MIT Amorphous Materials
|
|
|
- Jacob McCarthy
- 5 years ago
- Views:
Transcription
1 MIT Amorphous Maerials 7: Viscoelasiciy and Relaxaion Juejun (JJ) Hu 1
2 Afer-class reading lis Fundamenals of Inorganic Glasses Ch. 8, Ch. 13, Appendix A Inroducion o Glass Science and Technology Ch. 9 (does no cover relaxaion) Mahemaics is he language wih which God wroe he Universe. Galileo Galilei 2
3 Where and why does liquid end and glass begin? Wha don we know? Science 309, 83 (2005). 3
4 Is glass a solid or a viscous liquid? V Solid Elasiciy insananeous, ransien x E x G xy xy Glass Supercooled liquid Liquid Viscosiy ime-dependen, permanen xy xy E : Young s modulus G : shear modulus T m T : viscosiy (uni: Pa s or Poise) 4
5 Viscoelasiciy: complex shear modulus Consider a sinusoidally varying shear srain xy exp 0 i Elasic response: Viscous response: G G i xy exp xy i 0 i i In a general viscoelasic solid: xy xy 0 exp xy Loss modulus * G i G xy xy xy G * : complex shear modulus * G G i G ig ' " Shear/sorage modulus 5
6 Phenomenological models of viscoelasic maerials Elasiciy: Hookean spring G xy xy Viscosiy: Newonian dashpo xy xy Models assume linear maerial response or infiniesimal sress Each dashpo elemen corresponds o a relaxaion mechanism 6
7 The Maxwell elemen Serial connecion of a Hookean spring and a Newonian dashpo Toal sress: Toal srain: G E V E V E E V V 7
8 The Maxwell elemen E G Consan sress (creep): E V V consa n 0 E V E V G Consan srain (sress relaxaion): consa n 0 E V E exp V 1 exp G Relaxaion ime 8
9 The Maxwell elemen Oscillaory srain: G E E E V 0 expi V V G ' G" 2 2 G G 9
10 The Voig-Kelvin elemen E G E V V Oscillaory srain: G ' G G" G Parallel connecion of a Hookean spring and a Newonian dashpo Toal sress: Toal srain: Consan srain: Consan sress: 0 exp i E E 1 exp G V G V 10
11 Generalized Maxwell model For each Maxwell componen: G i 1 1 i Gi i 1 G 1 1 i i Gi i i G i 2 G i... Toal sress: G 1... i... 11
12 Generalized Maxwell model G Sress relaxaion: consan 0 Prony series: 1 2 G 1 G 2 G 1... i... G exp Gi i0 i G exp G R In real solids, a muliude of microscopic relaxaion processes give rise o dispersion of relaxaion ime (sreched exponenial) 12
13 Elasic, viscoelasic, and viscous responses Sress Sress Sress Elasic srain Viscoelasic srain Viscous srain 13
14 Viscoelasic maerials Mozzarella cheese Human skin Turbine blades Volcanic lava Memory foams Naval ship propellers Image of Naval ship propellers is in he public domain. Various images unknown. Lava image Lavapix on YouTube. All righs reserved. This conen is excludedfrom our Creaive Commons license. For more informaion, see hp://ocw.mi.edu/help/faq-fair-use/. 14
15 Bolzmann superposiion principle In he linear viscoelasic regime, he sress (srain) responses o successive srain (sress) simuli are addiive Srain Srain Srain Sress Sress Sress 1 1 G G
16 Bolzmann superposiion principle In he linear viscoelasic regime, he sress (srain) responses o successive srain (sress) simuli are addiive Srain i i Viscoelasic response is hisory-dependen d i 0 d ' d ' Relaxaion funcion dicaes ime-domain response Sress G i i i i i d i d d G 0 d 0 d ' ' d ' 16
17 V Srucural relaxaion in glass Glass ransiion Supercooled liquid Glassy sae Elasic regime Viscoelasic regime Viscous regime T re obs re ~ obs re obs All mechanical and Glass srucure and Srucural changes hermal effecs only affec aomic vibraions properies are hisorydependen are insananeous: equilibrium sae can be quickly reached Ergodiciy breakdown 17
18 V Srucural relaxaion in glass Glass ransiion Supercooled liquid Glassy sae Elasic regime DN 1 Viscoelasic regime DN ~1 Viscous regime DN 1 T Debroah number (DN): DN re obs he mounains flowed before he Lord Propheess Deborah (Judges 5:5) 18
19 Comparing sress relaxaion and srucural relaxaion V T Equilibrium sae Zero sress sae Driving force Residual sress Relaxaion kineics Exponenial decay wih a single relaxaion ime Relaxaion rae scales wih driving force Equilibrium sae Supercooled liquid sae Driving force Free volume Vf V V Relaxaion kineics Exponenial decay wih a single relaxaion ime Relaxaion rae scales wih driving force e 19
20 Free volume model of relaxaion (firs order kineic model) V T Relaxaion kineics a consan emperaure V f V V e exp Vf 0 re Vf Vf re Temperaure dependence of relaxaion E V f f E a V V a re exp 0 exp kt B 0 kt B 20
21 Model prediced relaxaion kineics Cooling rae: 10 C/s Varying reheaing rae Slope: volume CTE of supercooled liquid 1 C/s 0.1 C/s 10 C/s 21
22 Model prediced relaxaion kineics Varying cooling rae Reheaing rae: 1 C/s 100 C/s 10 C/s 1 C/s Ficive emperaure and glass srucure are funcions of cooling rae 22
23 Tool s Ficive emperaure heory V, H 3 Increasing cooling rae Supercooled liquid Ficive emperaure T f : he emperaure on he supercooled liquid curve a which he glass would find iself in equilibrium wih he supercooled liquid sae if brough suddenly o i Wih increasing cooling rae: 2 T f1 < T f2 < T f3 1 T f1 T f2 T f3 T A glass sae is fully described by hermodynamic parameers (T, P) and T f V V T, Tf H H T, Tf Glass properies are funcions of emperaure and T f (srucure) 23
24 Tool s Ficive emperaure heory V V, H Increasing cooling rae Supercooled liquid T f1 T f2 T f3 V T, Tf H H T, Tf T Glass propery change in he glass ransiion range consiss of wo componens Temperaure-dependen propery evoluion wihou modifying glass srucure V T Volume:, Propery change due o relaxaion (T f change) T f V g H T C Enhalpy:, T f Pg V T Volume:,, f T V e V g H T C C Enhalpy:,, f T P e P g 24
25 Tool s Ficive emperaure heory V V, H Increasing cooling rae Supercooled liquid T f1 T f2 T f3 V T, Tf H H T, Tf T Glass propery change in he glass ransiion range consiss of wo componens dv T, Tf V V dtf dt T T dt V, g V, e V, g Tf f P, g P, e P, g dt d f T dt d dh T, Tf H H dtf dt T T dt Tf f C C C dt d f T dt d 25
26 Predicing glass srucure evoluion due o relaxaion: Tool s equaion Consider a glass sample wih Ficive emperaure T f P (V, H, S, ec.) Glass P g - P e Supercooled liquid T T f T f - T P g P e P T P e g T Take ime derivaive: T dp P P dt d T T d g e g f dp P P d g g e re f T Assume firs-order relaxaion: dtf Tf T d re Tool s equaion 26
27 Tool s Ficive emperaure heory Line D'M: Tf T dtf Tf T d re dh T, Tf H H dtf dt T T dt Tf f dt CP, g CP, e CP, g d Tool s equaion f T dt d J. Am. Ceram. Soc. 29, 240 (1946) 27
28 Difficulies wih Tool s T f heory Riland experimen: wo groups of glass samples of idenical composiion were hea reaed o obain he same refracive index via wo differen roues Group A: kep a 530 C for 24 h Group B: cooled a 16 C/h hrough he glass ransiion range Boh groups were hen placed in a furnace sanding a 530 C and heir refracive indices were measured as a funcion of hea reamen ime Glass srucure canno be fully characerized by he single parameer T f J. Am. Ceram. Soc. 39, 403 (1956). 28
29 Explaining he Riland experimen n exp re,1 exp re,2 Srucural relaxaion in glass involves muliple srucural eniies and is characerized by a muliude of relaxaion ime scales Tool-Narayanaswamy-Moynihan (TNM) model 29
30 Wha is relaxaion? 30
31 Relaxaion: reurn of a perurbed sysem ino equilibrium Examples Sress and srain relaxaion in viscoelasic solids Free volume relaxaion in glasses near T g Glass srucural relaxaion (T f change) Time-dependen, occurs even afer simulus is removed Debroah Number: DN re obs DN >> 1: negligible relaxaion due o sluggish kineics DN << 1: sysem always in equilibrium DN ~ 1: sysem behavior dominaed by relaxaion 31
32 Modeling relaxaion Relaxaion rae P P P re e Driving force Maxwellian relaxaion models P P0 Pe exp P re P P0 Pe exp P re Bolzmann superposiion principle in linear sysems d G d ' ' d d G ' 0 d 0 d d d ' ' 0 R IRF S e e S R IRF ' ' d ' 32
33 The Naure of Glass Remains Anyhing bu Clear Free volume relaxaion heory V f a exp 0 kbt Tool s Ficive emperaure heory Uses a single parameer T f o label glass srucure Tool s equaion (of T f relaxaion) Srucural relaxaion in glass involves muliple srucural eniies and is characerized by a muliude of relaxaion ime scales V dtf Tf T d re E The Riland experimen 33
34 MIT OpenCourseWare hp://ocw.mi.edu Amorphous Maerials Fall 2015 For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.
Polymer Engineering (MM3POE)
Polymer Engineering (MM3POE) VISCOELASTICITY hp://www.noingham.ac.uk/~eazacl/mm3poe Viscoelasiciy 1 Conens Wha is viscoelasiciy? Fundamenals Creep & creep recovery Sress relaxaion Modelling viscoelasic
Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi
Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec
Relaxation in Glass. Transition
Relaxaion in Glass Lecure 2: he Glass ransiion as a Kineic Lecure 2: he Glass ransiion as a Kineic ransiion Enhalpy Changes in he Glass ransiion Range H decreases coninuously wih cooling Slope of he H
Summary of shear rate kinematics (part 1)
InroToMaFuncions.pdf 4 CM465 To proceed o beer-designed consiuive equaions, we need o know more abou maerial behavior, i.e. we need more maerial funcions o predic, and we need measuremens of hese maerial
ES240 Solid Mechanics Fall 2007
ES4 Solid Mechanics Fall 7 Viscoelasiciy References NG McCrum, CP Buckley, and CB Bucknall, Principles of Polymer Engineering, nd ediion, Oxford Universiy Press, 997 A good balance of heory and applicaion
Structural Dynamics and Earthquake Engineering
Srucural Dynamics and Earhquae Engineering Course 1 Inroducion. Single degree of freedom sysems: Equaions of moion, problem saemen, soluion mehods. Course noes are available for download a hp://www.c.up.ro/users/aurelsraan/
6.01: Introduction to EECS I Lecture 8 March 29, 2011
6.01: Inroducion o EES I Lecure 8 March 29, 2011 6.01: Inroducion o EES I Op-Amps Las Time: The ircui Absracion ircuis represen sysems as connecions of elemens hrough which currens (hrough variables) flow
where the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).
Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness
Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws
Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species
CHEMICAL KINETICS: 1. Rate Order Rate law Rate constant Half-life Temperature Dependence
CHEMICL KINETICS: Rae Order Rae law Rae consan Half-life Temperaure Dependence Chemical Reacions Kineics Chemical ineics is he sudy of ime dependence of he change in he concenraion of reacans and producs.
A First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18
A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly
Some Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
6.003 Homework 1. Problems. Due at the beginning of recitation on Wednesday, February 10, 2010.
6.003 Homework Due a he beginning of reciaion on Wednesday, February 0, 200. Problems. Independen and Dependen Variables Assume ha he heigh of a waer wave is given by g(x v) where x is disance, v is velociy,
2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
Andrei Tokmakoff, MIT Deparmen of Chemisry, 2/22/2007 2-17 2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaion of he dynamics of a quanum sysem is no unique. So far we have described
Analytic nonlinear elasto-viscosity of two types of BN and PI rubbers at large deformations
Bulgarian Chemical Communicaions, Volume 48, Special Issue E (pp. 59-64) 016 Analyic nonlinear elaso-viscosiy of wo ypes of BN and PI rubbers a large deformaions K. B. Hadjov, A. S. Aleksandrov, M. P.
2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts?
AP Chemisry Tes (Chaper 12) Muliple Choice (40%) 1) Which of he following is a kineic quaniy? A) Enhalpy B) Inernal Energy C) Gibb s free energy D) Enropy E) Rae of reacion 2) Of he following quesions,
Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
New effective moduli of isotropic viscoelastic composites. Part I. Theoretical justification
IOP Conference Series: Maerials Science and Engineering PAPE OPEN ACCESS New effecive moduli of isoropic viscoelasic composies. Par I. Theoreical jusificaion To cie his aricle: A A Sveashkov and A A akurov
Ch1: Introduction and Review
//6 Ch: Inroducion and Review. Soli and flui; Coninuum hypohesis; Transpor phenomena (i) Solid vs. Fluid No exernal force : An elemen of solid has a preferred shape; fluid does no. Under he acion of a
Mechanical Fatigue and Load-Induced Aging of Loudspeaker Suspension. Wolfgang Klippel,
Mechanical Faigue and Load-Induced Aging of Loudspeaker Suspension Wolfgang Klippel, Insiue of Acousics and Speech Communicaion Dresden Universiy of Technology presened a he ALMA Symposium 2012, Las Vegas
IV. Transport Phenomena. Lecture 19: Transient Diffusion. 1 Response to a Current Step. MIT Student. 1.1 Sand s Time
IV. Transpor Phenomena Lecure 19: Transien iffusion MIT Suden In his lecure we show how o use simple scaling argumens o approximae he soluion o ransien diffusion problems, which arise in elecrochemical
MECHANICS OF MATERIALS Poisson s Ratio
Poisson s Raio For a slender bar subjeced o axial loading: ε x x y 0 The elongaion in he x-direcion i is accompanied by a conracion in he oher direcions. Assuming ha he maerial is isoropic (no direcional
Examples of Dynamic Programming Problems
M.I.T. 5.450-Fall 00 Sloan School of Managemen Professor Leonid Kogan Examples of Dynamic Programming Problems Problem A given quaniy X of a single resource is o be allocaed opimally among N producion
Electrical and current self-induction
Elecrical and curren self-inducion F. F. Mende hp://fmnauka.narod.ru/works.hml mende_fedor@mail.ru Absrac The aricle considers he self-inducance of reacive elemens. Elecrical self-inducion To he laws of
( ) = b n ( t) n " (2.111) or a system with many states to be considered, solving these equations isn t. = k U I ( t,t 0 )! ( t 0 ) (2.
Andrei Tokmakoff, MIT Deparmen of Chemisry, 3/14/007-6.4 PERTURBATION THEORY Given a Hamilonian H = H 0 + V where we know he eigenkes for H 0 : H 0 n = E n n, we can calculae he evoluion of he wavefuncion
RC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
Traveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
FRACTIONAL THERMOVISCOELASTIC MATERIALS *G. Alba, *M. Di Paola
Meccanica dei Maeriali e delle Sruure Vol. 6 (26), no.2, pp. - ISSN: 235-679X Diparimeno di Ingegneria Civile, Ambienale, Aerospaziale, dei Maeriali - DICAM FRACIONAL HERMOVISCOELASIC MAERIALS *G. Alba,
Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
INDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
Viscoelastic Catenary
Viscoelasic Caenary Anshuman Roy 1 Inroducion This paper seeks o deermine he shape of a hin viscoelasic fluid filamen as i sags under is own weigh. The problem is an exension of he viscous caenary [1]
4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
Role of Material Functions in Rheological Analysis. Role of Material Functions in Rheological Analysis
Role of Maerial Funcions in Rheological Analysis QUALIY CONROL compare wih oher in-house daa on qualiaie basis unknown maerial QUALIAIVE ANALYSIS compare daa wih lieraure repors on arious fluids conclude
18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
ON LINEAR VISCOELASTICITY WITHIN GENERAL FRACTIONAL DERIVATIVES WITHOUT SINGULAR KERNEL
Gao F. e al.: On Linear Viscoelasiciy wihin General Fracional erivaives... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S335-S34 S335 ON LINEAR VISCOELASTICITY WITHIN GENERAL FRACTIONAL ERIVATIVES WITHOUT SINGULAR
6.003 Homework #9 Solutions
6.00 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 0 a 0 5 a k sin πk 5 sin πk 5 πk for k 0 a k 0 πk j
Class Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
Linear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
6.003: Signals and Systems
6.3: Signals and Sysems Lecure 7 April 8, 6.3: Signals and Sysems C Fourier ransform C Fourier ransform Represening signals by heir frequency conen. X(j)= x()e j d ( analysis equaion) x()= π X(j)e j d
Module 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
Failure of the work-hamiltonian connection for free energy calculations. Abstract
Failure of he work-hamilonian connecion for free energy calculaions Jose M. G. Vilar 1 and J. Miguel Rubi 1 Compuaional Biology Program, Memorial Sloan-Keering Cancer Cener, 175 York Avenue, New York,
Explaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015
Explaining Toal Facor Produciviy Ulrich Kohli Universiy of Geneva December 2015 Needed: A Theory of Toal Facor Produciviy Edward C. Presco (1998) 2 1. Inroducion Toal Facor Produciviy (TFP) has become
CHAPTER 6: FIRST-ORDER CIRCUITS
EEE5: CI CUI T THEOY CHAPTE 6: FIST-ODE CICUITS 6. Inroducion This chaper considers L and C circuis. Applying he Kirshoff s law o C and L circuis produces differenial equaions. The differenial equaions
arxiv:math-ph/ v1 9 Dec 2005
Physical inerpreaion of iniial condiions for fracional differenial equaions wih Riemann-Liouville fracional derivaives 1 Nicole Heymans (a) and Igor Podlubny (b) arxiv:mah-ph/0512028v1 9 Dec 2005 (a) Physique
THE EFFECT OF SUCTION AND INJECTION ON UNSTEADY COUETTE FLOW WITH VARIABLE PROPERTIES
Kragujevac J. Sci. 3 () 7-4. UDC 53.5:536. 4 THE EFFECT OF SUCTION AND INJECTION ON UNSTEADY COUETTE FLOW WITH VARIABLE PROPERTIES Hazem A. Aia Dep. of Mahemaics, College of Science,King Saud Universiy
( ) is the stretch factor, and x the
(Lecures 7-8) Liddle, Chaper 5 Simple cosmological models (i) Hubble s Law revisied Self-similar srech of he universe All universe models have his characerisic v r ; v = Hr since only his conserves homogeneiy
Lecture #11: Wavepacket Dynamics for Harmonic Oscillator
Lecure #11: Wavepacke Dynamics for Harmonic Oscillaor and PIB Las ime: Time Dependen Schrödinger Equaion Ψ HHΨ = iħ Express Ψ in complee basis se of eigenfuncions of ime independen H H {ψ n (x), E n }
Designing Information Devices and Systems I Spring 2019 Lecture Notes Note 17
EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive
Chapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
Reliability of Technical Systems
eliabiliy of Technical Sysems Main Topics Inroducion, Key erms, framing he problem eliabiliy parameers: Failure ae, Failure Probabiliy, Availabiliy, ec. Some imporan reliabiliy disribuions Componen reliabiliy
LAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
EECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits
EEE25 ircui Analysis I Se 4: apaciors, Inducors, and Firs-Order inear ircuis Shahriar Mirabbasi Deparmen of Elecrical and ompuer Engineering Universiy of Briish olumbia shahriar@ece.ubc.ca Overview Passive
Vehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
Oscillations. Periodic Motion. Sinusoidal Motion. PHY oscillations - J. Hedberg
Oscillaions PHY 207 - oscillaions - J. Hedberg - 2017 1. Periodic Moion 2. Sinusoidal Moion 3. How do we ge his kind of moion? 4. Posiion - Velociy - cceleraion 5. spring wih vecors 6. he reference circle
Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.
Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike
Viscoelastic Properties of Polymers a Short Course POLYCHAR 25
Viscoelasic Properies of Polymers a Shor Course POLYCHAR 5 Michael Hess Deparmen of Polymer Engineering & Science, Chosun Universiy, KwangJu, Souh Korea Deparmen of Maerials Science, Universiy of Norh
Keywords: thermal stress; thermal fatigue; inverse analysis; heat conduction; regularization
Proceedings Inverse Analysis for Esimaing Temperaure and Residual Sress Disribuions in a Pipe from Ouer Surface Temperaure Measuremen and Is Regularizaion Shiro Kubo * and Shoki Taguwa Deparmen of Mechanical
Basic notions of probability theory (Part 2)
Basic noions of probabiliy heory (Par 2) Conens o Basic Definiions o Boolean Logic o Definiions of probabiliy o Probabiliy laws o Random variables o Probabiliy Disribuions Random variables Random variables
Chapter 1 Fundamental Concepts
Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables
Finite Element Analysis of Structures
KAIT OE5 Finie Elemen Analysis of rucures Mid-erm Exam, Fall 9 (p) m. As shown in Fig., we model a russ srucure of uniform area (lengh, Area Am ) subjeced o a uniform body force ( f B e x N / m ) using
The field of mathematics has made tremendous impact on the study of
A Populaion Firing Rae Model of Reverberaory Aciviy in Neuronal Neworks Zofia Koscielniak Carnegie Mellon Universiy Menor: Dr. G. Bard Ermenrou Universiy of Pisburgh Inroducion: The field of mahemaics
Lab 10: RC, RL, and RLC Circuits
Lab 10: RC, RL, and RLC Circuis In his experimen, we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors. We will sudy he way volages and currens change in
Heat Transfer. Revision Examples
Hea Transfer Revision Examples Hea ransfer: energy ranspor because of a emperaure difference. Thermal energy is ransferred from one region o anoher. Hea ranspor is he same phenomena lie mass ransfer, momenum
CH.5. BALANCE PRINCIPLES. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
CH.5. BALANCE PRINCIPLES Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Balance Principles Convecive Flux or Flux by Mass Transpor Local and Maerial Derivaive of a olume Inegral Conservaion
1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
CHANGE IN THE RESISTANCE OF THE SEMICONDUCTOR IN THE VARIABLE DEFORMATION FIELD
CHANGE IN THE RESISTANCE OF THE SEMICONDUCTOR IN THE VARIABLE DEFORMATION FIELD M. AHMETOGLU (AFRAILOV) 1, G. GULYAMOV 2, S. H. SHAMIRZAEV 2, A. G. GULYAMOV 2, M. G. DADAMIRZAEV 2, N. APRAILOV 2, F. KOÇAK
Relaxation. T1 Values. Longitudinal Relaxation. dm z dt. = " M z T 1. (1" e "t /T 1 ) M z. (t) = M 0
Relaxaion Bioengineering 28A Principles of Biomedical Imaging Fall Quarer 21 MRI Lecure 2 An exciaion pulse roaes he magneiaion vecor away from is equilibrium sae (purely longiudinal). The resuling vecor
Initial Value Problems
Iniial Value Problems ChEn 2450 d d f(, ) (0) 0 6 ODE.key - November 26, 2014 Example - Cooking a Lobser Assumpions: The lobser remains a a uniform emperaure. This implies ha he hermal conduciviy of he
Second Law. first draft 9/23/04, second Sept Oct 2005 minor changes 2006, used spell check, expanded example
Second Law firs draf 9/3/4, second Sep Oc 5 minor changes 6, used spell check, expanded example Kelvin-Planck: I is impossible o consruc a device ha will operae in a cycle and produce no effec oher han
EECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11 Course Flow Diagram The arrows here show concepual flow beween
ψ ( t) = c n ( t) t n ( )ψ( ) t ku t,t 0 ψ I V kn
MIT Deparmen of Chemisry 5.74, Spring 4: Inroducory Quanum Mechanics II p. 33 Insrucor: Prof. Andrei Tokmakoff PERTURBATION THEORY Given a Hamilonian H ( ) = H + V ( ) where we know he eigenkes for H H
( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
TWO TEMPERATURE ELECTRO-MAGNETO THERMO-VISCO- ELASTIC RESPONSE WITH RHEOLOGICAL PROPERTIES AND TEMPERATURE DEPENDENT ELASTIC MODULI
Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp. 74-86/Mondal and Mukhopadhyay TWO TEMPERATURE ELECTRO-MAGNETO THERMO-VISCO- ELASTIC RESPONSE WITH RHEOLOGICAL
Fractal Aspects in Polymer Science
Fracals 3, 483-39 (1995) [also published in: "Fracal geomery and analysis - The Mandelbro Fesschrif, Curacao 1995", C. J. G. Eversz, H.-G. Peigen, and R. F. Voss (Eds.) (World Scienific, Singapore, 1996)
t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
Two- and Three Dimensional Solid Elements; Plane Stress, Plane Strain, and Axisymmetric Conditions
Topic 7 Two- and Three Dimensional Solid Elemens; Plane Sress, Plane Srain, and Axisymmeric Condiions Conens: soparameric inerpolaions of coordinaes and displacemens Consisency beween coordinae and displacemen
Solutions Problem Set 3 Macro II (14.452)
Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen.
Advanced Organic Chemistry
Lalic, G. Chem 53A Chemisry 53A Advanced Organic Chemisry Lecure noes 1 Kineics: A racical Approach Simple Kineics Scenarios Fiing Experimenal Daa Using Kineics o Deermine he Mechanism Doughery, D. A.,
Mathcad Lecture #8 In-class Worksheet Curve Fitting and Interpolation
Mahcad Lecure #8 In-class Workshee Curve Fiing and Inerpolaion A he end of his lecure, you will be able o: explain he difference beween curve fiing and inerpolaion decide wheher curve fiing or inerpolaion
Numerical investigation of Ranque-Hilsch energy separation effect A.S. Noskov 1,a, V.N. Alekhin 1,b, A.V. Khait 1,a
Applied Mechanics and Maerials Online: 2013-01-11 ISSN: 1662-7482, Vol. 281, pp 355-358 doi:10.4028/www.scienific.ne/amm.281.355 2013 Trans Tech Publicaions, Swizerland Numerical invesigaion of Ranque-Hilsch
5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
Continuous Time. Time-Domain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
Time-Domain Sysem Analysis Coninuous Time. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
From Particles to Rigid Bodies
Rigid Body Dynamics From Paricles o Rigid Bodies Paricles No roaions Linear velociy v only Rigid bodies Body roaions Linear velociy v Angular velociy ω Rigid Bodies Rigid bodies have boh a posiion and
CH.7. PLANE LINEAR ELASTICITY. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
CH.7. PLANE LINEAR ELASTICITY Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Plane Linear Elasici Theor Plane Sress Simplifing Hpohesis Srain Field Consiuive Equaion Displacemen Field The Linear
236 CHAPTER 3 Torsion. Strain Energy in Torsion
36 CHAPER 3 orsion Srain Energy in orsion Problem 3.9-1 A solid circular bar of seel (G 11. 1 6 psi) wih lengh 3 in. and diameer d 1.75 in. is subjeced o pure orsion by orques acing a he ends (see figure).
CHEMICAL KINETICS Rate Order Rate law Rate constant Half-life Molecularity Elementary. Complex Temperature dependence, Steady-state Approximation
CHEMICL KINETICS Rae Order Rae law Rae consan Half-life Moleculariy Elemenary Complex Temperaure dependence, Seady-sae pproximaion Chemical Reacions Kineics Chemical ineics is he sudy of ime dependence
Chapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
IB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
Sub Module 2.6. Measurement of transient temperature
Mechanical Measuremens Prof. S.P.Venkaeshan Sub Module 2.6 Measuremen of ransien emperaure Many processes of engineering relevance involve variaions wih respec o ime. The sysem properies like emperaure,
Learning Enhancement Team
Learning Enhancemen Team Model answers: Exponenial Funcions Exponenial Funcions sudy guide 1 i) The base rae of growh b is equal o 3 You can see his by noicing ha 1b 36 in his sysem, dividing boh sides
Empirical Process Theory
Empirical Process heory 4.384 ime Series Analysis, Fall 27 Reciaion by Paul Schrimpf Supplemenary o lecures given by Anna Mikusheva Ocober 7, 28 Reciaion 7 Empirical Process heory Le x be a real-valued
6.003: Signals and Systems. Relations among Fourier Representations
6.003: Signals and Sysems Relaions among Fourier Represenaions April 22, 200 Mid-erm Examinaion #3 W ednesday, April 28, 7:30-9:30pm. No reciaions on he day of he exam. Coverage: Lecures 20 Reciaions 20
Cosmic String Loop Distribution with a Gravitational Wave Cutoff
Ouline Inroducion Towards he Cosmological Aracor Cosmic Sring Loop Disribuion wih a Graviaional Wave Cuoff Larissa Lorenz Insiue of Mahemaics and Physics Cenre for Cosmology, Paricle Physics and Phenomenology
Activity 4 Solutions: Transfer of Thermal Energy
Aciviy 4 Soluions: Transfer of Thermal nergy 4.1 How Does Temperaure Differ from Thermal nergy? a) Temperaure Your insrucor will demonsrae molecular moion a differen emperaures. 1) Wha happens o molecular
Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.
Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since
arxiv:cond-mat/ May 2002
-- uadrupolar Glass Sae in para-hydrogen and orho-deuerium under pressure. T.I.Schelkacheva. arxiv:cond-ma/5538 6 May Insiue for High Pressure Physics, Russian Academy of Sciences, Troisk 49, Moscow Region,
AP Chemistry--Chapter 12: Chemical Kinetics
AP Chemisry--Chaper 12: Chemical Kineics I. Reacion Raes A. The area of chemisry ha deals wih reacion raes, or how fas a reacion occurs, is called chemical kineics. B. The rae of reacion depends on he