Electrodynamic Forces in Steel Strip during Induction Heating

Size: px
Start display at page:

Download "Electrodynamic Forces in Steel Strip during Induction Heating"

Transcription

1 Intenational Scientific Colloquium Modelling fo Electomagnetic Pocessing Hannove, Mach 4-6, 3 Electodynamic Foces in Steel Stip duing Induction Heating H. Kasjanow, H. Schülbe,. Nikanoov bstact Tangential component of foce density f t Fig. 1. Natue of electodynamic foces in metal stip duing induction heating B The foces, which act on cuent and/o field-conducting components of induction heating installation ae the esult of the ecipocal effect of the electomagnetic (EM) field with the cuent flowing in the system as well as with field-conducting elements [1]. These foces can cause oscillations of pats of the plant o even movement of the wokpiece. If the heating installation is designed asymmetically, developing foces can also lead to defomations of the stip (see fig. 1). Moeove a lage poblem could be the noise duing the tansvese flux heating. This noise esults mainly fom vibations of the stip. It is diffeentiated between tempoally constant and peiodically vaiable foces, which aise eithe individually o togethe. The tempoally constant component can lead paticulaly with thin stip to static defomations and cause in non-symmetical systems uncontolled movements of the volume. The peiodically vaiable foce components howeve cause vibations of the system elements as well as noise. 1. Models to calculate the electodynamic foces f t Nomal component of foce density f n B f n Geneally thee ae thee possibilities to calculate the electodynamic foces in tansvese flux heating systems []. The fist model is based on the ecipocal effect of cuent with magnetic field. The magnetic field is poduced by cuents in the inducto and in the othe elements of the system. The foce density f is a vecto poduct of cuent density S and flux density B. The foce density f coesponds to the integal of the volume foce density ove the stip thickness. The total foce, which affects the body, is an integal f ove the volume V: f = f dv = S B dv V. (1) V This model can be used fo calculation of electodynamic foces in non-magnetic elements of the induction systems. The second method is based on Maxwell s stess tensos. This model can be fomally used fo all kind of wokpiece. Cuent conducting and magnetic bodies cause a defomation 337

2 of the field, which leads to a esulting stength. In this case, the foce is detemined by ove a stess tenso expessed field at the suface. The thid model makes ove enegy calculations fo the total foce, which affects on the body o a pat of the body. This model can be used fo all kind of wokpiece also. The pinciple comes fom the enegy equilibium in electomagnetic field, whee the electomagnetic stength, which affects a body, poduces an elementay wok and an appopiate elementay change of electomagnetic enegy of the system. Hee the ule is used that the sum of all foces is in a closed system equal to zeo. The descibed methods make possible to calculate the foces, which affect the body if the distibutions of the cuent and the field ove the volume ae detemined.. Electodynamic foces in non-magnetic mateials Many numeical investigations ae made by an example of simple aangement. The won ealizations can be tansfeed howeve qualitatively to moe complicated systems. Because of an asymmety of the metal stip position, tangential field components ae tained (in the sheet metal), which usual esult in woking components to the sheet metal suface. The nomal component of the foce lining aises among the inducto leades. This foce ises with lage shift fom the symmetical position of the stip and is aanged in y- diection fo the cente position of the sheet metal. Theefoe it can be stated that with nonmagnetic sheet metals electodynamic foces wok stabilizing, i.e. these foces ty to cente the stip between the inductos. 3. Electodynamic foces in feomagnetic mateials The electodynamic foces, which affect a feomagnetic stip within the ai gap, consist of two pats: the Laplace foce and the magnetization foce. method to detemine this foce, exists in the integation of the electodynamic foces ove the volume foce density [3]: ( S B) dv ( H ² gadµ ) dv = ( S B 1 H ² gad ) F = 1 µ dv = f dv. () V V V V The volume density of the foce f is a sum of the Laplace component f L = S B and magnetization component f m = 1 H ² gad µ. These two foces act to opposite diections. That means that the Laplace foce woks tightening and the magnetization foce epulsively to the inducto. Then the total foce fo feomagnetic mateials is calculated by integating the volume foce density o by integating the Maxwell stess tenso. Electodynamic foces in tansvese flux induction system affect the metal stip within the ai gap between two inductos. The Laplace-component of the foce ties to cente vetically the stip within the ai gap. The magnetization component woks against it (at the same time the magnetization component centes the sheet metal in hoizontal diection). 4. Electomagnetic model fo calculation of the foces To analyse the electodynamic foces in tansvese flux heating systems, numeical computations wee accomplished using the softwae-package NSYS. In pinciple, the foces occu in all thee spatial diections. But thei epesentation is then extemely extensive. Nomal to the stip suface time-independent component of electodynamic foces is a subject investigated in the pape. In pactice these foces ae of significant impotance. It was 338

3 specified fo the calculations, that the inducto cuent I contained only one eal component ( Im( I ) = ). D appoach is used fo the calculations. The model consists of inductos, stip, magnetic coe if needed and fee space. Because the epesented model is symmetical along the y-axis, it can be educed to one half of the plant. In the othe pat of the pocess the foces look eflected alike. The stip has a thickness of d=1 mm. The dimensions of the stip ae alike in all calculations. The calculations ae accomp-lished fo two diffeent mateials: non-magnetic with µ = 1 and feomagnetic with a constant µ = 3. Electical esistivity of the stip mateial coesponds to the steel C45. Diffeent configuations of induction system with and without magnetic coe of diffeent shape wee investigated. 5. Paametic examinations The calculations wee conducted fo thee configuations of the induction system (see fig. ). Fo each type of the system the fequency, the pole pitch and the pemeability of the wokpiece wee vaied. In ode to eceive compaable esults, powe in the stip was kept constant on the level of P = 5 kw duing all calculations. Nomal to the stip suface foces ae a) System without magnetic coe equal to zeo only with accuate centeing of the sheet metal between uppe and lowe inducto, since in this case no tangential foce component aises and theefoe also the component nomally aanged to the suface is equal to zeo. In pactice an accuate centeing of the stip in the opeating gap is impossible due to inaccuate guidance of the stip o its b) System with magnetic coe without poles themal defomation. Theefoe an asymmetical deviation of the stip fom the middle position within the ai gap is accepted in the calculations. Fo the chosen geomety the optimal fequency should be selected, at which the maximum efficiency and the maximum powe facto ae eached. With ising fequency fist the electical efficiency of the tansvese flux c) System with magnetic coe with poles system inceases stongly until eaching a maximum value. ftewads it dops again easily, since the heat losses in the inducto become highe. The powe facto has a distinctive maximum. The maximum electical efficiency and the maximum powe facto appea at easily diffeent fequencies. To choose of the optimal fequency theefoe it is necessay to find a compomise between maximum efficiency and maximum powe Fig.. Types of investigated systems facto. To detemine the optimal fequency, efficiency and powe facto wee calculated fo a boad spectum of fequencies. These calculations wee accomplished using the pogam package HIHTEC D, which was developed at the Univesity of Hanove. 339

4 In the following the test esults fo non-magnetic and feomagnetic stip mateials ae compaed to each othe. Theefoe the cuves coesponding to the elative magnetic pemeability of µ = 1 and µ = 3 ae epesented in figues. The influence of pole pitch, fequency, ai gap and type of the system on the size of the total foce is examined. In fig. 3a the dependence of the esulting foce on the shift of the stip fom its cental position fo diffeent values of the pole pitch and the pemeability is epesented. The calculations wee accomplished fo optimal fequency and fo the system without magnetic coe. It is to be taken fom the pictue that the lage pole pitch povides moe homogeneous field and thus smalle foces, which wok stabilizing. t feomagnetic mateials the foces ae smalle compaed with non-magnetic mateials, since the magnetization potion educes the total foce. In this example all foces ae negative and thus stabilizing t= mm,mu()=1 t=4 mm,mu()=1 t= mm, mu()=3 t=4 mm, mu()= t= mm, mu()=1 t=4 mm, mu()=1 t= mm, mu()=3 t=4 mm, mu()= Fig. 3. Influence of pole pitch and µ fo system a) without MC and b) with MC with pole. t the system with magnetic coe with poles the foces, which affect the feomagnetic sheet metal, can become positive (see fig. 3b). In this case, the magnetization component is geate than the Laplace potion and the esulting foce becomes positive. The magnetic coe has influence on both foce components. Fo example, also the Laplace component is educed because of the concentation of the magnetic field. The fig. 4a and 4b show the distibution of the foce ove the length of non-magnetic stip fo diffeent shifts dh of the stip fom cental position. It can be distinguished at the system without magnetic coe (fig. 4a) that with appoximation of the sheet metal to the inducto the total foce becomes lage and emains always negative. Fyelemin N dh= mm dh=1 mm dh= mm dh=3 mm dh=4 mm dh=45 mm Fges = -115,5 N x-koodinate des Bleches in m Fyelemin N dh= mm dh=1 mm dh= mm dh=3 mm dh=4 mm dh=45 mm Fges = -49,69 N x-koodinate des Bleches in m Fig. 4. Distibution of nomal foces density in the stip fo the system a) without MC and b) with MC with pole. 34

5 Foces that affect the sheet metal ae concentated unde the inductos. The foce in nonfeous stip has also positive potions, although the esulting foce emains always negative. Fo the system with magnetic coe with poles (fig. 4b) the total foce is equal to zeo in the cental position and also becomes geate and woks moe stabilizing with appoximation of the stip to the inducto. But it is also shown in the pictue that the total foce becomes smalle compaed with the distibutions in the stip without magnetic coe. Thee ae moe positive foce components pesent, which lead in case of addition to a smalle magnitude of the total foce, which emains nevetheless negative. Fig. 5a, 5b and 6a epesent the dependence of the total foce on the shift of the stip fom the cental position fo diffeent fequency, when it is low o highe its optimum value. In fig. 5a fequency f and pemeability wee vaied. It was calculated with the pole pitch of t=4 mm, the ai gap of h=1 mm and without magnetic coe. It is to be taken fom the pictue that without magnetic coe the esulting foce emains always negative and thus stabilizing. The foces in feomagnetic stip ae smalle compaed with non-magnetic one, because of the magnetization component f=3 Hz, mu()=1 f= Hz, mu()=1 f=3 Hz, mu()=3 f= Hz, mu()= f=3 Hz, mu()=1 f= Hz, mu()=1 f=3 Hz, mu()=3 f= Hz, mu()= Fig. 5. Influence of fequency and µ fo system a) without MC and b) with MC with pole. Fig. 5b shows the same dependence fo the system with magnetic coe with poles. Similaly to the pevious pictue, the negative total foce is smalle at ising fequency. t non-magnetic mateials the foces fo both fequencies ae stabilizing. In contast to that, the esulting foces, which affect the feomagnetic stip, become positive, since the magnetic component outweighs the Laplace-potion. t optimal fequency the foces can go into the positive ange as well ewh f=15 Hz, mu()=1 f=3 Hz, mu()=1 f=15 Hz, mu()=3 f=3 Hz, mu()= h=15 mm, mu()=1 h=5 mm, mu()=3 h=5 mm, mu()=1 h=15 mm, mu()= Fig. 6. Influence a) f on system without MC, b) h on system with MC with pole. 341

6 The esults, which ae epesented in fig. 6a, wee calculated fo the system without magnetic coe and pole pitch of t= mm. s shown in the pictue, the negative total foce becomes lage and thus moe stongly stabilizing, at ising fequency. It is confimed again, that negative foces ae smalle with feomagnetic mateials than with non-magnetic ones. ll the time the magnetization component woks against the Laplace foce. Compaing fig. 5a and 6a it can be shown the influence of the pole pitch on non-magnetic and feomagnetic mateials. The smalle the pole pitch is the moe stongly stabilizing ae the esulting foces. Duing the calculations fo non-magnetic mateials it was shown that with enlagement of the pole pitch the magnetic field becomes moe homogeneous and the foce, which emains always negative, becomes smalle. The enlagement of the pole pitch, duing the usage of feomagnetic mateials, can lead to positive foces, which attact the stip to the one of inductos. The dependence of the foce on the shift of the stip fom its cental position fo diffeent ai gaps h is epesented in fig. 6b. The calculations wee accomplished with magnetic coe with poles, at a pole pitch of t=4 mm and at the optimal fequency. s shown in the pictue the lage the ai gap is the moe stabilizing the foces become. s befoe, the total foce in feomagnetic mateial is smalle than in non-magnetic one. With smalle opeating ai gap the foces in feomagnetic mateials become positive and thus no moe stabilizing, although with eduction of the gap the magnetic field becomes moe homogeneous. It goes without saying that the effot of the field homogenization and eduction of the foces in feomagnetic mateials can lead to positive foces. Conclusions The electodynamic foces in non-magnetic metal stip duing tansvese flux induction heating ae always negative and thus stabilizing. These foces have only the Laplace-component, which ties to bing the stip into the cental position. The esulting foces in feomagnetic mateials can become both negative and positive. That depends on which fom both potions outweighs. If the Laplace potion is lage, the foces ae negative. With lage magnetization component, which always woks against the Laplace potion, the foces become positive. But with exactly analyzed geomety of the system and opeating conditions (e.g. adjusted fequency) the heating of feomagnetic stip is possible, without any sticking of the wokpiece to the inductos. Refeences [1] Ruhnke,.: Systematische Untesuchung de induktiven Quefeld-Bandewämung. Dissetation, Univesität Hannove 1999, VDI-Velag, Düsseldof [] Nemkov,V., Demidovitch, V.: Theoie und Beechnung von induktiven Ewämungsanlagen. Enegoatomizdat, Leningad 1988, (uss.), p [3] Neau, Y., Paya, B., Tudoache, T., Fieteanu, V.; Numeical evaluation and expeimental validation of eddy cuents and elektomagnetic foces in tansvese flux induction heating of magnetic steel sheets. The 3d Intenational Symposium on Elektomagnetic Pocessing of Mateials,, Nagoya, Japan, ISIJ, p uthos Dipl.-Ing. Kasjanow, Helene Dipl.-Ing. Schülbe, Holge D.-Ing. Nikanoov, lexande Institute fo Electothemal Pocesses Univesity of Hannove Wilhelm-Busch-St. 4 D-3167 Hannove, Gemany ewh@ewh.uni-hannove.de 34

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum 2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known

More information

MAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS

MAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS The 8 th Intenational Confeence of the Slovenian Society fo Non-Destuctive Testing»pplication of Contempoay Non-Destuctive Testing in Engineeing«Septembe 1-3, 5, Potoož, Slovenia, pp. 17-1 MGNETIC FIELD

More information

THE INFLUENCE OF THE MAGNETIC NON-LINEARITY ON THE MAGNETOSTATIC SHIELDS DESIGN

THE INFLUENCE OF THE MAGNETIC NON-LINEARITY ON THE MAGNETOSTATIC SHIELDS DESIGN THE INFLUENCE OF THE MAGNETIC NON-LINEARITY ON THE MAGNETOSTATIC SHIELDS DESIGN LIVIU NEAMŢ 1, ALINA NEAMŢ, MIRCEA HORGOŞ 1 Key wods: Magnetostatic shields, Magnetic non-lineaity, Finite element method.

More information

Review: Electrostatics and Magnetostatics

Review: Electrostatics and Magnetostatics Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion

More information

Electrostatics (Electric Charges and Field) #2 2010

Electrostatics (Electric Charges and Field) #2 2010 Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when

More information

7.2. Coulomb s Law. The Electric Force

7.2. Coulomb s Law. The Electric Force Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electical and Compute Engineeing, Conell Univesity ECE 303: Electomagnetic Fields and Waves Fall 007 Homewok 8 Due on Oct. 19, 007 by 5:00 PM Reading Assignments: i) Review the lectue notes.

More information

ASTR415: Problem Set #6

ASTR415: Problem Set #6 ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal

More information

Analysis of high speed machining center spindle dynamic unit structure performance Yuan guowei

Analysis of high speed machining center spindle dynamic unit structure performance Yuan guowei Intenational Confeence on Intelligent Systems Reseach and Mechatonics Engineeing (ISRME 0) Analysis of high speed machining cente spindle dynamic unit stuctue pefomance Yuan guowei Liaoning jidian polytechnic,dan

More information

11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below.

11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below. Fall 2007 Qualifie Pat II 12 minute questions 11) A thin, unifom od of mass M is suppoted by two vetical stings, as shown below. Find the tension in the emaining sting immediately afte one of the stings

More information

Liquid gas interface under hydrostatic pressure

Liquid gas interface under hydrostatic pressure Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing,

More information

COMPUTATIONS OF ELECTROMAGNETIC FIELDS RADIATED FROM COMPLEX LIGHTNING CHANNELS

COMPUTATIONS OF ELECTROMAGNETIC FIELDS RADIATED FROM COMPLEX LIGHTNING CHANNELS Pogess In Electomagnetics Reseach, PIER 73, 93 105, 2007 COMPUTATIONS OF ELECTROMAGNETIC FIELDS RADIATED FROM COMPLEX LIGHTNING CHANNELS T.-X. Song, Y.-H. Liu, and J.-M. Xiong School of Mechanical Engineeing

More information

6.641 Electromagnetic Fields, Forces, and Motion Spring 2005

6.641 Electromagnetic Fields, Forces, and Motion Spring 2005 MIT OpenouseWae http://ocw.mit.edu 6.641 Electomagnetic Fields, Foces, and Motion Sping 2005 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems. 6.641 Electomagnetic

More information

Chapter 13 Gravitation

Chapter 13 Gravitation Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects

More information

LINEAR AND NONLINEAR ANALYSES OF A WIND-TUNNEL BALANCE

LINEAR AND NONLINEAR ANALYSES OF A WIND-TUNNEL BALANCE LINEAR AND NONLINEAR ANALYSES O A WIND-TUNNEL INTRODUCTION BALANCE R. Kakehabadi and R. D. Rhew NASA LaRC, Hampton, VA The NASA Langley Reseach Cente (LaRC) has been designing stain-gauge balances fo utilization

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

To Feel a Force Chapter 7 Static equilibrium - torque and friction

To Feel a Force Chapter 7 Static equilibrium - torque and friction To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on

More information

A NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM

A NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496

More information

Lecture 2 Date:

Lecture 2 Date: Lectue 2 Date: 5.1.217 Definition of Some TL Paametes Examples of Tansmission Lines Tansmission Lines (contd.) Fo a lossless tansmission line the second ode diffeential equation fo phasos ae: LC 2 d I

More information

J. Electrical Systems 1-3 (2005): Regular paper

J. Electrical Systems 1-3 (2005): Regular paper K. Saii D. Rahem S. Saii A Miaoui Regula pape Coupled Analytical-Finite Element Methods fo Linea Electomagnetic Actuato Analysis JES Jounal of Electical Systems In this pape, a linea electomagnetic actuato

More information

FI 2201 Electromagnetism

FI 2201 Electromagnetism FI 2201 Electomagnetism Alexande A. Iskanda, Ph.D. Physics of Magnetism and Photonics Reseach Goup Electodynamics ELETROMOTIVE FORE AND FARADAY S LAW 1 Ohm s Law To make a cuent flow, we have to push the

More information

Description of TEAM Workshop Problem 33.b: Experimental Validation of Electric Local Force Formulations.

Description of TEAM Workshop Problem 33.b: Experimental Validation of Electric Local Force Formulations. 1 Desciption of TEAM Wokshop Poblem 33.b: Expeimental Validation of Electic Local Foce Fomulations. Olivie Baé, Pascal Bochet, Membe, IEEE Abstact Unde extenal stesses, the esponse of any mateial is a

More information

7.2.1 Basic relations for Torsion of Circular Members

7.2.1 Basic relations for Torsion of Circular Members Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,

More information

AP Physics C: Electricity and Magnetism 2001 Scoring Guidelines

AP Physics C: Electricity and Magnetism 2001 Scoring Guidelines AP Physics C: Electicity and Magnetism 1 Scoing Guidelines The mateials included in these files ae intended fo non-commecial use by AP teaches fo couse and exam pepaation; pemission fo any othe use must

More information

Experiment I Voltage Variation and Control

Experiment I Voltage Variation and Control ELE303 Electicity Netwoks Expeiment I oltage aiation and ontol Objective To demonstate that the voltage diffeence between the sending end of a tansmission line and the load o eceiving end depends mainly

More information

The Millikan Experiment: Determining the Elementary Charge

The Millikan Experiment: Determining the Elementary Charge LAB EXERCISE 7.5.1 7.5 The Elementay Chage (p. 374) Can you think of a method that could be used to suggest that an elementay chage exists? Figue 1 Robet Millikan (1868 1953) m + q V b The Millikan Expeiment:

More information

you of a spring. The potential energy for a spring is given by the parabola U( x)

you of a spring. The potential energy for a spring is given by the parabola U( x) Small oscillations The theoy of small oscillations is an extemely impotant topic in mechanics. Conside a system that has a potential enegy diagam as below: U B C A x Thee ae thee points of stable equilibium,

More information

Current Balance Warm Up

Current Balance Warm Up PHYSICS EXPERIMENTS 133 Cuent Balance-1 Cuent Balance Wam Up 1. Foce between cuent-caying wies Wie 1 has a length L (whee L is "long") and caies a cuent I 0. What is the magnitude of the magnetic field

More information

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations

Phys-272 Lecture 17. Motional Electromotive Force (emf) Induced Electric Fields Displacement Currents Maxwell s Equations Phys-7 Lectue 17 Motional Electomotive Foce (emf) Induced Electic Fields Displacement Cuents Maxwell s Equations Fom Faaday's Law to Displacement Cuent AC geneato Magnetic Levitation Tain Review of Souces

More information

(read nabla or del) is defined by, k. (9.7.1*)

(read nabla or del) is defined by, k. (9.7.1*) 9.7 Gadient of a scala field. Diectional deivative Some of the vecto fields in applications can be obtained fom scala fields. This is vey advantageous because scala fields can be handled moe easily. The

More information

Physics 2212 GH Quiz #2 Solutions Spring 2016

Physics 2212 GH Quiz #2 Solutions Spring 2016 Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying

More information

( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.

( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx. 9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can

More information

Review Notes on Maxwell's Equations

Review Notes on Maxwell's Equations ELEC344 Micowave Engineeing, Sping 2002 Handout #1 Kevin Chen Review Notes on Maxwell's Equations Review of Vecto Poducts and the Opeato The del, gad o nabla opeato is a vecto, and can be pat of a scala

More information

MODULE 5 ADVANCED MECHANICS GRAVITATIONAL FIELD: MOTION OF PLANETS AND SATELLITES VISUAL PHYSICS ONLINE

MODULE 5 ADVANCED MECHANICS GRAVITATIONAL FIELD: MOTION OF PLANETS AND SATELLITES VISUAL PHYSICS ONLINE VISUAL PHYSICS ONLIN MODUL 5 ADVANCD MCHANICS GRAVITATIONAL FILD: MOTION OF PLANTS AND SATLLITS SATLLITS: Obital motion of object of mass m about a massive object of mass M (m

More information

Mathematical Analysis and Numerical Simulation of High Frequency Electromagnetic Field in Soft Contact Continuous Casting Mold

Mathematical Analysis and Numerical Simulation of High Frequency Electromagnetic Field in Soft Contact Continuous Casting Mold , pp. 974 981 Mathematical Analysis and Numeical Simulation of High Fequency Electomagnetic Field in Soft Contact Continuous Casting Mold Xianzhao NA, Xingzhong ZHANG and Yong GAN National Engineeing &

More information

Chem 453/544 Fall /08/03. Exam #1 Solutions

Chem 453/544 Fall /08/03. Exam #1 Solutions Chem 453/544 Fall 3 /8/3 Exam # Solutions. ( points) Use the genealized compessibility diagam povided on the last page to estimate ove what ange of pessues A at oom tempeatue confoms to the ideal gas law

More information

Analysis and Optimization of a Special Type of Dielectric Loaded Resonant Cavity for Mobile Communication Filters

Analysis and Optimization of a Special Type of Dielectric Loaded Resonant Cavity for Mobile Communication Filters 328 Analysis and Optimization of a Special Type of Dielectic Loaded Resonant Cavity fo Mobile Communication Filtes Haold S. Showes, Banmali S. Rawat *, Syam S. Challa Depatment of Electical and Biomedical

More information

A scaling-up methodology for co-rotating twin-screw extruders

A scaling-up methodology for co-rotating twin-screw extruders A scaling-up methodology fo co-otating twin-scew extudes A. Gaspa-Cunha, J. A. Covas Institute fo Polymes and Composites/I3N, Univesity of Minho, Guimaães 4800-058, Potugal Abstact. Scaling-up of co-otating

More information

On the integration of the equations of hydrodynamics

On the integration of the equations of hydrodynamics Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious

More information

A new force Magnetic force. Today. Force Fields: A disturbance of space. The correspondence of a loop of current and magnet.

A new force Magnetic force. Today. Force Fields: A disturbance of space. The correspondence of a loop of current and magnet. Today A new foce Magnetic foce Announcements HW#6 and HW#7 ae both due Wednesday Mach 18th. Thanks to my being WAY behind schedule, you 2nd exam will be a take-home exam! Stay tuned fo details Even if

More information

F g. = G mm. m 1. = 7.0 kg m 2. = 5.5 kg r = 0.60 m G = N m 2 kg 2 = = N

F g. = G mm. m 1. = 7.0 kg m 2. = 5.5 kg r = 0.60 m G = N m 2 kg 2 = = N Chapte answes Heinemann Physics 4e Section. Woked example: Ty youself.. GRAVITATIONAL ATTRACTION BETWEEN SMALL OBJECTS Two bowling balls ae sitting next to each othe on a shelf so that the centes of the

More information

OSCILLATIONS AND GRAVITATION

OSCILLATIONS AND GRAVITATION 1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,

More information

Physics: Work & Energy Beyond Earth Guided Inquiry

Physics: Work & Energy Beyond Earth Guided Inquiry Physics: Wok & Enegy Beyond Eath Guided Inquiy Elliptical Obits Keple s Fist Law states that all planets move in an elliptical path aound the Sun. This concept can be extended to celestial bodies beyond

More information

A Three-Dimensional Magnetic Force Solution Between Axially-Polarized Permanent-Magnet Cylinders for Different Magnetic Arrangements

A Three-Dimensional Magnetic Force Solution Between Axially-Polarized Permanent-Magnet Cylinders for Different Magnetic Arrangements Poceedings of the 213 Intenational Confeence on echanics, Fluids, Heat, Elasticity Electomagnetic Fields A Thee-Dimensional agnetic Foce Solution Between Axially-Polaied Pemanent-agnet Cylindes fo Diffeent

More information

ac p Answers to questions for The New Introduction to Geographical Economics, 2 nd edition Chapter 3 The core model of geographical economics

ac p Answers to questions for The New Introduction to Geographical Economics, 2 nd edition Chapter 3 The core model of geographical economics Answes to questions fo The New ntoduction to Geogaphical Economics, nd edition Chapte 3 The coe model of geogaphical economics Question 3. Fom intoductoy mico-economics we know that the condition fo pofit

More information

Physics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism

Physics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up

More information

Chapter 5 Force and Motion

Chapter 5 Force and Motion Chapte 5 Foce and Motion In Chaptes 2 and 4 we have studied kinematics, i.e., we descibed the motion of objects using paametes such as the position vecto, velocity, and acceleation without any insights

More information

Scattering in Three Dimensions

Scattering in Three Dimensions Scatteing in Thee Dimensions Scatteing expeiments ae an impotant souce of infomation about quantum systems, anging in enegy fom vey low enegy chemical eactions to the highest possible enegies at the LHC.

More information

Chapter 5 Force and Motion

Chapter 5 Force and Motion Chapte 5 Foce and Motion In chaptes 2 and 4 we have studied kinematics i.e. descibed the motion of objects using paametes such as the position vecto, velocity and acceleation without any insights as to

More information

Appendix B The Relativistic Transformation of Forces

Appendix B The Relativistic Transformation of Forces Appendix B The Relativistic Tansfomation of oces B. The ou-foce We intoduced the idea of foces in Chapte 3 whee we saw that the change in the fou-momentum pe unit time is given by the expession d d w x

More information

Lecture 8 - Gauss s Law

Lecture 8 - Gauss s Law Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

1.1 THE ELECTRIC CHARGE

1.1 THE ELECTRIC CHARGE 1.1 THE ELECTRIC CHARGE - In a dy day, one obseves "light spaks" when a wool pull is taken out o when the finges touch a metallic object. Aound the yea 1600, one classified these effects as electic phenomena.

More information

Mathematical Model of Magnetometric Resistivity. Sounding for a Conductive Host. with a Bulge Overburden

Mathematical Model of Magnetometric Resistivity. Sounding for a Conductive Host. with a Bulge Overburden Applied Mathematical Sciences, Vol. 7, 13, no. 7, 335-348 Mathematical Model of Magnetometic Resistivity Sounding fo a Conductive Host with a Bulge Ovebuden Teeasak Chaladgan Depatment of Mathematics Faculty

More information

Coupled Electromagnetic and Heat Transfer Simulations for RF Applicator Design for Efficient Heating of Materials

Coupled Electromagnetic and Heat Transfer Simulations for RF Applicator Design for Efficient Heating of Materials Coupled Electomagnetic and Heat Tansfe Simulations fo RF Applicato Design fo Efficient Heating of Mateials Jeni Anto 1 and Raj C Thiagaajan 2 * 1 Reseache, Anna Univesity, Chennai, 2 ATOA Scientific Technologies

More information

Right-handed screw dislocation in an isotropic solid

Right-handed screw dislocation in an isotropic solid Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We

More information

Black Body Radiation and Radiometric Parameters:

Black Body Radiation and Radiometric Parameters: Black Body Radiation and Radiometic Paametes: All mateials absob and emit adiation to some extent. A blackbody is an idealization of how mateials emit and absob adiation. It can be used as a efeence fo

More information

CHAPTER 10 ELECTRIC POTENTIAL AND CAPACITANCE

CHAPTER 10 ELECTRIC POTENTIAL AND CAPACITANCE CHAPTER 0 ELECTRIC POTENTIAL AND CAPACITANCE ELECTRIC POTENTIAL AND CAPACITANCE 7 0. ELECTRIC POTENTIAL ENERGY Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic

More information

12th WSEAS Int. Conf. on APPLIED MATHEMATICS, Cairo, Egypt, December 29-31,

12th WSEAS Int. Conf. on APPLIED MATHEMATICS, Cairo, Egypt, December 29-31, th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caio, Egypt, Decembe 9-3, 7 5 Magnetostatic Field calculations associated with thick Solenoids in the Pesence of Ion using a Powe Seies expansion and the Complete

More information

Handout: IS/LM Model

Handout: IS/LM Model Econ 32 - IS/L odel Notes Handout: IS/L odel IS Cuve Deivation Figue 4-4 in the textbook explains one deivation of the IS cuve. This deivation uses the Induced Savings Function fom Chapte 3. Hee, I descibe

More information

2 Governing Equations

2 Governing Equations 2 Govening Equations This chapte develops the govening equations of motion fo a homogeneous isotopic elastic solid, using the linea thee-dimensional theoy of elasticity in cylindical coodinates. At fist,

More information

Electric Forces: Coulomb s Law

Electric Forces: Coulomb s Law Electic Foces: Coulomb s Law All the matte aound you contains chaged paticles, and it is the electic foces between these chaged paticles that detemine the stength of the mateials and the popeties of the

More information

LC transfer of energy between the driving source and the circuit will be a maximum.

LC transfer of energy between the driving source and the circuit will be a maximum. The Q of oscillatos efeences: L.. Fotney Pinciples of Electonics: Analog and Digital, Hacout Bace Jovanovich 987, Chapte (AC Cicuits) H. J. Pain The Physics of Vibations and Waves, 5 th edition, Wiley

More information

Force between two parallel current wires and Newton s. third law

Force between two parallel current wires and Newton s. third law Foce between two paallel cuent wies and Newton s thid law Yannan Yang (Shanghai Jinjuan Infomation Science and Technology Co., Ltd.) Abstact: In this pape, the essence of the inteaction between two paallel

More information

AH Mechanics Checklist (Unit 2) AH Mechanics Checklist (Unit 2) Circular Motion

AH Mechanics Checklist (Unit 2) AH Mechanics Checklist (Unit 2) Circular Motion AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) Cicula Motion No. kill Done 1 Know that cicula motion efes to motion in a cicle of constant adius Know that cicula motion is conveniently descibed

More information

Contact impedance of grounded and capacitive electrodes

Contact impedance of grounded and capacitive electrodes Abstact Contact impedance of gounded and capacitive electodes Andeas Hödt Institut fü Geophysik und extateestische Physik, TU Baunschweig The contact impedance of electodes detemines how much cuent can

More information

A DETAILED STUDY OF THE HIGH ORDER SERIAL RESONANT INVERTER FOR INDUCTION HEATING

A DETAILED STUDY OF THE HIGH ORDER SERIAL RESONANT INVERTER FOR INDUCTION HEATING ELECTRONICS 005 1 3 Septembe, Sozopol, BULGARIA A DETAILED STUDY OF THE HIGH ORDER SERIAL RESONANT INVERTER FOR INDUCTION HEATING Evgeniy Ivanov Popov, Liliya Ivanova Pindeva, Elisaveta Histova Mileva,

More information

Lecture 5 Solving Problems using Green s Theorem. 1. Show how Green s theorem can be used to solve general electrostatic problems 2.

Lecture 5 Solving Problems using Green s Theorem. 1. Show how Green s theorem can be used to solve general electrostatic problems 2. Lectue 5 Solving Poblems using Geen s Theoem Today s topics. Show how Geen s theoem can be used to solve geneal electostatic poblems. Dielectics A well known application of Geen s theoem. Last time we

More information

Flux. Area Vector. Flux of Electric Field. Gauss s Law

Flux. Area Vector. Flux of Electric Field. Gauss s Law Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is

More information

Many Electron Atoms. Electrons can be put into approximate orbitals and the properties of the many electron systems can be catalogued

Many Electron Atoms. Electrons can be put into approximate orbitals and the properties of the many electron systems can be catalogued Many Electon Atoms The many body poblem cannot be solved analytically. We content ouselves with developing appoximate methods that can yield quite accuate esults (but usually equie a compute). The electons

More information

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1) EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq

More information

Phys-272 Lecture 18. Mutual Inductance Self-Inductance R-L Circuits

Phys-272 Lecture 18. Mutual Inductance Self-Inductance R-L Circuits Phys-7 ectue 8 Mutual nductance Self-nductance - Cicuits Mutual nductance f we have a constant cuent i in coil, a constant magnetic field is ceated and this poduces a constant magnetic flux in coil. Since

More information

COLD STRANGLING HOLLOW PARTS FORCES CALCULATION OF CONICAL AND CONICAL WITH CYLINDRICAL COLLAR

COLD STRANGLING HOLLOW PARTS FORCES CALCULATION OF CONICAL AND CONICAL WITH CYLINDRICAL COLLAR COLD STANGLING HOLLOW PATS OCES CALCULATION O CONICAL AND CONICAL WITH CYLINDICAL COLLA Lucian V. Sevein, Taian Lucian Sevein,, Stefan cel Mae Univesity of Suceava, aculty of Mechanical Engineeing, Mechatonics

More information

Chapter 3 Optical Systems with Annular Pupils

Chapter 3 Optical Systems with Annular Pupils Chapte 3 Optical Systems with Annula Pupils 3 INTRODUCTION In this chapte, we discuss the imaging popeties of a system with an annula pupil in a manne simila to those fo a system with a cicula pupil The

More information

Modelling Vortex Fields in Metal Smelting Furnaces

Modelling Vortex Fields in Metal Smelting Furnaces Int. Jnl. of Multiphysics Volume 4 Numbe 4 00 35 Modelling Votex Fields in Metal Smelting Funaces Oleg Kazak, Oleksand Semko Depatment Of Physics, Donetsk National Univesity, Donetsk, Ukaine Email: olegkazak@yandex.u;

More information

Annihilation of Relativistic Positrons in Single Crystal with production of One Photon

Annihilation of Relativistic Positrons in Single Crystal with production of One Photon Annihilation of Relativistic Positons in Single Cystal with poduction of One Photon Kalashnikov N.P.,Mazu E.A.,Olczak A.S. National Reseach Nuclea Univesity MEPhI (Moscow Engineeing Physics Institute),

More information

Eventually transatlantic signals! From Last Time. Electromagnetic Waves. The idea of electric fields. The electric field.

Eventually transatlantic signals! From Last Time. Electromagnetic Waves. The idea of electric fields. The electric field. Fom Last Time Electomagnetic waves Chages, cuent and foces: Coulomb s law. Acceleating chages poduce an electomagnetic wave The idea of the electic field. Today Electic fields, magnetic fields, and thei

More information

The physics of induction stoves

The physics of induction stoves The physics of uction stoves This is an aticle fom my home page: www.olewitthansen.dk Contents 1. What is an uction stove...1. Including self-uctance...4 3. The contibution fom the magnetic moments...6

More information

On the Sun s Electric-Field

On the Sun s Electric-Field On the Sun s Electic-Field D. E. Scott, Ph.D. (EE) Intoduction Most investigatos who ae sympathetic to the Electic Sun Model have come to agee that the Sun is a body that acts much like a esisto with a

More information

DonnishJournals

DonnishJournals DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş

More information

Introduction to Nuclear Forces

Introduction to Nuclear Forces Intoduction to Nuclea Foces One of the main poblems of nuclea physics is to find out the natue of nuclea foces. Nuclea foces diffe fom all othe known types of foces. They cannot be of electical oigin since

More information

AP Physics - Coulomb's Law

AP Physics - Coulomb's Law AP Physics - oulomb's Law We ve leaned that electons have a minus one chage and potons have a positive one chage. This plus and minus one business doesn t wok vey well when we go in and ty to do the old

More information

Modeling Fermi Level Effects in Atomistic Simulations

Modeling Fermi Level Effects in Atomistic Simulations Mat. Res. Soc. Symp. Poc. Vol. 717 Mateials Reseach Society Modeling Femi Level Effects in Atomistic Simulations Zudian Qin and Scott T. Dunham Depatment of Electical Engineeing, Univesity of Washington,

More information

Chapter Sixteen: Electric Charge and Electric Fields

Chapter Sixteen: Electric Charge and Electric Fields Chapte Sixteen: Electic Chage and Electic Fields Key Tems Chage Conducto The fundamental electical popety to which the mutual attactions o epulsions between electons and potons ae attibuted. Any mateial

More information

Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere

Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere Applied Mathematics, 06, 7, 709-70 Published Online Apil 06 in SciRes. http://www.scip.og/jounal/am http://dx.doi.og/0.46/am.06.77065 Absoption Rate into a Small Sphee fo a Diffusing Paticle Confined in

More information

Ultrafast effects in 3D Metamaterials

Ultrafast effects in 3D Metamaterials Ultafast effects in 3D Metamateials Nkoni Katte,2 and Philip G.Evans 3 Electical Engineeing Wilbefoce Univesity, 2 Sensos Laboatoy Wight State Univesity and 3 Oak idge National Laboatoy *Coesponding autho:

More information

APPLICATION OF MAC IN THE FREQUENCY DOMAIN

APPLICATION OF MAC IN THE FREQUENCY DOMAIN PPLICION OF MC IN HE FREQUENCY DOMIN D. Fotsch and D. J. Ewins Dynamics Section, Mechanical Engineeing Depatment Impeial College of Science, echnology and Medicine London SW7 2B, United Kingdom BSRC he

More information

Nuclear models: Shell model

Nuclear models: Shell model Lectue 3 Nuclea models: Shell model WS0/3: Intoduction to Nuclea and Paticle Physics,, Pat I Nuclea models Nuclea models Models with stong inteaction between the nucleons Liquid dop model α-paticle model

More information

Designing a Sine-Coil for Measurement of Plasma Displacements in IR-T1 Tokamak

Designing a Sine-Coil for Measurement of Plasma Displacements in IR-T1 Tokamak Designing a Sine-Coil fo Measuement of Plasma Displacements in IR-T Tokamak Pejman Khoshid, M. Razavi, M. Ghoanneviss, M. Molaii, A. TalebiTahe, R. Avin, S. Mohammadi and A. NikMohammadi Dept. of Physics,

More information

Stress, Cauchy s equation and the Navier-Stokes equations

Stress, Cauchy s equation and the Navier-Stokes equations Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted

More information

Swissmetro: design methods for ironless linear transformer

Swissmetro: design methods for ironless linear transformer Swissmeto: design methods fo ionless linea tansfome Nicolas Macabey GESTE Engineeing SA Scientific Pak PSE-C, CH-05 Lausanne, Switzeland Tel (+4) 2 693 83 60, Fax. (+4) 2 693 83 6, nicolas.macabey@geste.ch

More information

Math 2263 Solutions for Spring 2003 Final Exam

Math 2263 Solutions for Spring 2003 Final Exam Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate

More information

Levitation force analysis of ring and disk shaped permanent magnet-high temperature superconductor

Levitation force analysis of ring and disk shaped permanent magnet-high temperature superconductor Inn Jounal of Pue & Applied Physics Vol. 55, Apil 017, pp. 61-68 Levitation foce analysis of ing and disk shaped pemanent magnet-high tempeatue supeconducto Sinan Basaan & Selim Sivioglu* Depatment of

More information

Physics 122, Fall October 2012

Physics 122, Fall October 2012 Today in Physics 1: electostatics eview David Blaine takes the pactical potion of his electostatics midtem (Gawke). 11 Octobe 01 Physics 1, Fall 01 1 Electostatics As you have pobably noticed, electostatics

More information

IN SITU SOUND ABSORPTION COEFFICIENT MEASUREMENT OF VARIOUS SURFACES

IN SITU SOUND ABSORPTION COEFFICIENT MEASUREMENT OF VARIOUS SURFACES IN SITU SOUND ABSORPTION COEFFICIENT MEASUREMENT OF VARIOUS SURFACES PACS REFERENCES : 43.20.El, 43.20.Ye, 43.55.Ev, 43.58.Bh Michel Béengie 1 ; Massimo Gaai 2 1 Laboatoie Cental des Ponts et Chaussées

More information

Bifurcation Analysis for the Delay Logistic Equation with Two Delays

Bifurcation Analysis for the Delay Logistic Equation with Two Delays IOSR Jounal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 5 Ve. IV (Sep. - Oct. 05), PP 53-58 www.iosjounals.og Bifucation Analysis fo the Delay Logistic Equation with Two Delays

More information

1D2G - Numerical solution of the neutron diffusion equation

1D2G - Numerical solution of the neutron diffusion equation DG - Numeical solution of the neuton diffusion equation Y. Danon Daft: /6/09 Oveview A simple numeical solution of the neuton diffusion equation in one dimension and two enegy goups was implemented. Both

More information

EXAM NMR (8N090) November , am

EXAM NMR (8N090) November , am EXA NR (8N9) Novembe 5 9, 9. 1. am Remaks: 1. The exam consists of 8 questions, each with 3 pats.. Each question yields the same amount of points. 3. You ae allowed to use the fomula sheet which has been

More information

QUALITATIVE AND QUANTITATIVE ANALYSIS OF MUSCLE POWER

QUALITATIVE AND QUANTITATIVE ANALYSIS OF MUSCLE POWER QUALITATIVE AND QUANTITATIVE ANALYSIS OF MUSCLE POWER Jey N. Baham Anand B. Shetty Mechanical Kinesiology Laboatoy Depatment of Kinesiology Univesity of Nothen Coloado Geeley, Coloado Muscle powe is one

More information

A Double Exponential Function Fitting Algorithm for Optimize Parameter of µh Curve

A Double Exponential Function Fitting Algorithm for Optimize Parameter of µh Curve Advanced Mateials Reseach Online: 214-6-18 ISSN: 1662-8985, Vols. 96-961, pp 1146-115 doi:1.428/www.scientific.net/amr.96-961.1146 214 Tans Tech Publications, Switzeland A Double Exponential Function Fitting

More information