Five-axis NURBS Path Real-time Generation Method in CNC System

Size: px
Start display at page:

Download "Five-axis NURBS Path Real-time Generation Method in CNC System"

Transcription

1 Five-ais URBS Path Real-time Geeratio Metho i CC Sstem Che Liagji, Gao Chagi, Feg Xiazhag Five-ais URBS Path Real-time Geeratio Metho i CC Sstem Che Liagji, Gao Chagi, Feg Xiazhag Zhegzho Istitte of Aeroatical Istr Maagemet chejiaiclj@63.com, chagigao@zzia.e.c, phfz@zzia.e.c oi:.456/ijact.vol2.isse3.6 Abstract A real-time path geeratio metho with o Uiform Ratioal B Splie(URBS techolog i the Compter Aie Desig(CAD fiel is presete a realize i a five-ais Compter mericall Cotrolle(CC sstem. I this metho, the tool path was represete ito two URBS crves, tooltip a tool-orietatio splie crve, base o the same ot vector. The Ctter Locatio(CL ata was firstl calclate a the trasforme to five motio commas of five aes of machie tool throgh the real-time post-processio algorithm. The acceleratio/eceleratio cotrollig metho was also presete to avoi the impact of machie tool. The propose five-ais splie iterpolatio metho is realize i or evelopig five-ais CC machie tool a the reslt of machiig shows that the metho is feasible.. Itroctio Kewors: CC,Five-ais machiig, URBS, Path geeratio Comple parts sch as aeroatical trbie blaes, impellors, ies, mols are machie o five-ais Compter merical Cotrolle (CC machies. The core of the machies is the cotor cotrollig sstem which ecies machiig efficiec a accrac of a part. However, the crret five-ais CC sstem mostl provie ol lie iterpolators, that is, tool motio alog straight lies is spporte. To machie a srface, the tool paths, which are also ow as the ctter locatio (CL paths, are tpicall approimate with piecewise liear segmets b CAD/CAM sstems. The approimatel machiig metho has its ow rawbacs i avacig efficiec a accrac of a machie part. To overcome the isavatages, it mst be tae ito accot that the five-ais cotrol sstem has itelliget abilit to geerate the tool paths of the machie srfaces. o-uiform Ratioal B-Splie (URBS, as the ol staar of ata-echage of proct sactioe b ISO, has bee mostl se i almost all CAD/CAM sstems to escribe wor-piece s srface. Base o the realit, the CC sstem shol be provie with the abilit to irectl geerate five-ais URBS crves or srfaces i real-time. Some URBS iterpolatio methos ha bee propose b several ivestigators[-6]. Cheg et al.[2] ha propose a real-time URBS crve motio comma geerator for CC machies. Zhimig et al.[5] evelope a URBS crve iterpolator for CC machiig base o the geometric properties of the tool path. However, most of them ha cocetrate their attetios o 3-ais ball-e machiig, bt for five-ais URBS srface iterpolator, little has bee oe. Whe evelopig a realizig a five-ais CC sstem with fctio of URBS iterpolatio, relate st mst be carrie ot. I this paper, a real-time five-ais URBS crve geeratio metho is give. 2. URBS Crve Represetatio A p-egree URBS crve with parameter ca be efie as follows: C ( [ ( ( ( ] T i z i ( W P i i ( W i ( 64

2 Iteratioal joral of Avacemets i Comptig Techolog Volme 2, mber 3, Agst, 2 where P i is the i th 3-D cotrol poit; W i is the correspoig weight factor of P i ; (+ is the mber of cotrol poits; i,p (, B-splie basis fctio with egree of p, ca be calclate b the followig formla: ( i < i+ i, ( otherwise i i+ p+ (2 ( ( + i+, p ( i+ p i i+ p+ i+ (prescribe [,..., + + ] is the ot vector. More iformatio abot URBS ca be fo i [7]. where i i p 3. URBS Path Real-time Geeratio Metho The tas of real-time URBS splie iterpolatio is to calclate et iterpolatio perio s CL ata that iclig tool-tip locatio cooriatio a tool-ais orietatio a the trasfer the CL ata ito machie tool s motio commas sch as X,Y,Z,A,C throgh post-processio. 3.. Real-time Calclatio of CL Data As show i Figre, C ( a C ( is the ctter ceter poit splie a the orietatio splie, respectivel. The two splies are costrcte i the same ot vector. Assme that parameter is fctio of time t, that is, (t. B sig Talor s epasio of the parameter with respect to time t to obtai the first orer approimatio iterpolatio algorithm, the first orer approimatio p to the first erivatives is + + T s t t (3 t z C ( C ( C ( T + - e + T + C ( Figre. Priciple of five-ais URBS Iterpolatio + C ( + C ( + where T s is iterpolatio perio, a + are correspoig parameters of crret a et time t a t +. The feerate of tool-tip poit alog C ( is efie b Sice the crve spee V ( C( t C( t (4 65

3 Five-ais URBS Path Real-time Geeratio Metho i CC Sstem Che Liagji, Gao Chagi, Feg Xiazhag V ( The first erivative of with t is obtaie as t t t C ( t V ( C ( Therefore, the first-orer iterpolatio algorithm is obtaie b sbstittig Eq.(4 ito (5, (5 ca be processe as follows: + + The first a seco erivative of C ( with C( i i TV s ( C ( is obtaie as ( W ip i ( W i ( W ip i i i 2 ( W where the geeral algorithm for st orer erivative of i,p ( is C ( a ( + + ( i i i+, p p i+ p i i+ p i+ ( W i ( ( + ito C ( a ( C are obtaie b sbstittig the calclate Sppose the tool-ais it vector is + T +, so 3.2. Real-time Post-processio of CL Data C ( C ( C ( C ( T (5 (6 (7 (8 (9 C. B ow, we have the followig CL ata: C ( + (o, o,o z T + (t, t,t z For five-ais machie of table tiltig/rotatig tpe (Figre 2., the followig post-processio metho is presete to trasfer the CL ata ito the machie s motio commas i real-time. (a if t t, t, the (b else if (c else if ( else t, t t, t z,the,the A,C A arcsi( t, C A arcsi( t, C π /2 ( 66

4 Iteratioal joral of Avacemets i Comptig Techolog Volme 2, mber 3, Agst, 2 A arccos( t z arccos( t / si A t > C 2π arccos( t / si A t < Z C A X Y Figre 2. A Tpe of five-ais Machie with Tiltig/rotatig Table For the calclatios of the three traslatio cooriatio, the Cramer rle is se here. b c a b c X 2 b2 c2 a2 b2 c2 3 b3 c3 a3 b3 c3 a c a b c Y a2 2 c2 a2 b2 c2 a3 3 c3 a3 b3 c3 a b a b c Z a2 b2 2 a2 b2 c2 a3 b3 3 a3 b3 c3 where a cosc b cos A si C c t o ( l+ l2 t ( a2 si C b2 cos A cos C c2 t 2 o ( l+ l2 t a3 b3 si A c3 tz 3 oz + l2 ( l+ l2 tz I Eq.(, l a l 2 represet vertical istaces of tool-tip poit a itersectio poit of A a C ais to rotatio table. 4. Acceleraio /Deceleratio Cotrollig Metho Assme that the acceleratio a eceleratio is a a the estiatio feerate is v as show i Figre 3. V v a -a t Figre 3. Moel of liear acceleratio/eceleratio S D 67

5 Five-ais URBS Path Real-time Geeratio Metho i CC Sstem Che Liagji, Gao Chagi, Feg Xiazhag For the acceleratio process, the followig metho ca be se V ( at For the acceleratio process, the e problem to be solve is evalatio of eceleratio poit. I Figre 3, S D, istace of eceleratio, ca be calclate as follows S D v 2 ( 2a The istace of eceleratio, which is also the legth betwee eceleratio poit a estiatio poit alog the splie crve, ca be calclate as follows C ( S D (2 D where D is correspoig parameter of the eceleratio poit. The followig ewto-rapso metho is se to calclate D. If f( C( S (3 D The we have the followig reslt: f( D f( Assme that the iitial vale of D is, that is, -S D,the we have the followig eqatio i+ i f( i/ f ( i i (4 where i is a iteger. Eq.(4 ca be eqivalet to the followig formla f( i i + i C( i i (5 f( i i Eqs.(4 a (5 ca be obtaie sig Simpso itegral metho. 5. Realizatio i Five-ais CC Sstem The propose five-ais URBS iterpolatio metho has bee realize i the evelopig five-ais CC sstem. The tpe of the five-ais machie is table tiltig/rotatig. Figre 4 shows that the CC sstem is cotrollig cttig tool alog a URBS crve path to machie a impeller. a machiig process 68

6 Iteratioal joral of Avacemets i Comptig Techolog Volme 2, mber 3, Agst, 2 6. Coclsios b machiig reslt Figre 4. Five-ais URBS Path Geeratio O the basis of aalzig the efects of the eistig five-ais liear iterpolatio metho se i the machiig of sclptre srface, five-ais URBS iterpolatio metho with ACC/DEC cotrollig is presete a realize i a CC sstem. Differece from iscretizatio machiig metho of the fiveais liear iterpolatio, the propose metho irectl iterpolates a crve o a free-form srface. Coseqetl, the mber of the C coe for the same machiig path ca be greatl ecrease a machiig efficiec a accrac is also avace. 7. Acowlegmets This research is wholl spporte b the atioal atral Sciece Foatio of Chia (o throgh Zhegzho Istitte of Aeroatical Istr Maagemet. The athors tha for their spport for this research. 8. Refereces [] M. Tiho, et al, URBS iterpolator for costat material removal rate i ope C machie tools, Iteratioal Joral of Machie Tools & Mafactre, vol.44, pp , 24. [2] M. Cheg, M. Tsai, a J. Ko, Real-time URBS comma geerators for CC servo cotrollers, Iteratioal Joral of Machie Tools & Mafactre, vol. 42, pp. 8-83, 22. [3] G. Qii, R. Zhag, a B.Greewa, Developmet a implemetatio of a URBS crve motio iterpolator, Robotics a Compter-Itegrate Mafactrig, vol.4, pp.27-36, 998. [4] M. Tsai, C. Cheg, a M. Cheg, A real-time URBS srface iterpolator for precisio threeais CC machiig, Iteratioal Joral of Machie Tools & Mafactre, vol.43, pp ,23. [5] X. Zhimig, C. Jicheg, a F. Zhegji, Performace evalatio of a real-time iterpolatio algorithm for URBS crves, Iteratioal Joral of Avace Mafactrig Techolog, vol.2, pp , 22. [6] B. Bahr, X. Xiao, a K. Krisha, A real-time scheme of cbic parametric crve iterpolatios for CC sstems, Compters i Istr, vol. 45, pp , 2. [7] L. Piegl, O URBS: a srve, IEEE Compter Graphics & Applicatio, vol., o., pp. 55-7,

Five-axis Spline Interpolation Algorithm for Digital Manufacturing System

Five-axis Spline Interpolation Algorithm for Digital Manufacturing System 3r Iteratoal Coferece o Mechatrocs, Robotcs a Atomato (ICMRA 25 Fve-axs Sple Iterpolato Algorthm for Dgtal Mafactrg System LI Hyg, a a CHE Lagj 2,b * Zhegzho Isttte of Aeroatcal Istry Maagemet, Cha 2 Zhegzho

More information

Definition 2.1 (The Derivative) (page 54) is a function. The derivative of a function f with respect to x, represented by. f ', is defined by

Definition 2.1 (The Derivative) (page 54) is a function. The derivative of a function f with respect to x, represented by. f ', is defined by Chapter DACS Lok 004/05 CHAPTER DIFFERENTIATION. THE GEOMETRICAL MEANING OF DIFFERENTIATION (page 54) Defiitio. (The Derivative) (page 54) Let f () is a fctio. The erivative of a fctio f with respect to,

More information

Introduction. Question: Why do we need new forms of parametric curves? Answer: Those parametric curves discussed are not very geometric.

Introduction. Question: Why do we need new forms of parametric curves? Answer: Those parametric curves discussed are not very geometric. Itrodctio Qestio: Why do we eed ew forms of parametric crves? Aswer: Those parametric crves discssed are ot very geometric. Itrodctio Give sch a parametric form, it is difficlt to kow the derlyig geometry

More information

Calculus 2 Quiz 1 Review / Fall 2011

Calculus 2 Quiz 1 Review / Fall 2011 Calcls Qiz Review / Fall 0 () The fctio is f a the iterval is [, ]. Here are two formlas yo may ee. ( ) ( ) ( ) 6 (a.) Use a left-e, right-e, a mipoit sm of "" rectagles to approimate. The withs of all

More information

ADOMIAN DECOMPOSITION METHOD AND TAYLOR SERIES METHOD IN ORDINARY DIFFERENTIAL EQUATIONS

ADOMIAN DECOMPOSITION METHOD AND TAYLOR SERIES METHOD IN ORDINARY DIFFERENTIAL EQUATIONS IJRRS 6 () gst wwwarpapresscom/volmes/vol6isse/ijrrs_6 pf DOMI DECOMPOSITIO METHOD D TYOR SERIES METHOD I ORDIRY DIFFERETI EQUTIOS José lbeiro Sáchez Cao Uiversia EFIT Departameto e Ciecias ásicas Meellí-Colombia

More information

Course Outline. Course Outline. Computer Graphics (Fall 2008) Motivation. Outline of Unit. Bezier Curve (with HW2 demo)

Course Outline. Course Outline. Computer Graphics (Fall 2008) Motivation. Outline of Unit. Bezier Curve (with HW2 demo) Compter Graphics (Fall 2008) COMS 4160, Lectre 6: Crves 1 http://www.cs.colmbia.ed/~cs4160 3D Graphics Pipelie Modelig (Creatig 3D Geometry) Corse Otlie Rederig (Creatig, shadig images from geometry, lightig,

More information

Course Outline. Curves for Modeling. Graphics Pipeline. Outline of Unit. Motivation. Foundations of Computer Graphics (Spring 2012)

Course Outline. Curves for Modeling. Graphics Pipeline. Outline of Unit. Motivation. Foundations of Computer Graphics (Spring 2012) Fodatios of Compter Graphics (Sprig 0) CS 84, Lectre : Crves http://ist.eecs.bereley.ed/~cs84 3D Graphics Pipelie Corse Otlie Modelig Aimatio Rederig Graphics Pipelie Crves for Modelig I HW, HW, draw,

More information

Coordinate Systems. Things to think about:

Coordinate Systems. Things to think about: Coordiate Sstems There are 3 coordiate sstems that a compter graphics programmer is most cocered with: the Object Coordiate Sstem (OCS), the World Coordiate Sstem (WCS), ad the Camera Coordiate Sstem (CCS).

More information

THE CONSERVATIVE DIFFERENCE SCHEME FOR THE GENERALIZED ROSENAU-KDV EQUATION

THE CONSERVATIVE DIFFERENCE SCHEME FOR THE GENERALIZED ROSENAU-KDV EQUATION Zho, J., et al.: The Coservative Differece Scheme for the Geeralized THERMAL SCIENCE, Year 6, Vol., Sppl. 3, pp. S93-S9 S93 THE CONSERVATIVE DIFFERENCE SCHEME FOR THE GENERALIZED ROSENAU-KDV EQUATION by

More information

LINEARIZATION OF NONLINEAR EQUATIONS By Dominick Andrisani. dg x. ( ) ( ) dx

LINEARIZATION OF NONLINEAR EQUATIONS By Dominick Andrisani. dg x. ( ) ( ) dx LINEAIZATION OF NONLINEA EQUATIONS By Domiick Adrisai A. Liearizatio of Noliear Fctios A. Scalar fctios of oe variable. We are ive the oliear fctio (). We assme that () ca be represeted si a Taylor series

More information

Partial Differential Equations

Partial Differential Equations EE 84 Matematical Metods i Egieerig Partial Differetial Eqatios Followig are some classical partial differetial eqatios were is assmed to be a fctio of two or more variables t (time) ad y (spatial coordiates).

More information

Numerical Methods for Finding Multiple Solutions of a Superlinear Problem

Numerical Methods for Finding Multiple Solutions of a Superlinear Problem ISSN 1746-7659, Eglad, UK Joral of Iformatio ad Comptig Sciece Vol 2, No 1, 27, pp 27- Nmerical Methods for Fidig Mltiple Soltios of a Sperliear Problem G A Afrozi +, S Mahdavi, Z Naghizadeh Departmet

More information

A Note on the form of Jacobi Polynomial used in Harish-Chandra s Paper Motion of an Electron in the Field of a Magnetic Pole.

A Note on the form of Jacobi Polynomial used in Harish-Chandra s Paper Motion of an Electron in the Field of a Magnetic Pole. e -Joral of Sciece & Techology (e-jst) A Note o the form of Jacobi Polyomial se i Harish-Chara s Paper Motio of a Electro i the Fiel of a Magetic Pole Vio Kmar Yaav Jior Research Fellow (CSIR) Departmet

More information

Elementary Linear Algebra

Elementary Linear Algebra Elemetary Liear Algebra Ato & Rorres th Editio Lectre Set Chapter : Eclidea Vector Spaces Chapter Cotet Vectors i -Space -Space ad -Space Norm Distace i R ad Dot Prodct Orthogoality Geometry of Liear Systems

More information

Fluids Lecture 17 Notes

Fluids Lecture 17 Notes Flids Lectre 7 Notes. Obliqe Waves Readig: Aderso 9., 9. Obliqe Waves ach waves Small distrbaces created by a sleder body i a sersoic flow will roagate diagoally away as ach waves. These cosist of small

More information

Analysis of Infinite Beams on Elastic Foundation Using Meshfree Method

Analysis of Infinite Beams on Elastic Foundation Using Meshfree Method Joral of Civil Egieerig a Sciece March, Vol. Iss., PP. -9 Aalysis of Ifiite Beams o Elastic Foatio Usig Meshfree Metho K. B. ahae, S. P. Hajare, V. A. Sawat Departmet of Civil Egieerig, Iia Istitte of

More information

Approximation of the Likelihood Ratio Statistics in Competing Risks Model Under Informative Random Censorship From Both Sides

Approximation of the Likelihood Ratio Statistics in Competing Risks Model Under Informative Random Censorship From Both Sides Approimatio of the Lielihood Ratio Statistics i Competig Riss Model Uder Iformative Radom Cesorship From Both Sides Abdrahim A. Abdshrov Natioal Uiversity of Uzbeista Departmet of Theory Probability ad

More information

HWA CHONG INSTITUTION JC1 PROMOTIONAL EXAMINATION Wednesday 1 October hours. List of Formula (MF15)

HWA CHONG INSTITUTION JC1 PROMOTIONAL EXAMINATION Wednesday 1 October hours. List of Formula (MF15) HWA CHONG INSTITUTION JC PROMOTIONAL EXAMINATION 4 MATHEMATICS Higher 974/ Paper Wedesda October 4 hors Additioal materials: Aswer paper List of Formla (MF5) READ THESE INSTRUCTIONS FIRST Write or ame

More information

Applied Research on Clustering Algorithm in Basketball Training Aids

Applied Research on Clustering Algorithm in Basketball Training Aids 016 Iteratioal Cogress o Comptatio Algorithms i Egieerig (ICCAE 016) ISBN: 978-1-609-386-1 Applied Research o Clsterig Algorithm i Basketball Traiig Aids Shaoqig Li 1 & Yahi Zhag 1 Teachig ad Research

More information

CHAPTER 4 Integration

CHAPTER 4 Integration CHAPTER Itegratio Sectio. Atierivatives a Iefiite Itegratio......... 77 Sectio. Area............................. 8 Sectio. Riema Sums a Defiite Itegrals........... 88 Sectio. The Fuametal Theorem of Calculus..........

More information

Stability of Solution for Nonlinear Singular Systems with Delay

Stability of Solution for Nonlinear Singular Systems with Delay Sed Orders for Reprits to reprits@bethamscieceae he Ope Atomatio ad Cotrol Systems Joral 05 7 607-6 607 Stability of Soltio for Noliear Siglar Systems with Delay Ope Access Zhag Jig * ad Lig Chog Departmet

More information

EXISTENCE OF CONCENTRATED WAVES FOR THE DIFFERENT TYPE OF NONLINEARITY. V. Eremenko and N. Manaenkova

EXISTENCE OF CONCENTRATED WAVES FOR THE DIFFERENT TYPE OF NONLINEARITY. V. Eremenko and N. Manaenkova EXISTENCE OF CONCENTRATED WAVES FOR THE DIFFERENT TYPE OF NONLINEARITY. V. Eremeko ad N. Maaekova Istitte of Terrestrial Magetism, Ioosphere ad Radio Wave Propagatio Rssia Academy of Sciece E-mail: at_ma@mail.r

More information

The Mathematical Model and the Simulation Modelling Algoritm of the Multitiered Mechanical System

The Mathematical Model and the Simulation Modelling Algoritm of the Multitiered Mechanical System The Mathematical Model ad the Simulatio Modellig Algoritm of the Multitiered Mechaical System Demi Aatoliy, Kovalev Iva Dept. of Optical Digital Systems ad Techologies, The St. Petersburg Natioal Research

More information

Research on real time compensation of thermal errors of CNC lathe based on linear regression theory Qiu Yongliang

Research on real time compensation of thermal errors of CNC lathe based on linear regression theory Qiu Yongliang d Iteratioal Coferece o Machiery, Materials Egieerig, Chemical Egieerig ad Biotechology (MMECEB 015) Research o real time compesatio of thermal errors of CNC lathe based o liear regressio theory Qiu Yogliag

More information

too many conditions to check!!

too many conditions to check!! Vector Spaces Aioms of a Vector Space closre Defiitio : Let V be a o empty set of vectors with operatios : i. Vector additio :, v є V + v є V ii. Scalar mltiplicatio: li є V k є V where k is scalar. The,

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan

DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan Mathematical ad Computatioal Applicatios, Vol. 9, No. 3, pp. 30-40, 04 DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS Muhammad Aslam Noor, Khalida Iayat Noor ad Asif Waheed

More information

The z Transform. The Discrete LTI System Response to a Complex Exponential

The z Transform. The Discrete LTI System Response to a Complex Exponential The Trasform The trasform geeralies the Discrete-time Forier Trasform for the etire complex plae. For the complex variable is sed the otatio: jω x+ j y r e ; x, y Ω arg r x + y {} The Discrete LTI System

More information

MAT2400 Assignment 2 - Solutions

MAT2400 Assignment 2 - Solutions MAT24 Assigmet 2 - Soltios Notatio: For ay fctio f of oe real variable, f(a + ) deotes the limit of f() whe teds to a from above (if it eists); i.e., f(a + ) = lim t a + f(t). Similarly, f(a ) deotes the

More information

Research on fuzzy comprehensive evaluation model and its application to the evaluation of the public services

Research on fuzzy comprehensive evaluation model and its application to the evaluation of the public services Available olie www.jocpr.com Joral of Chemical ad Pharmacetical Research 0 6(7):07-0 Research Article ISSN : 0975-78 CODEN(SA) : JCPRC5 Research o fzzy comprehesive evalatio model ad its applicatio to

More information

Chapter 2 Transformations and Expectations

Chapter 2 Transformations and Expectations Chapter Trasformatios a Epectatios Chapter Distributios of Fuctios of a Raom Variable Problem: Let be a raom variable with cf F ( ) If we efie ay fuctio of, say g( ) g( ) is also a raom variable whose

More information

NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 595 NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS by Yula WANG *, Da TIAN, ad Zhiyua LI Departmet of Mathematics,

More information

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f,

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f, AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)

More information

Lecture #3. Math tools covered today

Lecture #3. Math tools covered today Toay s Program:. Review of previous lecture. QM free particle a particle i a bo. 3. Priciple of spectral ecompositio. 4. Fourth Postulate Math tools covere toay Lecture #3. Lear how to solve separable

More information

ME 375 FINAL EXAM Friday, May 6, 2005

ME 375 FINAL EXAM Friday, May 6, 2005 ME 375 FINAL EXAM Friay, May 6, 005 Divisio: Kig 11:30 / Cuigham :30 (circle oe) Name: Istructios (1) This is a close book examiatio, but you are allowe three 8.5 11 crib sheets. () You have two hours

More information

On Arithmetic Means of Sequences Generated by a Periodic Function

On Arithmetic Means of Sequences Generated by a Periodic Function Caad Math Bll Vol 4 () 1999 pp 184 189 O Arithmetic Meas of Seqeces Geerated by a Periodic Fctio Giovai Fiorito Abstract I this paper we prove the covergece of arithmetic meas of seqeces geerated by a

More information

THE LEGENDRE POLYNOMIALS AND THEIR PROPERTIES. r If one now thinks of obtaining the potential of a distributed mass, the solution becomes-

THE LEGENDRE POLYNOMIALS AND THEIR PROPERTIES. r If one now thinks of obtaining the potential of a distributed mass, the solution becomes- THE LEGENDRE OLYNOMIALS AND THEIR ROERTIES The gravitatioal potetial ψ at a poit A at istace r from a poit mass locate at B ca be represete by the solutio of the Laplace equatio i spherical cooriates.

More information

a is some real number (called the coefficient) other

a is some real number (called the coefficient) other Precalculus Notes for Sectio.1 Liear/Quadratic Fuctios ad Modelig http://www.schooltube.com/video/77e0a939a3344194bb4f Defiitios: A moomial is a term of the form tha zero ad is a oegative iteger. a where

More information

PC5215 Numerical Recipes with Applications - Review Problems

PC5215 Numerical Recipes with Applications - Review Problems PC55 Numerical Recipes with Applicatios - Review Problems Give the IEEE 754 sigle precisio bit patter (biary or he format) of the followig umbers: 0 0 05 00 0 00 Note that it has 8 bits for the epoet,

More information

+ve 10 N. Note we must be careful about writing If mass is not constant: dt dt dt

+ve 10 N. Note we must be careful about writing If mass is not constant: dt dt dt Force /N Moet is defied as the prodct of ass ad elocity. It is therefore a ector qatity. A ore geeral ersio of Newto s Secod Law is that force is the rate of chage of oet. I the absece of ay exteral force,

More information

AP Calculus BC Review Chapter 12 (Sequences and Series), Part Two. n n th derivative of f at x = 5 is given by = x = approximates ( 6)

AP Calculus BC Review Chapter 12 (Sequences and Series), Part Two. n n th derivative of f at x = 5 is given by = x = approximates ( 6) AP Calculus BC Review Chapter (Sequeces a Series), Part Two Thigs to Kow a Be Able to Do Uersta the meaig of a power series cetere at either or a arbitrary a Uersta raii a itervals of covergece, a kow

More information

Three-Step Iterative Methods with Sixth-Order Convergence for Solving Nonlinear Equations

Three-Step Iterative Methods with Sixth-Order Convergence for Solving Nonlinear Equations Article Three-Step Iteratie Methods with Sith-Order Coergece or Solig Noliear Eqatios Departmet o Mathematics, Kermashah Uiersity o Techology, Kermashah, Ira (Correspodig athor; e-mail: bghabary@yahoocom

More information

Diagonalization of Quadratic Forms. Recall in days past when you were given an equation which looked like

Diagonalization of Quadratic Forms. Recall in days past when you were given an equation which looked like Diagoalizatio of Qadratic Forms Recall i das past whe o were gie a eqatio which looked like ad o were asked to sketch the set of poits which satisf this eqatio. It was ecessar to complete the sqare so

More information

CHAPTER 4 Integration

CHAPTER 4 Integration CHAPTER Itegratio Sectio. Atierivatives a Iefiite Itegratio......... Sectio. Area............................. Sectio. Riema Sums a Defiite Itegrals........... Sectio. The Fuametal Theorem of Calculus..........

More information

Tutorial 4: FUNDAMENTAL SOLUTIONS: I-SIMPLE AND COMPOUND OPERATORS

Tutorial 4: FUNDAMENTAL SOLUTIONS: I-SIMPLE AND COMPOUND OPERATORS Boary Elemet Commicatios 00 Ttorial 4: FNDAMENTAL SOLTIONS: I-SIMPLE AND COMPOND OPERATORS YOSSEF F. RASHED Dept. o Strctral Egieerig Cairo iversity iza Egypt yosse@eg.c.e.eg Smmary a objectives I the

More information

μ are complex parameters. Other

μ are complex parameters. Other A New Numerical Itegrator for the Solutio of Iitial Value Problems i Ordiary Differetial Equatios. J. Suday * ad M.R. Odekule Departmet of Mathematical Scieces, Adamawa State Uiversity, Mubi, Nigeria.

More information

y = f x x 1. If f x = e 2x tan -1 x, then f 1 = e 2 2 e 2 p C e 2 D e 2 p+1 4

y = f x x 1. If f x = e 2x tan -1 x, then f 1 = e 2 2 e 2 p C e 2 D e 2 p+1 4 . If f = e ta -, the f = e e p e e p e p+ 4 f = e ta -, so f = e ta - + e, so + f = e p + e = e p + e or f = e p + 4. The slope of the lie taget to the curve - + = at the poit, - is - 5 Differetiate -

More information

Representing Functions as Power Series. 3 n ...

Representing Functions as Power Series. 3 n ... Math Fall 7 Lab Represetig Fuctios as Power Series I. Itrouctio I sectio.8 we leare the series c c c c c... () is calle a power series. It is a uctio o whose omai is the set o all or which it coverges.

More information

Approximate Solutions of Set-Valued Stochastic Differential Equations

Approximate Solutions of Set-Valued Stochastic Differential Equations Joral of Ucertai Systems Vol.7, No.1, pp.3-12, 213 Olie at: www.js.org.k Approximate Soltios of Set-Valed Stochastic Differetial Eqatios Jfei Zhag, Shomei Li College of Applied Scieces, Beijig Uiversity

More information

Calculus 2 Test File Fall 2013

Calculus 2 Test File Fall 2013 Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to

More information

Interactions of Soliton Waves for a Generalized Discrete KdV Equation

Interactions of Soliton Waves for a Generalized Discrete KdV Equation Comm. Theor. Phys. 68 (17) 6 1 Vol. 68, No. 1, Jly 1, 17 Iteractios of Solito Waves for a Geeralized Discrete KdV Eqatio Tog Zho ( 周统 ) 1 ad Zo-Nog Zh ( 朱佐农 ), 1 School of Statistics ad Iformatio, Shaghai

More information

which are generalizations of Ceva s theorem on the triangle

which are generalizations of Ceva s theorem on the triangle Theorems for the dimesioal simple which are geeralizatios of Ceva s theorem o the triagle Kazyi HATADA Departmet of Mathematics, Faclty of Edcatio, Gif Uiversity -, Yaagido, Gif City, GIFU 50-93, Japa

More information

k=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum

k=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum 0. L Hôpital s rule You alreay kow from Lecture 0 that ay sequece {s k } iuces a sequece of fiite sums {S } through S = s k, a that if s k 0 as k the {S } may coverge to the it k= S = s s s 3 s 4 = s k.

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

A COMPUTATIONAL STUDY UPON THE BURR 2-DIMENSIONAL DISTRIBUTION

A COMPUTATIONAL STUDY UPON THE BURR 2-DIMENSIONAL DISTRIBUTION TOME VI (year 8), FASCICULE 1, (ISSN 1584 665) A COMPUTATIONAL STUDY UPON THE BURR -DIMENSIONAL DISTRIBUTION MAKSAY Ştefa, BISTRIAN Diaa Alia Uiversity Politehica Timisoara, Faculty of Egieerig Hueoara

More information

FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES

FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES LECTURE Third Editio FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES A. J. Clark School of Egieerig Departmet of Civil ad Evirometal Egieerig Chapter 7.4 b Dr. Ibrahim A. Assakkaf SPRING 3 ENES

More information

Mathematics Extension 2

Mathematics Extension 2 009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard

More information

Theorem (Change of Variables Theorem):

Theorem (Change of Variables Theorem): Avance Higher Notes (Unit ) Prereqisites: Integrating (a + b) n, sin (a + b) an cos (a + b); erivatives of tan, sec, cosec, cot, e an ln ; sm/ifference rles; areas ner an between crves. Maths Applications:

More information

TECHNIQUES OF INTEGRATION

TECHNIQUES OF INTEGRATION 7 TECHNIQUES OF INTEGRATION Simpso s Rule estimates itegrals b approimatig graphs with parabolas. Because of the Fudametal Theorem of Calculus, we ca itegrate a fuctio if we kow a atiderivative, that is,

More information

Castiel, Supernatural, Season 6, Episode 18

Castiel, Supernatural, Season 6, Episode 18 13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio

More information

The structure of Fourier series

The structure of Fourier series The structure of Fourier series Valery P Dmitriyev Lomoosov Uiversity, Russia Date: February 3, 2011) Fourier series is costructe basig o the iea to moel the elemetary oscillatio 1, +1) by the expoetial

More information

A New Hybrid in the Nonlinear Part of Adomian Decomposition Method for Initial Value Problem of Ordinary Differential Equation

A New Hybrid in the Nonlinear Part of Adomian Decomposition Method for Initial Value Problem of Ordinary Differential Equation Joural of Matematics Researc; Vol No ; ISSN - E-ISSN - Publised b Caadia Ceter of Sciece ad Educatio A New Hbrid i te Noliear Part of Adomia Decompositio Metod for Iitial Value Problem of Ordiar Differetial

More information

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0, Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative

More information

The Development of Mathematical Model of. Components Drowned Jets Abrasive Working

The Development of Mathematical Model of. Components Drowned Jets Abrasive Working It. J. Cotemp. Math. cieces, Vol. 7, o., 3-3 The Developmet of Mathematical Model of Compoets Drowed Jets Abrasive Workig M. E. Matareh Departmet of Mechaical Egieerig Al-Balqa Applied Uiversity Al-Hso

More information

MATHEMATICS I COMMON TO ALL BRANCHES

MATHEMATICS I COMMON TO ALL BRANCHES MATHEMATCS COMMON TO ALL BRANCHES UNT Seqeces ad Series. Defiitios,. Geeral Proerties of Series,. Comariso Test,.4 tegral Test,.5 D Alembert s Ratio Test,.6 Raabe s Test,.7 Logarithmic Test,.8 Cachy s

More information

TEACHING THE IDEAS BEHIND POWER SERIES. Advanced Placement Specialty Conference. LIN McMULLIN. Presented by

TEACHING THE IDEAS BEHIND POWER SERIES. Advanced Placement Specialty Conference. LIN McMULLIN. Presented by Advaced Placemet Specialty Coferece TEACHING THE IDEAS BEHIND POWER SERIES Preseted by LIN McMULLIN Sequeces ad Series i Precalculus Power Series Itervals of Covergece & Covergece Tests Error Bouds Geometric

More information

1988 AP Calculus BC: Section I

1988 AP Calculus BC: Section I 988 AP Calculus BC: Sectio I 9 Miutes No Calculator Notes: () I this eamiatio, l deotes the atural logarithm of (that is, logarithm to the base e). () Uless otherwise specified, the domai of a fuctio f

More information

Chimica Inorganica 3

Chimica Inorganica 3 himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule

More information

Analytical Solution for One-Dimensional Finite Heat Conduction Problem with Heat Balance Integral Method

Analytical Solution for One-Dimensional Finite Heat Conduction Problem with Heat Balance Integral Method alytical Soltio for Oe-Diesioal Fiite Heat Coctio Proble with Heat alace Itegral Metho raga Walber F. * a Matelli Marcia. H. Heat Pipe aboratory - Feeral Uiversity of Sata Cataria Floriaópolis SC razil

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

Balancing. Rotating Components Examples of rotating components in a mechanism or a machine. (a)

Balancing. Rotating Components Examples of rotating components in a mechanism or a machine. (a) alacig NOT COMPLETE Rotatig Compoets Examples of rotatig compoets i a mechaism or a machie. Figure 1: Examples of rotatig compoets: camshaft; crakshaft Sigle-Plae (Static) alace Cosider a rotatig shaft

More information

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

More information

6.451 Principles of Digital Communication II Wednesday, March 9, 2005 MIT, Spring 2005 Handout #12. Problem Set 5 Solutions

6.451 Principles of Digital Communication II Wednesday, March 9, 2005 MIT, Spring 2005 Handout #12. Problem Set 5 Solutions 6.51 Priciples of Digital Commuicatio II Weesay, March 9, 2005 MIT, Sprig 2005 Haout #12 Problem Set 5 Solutios Problem 5.1 (Eucliea ivisio algorithm). (a) For the set F[x] of polyomials over ay fiel F,

More information

Honors Calculus Homework 13 Solutions, due 12/8/5

Honors Calculus Homework 13 Solutions, due 12/8/5 Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y

More information

FINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,

More information

SAMPLING LIPSCHITZ CONTINUOUS DENSITIES. 1. Introduction

SAMPLING LIPSCHITZ CONTINUOUS DENSITIES. 1. Introduction SAMPLING LIPSCHITZ CONTINUOUS DENSITIES OLIVIER BINETTE Abstract. A simple ad efficiet algorithm for geeratig radom variates from the class of Lipschitz cotiuous desities is described. A MatLab implemetatio

More information

Chapter 2 Feedback Control Theory Continued

Chapter 2 Feedback Control Theory Continued Chapter Feedback Cotrol Theor Cotiued. Itroductio I the previous chapter, the respose characteristic of simple first ad secod order trasfer fuctios were studied. It was show that first order trasfer fuctio,

More information

Indefinite Integral. Lecture 21 discussed antiderivatives. In this section, we introduce new notation and vocabulary. The notation f x dx

Indefinite Integral. Lecture 21 discussed antiderivatives. In this section, we introduce new notation and vocabulary. The notation f x dx 67 Iefiite Itegral Lecture iscusse atierivatives. I this sectio, we itrouce ew otatio a vocabulary. The otatio f iicates the geeral form of the atierivative of f a is calle the iefiite itegral. From the

More information

3. Calculus with distributions

3. Calculus with distributions 6 RODICA D. COSTIN 3.1. Limits of istributios. 3. Calculus with istributios Defiitio 4. A sequece of istributios {u } coverges to the istributio u (all efie o the same space of test fuctios) if (φ, u )

More information

Mathematics Extension 1

Mathematics Extension 1 016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved

More information

AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS

AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS http://www.paper.edu.c Iteratioal Joural of Bifurcatio ad Chaos, Vol. 1, No. 5 () 119 15 c World Scietific Publishig Compay AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC

More information

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =! .8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has

More information

d dx where k is a spring constant

d dx where k is a spring constant Vorlesug IX Harmoic Oscillator 1 Basic efiitios a properties a classical mechaics Oscillator is efie as a particle subject to a liear force fiel The force F ca be epresse i terms of potetial fuctio V F

More information

Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials

Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 15), PP 1-11 www.iosrjourals.org Numerical Solutios of Secod Order Boudary Value Problems

More information

Research Article. Asymmetry-width probabilistic fuzzy logic system for rigid-flexible manipulator modeling

Research Article. Asymmetry-width probabilistic fuzzy logic system for rigid-flexible manipulator modeling Available olie www.jocpr.com Joral of Chemical ad Pharmacetical Research, 4, 6(:753-758 Research Article ISSN : 975-7384 CODEN(USA : JCPRC5 Asymmetry-width probabilistic fzzy logic system for rigid-flexible

More information

BENDING FREQUENCIES OF BEAMS, RODS, AND PIPES Revision S

BENDING FREQUENCIES OF BEAMS, RODS, AND PIPES Revision S BENDING FREQUENCIES OF BEAMS, RODS, AND PIPES Revisio S By Tom Irvie Email: tom@vibratioata.com November, Itrouctio The fuametal frequecies for typical beam cofiguratios are give i Table. Higher frequecies

More information

Mechatronics II Laboratory Exercise 5 Second Order Response

Mechatronics II Laboratory Exercise 5 Second Order Response Mechatroics II Laboratory Exercise 5 Seco Orer Respose Theoretical Backgrou Seco orer ifferetial equatios approximate the yamic respose of may systems. The respose of a geeric seco orer system ca be see

More information

Application of Digital Filters

Application of Digital Filters Applicatio of Digital Filters Geerally some filterig of a time series take place as a reslt of the iability of the recordig system to respod to high freqecies. I may cases systems are desiged specifically

More information

Formation of A Supergain Array and Its Application in Radar

Formation of A Supergain Array and Its Application in Radar Formatio of A Supergai Array ad ts Applicatio i Radar Tra Cao Quye, Do Trug Kie ad Bach Gia Duog. Research Ceter for Electroic ad Telecommuicatios, College of Techology (Coltech, Vietam atioal Uiversity,

More information

(average number of points per unit length). Note that Equation (9B1) does not depend on the

(average number of points per unit length). Note that Equation (9B1) does not depend on the EE603 Class Notes 9/25/203 Joh Stesby Appeix 9-B: Raom Poisso Poits As iscusse i Chapter, let (t,t 2 ) eote the umber of Poisso raom poits i the iterval (t, t 2 ]. The quatity (t, t 2 ) is a o-egative-iteger-value

More information

Interval Intuitionistic Trapezoidal Fuzzy Prioritized Aggregating Operators and their Application to Multiple Attribute Decision Making

Interval Intuitionistic Trapezoidal Fuzzy Prioritized Aggregating Operators and their Application to Multiple Attribute Decision Making Iterval Ituitioistic Trapezoidal Fuzzy Prioritized Aggregatig Operators ad their Applicatio to Multiple Attribute Decisio Makig Xia-Pig Jiag Chogqig Uiversity of Arts ad Scieces Chia cqmaagemet@163.com

More information

Course Outline. Problem Identification. Engineering as Design. Amme 3500 : System Dynamics and Control. System Response. Dr. Stefan B.

Course Outline. Problem Identification. Engineering as Design. Amme 3500 : System Dynamics and Control. System Response. Dr. Stefan B. Course Outlie Amme 35 : System Dyamics a Cotrol System Respose Week Date Cotet Assigmet Notes Mar Itrouctio 8 Mar Frequecy Domai Moellig 3 5 Mar Trasiet Performace a the s-plae 4 Mar Block Diagrams Assig

More information

A New Simulation Model of Rician Fading Channel Xinxin Jin 1,2,a, Yu Zhang 1,3,b, Changyong Pan 4,c

A New Simulation Model of Rician Fading Channel Xinxin Jin 1,2,a, Yu Zhang 1,3,b, Changyong Pan 4,c 6 Iteratioal Coferece o Iformatio Egieerig ad Commuicatios Techology (IECT 6 ISB: 978--6595-375-5 A ew Simulatio Model of Ricia Fadig Chael Xixi Ji,,a, Yu Zhag,3,b, Chagyog Pa 4,c Tsighua atioal Laboratory

More information

dy ds dz ds dx ds ds ds ds ds ds 4-1 DEFORMATION OF A BODY Let there be a line segment PQ in the body with coordinates as:

dy ds dz ds dx ds ds ds ds ds ds 4-1 DEFORMATION OF A BODY Let there be a line segment PQ in the body with coordinates as: 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS 4- DEFORMAON OF A BODY Q Q Let there be a ie seget PQ i the bo with cooriates as: P(,,, Q(,, Legth of the ifferetia eeet: P P Uit taget vector aog PQ: e

More information

Damped Vibration of a Non-prismatic Beam with a Rotational Spring

Damped Vibration of a Non-prismatic Beam with a Rotational Spring Vibratios i Physical Systems Vol.6 (0) Damped Vibratio of a No-prismatic Beam with a Rotatioal Sprig Wojciech SOCHACK stitute of Mechaics ad Fudametals of Machiery Desig Uiversity of Techology, Czestochowa,

More information

A generalization of the Leibniz rule for derivatives

A generalization of the Leibniz rule for derivatives A geeralizatio of the Leibiz rule for erivatives R. DYBOWSKI School of Computig, Uiversity of East Loo, Docklas Campus, Loo E16 RD e-mail: ybowski@uel.ac.uk I will shamelessly tell you what my bottom lie

More information

MATH CALCULUS II Objectives and Notes for Test 4

MATH CALCULUS II Objectives and Notes for Test 4 MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius

More information

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist. Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si

More information

Gain scheduled observer state feedback controller for rational LPV systems

Gain scheduled observer state feedback controller for rational LPV systems Proceedigs of the 7th World Cogress he Iteratioal Federatio of Atomatic Cotrol Seol Korea Jly 6-008 Gai schedled observer state feedback cotroller for ratioal LPV systems Boali A Yagobi M Chevrel P Istitt

More information

2.3 Warmup. Graph the derivative of the following functions. Where necessary, approximate the derivative.

2.3 Warmup. Graph the derivative of the following functions. Where necessary, approximate the derivative. . Warmup Grap te erivative of te followig fuctios. Were ecessar, approimate te erivative. Differetiabilit Must a fuctio ave a erivative at eac poit were te fuctio is efie? Or If f a is efie, must f ( a)

More information