DYNAMICS AND STRUCTURAL LOADING IN WIND TURBINES

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "DYNAMICS AND STRUCTURAL LOADING IN WIND TURBINES"

Transcription

1 DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation and maintenance. An undestanding of the loadings on wind tubines and thei oigins, as well the tubines esponse to them is cucial to avoid thei catastophic failue. Fig. 1: ollapsed wind tubine in Januay 2007 in Gemany. Souce: AP. TYPES OF LOADS The types of loads a wind tubine is subject to duing sevice can be ckassified as: STATI LOADING This loading is constant in time and the esulting deflection of the stuctue is constant and popotional to its stiffness. YLI LOADING Two types of cycling loadings pesent themselves. In quasi static cycling, the loading vaies slowly enough wheeas the deflection of the stuctue is popotional to the loading. In dynamic cycling the loading esults in a deflection elated to the damping

2 foces of the stuctue, paticulaly when the load application fequency is close to the natual vibation fequency of the stuctue. STOHASTI LOADING This type of loading vaies in a andom manne. It esults pedominantly fom wind tubulence and is elevant to the fatigue esponse of the wind tubine stuctue. AERODYNAMI LOADING This is loading deived fom the foce of the wind. The oto of a wind tubine convets the wind's kinetic enegy into useful mechanical wok though aeodynamic effects. The basic undelying concept is the consevation of momentum. Most of the momentum exchange takes place in the wind flow diection, although the useful powe is poduced by foces in the oto plane, which is pependicula to the oncoming steam. A significant slowing down of the oncoming wind steam is necessay to poduce useful mechanical powe. Lage steady loads ae geneally not poblematic in tems of design, and they can be educed by blade coning, to be descibed late. The constant thust loads imposed by the wind may not be dangeous as long as they ae aptly accounted fo at the design stage. The cyclic and stochastic tubulence deived loads ae the ones that can cause most stuctual failues, paticulaly those due to fatigue. The ate of change of thust T on a tubine is popotional to the squae of the appaent wind velocity W: dt c = = ρ ( cosφ+ sin ) dt 2 φ (1) 2 T V L D whee: T V φ ρ c L D is the axial wind thust[ knewton] is the appaent wind speed [ m / s] is the inflow angle is the lift coefficient is the dag coefficient 3 [ / ] is theai density kg m is a constant Gusting involving apid wind change can be hazadous paticulaly if the natual fequency of the tubine stuctue lies within the fequency ange of the wind gusting.

3 Fig. 2: Lift and dag on the oto of a wind tubine. Wind shea coesponding to the incease of wind velocity with height can exacebate the effects of the wind speed due to gusting. A simple fom of the basic elation govening wind shea as a function of height is: H V( H) = Vef H ef α (2) whee: V ( H ) is the wind speed at the nacelle height H [ m / s] Vef isthe wind speed at a efeenceheight[ m / s] H is the elevation above gound [ m] α is the shea coefficient Fo a oto with a adius of 20 metes, the tip to tip distance is 40 metes and the wind speed at the top and bottom of the blade otation can diffe by as much as 30 pecent. If the shea effect coincides with sevee gusting the consequences on the oto can be seious possibly leading to failue. The aeodynamic loading is also affected fom the oientation of the tubine stuctue if it comes out of alignment with the diection of the wind. Yaw misalignment is a situation aising fom a discepancy between the wind diection and the oientation of the oto axis. Shaft tilt is a design featue used to incease the cleaance of the oto blades fom the towe in high winds. Howeve, it can cause sinusoidal cyclic loadings to be imposed on the oto and the dive tain.

4 These aeodynamic loadings can become significant in apidly veeing wind conditions whee a significant lag exists as the yaw mechanism of the tubine attempts to follow the wind diection. MEHANIAL LOADING Loadings that esult fom the mass o the momentum of the wind tubine's stuctue ae classified as mechanical loads. These include: 1. Gavity Loading Gavity can impose lage fatigue stesses on the moving oto, paticulaly at the egion of the oot and at any lamina joints. In the cases of coned and tilted otos, the load can also be out of the oto plane, leading to a flap wise bending aound the blade's chod. The bending moment at the oot of the blade can be expessed as: M = g m( ) sinψ d (3) b whee: m g ψ is theeffective massof the oto blade[ kg] is the distance fom the oto ' s oot[ m] is the gavity acceleation constant is the angleof the blade with the vetyical diection 2. oning Effect oning which is the bending of the oto blades in high winds intoduces centifugal foce loads which act against the aeodynamic steady thust loads, thus educing the mean blade loading, and is a desiable featue intoduced into the design of the oto blade. Howeve, the emoval of steady thust loads may also cause oscillations away fom the mean stess level, which could become vey damaging. 3. Yaw Foces The yaw o gyoscopic foces ae significant in fee yaw machines, whee instantaneous yaw velocities can cause sevee flapping of the oto blades. 4. Tansient Loads Tansient loads such as occu at the stat up and shut down times, ae dependent on the machine and contol system chaacteistics. Inheent flaws in cetain components such as a micoscopic void in a slow speed shaft may esult in fatigue damage unde tansient loading.

5 TURBINE DYNAMI RESPONSE The dynamic esponse of a wind tubine stuctue to the imposed loads affects the oto, the powe tain and the stuctual towe. Undestanding the behavio of these components unde both static and vaying loads is cucial to avoiding potentially dangeous esponses. It is impeative that the otational fequency of the tubine oto is diffeent fom any hamonics of the stuctue's natual vibation esonance fequency. The excitation of the esonant conditions in any dynamic stuctual component must be avoided, o passed though quickly at statup o shut down to avoid catastophic failue. The dynamic esponse of wind tubines encompasses diffeent situations. 1. Static Loads In the case of static loads the extent of deflection unde a load depends upon the stiffness of the stuctue and the size of the load. This can descibed as: whee: F k x F is the estoation foce is the stiffness facto is the deflection magnitude = kx (4) The estoing foce F acts in the opposite diection to the deflection x. This is the simplest model of the static esponse of a stuctue and it elies upon the assumptions that the esponse of the stuctue is linea and that the stuctue is allowed to each equilibium o its static condition balancing the applied loads and the intenal eactions in the stuctue.

6 Fig. 3: Static esponse of a wind tubine stuctue to an applied constant axial load. The static esponse of a wind tubine to an applied axial load F can be due to the thust of a steady wind on the powe tain. Fo a nacelle height h, the eaction at the foundation is given by a bending moment: M b = Fh. (5) An opposite eaction foce of magnitude F also aises at the foundation. Equilibium is eached fo both the extenal applied foce and the esulting moments at the foundation. The intenal eactions in the stuctue due to the elastic effects ae equal to the applied foces and moments. The esponse of most mateials is assumed to be linea so long as stesses in the mateial do not appoach the yield stess of the mateial. The lowe end of the stessstain cuve is a staight line fo most common stuctual mateials. 2. yclic Loads To estimate the dynamic esponse of the stuctue, seveal simplifications ae common: 1. The existence of a linea quasi static esponse 2. A linea modal esponse 3. A non linea quasi static esponse

7 Quasi static hee means that although the stuctue is moving, the motion may be subdivided into small time intevals so that at each inteval the system is teated as if it had eached equilibium. This is not totally accuate since the stiffness modulus of the stuctue will incease at high speeds; a situation designated as stain ate stiffening. Anothe eason is that the damping within the mateials of the stuctue geneates exta loads that ae popotional to the ate of defomation. In addition some load aise due to ai fiction caused by the motion These exta effects ae usually small and so it is easonable to neglect them in most cases. If a stuctue is teated as linea, the natual modes of vibation may be found using the technique of Modal Analysis. The mode shape of a natual mode is the shape of the stuctue at a point in a single peiod of the vibation when the defomation is lagest in magnitude. The fequency is the ecipocal of the time fo one peiod of vibation. Pactically, damping in the stuctue changes the theoetical mode shape and the fequency of vibation. The measued fequencies ae often vey close to those pedicted assuming a linea behavio model. Fig. 4: Mode shape of a natual vibation mode. The mode shape of a natual vibation mode looks like the static defomation of

8 the stuctue with the pactically simila. Howeve, the stuctue's defomation oscillates duing the vibation cycle fom its static shape to its maximum mode shape. Mathematical fomulae fo the natual modes and fequencies of simple systems can be analytically deived. 3. Dynamic Loading Thee possible esponses of a tubine towe to applied loads can occu: tosional, longitudinal and tansvese. In pactice, the actual esponse is a composite of the thee esponses. If the applied load vaies with a fequency appoaching that of the stuctue s natual fequency, then a mechanism exists wheeby enegy may be continuously added to the system. This leads to a pogessive incease in the vibation amplitude, which continues until stuctual failue occus and is called dynamic loading. Resonant wind conditions tansfeed enegy fom the wind and waves unde the Tacoma Naows Bidge in the State of Washington in the USA leading to its collapse, and tuning it into an example of esonant fequency failue. Moe infomation about the Tacoma Naows Bidge is coveed in the Appendices. 4. Vibation Damping The damping chaacteistics of the stuctue dictate how fast the system can estoe itself to the equilibium condition when the stess foces ae emoved. A stuctue is citically damped when the enegy dissipating damping foces acting on it ae just enough to soak up all the enegy impated to it by the applied load befoe the neutal position is eached. Tubine blades ae patially filled with a foam mateial, which helps to dampen the vibation of the blade unde tubulent wind conditions. This pevents the blade fom geneating a esonant esponse. At the citical damping, the citical damping coefficient is given by: = 4km (6) The citical damping atio is defined as: cit = (7) It is usually expessed as a pecentage. Thus a one pecent citical damping coesponds to = 0.01 o 1/100 of the value of the citical damping coefficient. Instability due to aeodynamic and mechanical loads becomes moe of a poblem as the stuctues become softe as thei stuctual modulus deceases. The designe must ensue that the esonant fequencies ae not excited excessively. Special attention must be given to the following situations and factos:

9 a. The stat up and shut down tansients. b. The stuctual towe shadow effects, paticulaly fo downwind machines. c. Excessive toque loadings on the hub and the dive tain. d. Roto speed vaiations paticulaly fo vaiable speed tubines designs. 5. Towe and blade vibations It is impotant to undestand the esponse of tubines with nacelles on slende wie-guyed towes. Fom the pespective of the stuctual towe and oto blades, esonances may cause unwelcome noise emissions and fatigue. The towe esonances may feed back to the electical output to the gid, causing powe suges. Stuctual towe and blade vibation may be studied by using the example of a simple cantileve beam. Fo the towe, the eal system includes othe factos such as inetial effects fom the nacelle and esonant effects fom the guy cables and the gin pole. These can be accounted fo by using a patial diffeential modeling technique. In this case we ae mainly concened with the simple case. In the moe ealistic analysis of the dynamic behavio of multiple degee of feedom cantileve systems, assumptions can be made to simplify the analysis: a) The stain enegy o potential enegy in the exteme position is consideed as equal to the kinetic enegy in the neutal position implying that thee ae no damping losses. b) The natual fequency is elated to the deflected shape of the cantileve implying that thee could be esonant vibation. c) The cantileve has seveal natual fequencies at which tue dynamic esponse is possible, whee the applied load vaies at the system's natual fequency. d) Each fequency is associated with a paticula deflected shape o the mode shape. Fo a oto blade o towe, and fo moe complex mode shapes, the stain enegy is highe fo a paticula deflection in compaison to simple mode shapes. This leads to a coesponding highe natual fequency. These mode shapes and natual fequencies ae fundamental to the system, and ae independent of the loading chaacteistics in a given vibation situation. 6. Dynamic esponse to stochastic andom load The stochastic andom loads occu pedominantly fom wind tubulence. In addition to fatigue, tubulence affects many othe design paametes such as the maximum load pediction, the stuctual excitation, and the contol and powe quality. The non linea natue of fatigue, thee esults that a doubling of the load amplitude has a vey stong effect on the fatigue life. The occuence of high load amplitudes esulting fom tubulent stochastic wind gusts educes a stuctue's fatigue esistance.

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

More information

VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

More information

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

PY1052 Problem Set 3 Autumn 2004 Solutions

PY1052 Problem Set 3 Autumn 2004 Solutions PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the

More information

12. Rolling, Torque, and Angular Momentum

12. Rolling, Torque, and Angular Momentum 12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

More information

Strength Analysis and Optimization Design about the key parts of the Robot

Strength Analysis and Optimization Design about the key parts of the Robot Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Pint): 2320-9356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.25-29 Stength Analysis and Optimization Design

More information

Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number.

Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number. Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 1-5 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE

More information

The Role of Gravity in Orbital Motion

The Role of Gravity in Orbital Motion ! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

More information

Revision Guide for Chapter 11

Revision Guide for Chapter 11 Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams

More information

Chapter 6. Gradually-Varied Flow in Open Channels

Chapter 6. Gradually-Varied Flow in Open Channels Chapte 6 Gadually-Vaied Flow in Open Channels 6.. Intoduction A stea non-unifom flow in a pismatic channel with gadual changes in its watesuface elevation is named as gadually-vaied flow (GVF). The backwate

More information

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

Lesson 2 Power Factor Improvement, Harmonic Reduction, Filter

Lesson 2 Power Factor Improvement, Harmonic Reduction, Filter Lesson 2 Powe Facto Impovement, Hamonic Reduction, Filte 1. Powe Facto Impovement Fo phase-contolled opeation in both single phase full wave half and full contolled bidge convetes, the displacement facto

More information

Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth

Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate

More information

PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013 PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

More information

Gravitation. AP Physics C

Gravitation. AP Physics C Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

More information

Solutions to Homework Set #5 Phys2414 Fall 2005

Solutions to Homework Set #5 Phys2414 Fall 2005 Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated

More information

Deflection of Electrons by Electric and Magnetic Fields

Deflection of Electrons by Electric and Magnetic Fields Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

More information

2008 Quarter-Final Exam Solutions

2008 Quarter-Final Exam Solutions 2008 Quate-final Exam - Solutions 1 2008 Quate-Final Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of

More information

Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of universal gravitation Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

More information

Exam 3: Equation Summary

Exam 3: Equation Summary MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Hamonic Motion Intoduction Simple hamonic motion occus when the net foce acting on an object is popotional to the object s displacement fom an equilibium position. When the object is at an equilibium

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

More information

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it. Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

More information

Lab #7: Energy Conservation

Lab #7: Energy Conservation Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

More information

Instituto Superior Técnico Av. Rovisco Pais, 1 1049-001 Lisboa E-mail: virginia.infante@ist.utl.pt

Instituto Superior Técnico Av. Rovisco Pais, 1 1049-001 Lisboa E-mail: virginia.infante@ist.utl.pt FATIGUE LIFE TIME PREDICTIO OF POAF EPSILO TB-30 AIRCRAFT - PART I: IMPLEMETATIO OF DIFERET CYCLE COUTIG METHODS TO PREDICT THE ACCUMULATED DAMAGE B. A. S. Seano 1, V. I. M.. Infante 2, B. S. D. Maado

More information

STABILITY ANALYSIS IN MILLING BASED ON OPERATIONAL MODAL DATA 1. INTRODUCTION

STABILITY ANALYSIS IN MILLING BASED ON OPERATIONAL MODAL DATA 1. INTRODUCTION Jounal of Machine Engineeing, Vol. 11, No. 4, 211 Batosz POWALKA 1 Macin CHODZKO 1 Kzysztof JEMIELNIAK 2 milling, chatte, opeational modal analysis STABILITY ANALYSIS IN MILLING BASED ON OPERATIONAL MODAL

More information

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27 Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

More information

PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary

PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC

More information

General Physics (PHY 2130)

General Physics (PHY 2130) Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:

More information

Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43 Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

More information

INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE

INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE 1 INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE ANATOLIY A. YEVTUSHENKO 1, ALEXEY N. KOCHEVSKY 1, NATALYA A. FEDOTOVA 1, ALEXANDER Y. SCHELYAEV 2, VLADIMIR N. KONSHIN 2 1 Depatment of

More information

Lab M4: The Torsional Pendulum and Moment of Inertia

Lab M4: The Torsional Pendulum and Moment of Inertia M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the

More information

10. Collisions. Before During After

10. Collisions. Before During After 10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: -F(t)

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

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

Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient:

Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient: Psychology 282 Lectue #2 Outline Review of Peason coelation coefficient: z z ( n 1) Measue of linea elationship. Magnitude Stength Sign Diection Bounded by +1.0 and -1.0. Independent of scales of measuement.

More information

Determining solar characteristics using planetary data

Determining solar characteristics using planetary data Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation

More information

Design of Wind Energy System on the Building Tower Applications

Design of Wind Energy System on the Building Tower Applications ISSN(Online): 39-8753 ISSN (Pint) :347-67 (An ISO 397: 7 Cetified Oganization) Vol. 4, Issue, Febuay 5 Design of Wind Enegy System on the Building owe Applications D.Anusha, L V Suesh Kuma, G.V. Nagesh

More information

mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !

mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2  2 = GM . Combining the results we get ! Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!

More information

An Introduction to Omega

An Introduction to Omega An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

More information

Chapter 4: Fluid Kinematics

Chapter 4: Fluid Kinematics Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian

More information

Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

More information

Complex Envelope Vectorization for the solution of mid-high frequency acoustic problems. A. Sestieri

Complex Envelope Vectorization for the solution of mid-high frequency acoustic problems. A. Sestieri Complex Envelope Vectoization fo the solution of mid-high fequency acoustic poblems A. Sestiei Depatment of Mechanical and Aeospace Engineeing Univesity of Rome la Sapienza Pesentation layout - Low fequency

More information

2. Orbital dynamics and tides

2. Orbital dynamics and tides 2. Obital dynamics and tides 2.1 The two-body poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body

More information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

Comparing Availability of Various Rack Power Redundancy Configurations

Comparing Availability of Various Rack Power Redundancy Configurations Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance the availability

More information

PY1052 Problem Set 8 Autumn 2004 Solutions

PY1052 Problem Set 8 Autumn 2004 Solutions PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

(3) Bipolar Transistor Current Sources

(3) Bipolar Transistor Current Sources B73 lectonics Analysis & Design (3) Bipola Tansisto Cuent Souces Leaning utcome Able to descibe and: Analyze and design a simple twotansisto BJT cuent-souce cicuit to poduce a given bias cuent. Analyze

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

More information

4.1 Cylindrical and Polar Coordinates

4.1 Cylindrical and Polar Coordinates 4.1 Cylindical and Pola Coodinates 4.1.1 Geometical Axisymmety A lage numbe of pactical engineeing poblems involve geometical featues which have a natual axis of symmety, such as the solid cylinde, shown

More information

Experiment 6: Centripetal Force

Experiment 6: Centripetal Force Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

More information

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Problem Set 6: Solutions

Problem Set 6: Solutions UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 16-4 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente

More information

Ilona V. Tregub, ScD., Professor

Ilona V. Tregub, ScD., Professor Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

Phys 2101 Gabriela González. cos. sin. sin

Phys 2101 Gabriela González. cos. sin. sin 1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe

More information

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

More information

Data Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation

Data Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation (213) 1 28 Data Cente Demand Response: Avoiding the Coincident Peak via Wokload Shifting and Local Geneation Zhenhua Liu 1, Adam Wieman 1, Yuan Chen 2, Benjamin Razon 1, Niangjun Chen 1 1 Califonia Institute

More information

Comparing Availability of Various Rack Power Redundancy Configurations

Comparing Availability of Various Rack Power Redundancy Configurations Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance

More information

Forces & Magnetic Dipoles. r r τ = μ B r

Forces & Magnetic Dipoles. r r τ = μ B r Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

More information

Modal Characteristics study of CEM-1 Single-Layer Printed Circuit Board Using Experimental Modal Analysis

Modal Characteristics study of CEM-1 Single-Layer Printed Circuit Board Using Experimental Modal Analysis Available online at www.sciencediect.com Pocedia Engineeing 41 (2012 ) 1360 1366 Intenational Symposium on Robotics and Intelligent Sensos 2012 (IRIS 2012) Modal Chaacteistics study of CEM-1 Single-Laye

More information

rotation -- Conservation of mechanical energy for rotation -- Angular momentum -- Conservation of angular momentum

rotation -- Conservation of mechanical energy for rotation -- Angular momentum -- Conservation of angular momentum Final Exam Duing class (1-3:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday

More information

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90 . Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal

More information

Learning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds.

Learning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds. Lectue #0 Chapte Atomic Bonding Leaning Objectives Descibe ionic, covalent, and metallic, hydogen, and van de Waals bonds. Which mateials exhibit each of these bonding types? What is coulombic foce of

More information

The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit

More information

Performance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation

Performance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation Cicuits and Systems, 013, 4, 117-1 http://dx.doi.og/10.436/cs.013.41017 Published Online Januay 013 (http://www.scip.og/jounal/cs) Pefomance Analysis of an Invese Notch Filte and Its Application to F 0

More information

Avoided emissions kgco 2eq /m 2 Reflecting surfaces 130

Avoided emissions kgco 2eq /m 2 Reflecting surfaces 130 ANALYSIS OF GLOBAL WARMING MITIGATION BY WHITE REFLECTING SURFACES Fedeico Rossi, Andea Nicolini Univesity of Peugia, CIRIAF Via G.Duanti 67 0615 Peugia, Italy T: +9-075-585846; F: +9-075-5848470; E: fossi@unipg.it

More information

Introduction to Electric Potential

Introduction to Electric Potential Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic

More information

THE NEW FIFTH EDITION OF API 618 FOR RECIPROCATING COMPRESSORS WHICH PULSATION AND VIBRATION CONTROL PHILOSOPHY SHOULD YOU USE?

THE NEW FIFTH EDITION OF API 618 FOR RECIPROCATING COMPRESSORS WHICH PULSATION AND VIBRATION CONTROL PHILOSOPHY SHOULD YOU USE? THE NEW FIFTH EDITION OF API 618 FOR RECIPROCATING COMPRESSORS WHICH PULSATION AND VIBRATION CONTROL PHILOSOPHY SHOULD YOU USE? by James D. Tison Senio Staff Enginee and Kenneth E. Atkins Senio Staff Enginee

More information

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1) Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

More information

Heat transfer has direction as well as magnitude. The rate of heat conduction

Heat transfer has direction as well as magnitude. The rate of heat conduction HEAT CONDUCTION EQUATION CHAPTER 2 Heat tansfe has diection as well as magnitude. The ate of heat conduction in a specified diection is popotional to the tempeatue gadient, which is the ate of change in

More information

SELF-INDUCTANCE AND INDUCTORS

SELF-INDUCTANCE AND INDUCTORS MISN-0-144 SELF-INDUCTANCE AND INDUCTORS SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. Self-Inductance L.........................................

More information

Do Vibrations Make Sound?

Do Vibrations Make Sound? Do Vibations Make Sound? Gade 1: Sound Pobe Aligned with National Standads oveview Students will lean about sound and vibations. This activity will allow students to see and hea how vibations do in fact

More information

Model Question Paper Mathematics Class XII

Model Question Paper Mathematics Class XII Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

More information

EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD

EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD 260 16-1. THEORY EXPERMENT 16 THE MAGNETC MOMENT OF A BAR MAGNET AND THE HORZONTAL COMPONENT OF THE EARTH S MAGNETC FELD The uose of this exeiment is to measue the magnetic moment μ of a ba magnet and

More information

Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Chapte 5. Foce and Motion In this chapte we study causes of motion: Why does the windsufe blast acoss the wate in the way he does? The combined foces of the wind, wate, and gavity acceleate him accoding

More information

Progress In Electromagnetics Research Letters, Vol. 11, , 2009

Progress In Electromagnetics Research Letters, Vol. 11, , 2009 Pogess In Electomagnetics Reseach Lettes, Vol. 11, 93 102, 2009 COMPUTATIONAL MODELING OF INDUCTION HEATING PROCESS M. H. Tavakoli, H. Kabaschi, and F. Samavat Physics Depatment Bu-Ali Sina Univesity Hamedan

More information

Rotation. Problem solving strategy is the same as for problems with linear motion... Rotation

Rotation. Problem solving strategy is the same as for problems with linear motion... Rotation Rotation The basic quantity in otation is the angula displacement simila ole to the linea displacement studied in chapte units: adians, eolutions, degees, θ Poblem soling stategy is the same as fo poblems

More information

The transport performance evaluation system building of logistics enterprises

The transport performance evaluation system building of logistics enterprises Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194-114 Online ISSN: 213-953 Pint ISSN: 213-8423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics

More information

Module 2: Dynamics of Electric and Hybrid vehicles

Module 2: Dynamics of Electric and Hybrid vehicles NPTEL Electical Engineeing Intoduction to Hybid and Electic Vehicles Module : Dynamics of Electic and Hybid vehicles Lectue 3: Motion and amic equations fo vehicles Motion and amic equations fo vehicles

More information

NUCLEAR MAGNETIC RESONANCE

NUCLEAR MAGNETIC RESONANCE 19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic

More information

Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2 F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

More information

Chapter 8, Rotational Kinematics. Angular Displacement

Chapter 8, Rotational Kinematics. Angular Displacement Chapte 8, Rotational Kinematics Sections 1 3 only Rotational motion and angula displacement Angula velocity and angula acceleation Equations of otational kinematics 1 Angula Displacement! B l A The length

More information

2 - ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1

2 - ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1 - ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation

More information

Classical Lifetime of a Bohr Atom

Classical Lifetime of a Bohr Atom 1 Poblem Classical Lifetime of a Boh Atom James D. Olsen and Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 85 (Mach 7, 5) In the Boh model of the hydogen atom s gound state,

More information

Software Engineering and Development

Software Engineering and Development I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining

More information

Gravity and the figure of the Earth

Gravity and the figure of the Earth Gavity and the figue of the Eath Eic Calais Pudue Univesity Depatment of Eath and Atmospheic Sciences West Lafayette, IN 47907-1397 ecalais@pudue.edu http://www.eas.pudue.edu/~calais/ Objectives What is

More information

8-1 Newton s Law of Universal Gravitation

8-1 Newton s Law of Universal Gravitation 8-1 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,

More information

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle

More information

Research on Risk Assessment of the Transformer Based on Life Cycle Cost

Research on Risk Assessment of the Transformer Based on Life Cycle Cost ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,

More information

Semipartial (Part) and Partial Correlation

Semipartial (Part) and Partial Correlation Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

More information

Resources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics

Resources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics 3/4/0 Resouces Cicula Motion: Fom Moto Racing to Satellites D Jeff McCallum School of Physics http://www.gap-system.og/~histoy/mathematicians/ Newton.html http://www.fg-a.com http://www.clke.com/clipat

More information

Chapter 5 1. Hydraulic Turbines (Gorla & Khan, pp )

Chapter 5 1. Hydraulic Turbines (Gorla & Khan, pp ) Electonic Notes apte 5. ydaulic Tubines (Gola & Kan, pp.9 4). Teminologies Related to A ydopowe lant oove Dam geneates moe tan 4 billion kilowatt-ous of electicity eac yea, enoug to seve.3 million people.

More information

Analytical Proof of Newton's Force Laws

Analytical Proof of Newton's Force Laws Analytical Poof of Newton s Foce Laws Page 1 1 Intouction Analytical Poof of Newton's Foce Laws Many stuents intuitively assume that Newton's inetial an gavitational foce laws, F = ma an Mm F = G, ae tue

More information

SIMULATION OF GAS TURBINES OPERATING IN OFF-DESIGN CONDITION

SIMULATION OF GAS TURBINES OPERATING IN OFF-DESIGN CONDITION SIMULAION OF GAS URBINES OPERAING IN OFF-DESIGN CONDIION Analdo Walte: awalte@fem.unicamp.b Univesity of Campinas Dept. of Enegy DE/FEM/Unicamp P.O. Box 6122 - ZIP code 13083-970 - Bazil Abstact. In many

More information