mthods. Zro ordr mthods rquir only th information of dynamic rsponss to construct optimal algorithms. A lot of nongradintbasd algorithms 4] hav bn dvlopd to solv th optimization problm of structurs subjctd to transint loads. First ordr mthods rquir th information of dynamic rsponss and thir first drivativs with rspct to dsign variabls to construct optimal algorithms. Thrfor, it is vry crucial to calculat fficintly th Corrsponding author: Qimao Liu, Dpartmnt of Civil and Structural Enginring, Aalto Univrsity, Espoo, FI Aalto, Finland. Tl.: ; Fax: ; ISSN /2/$ IOS Prss and th authors. All rights rsrvd
2 446 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs first drivativs of structural dynamic rsponss for gradintbasd algorithms. Howvr, many algorithms hav bn dvlopd to calculat dynamic rspons first drivativs with rspct to dsign variabls (also calld snsitivity analysis 5 2]). Today, th gradintbasd algorithms ar th mainstram pattrn 3] to solv th optimization problm of structurs subjctd to transint loads. Much rsarch 4 7] has bn don to solv th structural dynamic rspons optimization problm with th gradintbasd algorithms. Scond ordr mthods rquir th information of dynamic rsponss, thir first and scond drivativs with rspct to dsign variabls to construct optimal algorithms. Th Nwton s mthod and Quasi Nwton s mthod ar th commonly usd scond ordr mthod. Although many rsarchrs hav calculatd dynamic rspons first drivativs with rspct to dsign variabls, thr is littl litratur publishd on dynamic rspons scond drivativ analysis (also calld Hssian matrix analysis). Scond drivativ analysis is mor complicatd than first drivativanalysis. Howvr, th fficincy of th optimization using scond drivativ can b gratly improvd whn th gradint and Hssian matrix can b calculatd fficintly. In this papr, a gradinthssian matrixbasd algorithm for optimization problm of structurs subjctd to transint loads is dvlopd basd on gradint and Hssian matrix calculations. Th purpos of this papr is to dvlop a gradinthssian matrixbasd optimization mthod for structurs subjctd to dynamic loads. Th main work of this papr is as follows: First, w formulat an algorithm to calculat dynamic rsponssand thir first and scond drivativs with rspct to dsign variabls. Th algorithm is achivd by dirct diffrntiation and only a singl dynamics analysis, basd on Nwmarkβ mthod 8], is rquird. Scond, w formulat th timdpndnt structural optimization modl. In this modl, total mass of th structur is th objctiv function. Th dynamic rsponss including strsss of th bam lmnt and nodal displacmnts ar constraints. Third, th timdpndnt optimization modl is convrtd into a squnc of appropriatly formd unconstraind intgral mathmatic modls using th intrior pnalty function mthod. Th gradint and Hssian matrixs of th intrior pnalty functions ar also calculatd using th first and scond drivativs of dynamic rsponss. Fourth, Marquardt s mthod 9], a gradinthssian matrixbasd algorithm, is mployd to solv th unconstraind intgral mathmatic modl. Finally, as th illustration of th dvlopd approach, optimization dsigns of a plan fram subjctd to th horizontal dynamic loads ar dmonstratd. 2. Calculation of first and scond drivativs This sction prsnts two subjcts. Sction 2. introducs th calculations of first and scond drivativs of dynamic rsponss (i.., nodal displacmnt and normal strss) with rspct to dsign variabls of structurs. Sction 2.2 discusss how to calculat first and scond drivativs of structural mass with rspct to structural dsign variabls. 2.. Calculation of first and scond drivativs of dynamic displacmnt and strss Th first and scond drivativs of dynamic rspons of structurs ar th prrquisits of an fficint gradint Hssian matrixbasd algorithm. W will show hr that, in a singl structural dynamic analysis, th first and scond drivativs of a dynamic rspons can b drivd simultanously. Plan bam lmnt is usd xtnsivly in nginring. Thrfor, th dynamic optimal dsign mthod is dmonstratd with th plan bam lmnt in this work. Th lmnt crosssction shown in Fig. is H structuralstl shap. Th dsign variabls of lmnt ar d, d 2, d 3 and d 4. Point c is th cntroid of th lmnt crosssction. Axs y and z ar th principal cntroidal axs. Th crosssctional ara is A =2d d 3 + d 2 d 4 () Th crosssctional momnt of inrtia is Iz = d 4 d d d 3 3 ( ) ] d d 2 d 2 2 d 3 (2) Th farthst distanc y from th nutral axis is
3 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 447 Fig.. Elmnt crosssction and lmnt dsign variabls. ( ) = ± d 3 + d 2 2 Th first momnt of th shadd ara with rspct to th nutral axis is Sz = d d d 3 + d d 2 d 2 2 d Th structur is dividd with N lmnts. Th dsign variabl vctor is dfind as. (3) (4) d = d,d 2,d 3,d 4,,d,d 2,d 3,d 4,,d N,d N 2,d N 3,d N ] T 4 (5) Considr th quations of motion for a linar systm subjctd to dynamic forcs Mẍ + Cẋ + Kx = f (t) (6) with th following initial conditions: { x (0) = x0 (7) ẋ (0) = ẋ 0 whr K, M, andc ar stiffnss matrix, mass matrix, and damping matrix, rspctivly. x (t), ẋ (t) and ẍ (t) ar unknown nodal displacmnt, vlocity and acclration vctors, and f(t) is th load vctor. Suppos that th dynamic loads act on th nods. Rayligh damping is usd in this work, th structural damping matrix is C = α M + α 2 K (8) whr ( ) 2 ς ω ς2 ω 2 α = ω 2 ω2 2 (9) α 2 = 2(ς 2ω 2 ς ω ) ω 2 2 ω2 (0) whr ω and ω 2 ar th first and scond natural frquncy of th structur, rspctivly. ς and ς 2 ar th damping ratio. In this work, ς = ς 2 =0.02. α and α 2 ar constants. Equations (6) and (7) must b satisfid for all tim priod t 0, Γ]. Γ is th duration of th dynamic loads. In practic, th solution of this initialboundaryvalu problm (IBVP) rquirs intgration through tim. This is achivd numrically by discrtising in tim th continuous tmporal drivativs that appar in th quation. Any on of th tim intgration procdurs can b usd for this purpos. Th most widly usd family of dirct tim intgration mthods for solving Eq. (6) is th Nwmark family of mthods. Th Nwmark mthod can b formulatd by considring quilibrium at any discrt tim t +Δt, and is givn by th following quation: Mẍ +Cẋ +Kx =f ()
4 448 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Th nodal displacmnts, nodal acclrations and nodal vlocitis can b achivd at any tim t +Δt by solving Eq. (). Prhaps th most widly usd dirct mthod for th quation of motion (6) is th Nwmark mthod, an implicit tchniqu, which consists of th following finit diffrnc assumptions with rgard to th volution of th approximat solution: ( ) ] x =x (t)+δtẋ (t)+δt 2 2 β ẍ (t)+βẍ (2) ẋ =ẋ (t)+δt ( δ) ẍ (t)+δẍ ] (3) whr any particular choic of th paramtrs β and δ dtrmins th stability and accuracy charactristics of th solution. In this work, paramtrs δ 0.5 and β =0.25 (0.5+δ) 2.Wdfin th intgral constants: a 0 = βδt, ( 2 a = δ βδt, a 2 = βδt, a 3 = 2β, a 4 = δ β,a 5 = Δt δ 2 β ), 2 a 6 =Δt ( δ), a 7 = δδt. Th paramtrs β and δ will b rplacd by thos constants in th following formulas. In addition to Eqs (2) and (3) th quilibrium Eq. () at tim station t +Δt is considrd. This way a systm of quations is formd for th dtrmination of th thr unknowns x, ẋ and ẍ, assuming that th displacmnt, vlocity, and acclration vctors at th prvious tim station t hav alrady bn computd. Thus, th solution for th displacmnt vctor is K x =F (4) whr K = K + a 0 M + a C (5) and F =f +M a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] (6) Th acclrations, ẍ, which ar rquird for th computations at th nxt tim station, can b computd as ẍ =a 0 x x (t)] a 2 ẋ (t) a 3 ẍ (t) (7) whil th vlocitis, ẋ, can b obtaind dirctly from quation (3), as follows. ẋ =ẋ (t)+a 6 ẍ (t)+a 7 ẍ (8) Whn th dynamic loads act on th nods, w only nd to dtrmin th intrnal forcs and intrnal coupls at th cntr and two nds of th lmnt. Th lmnt nodal forc vctor, F,is F = K T δ (9) whr K is th lmnt stiffnss matrix in a local coordinat systm, T is th lmnt coordinat transformation matrix, δ is th lmnt nodal displacmnt vctor in a local coordinat systm. Th intrnal forc vctor at th cntr of lmnt, F M,is F M = F F 2 l 2 F 2 F 3 ] T (20) whr l is th lngth of lmnt. Th thr trms, i.., F, F 2 and l F 2 2 F 3, ar axial forc, shar forc and bnding momnt, rspctivly. F, F 2 and F 3 ar th first, scond and third trm of th lmnt nodal forc vctor, F, rspctivly. Th imum strss at th cntr and two nds, i.. th i nd and th j nd, of th lmnt ar calculatd as follows. Th imum normal strss at th i nd of th lmnt is σix = F A + F 3 y m (2)
5 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 449 Th imum normal strss at th j nd of th lmnt is σjx = F 4 A + F 6 y m (22) whr F 4 and F 6 ar th fourth and sixth trm of th lmnt nodal forc vctor, F, rspctivly. Th imum normal strss at th cntr of th lmnt is σmx = F M A + F M3 y m (23) whr F M and F M3 ar th first and third trm of th intrnal forc vctor at th cntr of lmnt, F M, rspctivly. Th imum shar strss at th i nd of th lmnt is τixy = F 2 Sz d (24) 4 I z Th imum shar strss at th j nd of th lmnt is τjxy = F 5 S d 4 I z whr F 5 is th fifth trm of th lmnt nodal forc vctor, F. Th imum shar strss at th cntr of th lmnt is τmxy = F M2 S d 4 I z whr F M2 is th scond trm of th intrnal forc vctor at th cntr of lmnt, F M. (25) (26) 2... Formulas for first drivativs of dynamic displacmnt and strss Now w will driv th formulas for th first drivativs of dynamic rspons, i.., dynamic displacmnt and strss. Diffrntiating Eq. (4) with rspct to th dsign variabl d i,whav x K = F K x (27) whr K and = K + a 0 M + a C (28) F a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + M = M + C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] + C a 0 x (t) ẋ (t) + a 2 ] x (t) ẋ (t) ẍ (t) a + a 4 + a 5 ] ẍ (t) + a 3 Aftr th first drivativs of displacmnt vctor at tim t +Δt is obtaind from Eq. (27), through diffrntiating Eq. (7) with rspct to dsign variabls d i,whav ] ẍ x x (t) ẋ (t) ẍ (t) = a 0 a 2 a 3 (30) Thn, th first drivativs of acclration vctor at tim t +Δt is obtaind from Eq. (30). Diffrntiating Eq. (8) with rspct to dsign variabl d i,whav ẋ = ẋ (t) + a 6 ẍ (t) (29) + a 7 ẍ (3)
6 450 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Diffrntiating Eq. (9) with rspct to th dsign variabl d i,whav F whr δ (t+δt) = K T δ + K T δ (32) can b calculatd from x(t+δt). Diffrntiating Eq. (20) with rspct to th dsign variabl d i,whav F M = d F (t+δt) i F 2 (t+δt) l 2 F 2 (t+δt) F ] T 3 (t+δt) (33) Diffrntiating Eq. (2) with rspct to th dsign variabl d i,whav σix = F + F 3 A F + F 3 A A 2 F 3 y m EI 2 z Diffrntiating Eq. (22) with rspct to th dsign variabl d i,whav σjx = F 4 + F 6 A F 4 + F 6 A A 2 F 6 y m EI 2 z Diffrntiating Eq. (23) with rspct to th dsign variabl d i,whav σmx = F M A F M A 2 A I z (34) I z (35) + F M3 + F M3 F M3 y m Iz 2 (36) Diffrntiating Eq. (24) with rspct to th dsign variabl d i,whav τixy = F 2 S d 4 I z + S F 2 d 4 I z d 4 F 2 S d 2 I z F 2 4 I z S d 4 I2 z Diffrntiating Eq. (25) with rspct to th dsign variabl d i,whav τjxy = F 5 d 4 F 5 S d 4 I z S d 2 I z F 5 4 I z + S F 5 d 4 I z S d 4 I2 z Diffrntiating Eq. (26) with rspct to th dsign variabl d i,whav τmxy d 4 F M2 = F M2 S d 4 I z S d 2 I z F M2 4 I z + S F M2 d 4 I z S d 4 I2 z (37) (38) (39)
7 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Formulas for scond drivativs of dynamic displacmnt and strss Now w will driv th formulas for th scond drivativs of dynamic rspons, i.., dynamic displacmnt and strss. Furthr diffrntiating Eq. (27) with rspct to th dsign variabl d j,whav K 2 x whr 2 K = and 2 F = 2 F 2 K x K x K x (40) 2 K 2 M 2 C + a 0 + a (4) x (t) ẋ (t) ẍ (t) a 0 + a 2 + a 3 ] = 2 M a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + M + M ] x (t) ẋ (t) ẍ (t) 2 x (t) 2 ẋ (t) 2 ẍ (t) a 0 + a 2 + a 3 + M a 0 + a 2 + a C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] + C ] x (t) ẋ (t) ẍ (t) a + a 4 + a 5 + C ] x (t) ẋ (t) ẍ (t) 2 x (t) 2 ẋ (t) 2 ] ẍ (t) a + a 4 + a 5 + C a + a 4 + a 5 Aftr 2 x(t+δt) is computd from Eq. (40), with furthr diffrntiating Eq. (30) with rspct to th dsign variabls d j,whav 2 ẍ 2 ] x = a 0 2 x (t) 2 ẋ (t) 2 ẍ (t) a 2 a 3 (43) Thn 2 ẍ(t+δt) is obtaind from Eq. (43). Furthr diffrntiating Eq. (3) with rspct to th dsign variabls d j, w obtain 2 ẋ = 2 ẋ (t) + a 6 2 ẍ (t) + a 7 2 ẍ (44) Furthr diffrntiating Eq. (32) with rspct to th dsign variabl d j,whav 2 F (t+δt) 2 = K T δ + K (45) + K T δ (t+δt) + K T 2 δ (t+δt) T δ (t+δt) Furthr diffrntiating Eq. (33) with rspct to th dsign variabl d j, w obtain 2 F M = F 2 (t+δt) 2 F 2 (t+δt) l 2 2 F 2 (t+δt) F ] T 2 3 (t+δt) (46) Furthr diffrntiating Eq. (34) with rspct to th dsign variabl d j,whav 2 σix + 2 F A A A 3 F 3 F 3 = 2 F A F A F A 2 Iz + F 3 Iz y m F 3 A F A 2 2 A + 2 F 3 EI z y m Iz 2 + I z I z F 3 EI 2 z A + F 3 y m + F 3 EI z ] 2 (42) (47) 2 F 3 y m Iz EI 3 F 3 y m 2 Iz EI 2 z z
8 452 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Furthr diffrntiating Eq. (35) with rspct to th dsign variabl d j, w obtain 2 σjx + 2 F 4 F 6 = 2 F 4 A A A A 3 F 4 F I z + F 6 I z y m A 2 F 4 A 2 A 2 A + 2 F 6 I z F 6 F 6 EI 2 z F 4 A 2 + F F 6 Iz + I z 2 F 6 EI 3 Furthr diffrntiating Eq. (36) with rspct to th dsign variabl d j,whav 2 σmx = 2 F M A F M A 2 z A + 2 F M A A A 3 F M 2 A A F M3 F M3 (t+δt) F M3 2 2 Iz + F M3 (t+δt) Iz y m F M3 EI z EI 2 z I z F M3 (t+δt) EI 2 I z z A EI z 2 (48) Iz F 6 2 Iz EI 2 F M A 2 y m z A + F M3 + F M3 (t+δt) 2 + I z 2 F M3 y m Iz 3 F M3 y m 2 Iz 2 (49) Furthr diffrntiating Eq. (37) with rspct to th dsign variabl d j, w obtain 2 τixy F 2 Iz S = 2 F 2 d 4 I2 z S d 4 F 2 d 2 4 I z d 4 F 2 Sz d 2 d 4 4 I z + d 4 Iz F 2 Sz d 2 4 I2 z Sz d + F 2 4 I z + 2 Sz F 2 d 4 I z S Iz S S d 4 I z + S F 2 F 2 d 4 I z F 2 d 2 d 4 F 2 S 4 I2 d 2 z F 2 d 2 + d 4 d 4 2 F 2 4 I z 2 Iz F 2 S d 4 I2 z 4 I z EI z d 4 Sz d 2 S d 3 4 I z I z F 2 S d 4 I2 z I z Sz F 2 d + I z d 4 F 2 S 4 I2 z d 2 + I z Iz 2 F 2 S 4 I2 z d 4 I3 z Furthr diffrntiating Eq. (38) with rspct to th dsign variabl d j,whav 2 τjxy F 5 Iz S = 2 F 5 d 4 I2 z S d 4 I z + F Sz F 5 d 4 I z S d 4 I z + S F 5 4 I z F 5 d 4 Sz d 2 d 4 I z 4 I z (50)
9 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 453 S d 4 d 4 F 5 + d 4 I z F 5 F 5 d 2 4 I z S Iz F 5 Sz d 2 d 4 4 I z S d 2 4 I2 z S 2 I z F 5 d 4 I2 z 2 d 4 F 5 F 5 d 2 + d 4 d 4 2 F 5 4 I z S d I z F 5 4 I2 z S d 2 4 I z S d 3 4 I z S d 4 I2 z I z Sz F 5 d + I z d 4 F 5 Sz 4 I2 z d 2 + I z Iz 2 F 5 Sz 4 I2 z d 4 I3 z Furthr diffrntiating Eq. (39) with rspct to th dsign variabl d j, w obtain 2 τ Mxy (t+δt) = 2 F M2 (t+δt) F M2 Iz S S d 4 F M2 d 4 I2 z d 2 4 I z d 4 F M2 + d 4 I z F M2 I z S Sz d 4 I z + F M2 (t+δt) + 2 Sz F M2 d 4 I z S Iz F M2 Sz d 2 d 4 4 I z S d 2 4 I2 z S 2 I z F M2 F M2 d + I z d 4 F M2 4 I2 z d 4 I2 z S d 4 I z + S F M2 (t+δt) d 4 F M2 2 d 4 F M2 F M2 d 2 + d 4 d 4 2 F M2 4 I z S d I z 4 I2 z S d 2 4 I2 z d 4 I z S d 2 4 I z S d 3 4 I z F M2 + I z I z 2 F M2 Sz d 2 S d 4 I2 z S d 4 I3 z Computation procdur of th first and scond drivativs of dynamic displacmnt and strss In this work, w suppos that th initial conditions ar x (0) = 0, ẋ (0) = 0, ẍ (0) = 0, and th dynamic rsponss (nodal displacmnts and strsss), thir first and scond drivativs with rspct to dsign variabls ar qual to zro. Sction 2..2 givs th formulas for calculating th first and scond drivativs of th dynamic rspons. This sction provids th dtaild computation procdur as follows. Procdur of calculating dynamic rspons first and scond drivativs: Stp Initial calculations: Stp. x (0) = 0,ẋ (0) = 0,ẍ (0) = 0,σix (0) = 0,σ jx (0) = 0,σ Mx (0) = 0,τ ixy (0) = 0,τ jxy (0) = 0, τ Mxy (0) = 0. Stp.2 x(0) =0, ẋ(0) τ jxy (0) =0, τ Mxy (0) =0. Stp.3 2 x(0) =0, 2 ẋ(0) =0, ẍ(0) =0, σ ix (0) =0, σ jx (0) =0, σ Mx (0) =0, τ ixy (0) =0, 2 τ ixy (0) =0, 2 τ jxy (0) =0, 2 τ Mxy (0) =0. =0, 2 ẍ(0) =0, 2 σ ix (0) =0, 2 σ jx (0) =0, 2 σ Mx (0) =0, 4 I z (5) (52)
10 454 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Stp.4 St Δt. Stp.5 Comput intgral constants: a 0 = βδt 2,a = δ βδt,a 2 = βδt,a 3 = 2β,a 4 = δ β,a 5 = Δt 2 a 7 = δδt. whr intgral paramtrs δ 0.5 and β =0.25 (0.5+δ) 2. Stp.6 Solv Eq. (5) K. ( ) δ β 2,a 6 =Δt ( δ), Stp 2 Calculations for ach tim stp t +Δt: Stp 2. Solv Eq. (4) x. Stp 2.2 Solv Eq. (7) ẍ, solv Eq. (2) σix, solv Eq. (22) σ jx, solv Eq. (23) σmx, solv Eq. (24) τ ixy, solv Eq. (25) τ jxy, and solv Eq. (26) τmxy. Stp 2.3 Solv Eq. (8) ẋ. Stp 2.4. Solv Eq. (27) x(t+δt). Stp 2.5 Solv Eq. (30) ẍ(t+δt) solv Eq. (35) σ jx (t+δt) solv Eq. (37) τ ixy (t+δt) and solv Eq. (39) τ Mxy (t+δt). Stp 2.6 Solv Eq. (3) ẋ(t+δt). Stp 2.7 Solv Eq. (40) 2 x(t+δt)., solv Eq. (34) σ ix (t+δt), solv Eq. (36) σ Mx (t+δt),,, solv Eq. (38) τ jxy (t+δt), Stp 2.8 Solv Eq. (43) 2 ẍ(t+δt), solv Eq. (47) 2 σix (t+δt), solv Eq. (48) 2 σ jx (t+δt), solv Eq. (49) 2 σ Mx (t+δt), solv Eq. (50) 2 τixy (t+δt), solv Eq. (5) 2 τjxy (t+δt), and solv Eq. (52) 2 τ Mxy (t+δt). Stp 2.9 Solv Eq. (44) 2 ẋ(t+δt). Stp 3 Rptition for th nxt tim stp. Rplac t by t +Δt and implmnt stps 2. to 2.9 for th nxt tim stp First and scond drivativs of structural mass W us structural mass as th objctiv function in optimization. Th objctiv function or th structural mass can b xprssd as w (d) = N ρa l = Th first drivativs of th structural mass ar obtaind by diffrntiating Eq. (53) with rspct to th dsign variabls, w (d) N A = ρl (54) = Th scond drivativs of th structural mass can b obtaind by furthr diffrntiating Eq. (54) with rspct to th dsign variabls, (53)
11 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs w (d) = N 2 A ρl = (55) 3. Optimization mathmatical modl In gnral, w aim to minimiz th total mass w (d) of th structur. At th sam tim, th normal strsss, shar strsss, and nodal displacmnts should satisfy th constraints in th duration of transint loads. W formulat th optimization problm of th structurs subjctd to transint loads as follows. Find d min w (d) s.t. σix + (d,t) ( =, 2,,N,t 0, Γ]) σ ix (d,t) ( =, 2,,N,t 0, Γ]) σ + jx (d,t) ( =, 2,,N,t 0, Γ]) σ jx (d,t) ( =, 2,,N,t 0, Γ]) σ + Mx (d,t) ( =, 2,,N,t 0, Γ]) σ Mx (d,t) ( =, 2,,N,t 0, Γ]) τ] τixy (d,t) τ] ( =, 2,,N,t 0, Γ]) τ] τjxy (d,t) τ] ( =, 2,,N,t 0, Γ]) τ] τmxy (d,t) τ] ( =, 2,,N,t 0, Γ]) x k ] x k (d,t) x k ] (k =, 2,,N f ) d J d J d J (J =, 2,, 4N) (56) whr + and ar tnsion and comprssion, rspctivly. is allowabl normal strss. τ] is allowabl shar strss. Γ is th duration of transint loads. N f is th numbr of dgr of frdom. x k ] is allowabl displacmnt on th kth dgr of frdom. d J is th lowr limit of th Jth dsign variabl. dj is th uppr limit of th Jth dsign variabl. Normalizing th constraints of th optimal modl Eq. (56), w obtain th nw quivalnt mathmatic modl as follows. Find d min w (d) s.t. g (d,t)= σ+ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +N (d,t)= σ+ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +2N (d,t)= σ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +3N (d,t)= σ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +4N (d,t)= σ+ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +5N (d,t)= σ+ jx (d,t) 0( =, 2,,N; t 0, Γ])
12 456 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs g +6N (d,t)= σ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +7N (d,t)= σ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +8N (d,t)= σ+ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +9N (d,t)= σ+ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +0N (d,t)= σ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +N (d,t)= σ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +2N (d,t)= τ ixy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +3N (d,t)= τ ixy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +4N (d,t)= τ jxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +5N (d,t)= τ jxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +6N (d,t)= τ Mxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +7N (d,t)= τ Mxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g k+8n (d,t)= x k(d,t) x k ] 0(k =, 2,,N f ) g k+nf +8N (d,t)= x k(d,t) x k ] 0(k =, 2,,N f ) g J (d) = dj d J 0(J =, 2,, 4N) g J+4N (d) = dj d J 0(J =, 2,, 4N) (57) 4. Transformation of mathmatical modl Th ky fatur of th transformation mthod is to transform a constraind problm into an unconstraind problm. Thus, w minimiz only on function in th transformation mthod. This is an attractiv aspct in that many timdpndnt constraints and an objctiv function can b mrgd into a singl function. Th rprsntativs of th transformation mthod ar th augmntd Lagrang multiplir mthod 6] and th xtrior pnalty function mthod 20] in structur optimal dsign undr dynamic loads. Howvr, th augmntd Lagrang multiplir function and th xtrior pnalty function ar discontinuous functions, thrfor, th gradint and Hssian matrix calculations of ths functions ar difficult whn th dirct diffrntiation mthod is mployd to obtain th first and scond drivativs. Compard to th augmntd Lagrang multiplir function and th xtrior pnalty function, th intrior pnalty function is a continuous function, so th gradint and Hssian matrix calculations of th intrior pnalty function ar rlativly asy whn th dirct diffrntiation mthod is usd to obtain th first and scond drivativs. Th intrior pnalty function mthod rquirs a fasibl initial dsign point. Typically, it may b difficult to obtain a fasibl initial dsign in a complx problm. Howvr, in structural optimization problms, a fasibl dsign point can b found in th structurs with th larg crosssctional aras. Thrfor, in this papr th intrior pnalty function is mployd to transform th inquality constraint optimization problm.
13 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Intrior pnalty function mthod In this papr th inquality constraint optimal mathmatic modl Eq. (57) is convrtd to a squnc of appropriatly formd unconstraind intgral mathmatic modl using th intrior pnalty function mthod. Th intrior pnalty function mthod is adoptd as follows. 2N f +8N Γ P (d,r k )=w (d) r k 8N g J (d,t) dt + (58) g I (d) J= 0 In Eq. (58), pnalty paramtr r k is a squnc of numbrs which ar dgrssiv. Whn r k 0, th minimum of pnalty function P (d,r k ) approachs th minimum of th constraint problm. So th solution of th inquality constraint optimization modl, Eq. (57), is transformd into a squnc of unconstraint problms: Find d 2N f +8N Γ min P (d,r k )=w (d) r k 8N g J (d,t) dt + (59) g I (d) J= 0 Initial pnalty paramtr r can b calculatd by th following quation, 2N f +8N Γ r 8N g J (d 0,t) dt + = p 0 g I (d 0 ) 00 w (d 0) (60) J= 0 I= whr d 0 is th initial dsign point and p 0 = 50. In this work, w choos p 0 = 50. Th pnalty paramtr r k dcrass according to th following rul: r k+ = r k (6) c whr c =0 50 and c = 0 in this work. k is th numbr of pnalty paramtr which will b usd in th procss of sarch Calculation of gradint and Hssian matrix of intrior pnalty function Now w calculat th first and scond drivativs of th pnalty function with rspct to th structural dsign variabls. Th tim stp and duration of dynamic loads ar Δt and Γ, rspctivly. Lt a = Γ Δt. Th first drivativs of pnalty function can b obtaind by diffrntiating Eq. (58) with rspct to th dsign variabl d i, P (d,r k ) = w (d) + r k 2N f +8N Γ J= 0 I= I= g J (d,t) 8N gj 2 (d,t) dt + I= g I (d) gi 2 (d) (62) Th scond drivativs of pnalty function is calculatd by furthr diffrntiating Eq. (62) with rspct to th dsign variabl d j, 2 P (d,r k ) 0 = 2 w (d) + r k 2N f +8N J= Γ 0 2 g J (d,t) gj 3 (d,t) g J (d,t) + 2 ] g J (d,t) gj 2 (d,t) dt 8N 2 g I (d) g I (d) + r k g 3 I= I (d) (63) Th intgral trms in Eqs (58), (59), (60), (62), (63) ar computd by using th trapzoidal form intgral formula: Γ a g J (d,t) dt = ] Δt 2 g J (d,zδt) + (64) g J (d, (z +)Δt) z=0
14 458 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Γ 0 g J (d,t) gj 2 (d,t) a a dt= z=0 Δt 2 Γ 2 g J (d,t) g J (d,t) 0 gj 3 (d,t) + { Δt 2 g J (d,zδt) 2 g 3 z=0 J (d,zδt) 2 g J (d, (z +)Δt) gj 3 (d, (z +)Δt) g 2 J (d,zδt) g J (d,zδt) + 2 ] g J (d,t) gj 2 (d,t) dt = g J (d,zδt) + g 2 c (d,zδt) g J (d, (z +)Δt) + ] g J (d, (z+)δt) gj 2 (d, (z+)δt) (65) 2 ] g J (d,zδt) + (66) ]} 2 g J (d, (z +)Δt) gj 2 (d, (z +)Δt) Th first and scond drivativs of th pnalty function with rspct to th structural variabls ar calculatd. Thn th gradint and Hssian matrix can b achivd. 5. Solving optimization problms Th inquality constraint optimization modl, Eq. (57), is convrtd into a squnc of th appropriatly formd unconstraind intgral modl, Eq. (59). Marquardt s mthod, a gradinthssian matrixbasd algorithm, is adoptd to solv th unconstraind problm, taking advantags that th gradint and Hssian matrixs ar fully usd in this optimal mthod. Marquardt s mthod combins Cauchy s and Nwton s mthods in a convnint mannr that xploits th strngths of both but dos rquir scondordr information. Th major mrit of Marquardt s mthod is its simplicity, dscnt proprty, xcllnt convrgnc rat nar th optimum, and absnc of a lin sarch. Basd on Marquardt s mthod, th computation procdur of solving th mathmatic modl Eq. (56) is as follows. Th computr procdur of solving th mathmatic modl Eq. (56): Stp. Chos th initial fasibl dsign point d 0, calculat r by solving Eq. (60), dfin convrgnc critrion ε,ltk =. Stp 2. Start from dsign point d k, solv th mathmatic modl Eq. (59) with Marquardt s mthod to obtain th optimum dsign d k.th stps of solving th mathmatic modl Eq. (59) with Marquardt s mthod is from Stp 2.. to Stp 2..: Stp 2.. Lt d (0) k = d k. Dfin M I = imum numbr of itrations allowd ε 2 =convrgnc critrion I =idntity matrix Stp 2.2. St i =0. λ (0) ( =0 5. ) Stp 2.3. Calculat P d (i) k,r k. ( Stp 2.4. Is P d (i) k k),r ε2? Ys: Go to stp 2.. No: Continu. Stp 2.5. Is i M I? Ys: Go to stp 2.. No: Continu. Stp 2.6. Calculat S Stp 2.7. St d (i+) k ( ) d (i) k = ( k + S = d(i) ( d (i+) k,r k Stp 2.8. Is P Ys: Go to stp 2.9. ) <P 2 P d (i) k ). ( d (i) k,r k ( ) ] ( ) d (i) k,r k + λ (i) I P d (i) k,r k. )?
15 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 459 No:Gotostp2.0. Stp 2.9. St λ (i+) = 2 λ(i) and i = i +.Gotostp2.3. Stp 2.0. St λ (i) =2λ (i).gotostp2.6. Stp 2.. St d k = d (i) k. Go to stp 3. Stp 3. Is P (d k,r k ) P (d k,r k ) ε? Ys: d k is th bst dsign. Printrsultandstop. No: Continu. Stp 4. Calculat r k+ by solving Eq. (6), k = k +.Gotostp2. 6. Numrical xampl In this sction, optimization dsign of th plan fram shown in Fig. 2 is prformd using th mthod proposd in this papr. Th plan fram is dividd with thr plan bam lmnts. Th crosssction of th lmnt is H structuralstl shap, shown in Fig.. W us typical stl matrials proprtis, i.., lastic modulus E = 20 GPa and matrial dnsity ρ = 7850 kg/m 3, for all lmnts. Th structural damping ratios ar ς = 0.02, ς 2 = Th horizontal dynamic load, F h (t) = 500 sin 3π 4 t (kn), acts on th nod 2. A duration of 3 s and an incrmntal tim stp of 0.0 s ar considrd in optimization procdurs. Th convrgnc critrion: ε = 0 3, ε 2 = 0 3 and th imum numbr of itrations allowd: M I = 5. Numrical tsts show that ths critria ar sufficint for achiving convrgnc in a rasonabl tim. Fig. 2. Plan fram. In th optimal mathmatic modl Eq. (57), th allowabl normal strss =200 MPa and allowabl shar strss τ] =00 MPa; th nodal allowabl displacmnts ar: x ]=0.00 m, x 2 ]=0.00 m, x 3 ]=0. rad, x 4 ] = 0.00 m, x 5 ] = 0.00 m, x 6 ] = 0. rad; th dsign variabl vctor is dfind as: d = d,d 2, d 3,d 4,d2,d2 2,d2 3,d2 4,d3,d3 2,d3 3,d3 4 ]T. Th dsign spac is shown in Tabl. Tabl Dsign spac of fram (unit: mm) Dsign variabls d d 2 d 3 d 4 d 2 d 2 2 d 2 3 d 2 4 d 3 d 3 2 d 3 3 d 3 4 Lowr limit Uppr limit Th optimum dsigns of th plan fram ar sarchd from two diffrnt initial fasibl dsign points. Squncs of optimum dsigns ar shown in Tabls 2 and 3. From th optimum rsults shown in Tabls 2 and 3, th masss of th optimum dsigns approach to th final targts, 26 kg and 6 kg, from th initial fasibl dsign (i.., 53 kg) and 2 (i.., 45 kg), rspctivly. It indicats that th optimization mthod prsntd in this papr is ffctiv. Howvr, th optimum dsigns ar diffrnt if th initial dsigns ar not sam. Thrfor, th optimum dsigns obtaind with th optimization mthod in this papr ar local solutions, but not global solutions. W should find th diffrnt local solutions from as many initial dsigns as possibl. Thn w can choos th bst dsign from th local dsigns as th ffctiv dsign.
16 460 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Tabl 2 Initial fasibl dsign and optimum dsign (unit: mm) Dsign variabls d d 2 d 3 d 4 d 2 d 2 2 d 2 3 d 2 4 d 3 d 3 2 d 3 3 d 3 4 Mass/kg Tim/s Initial dsign Mthod in this papr r = r 2 = r 3 = Augmntd Lagrang multiplir Tabl 3 Initial fasibl dsign 2 and optimum dsign (unit: mm) Dsign variabls d d 2 d 3 d 4 d 2 d 2 2 d 2 3 d 2 4 d 3 d 3 2 d 3 3 d 3 4 Mass/kg Tim/s Initial dsign Mthod in this papr r = r 2 = r 3 = r 4 = Augmntd Lagrang multiplir All simulations ar prformd on a prsonal computr with Window XP oprating systm. Th computr has a Pntium(R) 4 CPU, and frquncy of th CPU is 2.8 GHz. Th computr also has 52MB RAM. Th itration courss shown in Fig. 3 indicat that th algorithm is convrgnt. From initial fasibl dsign, th computational tim of achiving th local optimum dsign is about 749 s with th mthod in this papr, and about 8426 s with th Augmntd Lagrang multiplir mthod. From initial fasibl dsign 2, th computational tim of achiving th local optimum dsign is about 8896 s with th mthod in this papr, and about 042 s with th Augmntd Lagrang multiplir mthod. Th computational tim shows that th mthod proposd in this papr is somtims mor fficint than th Augmntd Lagrang multiplir mthod. Mass/kg Tim/s Fig. 3. Itration courss. Initial dsign Initial dsign 2 It should b notd that th gradint and Hssian matrix calculation is vry difficult for th dynamic optimization problm bcaus it is difficult to calculat th dynamic rspons first and scond drivativs. Gnrally, th gradint and Hssian matrix calculation rquirs much computational tim that can not b accptd in th structural optimization. Howvr, in this work w dvlop an algorithm, only a singl dynamics analysis is rquird, to obtain th gradint and Hssian matrix. In addition, w us th intrior pnalty function mthod to transform a constraind problm into an unconstraind problm. So many timdpndnt constraints and an objctiv function can b mrgd into a squnc of appropriatly formd unconstraind intgral singl timindpndnt functions. Thos unconstraind intgral singl timindpndnt functions ar continuous, so th gradint and Hssian matrix calculations ar asir than othr discontinuous transform functions, th augmntd Lagrang multiplir function and xtrior pnalty function.
17 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Conclusions In this papr, w dvlopd an optimization mthod of structurs subjctd to transint loads. Th conclusions ar as follows. () An algorithm is formulatd to calculat structural dynamic rsponss, i.., nodal displacmnts and strsss, and thir first and scond drivativs with rspct to dsign variabls. Th algorithm is achivd by dirct diffrntiation and only a singl dynamics analysis basd on Nwmarkβ mthod is rquird. (2) Th intrior pnalty function mthod is usd to transform a constraind problm into an unconstraind problm. So many timdpndnt constraints and an objctiv function can b mrgd into a squnc of appropriatly formd unconstraind intgral singl timindpndnt functions. Thos unconstraind intgral singl timindpndnt functions ar continuous, so th gradint and Hssian matrix calculations ar asir than othr discontinuous transform functions, th augmntd Lagrang multiplir function and xtrior pnalty function. (3) Th gradinthssian matrixbasd optimization mthod prsntd in this papr has th charactristics of its simplicity, dscnt proprty, xcllnt convrgnc rat nar th optimum, and absnc of a lin sarch. (4) Th numrical rsults show that th optimum dsigns obtaind with th optimization mthod prsntd in this papr ar local solutions, but not global solutions and that th optimization mthod is ffctiv. (5) Th numrical rsults also show that th optimal dsign mthod proposd in this papr is somtims mor fficint than th augmntd Lagrang multiplir mthod. Somtims, it dpnds on th chosn initial dsign. Rfrncs ] F.Y. Kocr and J.S. Arora, Optimal dsign of hfram transmission pols for arthquak loading, Journal of Structur Enginring 25 (999), ] A.E. Baumal, J.J. McPh and P.H. Calamai, Application of gntic algorithms to th dsign optimization of an activ vhicl suspnsion systm, Computr Mthods in Applid Mchanics and Enginring 63 (998), ] C.P. Pantlids and S.R. Tsan, Optimal dsign of dynamically constraints structurs, Computrs and Structurs 62 (997), ] I. Buchr, Paramtric optimization of structurs undr combind bas motion dirct forcs and static loading, Journal of Vibration and Acoustics Transactions of th ASME 24 (2002), ] F. van Kuln, R.T. Haftka and N.H. Kim, Rviw of options for structural dsign snsitivity analysis. Part : Linar systms, Computr Mthods in Applid Mchanics and Enginring 94 (2005), ] C.C. Hsih and J.S. Arora, Dsign snsitivity analysis and optimization of dynamic rspons, Computr Mthods in Applid Mchanics and Enginring 43 (984), ] J.L. Chn and J.S. Ho, Dirct variational mthod for sizing dsign snsitivity analysis of bam and fram structurs, Computrs and Structurs 42 (992), ] K. Kulkarni and A.K. Noor, Snsitivity analysis for th dynamic rspons of viscoplastic shlls of rvolution, Computrs and Structurs 55 (995), ] M. Bogomolni, U. Kirsch and I. Shinman, Efficint dsign snsitivitis of structurs subjctd to dynamic loading, Intrnational Journal of Solids and Structurs 43 (2006), ] U. Kirsch, M. Bogomolni and F. van Kuln, Efficint finitdiffrnc dsignsnsitivitis, AIAA Journal 43 (2005), ] U. Kirsch and P.Y. Papalambros, Accurat displacmnt drivativs for structural optimization using approximat ranalysis, Computr Mthods in Applid Mchanics and Enginring 90 (200), ] K.W. L and G.J. Park, Accuracy tst of snsitivity analysis in th smianalytic mthod with rspct to configuration variabls, Computrs and Structurs 63 (997), ] B.S. Kang, G.J. Park and J.S. Arora, A rviw of optimization of structurs subjctd to transint loads, Structural and Multidisciplinary Optimization 3 (2006), ] J.S. Arora and J.E.B. Cardoso, A dsign snsitivity analysis principl and its implmntation into ADINA, Computrs and Structurs 32 (989), ] C.C. Hsih and J.S. Arora, A hybrid formulation for tratmnt of pointwis stat variabl constraints in dynamic rspons optimization, Computr Mthods in Applid Mchanics and Enginring 48 (985), ] A.I. Chahand and J.S. Arora, Dvlopmnt of a multiplir mthod for dynamic rspons optimization problm, Structural Optimization 6 (993), ] C.P. Pantlids and S.R. Tsan, Optimal dsign of dynamically constraints structurs, Computrs and Structurs 62 (997), ] N.M. Nwmark, A Mthod of Computation for structural dynamics, Journal of Enginr Mchanics Division 85 (959), ] G.V. Rklaitis, A. Ravindran and K.M. Ragsdll, Enginring Optimization Mthods and Applications, John Wily and Sons, Nw York, ] J.H. Cassis and L.A. Schmit, Optimum structural dsign with dynamic constraints, Journal of Structural Enginring Procdings ASCE 02 (976),
Introduction to Finite Element Modeling
Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation
More informationEFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS
25 Vol. 3 () JanuaryMarch, pp.375/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut
More informationNonHomogeneous Systems, Euler s Method, and Exponential Matrix
NonHomognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous firstordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach
More information(Analytic Formula for the European Normal Black Scholes Formula)
(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually
More informationA Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 4549 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) 92.222  Linar Algbra II  Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial
More informationQUANTITATIVE METHODS CLASSES WEEK SEVEN
QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.
More informationME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
More informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. GangLn Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationMathematics. Mathematics 3. hsn.uk.net. Higher HSN23000
hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails
More informationNew Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ralvalud Fourir sris is xplaind and formula ar givn for convrting
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More informationAdverse Selection and Moral Hazard in a Model With 2 States of the World
Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,
More informationThe Normal Distribution: A derivation from basic principles
Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn
More informationPolicies for Simultaneous Estimation and Optimization
Policis for Simultanous Estimation and Optimization Migul Sousa Lobo Stphn Boyd Abstract Policis for th joint idntification and control of uncrtain systms ar prsntd h discussion focuss on th cas of a multipl
More informationLecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
More informationThe example is taken from Sect. 1.2 of Vol. 1 of the CPN book.
Rsourc Allocation Abstract This is a small toy xampl which is wllsuitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of Cnts. Hnc, it can b rad by popl
More informationIncomplete 2Port Vector Network Analyzer Calibration Methods
Incomplt Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 1213)
con 37: Answr Ky for Problm St (Chaptr 23) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc
More informationICES REPORT 1501. January 2015. The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 78712
ICES REPORT 1501 January 2015 A lockingfr modl for RissnrMindlin plats: Analysis and isogomtric implmntation via NURBS and triangular NURPS by L. Birao da Viga, T.J.R. Hughs, J. Kindl, C. Lovadina,
More informationQuestion 3: How do you find the relative extrema of a function?
ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating
More informationby John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia
Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs
More informationCPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.
Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by
More informationStatistical Machine Translation
Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm
More informationWhole Systems Approach to CO 2 Capture, Transport and Storage
Whol Systms Approach to CO 2 Captur, Transport and Storag N. Mac Dowll, A. Alhajaj, N. Elahi, Y. Zhao, N. Samsatli and N. Shah UKCCS Mting, July 14th 2011, Nottingham, UK Ovrviw 1 Introduction 2 3 4 Powr
More informationJournal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi
Journal of Enginring and Natural Scincs Mühndisli v Fn Bilimlri Drgisi Sigma 4/ Invitd Rviw Par OPTIMAL DESIGN OF NONLINEAR MAGNETIC SYSTEMS USING FINITE ELEMENTS Lvnt OVACIK * Istanbul Tchnical Univrsity,
More informationA MultiHeuristic GA for Schedule Repair in Precast Plant Production
From: ICAPS03 Procdings. Copyright 2003, AAAI (www.aaai.org). All rights rsrvd. A MultiHuristic GA for Schdul Rpair in Prcast Plant Production WngTat Chan* and Tan Hng W** *Associat Profssor, Dpartmnt
More informationArchitecture of the proposed standard
Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th
More informationGold versus stock investment: An econometric analysis
Intrnational Journal of Dvlopmnt and Sustainability Onlin ISSN: 2688662 www.isdsnt.com/ijds Volum Numbr, Jun 202, Pag 7 ISDS Articl ID: IJDS20300 Gold vrsus stock invstmnt: An conomtric analysis Martin
More informationFACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data
FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among
More informationCategory 7: Employee Commuting
7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil
More informationLecture 20: Emitter Follower and Differential Amplifiers
Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.
More information811ISD Economic Considerations of Heat Transfer on Sheet Metal Duct
Air Handling Systms Enginring & chnical Bulltin 811ISD Economic Considrations of Hat ransfr on Sht Mtal Duct Othr bulltins hav dmonstratd th nd to add insulation to cooling/hating ducts in ordr to achiv
More informationProceedings of the 6th WSEAS International Conference on Simulation, Modelling and Optimization, Lisbon, Portugal, September 2224, 2006 246
Procdings of th 6th WSEAS Intrnational Confrnc on Simulation, Modlling and Optimization, Lisbon, Portugal, Sptmbr 2224, 2006 246 Larg dformation modling in soiltillag tool intraction using advancd 3D
More informationUpper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing
Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif
More informationHardware Modules of the RSA Algorithm
SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 11, No. 1, Fbruary 2014, 121131 UDC: 004.3`142:621.394.14 DOI: 10.2298/SJEE140114011S Hardwar Moduls of th RSA Algorithm Vlibor Škobić 1, Branko Dokić 1,
More informationCHAPTER 4c. ROOTS OF EQUATIONS
CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03  Computation Mthod in Civil Enginring II Dpartmnt o Civil
More information7 Timetable test 1 The Combing Chart
7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing
More informationImproving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost
Economy Transdisciplinarity Cognition www.ugb.ro/tc Vol. 16, Issu 1/2013 5054 Improving Managrial Accounting and Calculation of Labor Costs in th Contxt of Using Standard Cost Lucian OCNEANU, Constantin
More informationSPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM
RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs
More informationWORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769
081685 WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769 Summary of Dutis : Dtrmins City accptanc of workrs' compnsation cass for injurd mploys; authorizs appropriat tratmnt
More informationVector Network Analyzer
Cours on Microwav Masurmnts Vctor Ntwork Analyzr Prof. Luca Prrgrini Dpt. of Elctrical, Computr and Biomdical Enginring Univrsity of Pavia mail: luca.prrgrini@unipv.it wb: microwav.unipv.it Microwav Masurmnts
More informationC H A P T E R 1 Writing Reports with SAS
C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd
More information5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power
Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim
More informationKeywords Cloud Computing, Service level agreement, cloud provider, business level policies, performance objectives.
Volum 3, Issu 6, Jun 2013 ISSN: 2277 128X Intrnational Journal of Advancd Rsarch in Computr Scinc and Softwar Enginring Rsarch Papr Availabl onlin at: wwwijarcsscom Dynamic Ranking and Slction of Cloud
More informationFar Field Estimations and Simulation Model Creation from Cable Bundle Scans
Far Fild Estimations and Simulation Modl Cration from Cabl Bundl Scans D. Rinas, S. Nidzwidz, S. Fri Dortmund Univrsity of Tchnology Dortmund, Grmany dnis.rinas@tudortmund.d stphan.fri@tudortmund.d Abstract
More informationSharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means
Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s166001507411 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans WiMao Qian
More informationVersion 1.0. General Certificate of Education (Alevel) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.
Vrsion.0 Gnral Crtificat of Education (Alvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,
More informationMEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM. Hans G. Jonasson
MEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM Hans G. Jonasson SP Tchnical Rsarch Institut of Swdn Box 857, SE501 15 Borås, Swdn hans.jonasson@sp.s ABSTRACT Drum sound, that is th walking
More informationThe international Internet site of the geoviticulture MCC system Le site Internet international du système CCM géoviticole
Th intrnational Intrnt sit of th goviticultur MCC systm L sit Intrnt intrnational du systèm CCM géoviticol Flávio BELLO FIALHO 1 and Jorg TONIETTO 1 1 Rsarchr, Embrapa Uva Vinho, Caixa Postal 130, 95700000
More informationAbstract. Introduction. Statistical Approach for Analyzing Cell Phone Handoff Behavior. Volume 3, Issue 1, 2009
Volum 3, Issu 1, 29 Statistical Approach for Analyzing Cll Phon Handoff Bhavior Shalini Saxna, Florida Atlantic Univrsity, Boca Raton, FL, shalinisaxna1@gmail.com Sad A. Rajput, Farquhar Collg of Arts
More informationConstraintBased Analysis of Gene Deletion in a Metabolic Network
ConstraintBasd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFGRsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany
More informationAn Adaptive Clustering MAP Algorithm to Filter Speckle in Multilook SAR Images
An Adaptiv Clustring MAP Algorithm to Filtr Spckl in Multilook SAR Imags FÁTIMA N. S. MEDEIROS 1,3 NELSON D. A. MASCARENHAS LUCIANO DA F. COSTA 1 1 Cybrntic Vision Group IFSC Univrsity of São Paulo Caia
More informationWaves and Vibration in Civil Engineering
Wavs and Vibration An ntrodction to Wavs and Vibration in ivil Enginring ntrodction to spctral lmnts and soilstrctr intraction Matthias Baitsch Vitnams Grman Univrsity Ho hi Min ity Yvona olová lova Tchnical
More informationSOFTWARE ENGINEERING AND APPLIED CRYPTOGRAPHY IN CLOUD COMPUTING AND BIG DATA
Intrnational Journal on Tchnical and Physical Problms of Enginring (IJTPE) Publishd by Intrnational Organization of IOTPE ISSN 077358 IJTPE Journal www.iotp.com ijtp@iotp.com Sptmbr 015 Issu 4 Volum 7
More informationPerformance Evaluation
Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Baylik rputation systms: Analysis, charactrization and insuranc mchanism
More informationInternational Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)
Intrnational Association of Scintific Innovation and Rsarch (IASIR) (An Association Unifing th Scincs, Enginring, and Applid Rsarch) ISSN (Print): 79000 ISSN (Onlin): 79009 Intrnational Journal of Enginring,
More informationSUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* RostovonDon. Russia
SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* RostovonDon. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs
More informationSection 7.4: Exponential Growth and Decay
1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 117 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
More informationFinite Elements from the early beginning to the very end
Finit Elmnts from th arly bginning to th vry nd A(x), E(x) g b(x) h x =. x = L An Introduction to Elasticity and Hat Transfr Applications x Prliminary dition LiUIEIS8/535SE Bo Torstnflt Contnts
More informationTheoretical approach to algorithm for metrological comparison of two photothermal methods for measuring of the properties of materials
Rvista Invstigación Cintífica, ol. 4, No. 3, Nuva época, sptimbr dicimbr 8, IN 187 8196 Thortical approach to algorithm for mtrological comparison of two photothrmal mthods for masuring of th proprtis
More informationThe Constrained SkiRental Problem and its Application to Online Cloud Cost Optimization
3 Procdings IEEE INFOCOM Th Constraind SkiRntal Problm and its Application to Onlin Cloud Cost Optimization Ali Khanafr, Murali Kodialam, and Krishna P. N. Puttaswam Coordinatd Scinc Laborator, Univrsit
More informationAn Broad outline of Redundant Array of Inexpensive Disks Shaifali Shrivastava 1 Department of Computer Science and Engineering AITR, Indore
Intrnational Journal of mrging Tchnology and dvancd nginring Wbsit: www.ijta.com (ISSN 22502459, Volum 2, Issu 4, pril 2012) n road outlin of Rdundant rray of Inxpnsiv isks Shaifali Shrivastava 1 partmnt
More informationVibrational Spectroscopy
Vibrational Spctroscopy armonic scillator Potntial Enrgy Slction Ruls V( ) = k = R R whr R quilibrium bond lngth Th dipol momnt of a molcul can b pandd as a function of = R R. µ ( ) =µ ( ) + + + + 6 3
More informationMETHODS FOR HANDLING TIED EVENTS IN THE COX PROPORTIONAL HAZARD MODEL
STUDIA OECONOMICA POSNANIENSIA 204, vol. 2, no. 2 (263 Jadwiga Borucka Warsaw School of Economics, Institut of Statistics and Dmography, Evnt History and Multilvl Analysis Unit jadwiga.borucka@gmail.com
More informationPlanning and Managing Copper Cable Maintenance through Cost Benefit Modeling
Planning and Managing Coppr Cabl Maintnanc through Cost Bnfit Modling Jason W. Rup U S WEST Advancd Tchnologis Bouldr Ky Words: Maintnanc, Managmnt Stratgy, Rhabilitation, Costbnfit Analysis, Rliability
More informationTIME MANAGEMENT. 1 The Process for Effective Time Management 2 Barriers to Time Management 3 SMART Goals 4 The POWER Model e. Section 1.
Prsonal Dvlopmnt Track Sction 1 TIME MANAGEMENT Ky Points 1 Th Procss for Effctiv Tim Managmnt 2 Barrirs to Tim Managmnt 3 SMART Goals 4 Th POWER Modl In th Army, w spak of rsourcs in trms of th thr M
More informationDevelopment of Financial Management Reporting in MPLS
1 Dvlopmnt of Financial Managmnt Rporting in MPLS 1. Aim Our currnt financial rports ar structurd to dlivr an ovrall financial pictur of th dpartmnt in it s ntirty, and thr is no attmpt to provid ithr
More informationUniversity of Pisa, Department of Civil and Industrial Engineering, Italy
Strojniši vstni  Journal of Mchanical Enginring 60(2014)5, 363372 Rcivd for rviw: 20131213 2014 Journal of Mchanical Enginring. All rights rsrvd. Rcivd rvisd form: 20140214 DOI:10.5545/svjm.2014.1837
More informationKey Management System Framework for Cloud Storage Singa Suparman, Eng Pin Kwang Temasek Polytechnic {singas,engpk}@tp.edu.sg
Ky Managmnt Systm Framwork for Cloud Storag Singa Suparman, Eng Pin Kwang Tmask Polytchnic {singas,ngpk}@tp.du.sg Abstract In cloud storag, data ar oftn movd from on cloud storag srvic to anothr. Mor frquntly
More informationDeveloping Software Bug Prediction Models Using Various Software Metrics as the Bug Indicators
Dvloping Softwar Bug Prdiction Modls Using Various Softwar Mtrics as th Bug Indicators Varuna Gupta Rsarch Scholar, Christ Univrsity, Bangalor Dr. N. Ganshan Dirctor, RICM, Bangalor Dr. Tarun K. Singhal
More informationA Theoretical Model of Public Response to the Homeland Security Advisory System
A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL 60607 wxding@uicdu Using a diffrntial
More information5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:
.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This
More informationthe socalled KOBOS system. 1 with the exception of a very small group of the most active stocks which also trade continuously through
Liquidity and InformationBasd Trading on th Ordr Drivn Capital Markt: Th Cas of th Pragu tock Exchang Libor 1ÀPH³HN Cntr for Economic Rsarch and Graduat Education, Charls Univrsity and Th Economic Institut
More informationCombinatorial Analysis of Network Security
Combinatorial Analysis of Ntwork Scurity Stvn Nol a, Brian O Brry a, Charls Hutchinson a, Sushil Jajodia a, Lynn Kuthan b, and Andy Nguyn b a Gorg Mason Univrsity Cntr for Scur Information Systms b Dfns
More informationLG has introduced the NeON 2, with newly developed Cello Technology which improves performance and reliability. Up to 320W 300W
Cllo Tchnology LG has introducd th NON 2, with nwly dvlopd Cllo Tchnology which improvs prformanc and rliability. Up to 320W 300W Cllo Tchnology Cll Connction Elctrically Low Loss Low Strss Optical Absorption
More informationFleet vehicles opportunities for carbon management
Flt vhicls opportunitis for carbon managmnt Authors: Kith Robrtson 1 Dr. Kristian Stl 2 Dr. Christoph Hamlmann 3 Alksandra Krukar 4 Tdla Mzmir 5 1 Snior Sustainability Consultant & Lad Analyst, Arup 2
More information1754 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007
1754 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 007 On th Fasibility of Distributd Bamforming in Wirlss Ntworks R. Mudumbai, Studnt Mmbr, IEEE, G. Barriac, Mmbr, IEEE, and U. Madhow,
More informationEssays on Adverse Selection and Moral Hazard in Insurance Market
Gorgia Stat Univrsity ScholarWorks @ Gorgia Stat Univrsity Risk Managmnt and Insuranc Dissrtations Dpartmnt of Risk Managmnt and Insuranc 800 Essays on Advrs Slction and Moral Hazard in Insuranc Markt
More informationThroughput and Buffer Analysis for GSM General Packet Radio Service (GPRS)
Throughput and Buffr Analysis for GSM Gnral Packt Radio Srvic (GPRS) Josph Ho, Yixin Zhu, and Sshu Madhavapddy Nortl Ntorks 221 Laksid Blvd. Richardson, TX 7582 EMail: joho@nortlntorks.com Abstract 
More informationDENTAL CAD MADE IN GERMANY MODULAR ARCHITECTURE BACKWARD PLANNING CUTBACK FUNCTION BIOARTICULATOR INTUITIVE USAGE OPEN INTERFACE. www.smartoptics.
DENTAL CAD MADE IN GERMANY MODULAR ARCHITECTURE BACKWARD PLANNING CUTBACK FUNCTION BIOARTICULATOR INTUITIVE USAGE OPEN INTERFACE www.smartoptics.d dntprogrss an b rsion c v o m d ss.d! A fr ntprog.d w
More informationGOAL SETTING AND PERSONAL MISSION STATEMENT
Prsonal Dvlopmnt Track Sction 4 GOAL SETTING AND PERSONAL MISSION STATEMENT Ky Points 1 Dfining a Vision 2 Writing a Prsonal Mission Statmnt 3 Writing SMART Goals to Support a Vision and Mission If you
More informationProduction Costing (Chapter 8 of W&W)
Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary
More informationI. INTRODUCTION. Figure 1, The Input Display II. DESIGN PROCEDURE
Ballast Dsign Softwar Ptr Grn, Snior ighting Systms Enginr, Intrnational Rctifir, ighting Group, 101S Spulvda Boulvard, El Sgundo, CA, 9045438 as prsntd at PCIM Europ 0 Abstract: W hav dvlopd a Windows
More informationAn IAC Approach for Detecting Profile Cloning in Online Social Networks
An IAC Approach for Dtcting Profil Cloning in Onlin Social Ntworks MortzaYousfi Kharaji 1 and FatmhSalhi Rizi 2 1 Dptartmnt of Computr and Information Tchnology Enginring,Mazandaran of Scinc and Tchnology,Babol,
More informationVoice Biometrics: How does it work? Konstantin Simonchik
Voic Biomtrics: How dos it work? Konstantin Simonchik Lappnranta, 4 Octobr 2012 Voicprint Makup Fingrprint Facprint Lik a ingrprint or acprint, a voicprint also has availabl paramtrs that provid uniqu
More informationTheoretical aspects of investment demand for gold
Victor Sazonov (Russia), Dmitry Nikolav (Russia) Thortical aspcts of invstmnt dmand for gold Abstract Th main objctiv of this articl is construction of a thortical modl of invstmnt in gold. Our modl is
More information[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lbinsec^2)
MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (voltssc/rad Motor torqu constant (lbin/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris
More informationAn Analysis of Synergy Degree of PrimaryTertiary Industry System in Dujiangyan City
www.ccsnt.org/ijbm Intrnational Journal of Businss and Managmnt Vol. 6, No. 8; August An Analysis of Synrgy Dgr of PrimaryTrtiary Industry Systm in Dujiangyan City Qizhi Yang School of Tourism, Sichuan
More informationResearch Progress in Acoustical Application to Petroleum Logging and Seismic Exploration
Snd Ordrs of Rprints at rprints@bnthamscinc.nt Th Opn Acoustics Journal 23 6  Opn Accss Rsarch Progrss in Acoustical Application to Ptrolum Logging and Sismic Exploration Lin Fa * Li Wang Yuan Zhao 2
More informationChapter 3: Capacitors, Inductors, and Complex Impedance
haptr 3: apacitors, Inductors, and omplx Impdanc In this chaptr w introduc th concpt of complx rsistanc, or impdanc, by studying two ractiv circuit lmnts, th capacitor and th inductor. W will study capacitors
More informationJune 2012. Enprise Rent. Enprise 1.1.6. Author: Document Version: Product: Product Version: SAP Version: 8.81.100 8.8
Jun 22 Enpris Rnt Author: Documnt Vrsion: Product: Product Vrsion: SAP Vrsion: Enpris Enpris Rnt 88 88 Enpris Rnt 22 Enpris Solutions All rights rsrvd No parts of this work may b rproducd in any form or
More informationInstallation Saving Spaceefficient Panel Enhanced Physical Durability Enhanced Performance Warranty The IRR Comparison
Contnts Tchnology Nwly Dvlopd Cllo Tchnology Cllo Tchnology : Improvd Absorption of Light Doublsidd Cll Structur Cllo Tchnology : Lss Powr Gnration Loss Extrmly Low LID Clls 3 3 4 4 4 Advantag Installation
More informationREPORT' Meeting Date: April 19,201 2 Audit Committee
REPORT' Mting Dat: April 19,201 2 Audit Committ For Information DATE: March 21,2012 REPORT TITLE: FROM: Paul Wallis, CMA, CIA, CISA, Dirctor, Intrnal Audit OBJECTIVE To inform Audit Committ of th rsults
More informationOn the moments of the aggregate discounted claims with dependence introduced by a FGM copula
On th momnts of th aggrgat discountd claims with dpndnc introducd by a FGM copula  Mathiu BARGES Univrsité Lyon, Laboratoir SAF, Univrsité Laval  Hélèn COSSETTE Ecol Actuariat, Univrsité Laval, Québc,
More informationPrinciples of Humidity Dalton s law
Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid
More informationSTATEMENT OF INSOLVENCY PRACTICE 3.2
STATEMENT OF INSOLVENCY PRACTICE 3.2 COMPANY VOLUNTARY ARRANGEMENTS INTRODUCTION 1 A Company Voluntary Arrangmnt (CVA) is a statutory contract twn a company and its crditors undr which an insolvncy practitionr
More informationFactorials! Stirling s formula
Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical
More informationExamples. Epipoles. Epipolar geometry and the fundamental matrix
Epipoar gomtry and th fundamnta matrix Epipoar ins Lt b a point in P 3. Lt x and x b its mapping in two imags through th camra cntrs C and C. Th point, th camra cntrs C and C and th (3D points corrspon
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scic March 15, 005 Srii Dvadas ad Eric Lhma Problm St 6 Solutios Du: Moday, March 8 at 9 PM Problm 1. Sammy th Shar is a fiacial srvic providr who offrs loas o th followig
More information