5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.

Save this PDF as:

Size: px
Start display at page:

Download "5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance."

Transcription

1 5.8 Resonance Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) () + ω 2 () = F cos(ω) is he eisence of a soluion ha is unbounded as. We already know (page 224) ha for ω ω, he general soluion of (1) is he sum of wo harmonic oscillaions, hence i is bounded. Equaion (1) for ω = ω has by he mehod of undeermined coefficiens he unbounded oscillaory soluion () = F 2ω sin(ω ). To summarize: Pure resonance occurs eacly when he naural inernal frequency ω maches he naural eernal frequency ω, in which case all soluions of he differenial equaion are unbounded. In Figure 2, his is illusraed for () + 16() = 8cos 4, which in (1) corresponds o ω = ω = 4 and F = 8. Figure 2. Pure resonance for () + 16() = 8 cosω. Graphed are he soluion () = sin 4 for ω = 4 and he envelope curves = ±. Resonance and undeermined coefficiens. An eplanaion of resonance can be based upon he heory of undeermined coefficiens. An iniial rial soluion of () + 16() = 8cos ω is = d 1 cos ω + d 2 sin ω. The homogeneous soluion h = c 1 cos 4 + c 2 sin4 considered in he fiup rule has duplicae erms eacly when he naural frequencies mach: ω = 4. Then he final rial soluion is (2) () = { d1 cos ω + d 2 sin ω ω 4, (d 1 cos ω + d 2 sin ω) ω = 4. Even before he undeermined coefficiens d 1, d 2 are evaluaed, we can decide ha unbounded soluions occur eacly when frequency maching

2 232 ω = 4 occurs, because of he ampliude facor. If ω 4, hen p () is a pure harmonic oscillaion, hence bounded. If ω = 4, hen p () equals a ime varying ampliude C imes a pure harmonic oscillaion, hence i is unbounded. The Wine Glass Eperimen. Equaion (1) is adverised as he basis for a physics eperimen which has appeared ofen on Public Television, called he wine glass eperimen. A famous physicis, in fron of an audience of physics sudens, equips a lab able wih a frequency generaor, an amplifier and an audio speaker. The valuable wine glass is replaced by a glass beaker. The frequency generaor is uned o he naural frequency of he glass beaker (ω ω ), hen he volume knob on he amplifier is suddenly urned up (F adjused larger), whereupon he sound waves emied from he speaker break he glass beaker. The glass iself will vibrae a a cerain frequency, as can be deermined eperimenally by pinging he glass rim. This vibraion operaes wihin elasic limis of he glass and he glass will no break under hese circumsances. A physical eplanaion for he breakage is ha an incoming sound wave from he speaker is imed o add o he glass rim ecursion. Afer enough ampliude addiions, he glass rim moves beyond he elasic limi and he glass breaks. The eplanaion implies ha he eernal frequency from he speaker has o mach he naural frequency of he glass. Bu here is more o i: he glass has some naural damping ha nullifies feeble aemps o increase he glass rim ampliude. The physicis uses o grea advanage his naural damping o une he eernal frequency o he glass. The reason for urning up he volume on he amplifier is o nullify he damping effecs of he glass. The ampliude addiions hen build rapidly and he glass breaks. Soldiers Breaking Cadence. The collapse of he Broughon bridge near Mancheser, England in 1831 is blamed for he now sandard pracise of breaking cadence when soldiers cross a bridge. Bridges like he Broughon bridge have many naural low frequencies of vibraion, so i is possible for a column of soldiers o vibrae he bridge a one of he bridge s naural frequencies. The bridge locks ono he frequency while he soldiers coninue o add o he ecursions wih every sep, causing larger and larger bridge oscillaions. Pracical Resonance The noion of pure resonance is easy o undersand boh mahemaically and physically, because frequency maching characerizes he even. This ideal siuaion never happens in he physical world, because damping is

3 5.8 Resonance 233 always presen. In he presence of damping c >, i will be esablished below ha only bounded soluions eis for he forced spring-mass sysem (3) m () + c () + k() = F cos ω. Our inuiion abou resonance seems o vaporize in he presence of damping effecs. Bu no compleely. Mos would agree ha he undamped inuiion is correc when he damping effecs are nearly zero. Pracical resonance is said o occur when he eernal frequency ω has been uned o produce he larges possible soluion (a more precise definiion appears below). I will be shown ha his happens for he condiion (4) ω = k/m c 2 /(2m 2 ), k/m c 2 /(2m 2 ) >. Pure resonance ω = ω k/m is he limiing case obained by seing he damping consan c o zero in condiion (4). This srange bu predicable ineracion eiss beween he damping consan c and he size of soluions, relaive o he eernal frequency ω, even hough all soluions remain bounded. The decomposiion of () ino homogeneous soluion h () and paricular soluion p () gives some inuiion ino he comple relaionship beween he inpu frequency ω and he size of he soluion (). The homogeneous soluion. For posiive damping, c >, equaion (3) has homogeneous soluion h () = c 1 1 () + c 2 2 () where according o he recipe he basis elemens 1 and 2 are given in erms of he roos of he characerisic equaion mr 2 + cr + k =, as classified by he discriminan D = c 2 4mk, as follows: Case 1, D > Case 2, D = 1 = e r 1, 2 = e r 2 wih r 1 and r 2 negaive. 1 = e r 1, 2 = e r 1 wih r 1 negaive. Case 3, D < 1 = e α cos β, 2 = e α sin β wih β > and α negaive. I follows ha h () conains a negaive eponenial facor, regardless of he posiive values of m, c, k. A soluion () is called a ransien soluion provided i saisfies he relaion lim () =. The conclusion: The homogeneous soluion h () of he equaion m () + c () + k() = is a ransien soluion for all posiive values of m, c, k. A ransien soluion graph () for large lies aop he ais =, as in Figure 21, because lim () =.

4 π Figure 21. Transien oscillaory soluion = 2e (cos + sin ) of he differenial equaion =. The paricular soluion. The mehod of undeermined coefficiens gives a rial soluion of he form () = Acos ω + B sinω wih coefficiens A, B saisfying he equaions (5) (k mω 2 )A + (cω)b = F, ( cω)a + (k mω 2 )B =. Solving (5) wih Cramer s rule or eliminaion produces he soluion (6) A = (k mω 2 )F (k mω 2 ) 2 + (cω) 2, B = cωf (k mω 2 ) 2 + (cω) 2. The seady sae soluion, periodic of period 2π/ω, is given by (7) p () = = F ( ) (k mω 2 ) 2 + (cω) 2 (k mω 2 )cos ω + (cω)sin ω F cos(ω α), (k mω 2 ) 2 + (cω) 2 where α is defined by he phase ampliude relaions (see page 216) (8) C cos α = k mω 2, C sin α = cω, C = F / (k mω 2 ) 2 + (cω) 2. The erminology seady sae refers o ha par ss() of he soluion () ha remains when he ransien porion is removed, ha is, when all erms conaining negaive eponenials are removed. As a resul, for large T, he graphs of () and ss() on T are he same. This feaure of ss() allows us o find is graph direcly from he graph of (). We say ha ss() is observable, because i is he soluion visible in he graph afer he ransiens (negaive eponenial erms) die ou. Readers may be mislead by he mehod of undeermined coefficiens, in which i urns ou ha p () and ss() are he same. Alernaively, a paricular soluion p () can be calculaed by variaion of parameers, a mehod which produces in p () era erms conaining negaive eponenials. These era erms come from he homogeneous soluion heir appearance canno always be avoided. This jusifies he careful definiion of seady sae soluion, in which he ransien erms are removed from p () o produce ss(). Pracical resonance is said o occur when he eernal frequency ω has been uned o produce he larges possible seady sae ampliude.

5 5.8 Resonance 235 Mahemaically, his happens eacly when he ampliude funcion C = C(ω) defined in (8) has a maimum. If a maimum eiss on < ω <, hen C (ω) = a he maimum. The derivaive is compued by he power rule: (9) C (ω) = F 2(k mω 2 )( 2mω) + 2c 2 ω 2 ((k mω 2 ) 2 + (cω) 2 ) 3/2 = ω (2mk c 2 2m 2 ω 2) C(ω) 3 F 2 If 2km c 2, hen C (ω) does no vanish for < ω < and hence here is no maimum. If 2km c 2 >, hen 2km c 2 2m 2 ω 2 = has eacly one roo ω = k/m c 2 /(2m 2 ) in < ω < and by C( ) = i follows ha C(ω) is a maimum. In summary: Pracical resonance for m ()+c ()+k() = F cosω occurs precisely when he eernal frequency ω is uned o ω = k/m c 2 /(2m 2 ) and k/m c 2 /(2m 2 ) >. In Figure 22, he ampliude of he seady sae periodic soluion is graphed agains he eernal naural frequency ω, for he differenial equaion + c + 26 = 1cos ω and damping consans c = 1,2,3. The pracical resonance condiion is ω = 26 c 2 /2. As c increases from 1 o 3, he maimum poin (ω,c(ω)) saisfies a monooniciy condiion: boh ω and C(ω) decrease as c increases. The maima for he hree curves in he figure occur a ω = 25.5, 24, Pure resonance occurs when c = and ω = 26. C c = 1 c = 2 c = 3 ω Figure 22. Pracical resonance for + c + 26 = 1 cosω: ampliude C = 1/ (26 ω 2 ) 2 + (cω) 2 versus eernal frequency ω for c = 1, 2, 3. Uniqueness of he Seady Sae Periodic Soluion. Any wo soluions of he nonhomogeneous differenial equaion (3) which are periodic of period 2π/ω mus be idenical. The vehicle of proof is o show ha heir difference () is zero. The difference () is a soluion of he homogeneous equaion, i is 2π/ω periodic and i has limi zero a infiniy. A periodic funcion wih limi zero mus be zero, herefore he wo soluions are idenical. A more general saemen is rue: Consider he equaion m () + c () + k() = f() wih f( + T) = f() and m, c, k posiive. Then a T periodic soluion is unique.

6 236 In Figure 23, he unique seady sae periodic soluion is graphed for he differenial equaion = sin +2cos. The ransien soluion of he homogeneous equaion and he seady sae soluion appear in Figure 24. In Figure 25, several soluions are shown for he differenial equaion = sin + 2cos, all of which reproduce evenually he seady sae soluion = sin π 6π π Figure 23. Seady-sae periodic soluion () = sin of he differenial equaion = sin + 2 cos. Figure 24. Transien soluion of = and he seady-sae soluion of = sin + 2 cos. Figure 25. Soluions of = sin + 2 cos wih () = 1 and () = 1, 2, 3, all of which graphically coincide wih he seady-sae soluion = sin for π. Pseudo Periodic Soluion. Resonance gives rise o soluions of he form () = A()sin(ω α) where A() is a ime varying ampliude. Figure 26 shows such a soluion, which is called a pseudo periodic soluion because i has a naural period 2π/ω arising from he rigonomeric facor sin(ω α). The only requiremen on A() is ha i be non vanishing, so ha i acs like an ampliude. The pseudo period of a pseudo periodic soluion can be deermined graphically, by compuing he lengh of ime i akes for () o vanish hree imes. 4 e Eercises Figure 26. The pseudo-periodic soluion = e /4 sin(3) of = 96e /4 cos3 and is envelope curves = ±e /4. Resonance and Beas. Classify for resonance or beas. In he case of resonance, find an unbounded soluion. In he case of beas, find he soluion for () =, () =, hen graph i hrough a full period of he slowlyvarying envelope = 1 sin = 5 sin2

7 5.8 Resonance = 1 sin = 5 sin = 5 sin = 5 cos = 5 sin = 1 sin4 Resonan Frequency and Ampliude. Consider m + c + k = F cos(ω). Compue he seady-sae oscillaion A cos(ω) + B sin(ω), is ampliude C = A 2 + B 2, he uned pracical resonance frequency ω, he resonan ampliude C and he raio 1C/C. 9. m = 1, β = 2, k = 17, F = 1, ω = m = 1, β = 2, k = 1, F = 1, ω = m = 1, β = 4, k = 5, F = 1, ω = m = 1, β = 2, k = 6, F = 1, ω = m = 1, β = 4, k = 5, F = 5, ω = m = 1, β = 2, k = 5, F = 5, ω = 3/2.

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,

Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..

Chapter 7. Response of First-Order RL and RC Circuits

Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

Newton's second law in action

Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

Fourier series. Learning outcomes

Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

11. Properties of alternating currents of LCR-electric circuits

WS. Properies of alernaing currens of L-elecric circuis. Inroducion So-called passive elecric componens, such as ohmic resisors (), capaciors () and inducors (L), are widely used in various areas of science

The Transport Equation

The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

Math 201 Lecture 12: Cauchy-Euler Equations

Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

Fourier Series Solution of the Heat Equation

Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

2. Waves in Elastic Media, Mechanical Waves

2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in

A Mathematical Description of MOSFET Behavior

10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

LAB 6: SIMPLE HARMONIC MOTION

1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion:

Circuit Types. () i( t) ( )

Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Whole-range Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in

Graphing the Von Bertalanffy Growth Equation

file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

Differential Equations and Linear Superposition

Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

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.

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

Understanding Sequential Circuit Timing

ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

Basic Assumption: population dynamics of a group controlled by two functions of time

opulaion Models Basic Assumpion: populaion dynamics of a group conrolled by wo funcions of ime Birh Rae β(, ) = average number of birhs per group member, per uni ime Deah Rae δ(, ) = average number of

Chapter 2: Principles of steady-state converter analysis

Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

Chapter 7: Estimating the Variance of an Estimate s Probability Distribution

Chaper 7: Esimaing he Variance of an Esimae s Probabiliy Disribuion Chaper 7 Ouline Review o Clin s Assignmen o General Properies of he Ordinary Leas Squares (OLS) Esimaion Procedure o Imporance of he

RC (Resistor-Capacitor) Circuits. AP Physics C

(Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

Inductance and Transient Circuits

Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

CHARGE AND DISCHARGE OF A CAPACITOR

REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

Module 3. R-L & R-C Transients. Version 2 EE IIT, Kharagpur

Module 3 - & -C Transiens esson 0 Sudy of DC ransiens in - and -C circuis Objecives Definiion of inducance and coninuiy condiion for inducors. To undersand he rise or fall of curren in a simple series

Chapter 15: Superposition and Interference of Waves

Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when

Relative velocity in one dimension

Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

Renewal processes and Poisson process

CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

Acceleration Lab Teacher s Guide

Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

MTH6121 Introduction to Mathematical Finance Lesson 5

26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

RC Circuit and Time Constant

ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

AP Calculus AB 2007 Scoring Guidelines

AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

Optimal Investment and Consumption Decision of Family with Life Insurance

Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67 - FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1 - TRANSIENTS

EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67 - FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1 - RANIEN Uni conen 2 Undersand he ransien behaviour of resisor-capacior (R) and resisor-inducor (RL) D circuis

Transient Analysis of First Order RC and RL circuits

Transien Analysis of Firs Order and iruis The irui shown on Figure 1 wih he swih open is haraerized by a pariular operaing ondiion. Sine he swih is open, no urren flows in he irui (i=0) and v=0. The volage

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

1. The graph shows the variation with time t of the velocity v of an object.

1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

Vector Autoregressions (VARs): Operational Perspectives

Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

Rotational Inertia of a Point Mass

Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur

Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

Damped Harmonic Motion Closing Doors and Bumpy Rides

Prerequisies and Goal Damped Harmonic Moion Closing Doors and Bumpy Rides Andrew Forreser May 4, 21 Assuming you are familiar wih simple harmonic moion, is equaion of moion, and is soluions, we will now

Lecture 18. Serial correlation: testing and estimation. Testing for serial correlation

Lecure 8. Serial correlaion: esing and esimaion Tesing for serial correlaion In lecure 6 we used graphical mehods o look for serial/auocorrelaion in he random error erm u. Because we canno observe he u

Cointegration: The Engle and Granger approach

Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

Part II Converter Dynamics and Control

Par II onverer Dynamics and onrol 7. A equivalen circui modeling 8. onverer ransfer funcions 9. onroller design 1. Inpu filer design 11. A and D equivalen circui modeling of he disconinuous conducion mode

Solution of a differential equation of the second order by the method of NIGAM

Tire : Résoluion d'une équaion différenielle du second[...] Dae : 16/02/2011 Page : 1/6 Soluion of a differenial equaion of he second order by he mehod of NIGAM Summarized: We presen in his documen, a

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

and Decay Functions f (t) = C(1± r) t / K, for t 0, where

MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

Part 1: White Noise and Moving Average Models

Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

Capacitors and inductors

Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear

The Torsion of Thin, Open Sections

EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

Laboratory #3 Diode Basics and Applications (I)

Laboraory #3 iode asics and pplicaions (I) I. Objecives 1. Undersand he basic properies of diodes. 2. Undersand he basic properies and operaional principles of some dioderecifier circuis. II. omponens

Communication Networks II Contents

3 / 1 -- Communicaion Neworks II (Görg) -- www.comnes.uni-bremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP

Intro to Fourier Series

Inro o Fourier Series Vecor decomposiion Even and Odd funcions Fourier Series definiion and examples Copyrigh 27 by M.H. Perro All righs reserved. M.H. Perro 27 Inro o Fourier Series, Slide 1 Review of

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

2 Electric Circuits Concepts Durham

Chaper 3 - Mehods Chaper 3 - Mehods... 3. nroducion... 2 3.2 Elecrical laws... 2 3.2. Definiions... 2 3.2.2 Kirchhoff... 2 3.2.3 Faraday... 3 3.2.4 Conservaion... 3 3.2.5 Power... 3 3.2.6 Complee... 4

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor

A Re-examination of the Joint Mortality Functions

Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)

THE PRESSURE DERIVATIVE

Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.

Basic Circuit Elements - Prof J R Lucas

Basic Circui Elemens - Prof J ucas An elecrical circui is an inerconnecion of elecrical circui elemens. These circui elemens can be caegorized ino wo ypes, namely acive elemens and passive elemens. Some

Impact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences

S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough

Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )

Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y

a. Defining set of equations. Create separate m-file nanme.m containing the set of first order equations.

Signals and Sysems. Eperimen. Hins Problems: a) Solving sae equaions using sandard Malab 5. procedures b) Solving ordinary differenial n-h order equaions wihou Symbolic Mah Toolbo (Malab 5.) a.) Solving

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm

Fourier Series Approximation of a Square Wave

OpenSax-CNX module: m4 Fourier Series Approximaion of a Square Wave Don Johnson his work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License. Absrac Shows how o use Fourier

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

Chapter 4: Exponential and Logarithmic Functions

Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion

Lecture III: Finish Discounted Value Formulation

Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal

Time variant processes in failure probability calculations

Time varian processes in failure probabiliy calculaions A. Vrouwenvelder (TU-Delf/TNO, The Neherlands) 1. Inroducion Acions on srucures as well as srucural properies are usually no consan, bu will vary

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.

4. The Poisson Distribution

Virual Laboraories > 13. The Poisson Process > 1 2 3 4 5 6 7 4. The Poisson Disribuion The Probabiliy Densiy Funcion We have shown ha he k h arrival ime in he Poisson process has he gamma probabiliy densiy

Lectures # 5 and 6: The Prime Number Theorem.

Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζ-funcion o skech an argumen which would give an acual formula for π( and sugges