Analysis for Transverse Sensitivity of the Microaccelerometer
|
|
- Donald McDonald
- 5 years ago
- Views:
Transcription
1 Engineering, 009, 1, doi:10.436/eng Published Online November 009 ( Anlsis for Trnsverse ensitivit of the Microccelerometer Abstrct Yu LIU 1,,3, Guocho WANG 1,,3, Chngwen GUO 1,,3 1 Automobile College, Chongqing Universit of Technolog, Chongqing, Chin e Lbortor of Automobile Prts & Test Technique in Chongqing, Chongqing, Chin 3 Chongqing Engineering Reserch Center for Automobile Power stem nd Control, Chongqing, Chin E-mil: liuu_cq@16.com Received Jnur 10, 009; revised Februr 1, 009; ccepted Februr 3, 009 For the microccelerometer, strong il response nd wek cross-il one re lws epected. This pper presents generl nlsis bout trnsverse sensitivit of the microccelerometer. The nlsis model is developed, where the influence of response stiffness nd dmping in different es, s well s smmetricl decline ngles of 3 degrees of freedom sstem is considered. Moreover, multi-freedom vibrtion equtions bsed on the nlsis model re estblished. And the equtions re solved on condition tht dmping force is ignored. Finll, the theoreticl nlsis bout trnsverse sensitivit is ccomplished, nd some effective methods, which re beneficil to reduce cross disturbnce, re provided. ewords: MEM, Accelerometers, Trnsverse ensitivit, Multi-Freedom Vibrtion Eqution 1. Introduction For the microccelerometer, there should be no output if the input ccelertion is long the cross is. In fct, however, the output creted b forces induced in orthogonl is is not equl to ero. This phenomen is clled cross coupling, which is mesured b trnsverse sensitivit [1 3]. In this pper, the nlsis model for cross disturbnce of the microccelerometer is developed, where the influence of response stiffness nd dmping in different es, s well s smmetricl decline ngles of 3 degrees of freedom sstem is considered. Moreover, multi- freedom vibrtion equtions bsed on the nlsis model re estblished. And the equtions re solved on condition tht dmping force is ignored. Finll, the theoreticl nlsis bout trnsverse sensitivit is ccomplished, nd some effective methods, which re beneficil to reduce cross disturbnce, re provided.. Trnsverse ensitivit Trnsverse sensitivit is the rtio of the output cused b ccelertion perpendiculr to the min sensitivit is divided b the bsic sensitivit in the min direction. It is n importnt chrcteristic of the microccelerometer, nd is primril cused b two fctors [4 6]. One is from the inherent microstructure, which m be eliminted b dopting the pproprite working principle nd optimiing the design prmeters. The other is from inccurcies in fbriction process, pckge orienttion nd mislignment, which is onl to be reduced s possible s we cn. For emple, -is ccelerometer, due to inevitbilit of errors in fbriction nd mislignments, the pplied ccelertion cn be epressed s ccelertion long the -is nd ccelertions perpendiculr to the min sensitivit is, denoted s,, respectivel. Therefore, the output is given b V.(1) out where or is trnsverse sensitivit of -is in or direction. Unfortuntel, the ccelerometer cnnot distinguish the chnge in voltge cused b ccelertions nd, which results in difference of +. Disturbnce nd coupling from different es hve importnt influences on the performnce of the microccelerometer. o strong il response nd wek crossil one re lws epected. And the trnsverse sensitivit is lws epected to smll enough, even close to ero. Copright 009 cires.
2 Y. LIU ET AL Anlsis of Trnsverse ensitivit 3.1. Model In most cses, the microstructure of the ccelerometer cn be represented s mss-spring-dmper sstem. Figure 1 shows the mechnicl model of the microccelerometer with single - degree of freedom. In perfect condition, elstic deformtion of the spring induced b the inertil force is lws long the -is no mtter where ccelertion signl is from. In fct, however, the phenomenon of cross coupling eists inevitbl. On the one hnd, the elstic deformtion of the equivlent spring occurs not onl in primr -is but lso in orthogonl -is nd -is, nd on the other hnd, the displcement of the microstructure under ccelertion is not lws long the primr -is, which m be t n ngle with the idel sensitive is [7]. Figure illustrtes the mechnicl model of microccelerometer with three degrees of freedom, where elstic deformtion is long -is, -is nd -is. o the ccelertion pondernce,, re detected b the corresponding degree of freedom sstem. Becuse the microstructure hs the s me proof mss, the different equivlent stiffness nd dmping coefficients, denoted s,, nd B, B, B respectivel, the model in Figure is the nlsis model of cross disturbnce resulted from stiffness nd dmping in different es. Figure 1. implified mechnicl model of the microccelerometer with single - degree of freedom. Figure. Anlsis model of cross disturbnce resulted from stiffness nd dmping in different es. Figure 3 shows the other model of the microccelerometer, where the spring nd the dmper re not long the primr is but tht t n ngle with the corresponding idel is. For emple, - degree of freedom sstem, s illustrted in Figure 3(), due to inevitbilit of errors in fbriction process, pckge orienttion s well s mislignment, the spring nd dmper B re ll t n ngle with -is, which is clled smmetricl decline ngle nddenoted s. At the sme time, the spring nd dmper B re lso t n ngle with -is or -is, denoted s, respectivel. Most often, is quite smll, nd, re ll close to. Therefore, cos cos cos 1 () imilrl,, re the smmetricl decline ngles of - nd - degree of freedom sstems respectivel, s shown in Figu re 3 (b) nd (c). o the model in Figure 3 is the nlsis model of cross disturbnce resulted from the smmetricl decline ngles. 3.. olution In order to nle the influence of cross disturbnce, the multi-freedom vibrtion equtions bsed on the bovementioned models should be estblished. Here sinusoidl signl is considered. sin t, sin t, sin t denote three projections of vec- tor ccelertion respectivel. Furthermore, ssume the displceme nt function is s follows: w W sint w W sint w W sint where W, W, W re the mplitudes long the, nd -direction respectivel. For -degree of freedom sstem, the vibrtion eqution responded to ccelertion sign l sin t is given b: w B w msin t B B w mw B B Bw w0 w B B B 0 w w (4) where w, w, w re the displcements of proof mss long the, nd -direction respectivel. w, w, w nd w, w, w denote the first nd the second deriv- displce ment w, w, w with respect to time t re- tive of spective l.,, re the self-stiffness of equiv- (3) Copright 009 cires.
3 198 Y. LIU ET AL. Figure 3. Anlsis model of cross disturbnce resulted from the smmetricl decline ngles. () -degree of freedom sstem. (b) -degree of freedom sstem. (c) -degree of freedom sstem. lent spring, which reflect responsibilit of spring in three orthogonl es.,, re the coupling stiffness of equivlent spring, which reflect responsibilit of spring in three coupling orthogonl es. And B, B, B re the self-dmping of equivlent coefficient B, which reflect the dmping effect of B in three orthogonl es. B, B, B re the coupling dmping of equivlent coefficient B, which reflect the dmping effect of B in three coupling orthogonl es. ubstituting Eqution (3) into Eqution (4), we get the sstem of three liner equtions in three vribles W, W, W : ( ) ( ) ( sintbcostm sin t) W ( sintbcos t) W ( sintbcos t) W m sint ( sintbcos t) W ( sintbcostm sin t) W ( sintbcos t) W 0 ( sint BcostW ) ( sintbcos tw ) ( sintbcostm sin tw ) 0 Usull, there re three pondernces of vector ccelertion, denoted s,,. The re responded b the respective degree o f freedom sstem. Furthermore, response long the, nd -direction eist in ech degree of freedom sstem. o there re nine mplitude epressions responded to the ccelertion signl. In order to simplif the nlsis, dmping force is ignored. Therefore, Eqution (5) is simplified to: (5) Copright 009 cires.
4 Y. LIU ET AL. 199 ( m ) W W W m W ( m ) W W 0 W W ( m ) W 0 Hence, W W W sin m ( m ) cos cos ( m ) cos cos ( m ) Likewise, the mplitudes responded to ccelertion signl sin t, sin t re respectivel given b And 3.3. Anlsis W W W W W W cos cos ( m ) sin m ( m ) cos cos ( m ) cos cos ( m ) cos cos ( m ) sin m ( m ) Building on the previous nlsis, we get the nine m- responded to the sinusoidl cceler- plitude epressions tion signl, s listed in Tble 1. For 3-is microccelerometer, wht we need is the mplitude response in leding digonl of Tble 1. The should be the strongest, wheres the others re the cross disturbnce. Therefore, trnsverse sensitivit of -is in - nd -direction re given b cos cos m, m m cross il sin cross cos cos il sin m m m imilrl,. (6) (7) (8) (9) (10) cos cos m, m m cross il sin cos cos m. m m cross il sin cross coscos m, il m sin m cos cos m. m m cross il sin (11) (1) If the microccelerometer sensitive to the chnge in displcement hs three primr es, then the eqution = = is lws epected for uniform sensitivit in three sense directions. o the epressions of trnsverse sensitivit in Eqution (10) (1) could be simplified. Considering constrints for smmetricl decline ngles of -, - nd - degrees of freedom sstem, tht is,,, re ll close to ero, it s not difficult to find tht t rnsverse sensitivit onl reltes to the coupling ngles nd the w for smll trnsverse sensitivit is to mke the coupling ngles equl to. Of course,, re the coupling ngles of - sstem, while, nd, re those of - nd - sstems respectivel. If the co upling ngles re ll equl to, then trnsverse sensitivit =0. It should be pointed out if the microccelerometer hs onl one or two primr es, then the bove si epressions bout trnsverse sensitivit will be decresed b four or two. For instnce, there re onl, for -is ccelerometer. o trnsverse sensitivit reltes not onl to the coupling ngles but lso to the response stiffness in different es. Therefore, two methods re recommended to reduce trnsverse sensitivit. One is to mke the coupling ngles equl to. The other is to ensure the response stiffness in the prim r is fr less thn tht in the cross is. 4. ummr This pper presents generl nlsis bout trnsverse sensitivit of the microccelerometer. Firstl, the nlsis model for cross disturbnce of the microccelerometer is developed, where the influence of response stiffness nd dmping in different es, s well s smmetricl decline ngles of 3 degrees of freedom sstem re considered. econdl, multi-freedom vibrtion equtions bsed on the nlsis model re estblished. And the equtions re solved on condition tht dmping force is ignored. Finll, some effective methods, which re beneficil to reduce cross disturbnce, re provided. For the microcceleromesensitive to the chnge in displcement, if it hs Copright 009 cires.
5 00 Y. LIU ET AL. Tble 1. Amplitude responded to the sinusoidl ccelertion signl. Accelertion Amplitude -direction -direction -direction sin t sin m ( m ) cos cos m cos cos m coscos sin t ( m ) coscos sin t ( m ) sin m ( m ) cos cos m cos cos m sin m ( m ) three primr es, then trnsverse sensitivit onl reltes to the coupling ngle nd the w for smll trnsverse sensitivit is to mke the coupling ngles equl to. If it hs one or two primr es, then trnsverse sensitivit reltes not onl to the coupling ngle but lso to the response stiffness in different es. o in order to reduce trnsverse sensitivit, two methods re recommended. One is to mke the coupling ngles equl to. The other is to ensure the response stiffness in the primr is fr less thn tht in the cross is. 5. References [1] L. Fei, X. X. Zhong, Z. Y. Wen, et l, Methods for reducing sensitivit of micromchined silicon ccelerometer, Optics nd Precision Engineering, No. 3, pp , 1995 [] J.. Wng, Q. Wng, nd. H. un, Effect of crosscoupling coefficient of ccelerometer on gros free iner til mesurement unit, Journl of Chinese Inertil Technolog, No. 11, pp. 9 33, 003. [3] L. Zhong., J. G. Liu, nd H. Y. Zho, A solution to the problem of ecessive error of pieoelectricl ccelerometer s cross is sensitivit, Electro-Mechnicl Engineering, Vol. 0, pp. 4 5, 004. [4] Y Liu, Z. Y. Wen, Z. Q. Wen, et l, Design nd fbriction of high-sensitive cpcitive biil microccelerometer, J. Micromech. Microeng, Vol. 17, pp , 007. [5] Y Liu, Z. Y. Wen, nd H. Y. Yng, Effect of fbriction chrcteristic of d microccelerometer on performnce, Journl of Functionl Mterils nd Devices, Vol. 14, pp , 008. [6] Y. Liu, Z. Y. Wen, L. Q. Zhng, et l, tructure design nd sstem simultion of d microccelerometer, ITM05, Vol. 3, pp [7] N. Zeng, Reserch on the ke technolog of fiber optic ccelerometers, Doctor Thesis, Tsinghu Universit, pp , 005. Copright 009 cires.
Fig. 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 informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationSUPPLEMENTARY INFORMATION
DOI:.38/NMAT343 Hybrid Elstic olids Yun Li, Ying Wu, Ping heng, Zho-Qing Zhng* Deprtment of Physics, Hong Kong University of cience nd Technology Cler Wter By, Kowloon, Hong Kong, Chin E-mil: phzzhng@ust.hk
More informationMath 124A October 04, 2011
Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More informationApplied Physics Introduction to Vibrations and Waves (with a focus on elastic waves) Course Outline
Applied Physics Introduction to Vibrtions nd Wves (with focus on elstic wves) Course Outline Simple Hrmonic Motion && + ω 0 ω k /m k elstic property of the oscilltor Elstic properties of terils Stretching,
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationSolutions to Physics: Principles with Applications, 5/E, Giancoli Chapter 16 CHAPTER 16
CHAPTER 16 1. The number of electrons is N = Q/e = ( 30.0 10 6 C)/( 1.60 10 19 C/electrons) = 1.88 10 14 electrons.. The mgnitude of the Coulomb force is Q /r. If we divide the epressions for the two forces,
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 informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationDriving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d
Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,
More information1. Extend QR downwards to meet the x-axis at U(6, 0). y
In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions
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 informationINTERFACE DESIGN OF CORD-RUBBER COMPOSITES
8 TH INTENATIONAL CONFEENCE ON COMPOSITE MATEIALS INTEFACE DESIGN OF COD-UBBE COMPOSITES Z. Xie*, H. Du, Y. Weng, X. Li Ntionl Ke Lbortor of Science nd Technolog on Advnced Composites in Specil Environment,
More informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationPlate Theory. Section 13: PLATE BENDING ELEMENTS
Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s
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 informationPhysics 207 Lecture 7
Phsics 07 Lecture 7 Agend: Phsics 07, Lecture 7, Sept. 6 hpter 6: Motion in (nd 3) dimensions, Dnmics II Recll instntneous velocit nd ccelertion hpter 6 (Dnmics II) Motion in two (or three dimensions)
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 informationGraduate Students do all problems. Undergraduate students choose three problems.
OPTI 45/55 Midterm Due: Februr, Grdute Students do ll problems. Undergrdute students choose three problems.. Google Erth is improving the resolution of its globl mps with dt from the SPOT5 stellite. The
More information16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings
Chpter 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings When, in the cse of tilted coordinte system, you brek up the
More informationMA Lesson 21 Notes
MA 000 Lesson 1 Notes ( 5) How would person solve n eqution with vrible in n eponent, such s 9? (We cnnot re-write this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this
More informationPlate Theory. Section 11: PLATE BENDING ELEMENTS
Section : PLATE BENDING ELEMENTS Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s the thickness of the plte. A
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationSECTION 9-4 Translation of Axes
9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationFlutter frequency based on bending - torsion coupling theory
Flutter frequency bsed on bending - torsion coupling theory *ZHENG Xin ¹, LIU Yu-Bin¹, CHEN Pu, SHEN Feng 3,ZHANG Sheng-Jun 3,nd FU Xing-Rong ¹ 1 College of Wter Conservncy nd Civil Engineering, Chin Agriculturl
More informationTHREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationMotion of Electrons in Electric and Magnetic Fields & Measurement of the Charge to Mass Ratio of Electrons
n eperiment of the Electron topic Motion of Electrons in Electric nd Mgnetic Fields & Mesurement of the Chrge to Mss Rtio of Electrons Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1.
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationThe Velocity Factor of an Insulated Two-Wire Transmission Line
The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationZero-Sum Magic Graphs and Their Null Sets
Zero-Sum Mgic Grphs nd Their Null Sets Ebrhim Slehi Deprtment of Mthemticl Sciences University of Nevd Ls Vegs Ls Vegs, NV 89154-4020. ebrhim.slehi@unlv.edu Abstrct For ny h N, grph G = (V, E) is sid to
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More informationAN020. a a a. cos. cos. cos. Orientations and Rotations. Introduction. Orientations
AN020 Orienttions nd Rottions Introduction The fct tht ccelerometers re sensitive to the grvittionl force on the device llows them to be used to determine the ttitude of the sensor with respect to the
More information( ) ( ) Chapter 5 Diffraction condition. ρ j
Grdute School of Engineering Ngo Institute of Technolog Crstl Structure Anlsis Tkshi Id (Advnced Cermics Reserch Center) Updted Nov. 3 3 Chpter 5 Diffrction condition In Chp. 4 it hs been shown tht the
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationStress distribution in elastic isotropic semi-space with concentrated vertical force
Bulgrin Chemicl Communictions Volume Specil Issue pp. 4 9 Stress distribution in elstic isotropic semi-spce with concentrted verticl force L. B. Petrov Deprtment of Mechnics Todor Kbleshkov Universit of
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 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 informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationCredibility Hypothesis Testing of Fuzzy Triangular Distributions
666663 Journl of Uncertin Systems Vol.9, No., pp.6-74, 5 Online t: www.jus.org.uk Credibility Hypothesis Testing of Fuzzy Tringulr Distributions S. Smpth, B. Rmy Received April 3; Revised 4 April 4 Abstrct
More informationForces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direc
Physics 170 Summry of Results from Lecture Kinemticl Vribles The position vector ~r(t) cn be resolved into its Crtesin components: ~r(t) =x(t)^i + y(t)^j + z(t)^k. Rtes of Chnge Velocity ~v(t) = d~r(t)=
More informationEquations of Motion. Figure 1.1.1: a differential element under the action of surface and body forces
Equtions of Motion In Prt I, lnce of forces nd moments cting on n component ws enforced in order to ensure tht the component ws in equilirium. Here, llownce is mde for stresses which vr continuousl throughout
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More informationOn the Uncertainty of Sensors Based on Magnetic Effects. E. Hristoforou, E. Kayafas, A. Ktena, DM Kepaptsoglou
On the Uncertinty of Sensors Bsed on Mgnetic Effects E. ristoforou, E. Kyfs, A. Kten, DM Kepptsoglou Ntionl Technicl University of Athens, Zogrfou Cmpus, Athens 1578, Greece Tel: +3177178, Fx: +3177119,
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationES.182A Topic 30 Notes Jeremy Orloff
ES82A opic 3 Notes Jerem Orloff 3 Non-independent vribles: chin rule Recll the chin rule: If w = f, ; nd = r, t, = r, t then = + r t r t r t = + t t t r nfortuntel, sometimes there re more complicted reltions
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 informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
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 informationSimple Harmonic Motion I Sem
Simple Hrmonic Motion I Sem Sllus: Differentil eqution of liner SHM. Energ of prticle, potentil energ nd kinetic energ (derivtion), Composition of two rectngulr SHM s hving sme periods, Lissjous figures.
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationMiscellaneous Problems. pinned to the ground
Miscellneous Problems Problem. Use the mobilit formul to determine the number of degrees of freedom for this sstem. pinned to the ground pinned to the ground Problem. For this mechnism: () Define vectors
More informationSet up Invariable Axiom of Force Equilibrium and Solve Problems about Transformation of Force and Gravitational Mass
Applied Physics Reserch; Vol. 5, No. 1; 013 ISSN 1916-9639 E-ISSN 1916-9647 Published by Cndin Center of Science nd Eduction Set up Invrible Axiom of orce Equilibrium nd Solve Problems bout Trnsformtion
More informationExplain shortly the meaning of the following eight words in relation to shells structures.
Delft University of Technology Fculty of Civil Engineering nd Geosciences Structurl Mechnics Section Write your nme nd study number t the top right-hnd of your work. Exm CIE4143 Shell Anlysis Tuesdy 15
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
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 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 informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationNetwork Analysis and Synthesis. Chapter 5 Two port networks
Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on
More informationMultiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
More information4 The dynamical FRW universe
4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which
More informationMeasuring Electron Work Function in Metal
n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationBasic Derivative Properties
Bsic Derivtive Properties Let s strt this section by remining ourselves tht the erivtive is the slope of function Wht is the slope of constnt function? c FACT 2 Let f () =c, where c is constnt Then f 0
More information0.1 THE REAL NUMBER LINE AND ORDER
6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
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 informationA-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)
A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationLecture 8. Newton s Laws. Applications of the Newton s Laws Problem-Solving Tactics. Physics 105; Fall Inertial Frames: T = mg
Lecture 8 Applictions of the ewton s Lws Problem-Solving ctics http://web.njit.edu/~sireno/ ewton s Lws I. If no net force ocects on body, then the body s velocity cnnot chnge. II. he net force on body
More informationProblems for HW X. C. Gwinn. November 30, 2009
Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object
More information#6A&B Magnetic Field Mapping
#6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by
More informationBend 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 informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
More informationINTRODUCTION. The three general approaches to the solution of kinetics problems are:
INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The
More informationMath 113 Exam 1-Review
Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More information, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
More informationPerformance limit of a passive vertical isolator using a negative stiffness mechanism
Journl of Mechnicl Science nd Technolog (008) 357~364 Journl of Mechnicl Science nd Technolog www.springerlin.com/content/1738-494x DOI: 10.1007/s106-008-0930-7 Performnce limit of pssive verticl isoltor
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 bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationBoolean Algebra. Boolean Algebras
Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with: - two binry opertions, commonly denoted by + nd, - unry opertion, usully denoted by or ~ or, - two elements usully clled zero
More informationR. I. Badran Solid State Physics
I Bdrn Solid Stte Physics Crystl vibrtions nd the clssicl theory: The ssmption will be mde to consider tht the men eqilibrim position of ech ion is t Brvis lttice site The ions oscillte bot this men position
More informationA Case Study on Simple Harmonic Motion and Its Application
Interntionl Journl of Ltest Engineering nd Mngement Reserch IJLEMR ISSN: 55-87 Volume 0 - Issue 08 August 07 PP. 5-60 A Cse Stud on Simple Hrmonic Motion nd Its Appliction Gowri.P, Deepik.D, Krithik.S
More informationSatellite Retrieval Data Assimilation
tellite etrievl Dt Assimiltion odgers C. D. Inverse Methods for Atmospheric ounding: Theor nd Prctice World cientific Pu. Co. Hckensck N.J. 2000 Chpter 3 nd Chpter 8 Dve uhl Artist depiction of NAA terr
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 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 information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More information