Inverse Stress Relaxation and Viscoelastic Recovery of Multifilament Textile Yarns in Different Test Cycles
|
|
- Abner Stanley Cunningham
- 5 years ago
- Views:
Transcription
1 ISSN 9 MATERIALS SCIENCE (MEDŽIAGOTYRA). Vol., No.. Inverse Stress Relxtion nd Viscoelstic Recovery of Multifilment Textile Yrns in Different Test Cycles Rit POCIENĖ, Arvyds VITKAUSKAS Deprtment of Textile Technology, Kuns University of Technology, Studentų, LT- Kuns, Lithuni Received August ; ccepted Jnury The rticle presents the results of the experimentl reserch of inverse stress relxtion (IR) nd viscoelstic recovery (VR) tht tke plce in cette nd polyester multifilment yrns in dependnce on the mechnicl pre-history. The ove-mentioned time-effects re investigted in two different stress relxtion (R-) nd creep (C-) testing cycles. The fct tht inverse stress relxtion process tkes plce in C- test cycle, i.e. fter previous sustining the specimen t constnt lod is experimentlly confirmed. It is shown tht viscoelstic recovery is the slower process thn the inverse stress relxtion. At identicl elongtions of the yrns t the end of loding period the inverse relxtion nd viscoelstic recovery processes go on similrly regrdless of the chrcter of testing cycle. Keywords: relxtion, inverse relxtion, creep, recovery, yrn, viscoelsticity. INTRODUCTION Strength nd deformility of textile yrns re importnt only mechnicl properties determining the ehviour of yrns in woven or knitted frics mking-up s well s the ehviour of fric in end-use. The ehviour of textile mterils s of ny polymeric odies is mostly viscoelstic. The response of mteril to the specific mechnicl ction depends not only on the ction itself ut lso on the former ctions undergone, i. e., it depends on mechnicl pre-history of mteril [, ]. This implies tht time-dependence of the response of ny textile mteril opposing the pplied forces, should e tken into ccount [ ]. Viscoelstic properties cn serve s n index of gretly vrious purposes, e. g. for comprtive evlution of mterils or s criterion t the control of the specific process. Experimentl investigtion of viscoelstic properties in textile fires nd yrns is commonly sed on the results of tensile tests tht cn e relted to two min groups of testing cycles, tking into considertion the conditions of mechnicl ction on the mteril nd the prmeters mesured. The tests t constnt elongtion ε t (Fig., ) re relted to the first group. In these tests stress relxtion (R) s well s inverse stress relxtion (IR) (smooth lines) or viscoelstic recovery (VR) in length of specimen (dotted lines) cn e oserved. The tests t constnt force or lod F c (Fig., ) re relted to the second group. In this cse creep (C) s well s viscoelstic creep recovery (CR) (dotted lines) or inverse stress relxtion (IR) (smooth lines) cn e oserved in the tests. Numerous works re done reveling the fetures of creep, recovery nd stress relxtion in textiles: the pulictions [, 7 ] cn serve s the proper exmples. The phenomenon of inverse stress relxtion ws first mentioned in 9 y Stein, Hlsey nd Eyring [], i. e. Corresponding uthor. Tel.: +7-7-; fx.: E-mil ddress: rit.pociene@ktu.lt (R. Pocienė) sustntilly lter thn the ove mentioned phenomen. Inverse relxtion is often met with under prcticl conditions [ 7], so its study is mtter of gret interest. Despite of the fct tht the studies of the inverse relxtion ecme more intensive during lst decdes [ - ], knowledge on the regulrities of the phenomenon re oviously insufficient. Except for theoreticl model proposed in [], up to now there re no experimentl dt on the inverse relxtion mnifesting in the testing cycle of the second group (Fig., ). Strin (ε ), stress (F ) Strin (ε ), stress (F ) R F t ε t θ t F θ F ε VR IR t t t θ t Time (t ) ε c C θ t F c F ε θ ε CR IR t t t θ t Time (t ) Fig.. The testing cycles: test t constnt elongtion; test t constnt force
2 In some pulictions the ttempts hve een mde to predict the development of one viscoelstic ftereffect y the dt on the nother one [, 7, ]. Presumptively positive results could e otined predicting the inverse relxtion y viscoelstic recovery, for the oth phenomen strt to develop t the solutely identicl mechnicl pre-history (points F in Fig., nd ). Such prediction would e especilly useful ecuse the inverse relxtion test implictes much more prolems thn the comprtively simple recovery test [7]. The im of this study is to investigte experimentlly the chrcter of oth inverse relxtion nd viscoelstic recovery in different textile yrns t identicl mechnicl pre-history. A specil considertion is pid to the inverse relxtion development following the creep in the yrns ecuse there re no ny experimentl dt on the effect in the yrn eing undergone y constnt force. EXPERIMENTAL Two different types of multifilment yrns were tken for the experimentl investigtion: cette (CA). tex nd polyester (PES). tex. All specimens were preconditioned, then conditioned nd tested in the tmospheres ccording to ISO 9. The experiments were provided on Zwick/Z universl testing mchine. To ensure the setting of testing cycle prmeters with s possile higher ccurcy the guge length ws tken 7 mm, i. e., lrger thn it is customry used in yrn testing. Ech specimen ws pretensioned to. mn/tex efore the testing. The testing cycle t constnt elongtion R- (for symols used elow in Fig., ) consisting of four phses ws provided in two different regimes (R-IR nd R-VR). The first three phses were identicl for oth regimes: ) constnt rte (v t ) extension up to the strin ε t, ) sustining t ε t up to the time t* = (t t t ) = θ t, while stress relxtion (R) ws going on nd mesured, ) constnt rte (v ) retrction down to pretension (F ). The fourth phse ws different for the prticulr regime. For the regime R-IR the specimen ws sustined t strin ε corresponding to F, nd the inverse relxtion (IR) ws going on nd mesured in time t** = (t t ). For the regime R-VR, in the lst fourth phse viscoelstic recovery VR, i.e the decrese in specimen strin ε ws going on nd mesured in time t** = (t t ) t constnt pretension F. The testing cycle t constnt force (for symols used elow look t Fig., ) ws lso provided in two regimes (C-IR nd C-CR). The first three phses were identicl for oth regimes: ) constnt rte (v t ) extension up to the force F c corresponding to the strin ε c, ) sustining t force F c up to the time t* = (t t t ) = θ t, while creep (C) ws going on nd mesured, c) constnt rte (v ) retrction down to pretension (F ). For the regime C-IR the fourth phse ws identicl to tht of the regime R-IR, while for the regime C-CR the fourth phse ws identicl to tht of the regime R-VR. In ll tests the rte of extension (v t ) ws mm/min (. % / s) nd it ws equl to the rte of retrction (v ). The limits of extension (ε t, F c ) during the first phses of the regimes, nd the sustining times (θ t ) during the second phses were vried. The inverse relxtion (IR) nd the viscoelstic recovery (VR, CR) were minly mesured during the time t** = s. In some cses the mesuring time ws prolonged up to t** = s. To exmine the chrcters of oth IR nd CR processes running in the C- testing cycle s distinct from the corresponding IR nd VR processes running in the R- testing cycle the tests were provided in two different modes: Mode I: The level of force F c in C- testing cycle is equl to the corresponding level of force F θ in R- testing cycle. Mode II: The level of strin ε t in R- testing cycle is equl to the corresponding level of strin ε θ in C- testing cycle. RESULTS AND DISCUSSIONS Stress relxtion curves of CA nd PES yrns re shown in Fig.. Stress relxtion process is very distinctive in CA yrn, where the ovious yield point is oserved in its stress-strin curve. With increse of the set extension level (ε t ) the vlues of stress increse s well. However, when the extension level of CA yrn is in the yield zone (ε t = 9. %), from time t* s the vlues of stress re lower thn t elongtion ε t = %. This nomlous ehviour of cette yrn ws formerly noticed y Meredith [], who explined it s kind of ditic sudden stretch. In our cse, the extension rte is not so high to rise the temperture of the specimen. It is possile tht the ove-mentioned nomly of the ehviour is relted to complex entropy chnges in stretched yrn. Extending the CA yrn over the yield point, the relxtion curves ecome lmost prllel. The yield zone is not chrcteristic for PES yrn so s extension level increses, relxtion is going on t higher stresses throughout the whole oservtion time θ t. Creep curves of the yrns t constnt forces in the C- test cycle re presented in Fig.. With increse of the force (F c ), elongtions of the yrns during creep process increse s well. Creep of CA yrns is especilly intense t yield zone nd over it, while in PES yrn t the zone where the slope of its stress-strin curve mrkedly grows up through the most structure chnge, i.e. when ε c > %. Due to chrcteristic yield zone in stress-strin curve of CA yrn the dt otined in R- nd C- cycles when testing in mode I re unlikely comprle. It is seen in Fig., tht in R- test cycle t t* ( ) s stresses of CA yrn during relxtion process nrrowly differ etween themselves while the relxtion is going on t completely different elongtions ε t. Stress development (IR) curves of the yrns in oth test cycles of mode I (the vlues of F c, mn/tex for CA/PES yrn re: 9.7/.9 t θ t = s,./7. t θ t = s,./. t θ t = s,./99.7 t θ t = s) re shown in Figure, nd,. In R- test cycle of CA yrn the mount of stress increse is slightly dependent on the sustining time efore the retrction. Due to superposition of inverse relxtion nd ordinry stress relxtion processes, the distinct mximum in the curves is oserved. With increse of the sustining time θ t, the inverse 9
3 F, mn/tex 9 7. t*, s F, mn/tex. t*, s Fig.. Stress relxtion curves of the yrns. CA yrn; ε t : %, %, 9. %,. % (mode II); PES yrn; ε t : %, %, %,. %,. % (mode II) ε, % ε, %. t*, s. t *, s Fig.. Creep curves of the yrns. CA yrn; F c, mn/tex :., 7.,., 7.; PES yrn ; F c, mn/tex : 7., 99., 99.7,., 9. relxtion process ecomes slower nd less ffected y the stress relxtion process. Therefore, the mximum in the curves moves towrds longer time of oservtion (t**). After θ t = s the mximum is moved so considerly tht it could not e reched even till oservtion time t** = s. The IR process is more slow s the sustining time is incresed. In C- test cycle the mximum in IR curves moves towrds time t** xis in dependence on the sustining time θ t in the similr wy s in the R- cycle. Nevertheless, the increse in stress (IR) is pproximtely four times lower thn in R- test cycle. In the tests y mode I the vlues of strin ε t in R- cycle is 9. %, i.e. ove the yield point, while ll vlues of strin ε θ in C- cycle t constnt force re elow the yield point (ε θ =. % t θ t = s, ε θ =. % t θ t = s, ε θ =. % t θ t = s, ε θ =. % t θ t = s). So, the increse in stress depends more on the sustining time thn on the yrn elongtion: the increse in stress t time θ t = s is the highest despite the mount of strin (ε θ ) efore the retrction is the lowest (ε θ =. %). It is supposed tht this fct is ssocited with more pronounced orienttion of polymer chins during creep if compred to tht during relxtion t constnt elongtion. For the PES yrn the mximum in the stress development (IR) curves is eyond the oservtion time, ut the tendency of its movement towrds longer time of oservtion (t**) cn e distinguished. The chrcter of stress increse dependence on test regime is nlogous to tht of CA yrn. However, for PES yrn the difference in the vlues of strins when testing in the different cycles is not so distinct s for CA yrn: in R- cycle ε t =. %, while in C- cycle t θ t = s ε θ =. %, t θ t = s ε θ =. %, t θ t = s ε θ =. %, nd t θ t = s ε θ =. %. Therefore, inverse relxtion is quite comprle in its mount when testing in the different cycles. The results oviously showed tht the presupposition of the inverse stress relxtion process s tking plce in C- test cycle, i.e. fter previous sustining the specimen t constnt lod is proven out. The effect must e lso credily chrcteristic for ny for ny viscoelstic polymeric mteril. The viscoelstic recovery (VR, CR) of the yrns in mode I depends on the test regime similrly to the IR process (Fig., nd Fig., ). Due to the resons discussed ove creep recovery vlues of the yrns in the C- testing cycle re lower thn in R- testing cycle: pproximtely ten times lower for CA yrn nd pproximtely. times lower for PES yrn. Much more comprle results re otined in R- nd C- test cycles, when testing in mode II, i.e. when identicl elongtions vlues t time instnt t θ re mintined in oth test cycles (Fig., nd Fig. ). The vlues of strin ε t = ε θ, % for CA/PES yrn re the following: 9./. t 7
4 F F, mn/tex. 7 ε ε, % Fig.. Inverse stress relxtion () nd viscoelstic recovery () curves of CA yrn in R- (, left scle) nd C- ( , right scle) test cycles, mode I (ε t = 9. %); θ t : s, s, s, s F F, mn/tex 9 7. ε ε, % F F, mn/tex. ε ε, % Fig.. Inverse stress relxtion (IR, left scle) nd viscoelstic recovery (VR (CR), right scle) curves of PES yrn in R- ( ) nd C- ( ) test cycles. mode I, (ε t =. %); mode II, (F c = 9. mn/tex); IR curves s θ t : s, s, s, s; VR nd CR curves s θ t : s, s F F, mn/tex. ε ε, % Fig.. Inverse stress relxtion () nd viscoelstic recovery () curves of CA yrn in R- (, left scle) nd C- ( , right scle) test cycles, mode II (F c = 7. mn/tex); θ t : s, s, s, s 7
5 θ t = s,./7. t θ t = s, 9.7/.7 t θ t = s, nd./. t θ t = s. It is seen tht when the sme strin vlue is held on efore the retrction the mount of stress increse nd of viscoelstic contrction in R- nd C- cycles re similr nd tht with increse of sustining time efore the retrction the processes go on slower similrly to mode I. As result, it is oviously seen tht t identicl elongtions of the yrns t the end of loding period the inverse relxtion nd viscoelstic recovery processes go on similrly regrdless of the chrcter of testing cycle. CONCLUSIONS The inverse stress relxtion process is proven out s tking plce in the yrns in C- test cycle, i.e. fter previous sustining the specimen t constnt lod. The effect must e lso credily chrcteristic for textile frics nd for ny polymeric mteril. Viscoelstic recovery is the slower process thn the inverse stress relxtion. The time during which the yrns re undergone y specified constnt elongtion or constnt lod is the most effective fctor influencing the mount nd chrcter of oth inverse relxtion nd viscoelstic recovery in the yrns. At identicl elongtions of the yrns t the end of loding period the inverse relxtion nd viscoelstic recovery processes go on similrly regrdless of the chrcter of testing cycle. REFERENCES. Ledermn, H. Elstic nd Creep Properties of Filmentous Mterils nd other High Polymers. nd print. Wshington: The Textile Foundtion, 9.. Meredith, R. The Mechnicl Properties of Textile Fires. Amsterdm: North-Hollnd Pul. Co, 99.. Nchne, R. P., Sundrm, V. Anlysis or Relxtion Phenomen in Textile Fires Prt I: Stress Relxtion The Journl of The Textile Institute () 99: pp. 9.. Mnich, A. M., Ussmn, M. H., Brell, A. Viscoelstic Behvior of Polypropylene Fiers Textile Reserch Journl 9 () 999: pp... Menrd, K. P. Dynmic Mechnicl Anlysis: A Prcticl Introduction. CRC Press LLC, Wu, X., Wng, F., Wng, S. Properties of Wool/PET Composite Yrns Textile Reserch Journl 7 () : pp Aott, N. J. Extension nd Relxtion of Nylon Filments Textile Reserch Journl () 9: pp. 7.. Guthrie, J. C., Wierley, J. The Effect of Time on the Elstic Recovery of Fires Journl of the Textile Institute () 9: pp Gupt, V. B., Gopl Krishnn, Y. Creep nd Recovery Behviour of Oriented Nylon Filments Indin Journl of Textile Reserch () 97: pp Meredith, R. Relxtion of Stress in Stretched Cellulose Fires Journl of the Textile Institute () 9: pp. T T.. Stlevich, A. M., Tirnov, V. G., Meshchninov, Yu. N., Vol f, L. A. Anlyticl Description of the Process of Stress Relxtion in Synthetic Yrn Technology of the Textile Industry U.S.S.R Mnchester, The Textile Institute 97: pp... Vng, X., Yu, L. Y. The Stress Relxtion of Wool t High Strining Rte Journl of the Textile Institute () 99: pp. 9.. Inoue, M., Niw, M. Tensile nd Tensile Stress Relxtion Properties of Wool/Cotton Plied Yrns Textile Reserch Journl 7 () 997: pp. 7.. Stein, R., Hlsey, G., Eyring, H. Mechnicl Properties of Textile. IV. Textile Reserch Journl () 9: pp... Vitkusks, A., Mtukonis, A. Phenomenologicl Presenttion of the Inverse Stress Relxtion of Yrn Under Tension Lod Technology of the Textile Industry U.S.S.R Mnchester, The Textile Institute 9: pp. 9.. Vngheluwe, L., Kiekens, P. Simultion of Procedures to Avoid Set Mrks in Weving Cused y Relxtion Textile Reserch Journl 7 () 997: pp Vitkusks, A. Viscoelstic Properties of Textile Yrns. Reserch Prolems Fires & Textiles in Estern Europe () 99: pp... Mnich, A. M., de Cstellr, M. D. Elstic Recovery nd Inverse Relxtion of Polyester Stple Fier Rotor Spun Yrns Textile Reserch Journl () 99: pp Vngheluwe, L. Relxtion nd Inverse Relxtion of Yrns After Dynmic Loding. Textile Reserch Journl, vol., No. 9, 99: pp... Nchne, R. P., Sundrm, V. Anlysis or Relxtion Phenomen in Textile Fires Prt II: Inverse Relxtion The Journl of the Textile Institute () 99: pp... Vitkusks, A. Simultion of Inverse Stress Relxtion Following Creep in Textile Mterils Mterils Science (Medžigotyr) () 99: pp. 7.. Kothri, V. K., Rjkhow, R., Gupt, V. B. Stress Relxtion nd Inverse Stress Relxtion in Silk Fiers Journl of Applied Polymer Science : pp. 7.. Milšius, V. Investigtion of the Stress Relxtion nd the Inverse Stress Relxtion of the Chemicl Yrns nd the Woven Fric when the Relxtion Processes re Not Liner Behviour Tekhnologij tekstil noi promyshlennosti (Technology of the Textile Industry U.S.S.R.) 97: pp. 9 (in Russin).. Ymguchi, T., Kitgw, T., Yngw, T., Kimur, H. Reltionship Between Stress Relxtion nd Tensile Recovery of Filment Yrns Journl of the Textile Mchinery Society of Jpn 7 () 9: pp. 9.. Vitkusks, A. Inverse Relxtion nd Delyed Elstic Recovery of Textile Yrns: Resemlnt or Disprte? Textiles nd The Informtion Society Ppers Presented t The 7 th World Conference of The Textile Institute (My, 997, Thessloniki, Greece), vol. III, pp
Bend Forms of Circular Saws and Evaluation of their Mechanical Properties
ISSN 139 13 MATERIALS SCIENCE (MEDŽIAGOTYRA). Vol. 11, No. 1. 5 Bend Forms of Circulr s nd Evlution of their Mechnicl Properties Kristin UKVALBERGIENĖ, Jons VOBOLIS Deprtment of Mechnicl Wood Technology,
More informationMechanical Properties and a Physical-Chemical Analysis of Acetate Yarns
ISSN 392 320 MATERIALS SCIENCE (MEDŽIAGOTYRA). Vol. 0, No.. 2004 Mechnicl Properties nd Physicl-Chemicl Anlysis of Acette Yrns R. Pocienė, R. Žemititienė 2, A. Vitkusks Deprtment of Textile Technology,
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationAvailable online at ScienceDirect. Procedia Engineering 172 (2017 )
Aville online t www.sciencedirect.com ScienceDirect Procedi Engineering 172 (2017 ) 218 225 Modern Building Mterils, Structures nd Techniques, MBMST 2016 Experimentl nd Numericl Anlysis of Direct Sher
More informationThis chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2
1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion
More informationDesigning Information Devices and Systems I Spring 2018 Homework 7
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should
More informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationANALYSIS OF MECHANICAL PROPERTIES OF COMPOSITE SANDWICH PANELS WITH FILLERS
ANALYSIS OF MECHANICAL PROPERTIES OF COMPOSITE SANDWICH PANELS WITH FILLERS A. N. Anoshkin *, V. Yu. Zuiko, A.V.Glezmn Perm Ntionl Reserch Polytechnic University, 29, Komsomolski Ave., Perm, 614990, Russi
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 17
CS 70 Discrete Mthemtics nd Proility Theory Summer 2014 Jmes Cook Note 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion, y tking
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationCompiler Design. Fall Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz
University of Southern Cliforni Computer Science Deprtment Compiler Design Fll Lexicl Anlysis Smple Exercises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sciences Institute 4676 Admirlty Wy, Suite
More informationLecture 09: Myhill-Nerode Theorem
CS 373: Theory of Computtion Mdhusudn Prthsrthy Lecture 09: Myhill-Nerode Theorem 16 Ferury 2010 In this lecture, we will see tht every lnguge hs unique miniml DFA We will see this fct from two perspectives
More informationSpecial Relativity solved examples using an Electrical Analog Circuit
1-1-15 Specil Reltivity solved exmples using n Electricl Anlog Circuit Mourici Shchter mourici@gmil.com mourici@wll.co.il ISRAE, HOON 54-54855 Introduction In this pper, I develop simple nlog electricl
More informationAMPERE CONGRESS AMPERE on Magnetic Resonance and Related Phenomena. Under the auspices of The GROUPEMENT AMPERE
AMPERE 2000 th 30 CONGRESS AMPERE on Mgnetic Resonnce nd Relted Phenomen Lison, Portugl, 23-2 July 2000 Under the uspices of The GROUPEMENT AMPERE Edited y: A.F. MARTINS, A.G. FEIO nd J.G. MOURA Sponsoring
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More informationAN IMPROVED SMALL CLOSED DRIFT THRUSTER WITH BOTH CONDUCTING AND DIELECT RIC CHANNELS
AN IMPROVED SMALL CLOSED DRIFT THRUSTER WITH BOTH CONDUCTING AND DIELECT RIC CHANNELS A.I.Bugrov, A.D.Desitskov, H.R.Kufmn, V.K.Khrchevnikov, A.I.Morozov c, V.V.Zhurin d Moscow Institute of Rdioelectronics,
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationLab 11 Approximate Integration
Nme Student ID # Instructor L Period Dte Due L 11 Approximte Integrtion Ojectives 1. To ecome fmilir with the right endpoint rule, the trpezoidl rule, nd Simpson's rule. 2. To compre nd contrst the properties
More informationAnalytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationHamiltonian Cycle in Complete Multipartite Graphs
Annls of Pure nd Applied Mthemtics Vol 13, No 2, 2017, 223-228 ISSN: 2279-087X (P), 2279-0888(online) Pulished on 18 April 2017 wwwreserchmthsciorg DOI: http://dxdoiorg/1022457/pmv13n28 Annls of Hmiltonin
More informationCOUPLING OF DAMAGE MECHANICS AND PROBABILISTIC APPROACH FOR LIFE-TIME PREDICTION OF COMPOSITE STRUCTURES
ORAL/POSTER REFERENCE : COUPLING OF DAMAGE MECHANICS AND PROBABILISTIC APPROACH FOR LIFE-TIME PREDICTION OF COMPOSITE STRUCTURES Y. Bruner, J. Renrd, D. Jeulin nd A. Thionnet Centre des Mtériux Pierre-Mrie
More informationA027 Uncertainties in Local Anisotropy Estimation from Multi-offset VSP Data
A07 Uncertinties in Locl Anisotropy Estimtion from Multi-offset VSP Dt M. Asghrzdeh* (Curtin University), A. Bon (Curtin University), R. Pevzner (Curtin University), M. Urosevic (Curtin University) & B.
More informationADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS
ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS F. Tkeo 1 nd M. Sk 1 Hchinohe Ntionl College of Technology, Hchinohe, Jpn; Tohoku University, Sendi, Jpn Abstrct:
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationEffects of peripheral drilling moment on delamination using special drill bits
journl of mterils processing technology 01 (008 471 476 journl homepge: www.elsevier.com/locte/jmtprotec Effects of peripherl illing moment on delmintion using specil ill bits C.C. Tso,, H. Hocheng b Deprtment
More informationLesson 8. Thermomechanical Measurements for Energy Systems (MENR) Measurements for Mechanical Systems and Production (MMER)
Lesson 8 Thermomechnicl Mesurements for Energy Systems (MEN) Mesurements for Mechnicl Systems nd Production (MME) A.Y. 205-6 Zccri (ino ) Del Prete Mesurement of Mechnicl STAIN Strin mesurements re perhps
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationImpact of the tribological characteristics on the dynamics of the ultrasonic piezoelectric motor
51 ISSN 1392 1207. MECHANIKA. 2015 Volume 21(1): 51 55 Impct of the triologicl chrcteristics on the dynmics of the ultrsonic piezoelectric motor J. Pdgursks*, R. Rukuiž**, R. Bnsevičius***, V. Jūrėns****,
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationAn Overview of Integration
An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is
More informationFully Kinetic Simulations of Ion Beam Neutralization
Fully Kinetic Simultions of Ion Bem Neutrliztion Joseph Wng University of Southern Cliforni Hideyuki Usui Kyoto University E-mil: josephjw@usc.edu; usui@rish.kyoto-u.c.jp 1. Introduction Ion em emission/neutrliztion
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationIndustrial Electrical Engineering and Automation
CODEN:LUTEDX/(TEIE-719)/1-7/(7) Industril Electricl Engineering nd Automtion Estimtion of the Zero Sequence oltge on the D- side of Dy Trnsformer y Using One oltge Trnsformer on the D-side Frncesco Sull
More information1 Bending of a beam with a rectangular section
1 Bending of bem with rectngulr section x3 Episseur b M x 2 x x 1 2h M Figure 1 : Geometry of the bem nd pplied lod The bem in figure 1 hs rectngur section (thickness 2h, width b. The pplied lod is pure
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationCS 311 Homework 3 due 16:30, Thursday, 14 th October 2010
CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w
More informationAn Alternative Approach to Estimating the Bounds of the Denominators of Egyptian Fractions
Leonrdo Journl of Sciences ISSN 58-0 Issue, Jnury-June 0 p. -0 An Alterntive Approch to Estimting the Bounds of the Denomintors of Egyptin Frctions School of Humn Life Sciences, University of Tsmni, Locked
More informationApplications of Bernoulli s theorem. Lecture - 7
Applictions of Bernoulli s theorem Lecture - 7 Prcticl Applictions of Bernoulli s Theorem The Bernoulli eqution cn be pplied to gret mny situtions not just the pipe flow we hve been considering up to now.
More informationDesigning finite automata II
Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationDETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE
Determintion RevAdvMterSci of mechnicl 0(009) -7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationShear and torsion interaction of hollow core slabs
Competitive nd Sustinble Growth Contrct Nº G6RD-CT--6 Sher nd torsion interction of hollow core slbs HOLCOTORS Technicl Report, Rev. Anlyses of hollow core floors December The content of the present publiction
More informationShear Degradation and Possible viscoelastic properties of High Molecular Weight Oil Drag Reducer Polymers
ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 3, 2005 Sher Degrdtion nd Possible viscoelstic properties of High Moleculr Weight Oil Drg Reducer Polymers A.A. Hmoud, C. Elissen, C. Idsøe nd T.
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd
More informationAcceptance Sampling by Attributes
Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire
More informationA New Grey-rough Set Model Based on Interval-Valued Grey Sets
Proceedings of the 009 IEEE Interntionl Conference on Systems Mn nd Cybernetics Sn ntonio TX US - October 009 New Grey-rough Set Model sed on Intervl-Vlued Grey Sets Wu Shunxing Deprtment of utomtion Ximen
More informationA study of Pythagoras Theorem
CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the est-known mthemticl theorem. Even most nonmthemticins
More informationDesigning Information Devices and Systems I Discussion 8B
Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V
More informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
More informationChapter 4: Techniques of Circuit Analysis. Chapter 4: Techniques of Circuit Analysis
Chpter 4: Techniques of Circuit Anlysis Terminology Node-Voltge Method Introduction Dependent Sources Specil Cses Mesh-Current Method Introduction Dependent Sources Specil Cses Comprison of Methods Source
More informationTHERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION
XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es
More informationCS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata
CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or
More informationData Structures and Algorithms CMPSC 465
Dt Structures nd Algorithms CMPSC 465 LECTURE 10 Solving recurrences Mster theorem Adm Smith S. Rskhodnikov nd A. Smith; bsed on slides by E. Demine nd C. Leiserson Review questions Guess the solution
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationdu = C dy = 1 dy = dy W is invertible with inverse U, so that y = W(t) is exactly the same thing as t = U(y),
29. Differentil equtions. The conceptul bsis of llometr Did it occur to ou in Lecture 3 wh Fiboncci would even cre how rpidl rbbit popultion grows? Mbe he wnted to et the rbbits. If so, then he would be
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More information9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes
The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationDetermination of the activation energy of silicone rubbers using different kinetic analysis methods
Determintion of the ctivtion energy of silicone rubbers using different kinetic nlysis methods OU Huibin SAHLI ohmed BAIEE Thierry nd GELIN Jen-Clude FETO-ST Institute / Applied echnics Deprtment, 2 rue
More informationDerivations for maximum likelihood estimation of particle size distribution using in situ video imaging
2 TWMCC Texs-Wisconsin Modeling nd Control Consortium 1 Technicl report numer 27-1 Derivtions for mximum likelihood estimtion of prticle size distriution using in situ video imging Pul A. Lrsen nd Jmes
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More informationVorticity. curvature: shear: fluid elements moving in a straight line but at different speeds. t 1 t 2. ATM60, Shu-Hua Chen
Vorticity We hve previously discussed the ngulr velocity s mesure of rottion of body. This is suitble quntity for body tht retins its shpe but fluid cn distort nd we must consider two components to rottion:
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationChapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1
Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more
More informationFormulae For. Standard Formulae Of Integrals: x dx k, n 1. log. a dx a k. cosec x.cot xdx cosec. e dx e k. sec. ax dx ax k. 1 1 a x.
Forule For Stndrd Forule Of Integrls: u Integrl Clculus By OP Gupt [Indir Awrd Winner, +9-965 35 48] A B C D n n k, n n log k k log e e k k E sin cos k F cos sin G tn log sec k OR log cos k H cot log sin
More informationCHAPTER 20: Second Law of Thermodynamics
CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het
More informationBasic model for traffic interweave
Journl of Physics: Conference Series PAPER OPEN ACCESS Bsic model for trffic interweve To cite this rticle: Ding-wei Hung 25 J. Phys.: Conf. Ser. 633 227 Relted content - Bsic sciences gonize in Turkey!
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationKINEMATICS OF RIGID BODIES
KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description
More informationTrigonometric Functions
Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds
More informationDate Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )
UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4
More informationDesigning Information Devices and Systems I Spring 2018 Homework 8
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission
More informationFig. 1. Open-Loop and Closed-Loop Systems with Plant Variations
ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses
More informationQUADRATURE is an old-fashioned word that refers to
World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationENERGY-BASED METHOD FOR GAS TURBINE ENGINE DISK BURST SPEED CALCULATION
28 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES ENERGY-BASED METHOD FOR GAS TURBINE ENGINE DISK BURST SPEED CALCULATION Anton N. Servetnik Centrl Institute of Avition Motors, Moscow, Russi servetnik@cim.ru
More information