TECHNISCHE UNIVERSITEIT EINDHOVEN Faculteit Biomedische Technologie, groep Cardiovasculaire Biomechanica
|
|
- Laureen Green
- 6 years ago
- Views:
Transcription
1 TECHNISCHE UNIVERSITEIT EINDHOVEN Faculteit Biomedische Technologie, groep Cardiovasculaire Biomechanica Tentamen Cardiovasculaire Stromingsleer (8W090) blad /4 dinsdag 8 mei 2007, 9-2 uur Maximum score is 30 points. The partial scores are indicated in the margin of each question.. Answer the following questions with yes or no and give a short motivation. a. Is it correct that in the determination of the parameters of a 3-component windkessel model of the systemic circulation (resistance, inertance and compliance), the resistance can be obtained through division of the mean blood pressure by the cardiac output? b. Is it correct that the longitudinal impedance for flow in a straight ridid tube for large values of the Womersley parameter α becomes imaginary and linearly proportional to α? c. Is the inlet length to obtain fully developed tube flow for a stationary flow generally much smaller than the inlet length for a pulsating flow, in the case the latter has the same time-averaged flow rate? d. Is, in the analysis of entrance flow in a curved tube, it allowed to apply Bernoulli s law because the flow is friction dominated? e. Is it correct that in the case of a stationary fully developed flow in a curved tube the wall shear stress in the outer bend will increase with decreasing Dean-number? f. Does the non-linear stress-strain relation of collagen fibers in the arterial wall lead to a decrease of the compliance if pressure increases? g. Is it true that the imaginary part of the wave number k is a measure for the attenuation of the wave? h. In the figure on the right a part of the velocity profile of a layered flow of blood (high viscosity) and plasma (low viscosity) is given. r 2 Is layer the plasma-layer? z i. Two elastic tubes with admittance Y and Y 2 respectively are connected (end to end). Is it correct that the coefficient of transmission for pressure waves is invariant for exchange of the two tubes? j. The viscosity model of Carreau-Yasuda is given by: η η =[+(λ γ) a ] (n )/a. η 0 η Is it true that for lower values of the parameter a the viscosity decreases faster with increasing shear rate γ than for higher values of a?
2 Tentamen Cardiovasculaire Stromingsleer (8W090) blad 2/4 dinsdag 8 mei 2007, 9-2 uur 2. Endothelial cels that line the luminal side of the arterial wall appear to orient in the direction of the shear stress they experience from the blood flow. An experimental setup as depicted in the figure below in which a monolayer of endothelial cells on a rigid substrate is sheared with an oscillating shear stress τ w, is used to mimic the physiological situation. r A z a 0 z L 0 monolayer L Theset-upconsistsofarigidtubewithinternalradiusa 0 and length 2L. Atlocation z = L a piston is moving with an oscillating velocity given by v z = v z0 cos(ωt) and generates a flow q(t). At location z = 0, one can measure an harmonic pressure p m = p m0 cos(ωt + φ) independent from the radial position in the tube. For z>0 the flow is assumed to be fully developed. At z = L the tube is connected to a large reservoir with a constant pressure p z=l = 0. For fully developed flow in a tube, the Navier Stokes equations simplify to: v = ρ z + ν v (r r r r ) 0 = ρ r a. Give an expression for the pressure p(z, t) andtheflowq(t) intermsofv z0 and p m0 for z>0. b. Show that in complex notation (e ix =cosx + i sin x) we find: z = ˆp z eiωt with q =ˆqe iωt with ˆq = π v z0 ˆp z = p m0 L eiφ c. Scale the Navier-Stokes equation given above with v = v/v z0, r = r/a 0, z = z/l, t = ωt and p = p/p 0 and show that for p 0 = ρlνv z0 / the Womersley parameter α = a 0 ω/ν is the only dimensionless group. d. Show that for a frictionless (inviscid) flow the velocity profile can be assumed to be a flat profile given by: v(r, t) = p m0 ρωl cos(ωt) 2
3 Show also that in that case φ = π/2 andp m0 = ρωlv z0. What will be the wall shear stress for this situation. Is this rational for a frictionless flow? e. Show that for a friction dominated flow the velocity profile will be parabolic given by: v(r, t) = 4 ρν ( ) r2 z and show also that in that case we have φ =0andp m0 =8ρνLv z0 /. What, based on this profile, will be the wall shear stress? f. For arbitrary values of α (take α =5),forz>0aninstationary boundary layer with thickness δ a 0 α will exist. Show this and make a sketch of the corresponding velocity profiles at a few instants of time during the flow cycle. g. Show, making use of force equibrium on a slice with thickness dz that one can find: τ w = a ( ) 0 pm0 2 L cos(ωt + φ)+ρωv z0 sin ωt so that for given motion of the piston τ w can be determined by a pressure measurment at z =0. 3
4 Tentamen Cardiovasculaire Stromingsleer (8W090) blad 3/4 dinsdag 8 mei 2007, 9-2 uur 3. Endotheleal cells also orient perpendicular to the direction in which they are stretched. To investigate this, a thin walled straight elastic tube with wall thickness h, radiusa 0 and length 2L is locally covered with a monolayer of endothelial cells. The Young s modulus of the tube is E and the Poisson ratio is μ. The elastic tube is connected to a rigid tube with a piston with cross-sectional area A 0 and an oscillatory motion given by the velocity v z = v z0 cos(ωt). A z a 0 r z h L L 0 monolayer At the location z = 0 one can measure a pressure according to p m = p m0 cos(ωt + φ). The end of the tube is terminated with a characteristic impedance so at that point we have q z=l =0. a. At the location of the pressure sensor z = 0 we find a wall motion in radial direction give by: u r = u r0 cos(ωt+ φ). Show, starting from force equilibrium, that in case we neglect inertia forces in the tube wall, we have: p = σ φφh and u r = ( μ2 ) p a 0 he Make use of the fact that for thin walled tubes the circumferential stress is given by σ φφ =(E/( μ 2 ))(u r /a 0 ). b. The compliance per unit of length of the tube is defined as C =(da/dp) witha the instantaneous cross-sectioal area. Show that: C = 2πa3 0 μ 2 h E c. Due to the motion of the piston pressure and flow waves will travel along the tube. Show, starting from conservation of mass for a slice with thicknes dz and cross-sectional area A that we have: C + q z =0 d. Show, starting from equilibrium of forces in axial direction that we have: ρ q + A 0 z =0 if friction can be neglected. e. Give for the experimental situation we consider here the wave equation for the pressure and an expression for the wave speed and the wave length. 4
5 TECHNISCHE UNIVERSITEIT EINDHOVEN Faculteit Biomedische Technologie, groep Cardiovasculaire Biomechanica Antwoorden by het tentamen Cardiovasculaire Stromingsleer (8W090) dinsdag 8 mei 2007, 9-2 uur. Answers: a. Yes. The mean pressure and flow (zeroth harmonic) are related according to p 0 = q 0 R.() b. No, the longitudinal impedance increases with α 2 since α 2 v ( pnt) = z for large α. c. No, when the time-averaged flow is not equal to zero the entrance length of pulsating flow will be determined by the steady component. Hence, it will be equal to the entance length in steady flow. ( pnt) d. No, this is because the boundary layer is not developed and the flow may be considered frictionless. ( pnt) e. No. The boundary layer thicikness of the scondary flow will decrease δ s /a = O(Dn /2 ). Hence the shear rate will increase. ( pnt) f. Yes. C = A decreases with increasing pressure. ( pnt) g. Yes. p =ˆpe i(ωt kz) =ˆpe iωt e ikrz e kiz. Attenuation e ikiz.() h. The stress η u is continuous. Layer shows the largest shear rate so the lowest r viscosity. Yes. ( pnt) i. No. Continuity of pressure and flow at the interfavce: p i + p r = p t q i + q r = q t Y p i + Y p r = Y p t 2Y p i =(Y + Y 2 )p t Y p i Y p r = Y 2 p t so T 2 = 2Y T 2.() Y + Y 2 j. No, the rate of decrease is related to (λ γ) n. The parameter a determines at what value of the shear rate the decrease of the viscosity is maximal. ( pnt) 2. Answers: a. Rigid tube so q = q 0 cos ωt anywhere. With q 0 = A z v z0. For z > 0wehave fully developd flow so =constant cos(ωt + φ) = (p z m0/l)cos(ωt + φ). So L z p = p m0 cos(ωt + φ). ( pnt) L b. = p m0 [cos(ωt + φ)+isin(ωt + φ)] = p m0 z L L ei(ωt+φ) = p m0 L eiφ e iωt = ˆp z eiωt q = q 0 [cos ωt + i sin ωt] =ˆqe iωt = π v z0. () v c. ωv z0 = p 0 ρl r r (r v + νv z z0 r r ). Dus p 0 = ρlνv z0 (r v ). Division by νv r r z0 / gives a2 0 ω v = p 0a 2 ν 0 ρlνv z0.() z + 5
6 d. α large so v =. substitution of v ρ z =ˆveiωt and p =ˆpe iωt gives ˆv = i ˆp = ρω z i ( p m0 ρω L eiφ )sov = p m0 sin(ωt + φ) =v ρωl z0 cos ωt. Soφ = π/2 andp m0 = ρωlv z0. ( pnt) z + ν r r e. α small so 0 = v (r ). Integration (2 times) + v(a) =0givesv(r, t) = ρ r ( 4 ρν r2 /a 2 ). Integration over cross-sectional area gives q = πa4 0 p m0 cos ωt. z 8ρν Also q = v z0 π cos ωt so p m0 = 8ρνLv z0.() f. The boundary layer thickness is determined by the transition where viscous forces are of the smae order of magnitude as the intertial forces. So O( νv z0 δ 2 )=O(ωv z0 ). Hence δ = O(a/α). (2 pnt) g. For the segment we have ΣF = ma so: [p(z) p(z + dz)]a 2πa 0 dzτ w = ρadz v or 2 z a 0 τ w = ρ v (2 pnt) 3. Answers: a. Equilibrium of forces yields 2pa 0 =2σ φφ h. Substitution of σ φφ gives the answer. (2 pnt) b. Gebruik da = π(a 0 + u r ) 2 π 2πa 0u r en antwoord b. (2 pnt) c. Mass conservation: [q(z + dz) q(z)]dt +[A(t + dt) A(t)]dz =0so A A + q = C + q z z (2 pnt) d. Equilibrium of forces: A 0 [p(z + dz) p(z)] = ρa 0 dz v e. Cross differentiation gives C 2 p + 2 q =0and 2 q + A 2 0 z z with c 0 = A 0 /(ρc). (2 pnt) 2 p ρ z 2 + q = z q so ρ + A 0 = 0 (2 pnt) z =0so: 2 p c p =0 z 2 6
Exam Cardiovascular Fluid Mechanics 8W090 sheet 1/4 on Thursday 5th of june 2005, 9-12 o clock
EINDHOVEN UNIVERSITY OF TECHNOLOGY DEPARTMENT OF PHYSICAL TECHNOLOGY, Fluid Mechanics group DEPARTMENT OF BIOMEDICAL ENGINEERING, Cardiovascular Biomechanics group Exam Cardiovascular Fluid Mechanics 8W090
More informationTECHNISCHE UNIVERSITEIT EINDHOVEN Department of Biomedical Engineering, section Cardiovascular Biomechanics
TECHNISCHE UNIVERSITEIT EINDHOVEN Department of Biomedical Engineering, section Cardiovascular Biomechanics Exam Cardiovascular Fluid Mechanics (8W9) page 1/4 Monday March 1, 8, 14-17 hour Maximum score
More information31545 Medical Imaging systems
31545 Medical Imaging systems Lecture 5: Blood flow in the human body Jørgen Arendt Jensen Department of Electrical Engineering (DTU Elektro) Biomedical Engineering Group Technical University of Denmark
More informationBME 419/519 Hernandez 2002
Vascular Biology 2 - Hemodynamics A. Flow relationships : some basic definitions Q v = A v = velocity, Q = flow rate A = cross sectional area Ohm s Law for fluids: Flow is driven by a pressure gradient
More informationArterial Macrocirculatory Hemodynamics
Arterial Macrocirculatory Hemodynamics 莊漢聲助理教授 Prof. Han Sheng Chuang 9/20/2012 1 Arterial Macrocirculatory Hemodynamics Terminology: Hemodynamics, meaning literally "blood movement" is the study of blood
More informationLongitudinal Waves. waves in which the particle or oscillator motion is in the same direction as the wave propagation
Longitudinal Waves waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal waves propagate as sound waves in all phases of matter, plasmas, gases,
More informationRutgers University Department of Physics & Astronomy. 01:750:271 Honors Physics I Fall Lecture 19. Home Page. Title Page. Page 1 of 36.
Rutgers University Department of Physics & Astronomy 01:750:271 Honors Physics I Fall 2015 Lecture 19 Page 1 of 36 12. Equilibrium and Elasticity How do objects behave under applied external forces? Under
More informationUNIVERSITY OF EAST ANGLIA
UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2007 2008 FLUIDS DYNAMICS WITH ADVANCED TOPICS Time allowed: 3 hours Attempt question ONE and FOUR other questions. Candidates must
More informationSummary PHY101 ( 2 ) T / Hanadi Al Harbi
الكمية Physical Quantity القانون Low التعريف Definition الوحدة SI Unit Linear Momentum P = mθ be equal to the mass of an object times its velocity. Kg. m/s vector quantity Stress F \ A the external force
More informationSound Pressure Generated by a Bubble
Sound Pressure Generated by a Bubble Adrian Secord Dept. of Computer Science University of British Columbia ajsecord@cs.ubc.ca October 22, 2001 This report summarises the analytical expression for the
More informationPREDICTION OF PULSATILE 3D FLOW IN ELASTIC TUBES USING STAR CCM+ CODE
11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver
More informationCardiovascular Fluid Mechanics - lecture notes 8W090 -
Cardiovascular Fluid Mechanics - lecture notes 8W9 - F.N. van de Vosse (23) Eindhoven University of Technology department of Biomedical Engineering Preface i Preface As cardiovascular disease is a major
More informationAnswers to questions in each section should be tied together and handed in separately.
EGT0 ENGINEERING TRIPOS PART IA Wednesday 4 June 014 9 to 1 Paper 1 MECHANICAL ENGINEERING Answer all questions. The approximate number of marks allocated to each part of a question is indicated in the
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationMechanical Design in Optical Engineering
Torsion Torsion: Torsion refers to the twisting of a structural member that is loaded by couples (torque) that produce rotation about the member s longitudinal axis. In other words, the member is loaded
More informationExam in Fluid Mechanics 5C1214
Eam in Fluid Mechanics 5C1214 Final eam in course 5C1214 13/01 2004 09-13 in Q24 Eaminer: Prof. Dan Henningson The point value of each question is given in parenthesis and you need more than 20 points
More informationPolymer Dynamics and Rheology
Polymer Dynamics and Rheology 1 Polymer Dynamics and Rheology Brownian motion Harmonic Oscillator Damped harmonic oscillator Elastic dumbbell model Boltzmann superposition principle Rubber elasticity and
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationPROBLEM SET 6. SOLUTIONS April 1, 2004
Harvard-MIT Division of Health Sciences and Technology HST.54J: Quantitative Physiology: Organ Transport Systems Instructors: Roger Mark and Jose Venegas MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departments
More informationIran University of Science & Technology School of Mechanical Engineering Advance Fluid Mechanics
1. Consider a sphere of radius R immersed in a uniform stream U0, as shown in 3 R Fig.1. The fluid velocity along streamline AB is given by V ui U i x 1. 0 3 Find (a) the position of maximum fluid acceleration
More informationNumerical Model of the Influence of Shear Stress on the Adaptation of a Blood Vessel BMT 03-35
Numerical Model of the Influence of Shear Stress on the Adaptation of a Blood Vessel BMT 03-35 Mirjam Yvonne van Leeuwen Supervisor: Dr. Ir. M.C.M. Rutten Ir. N.J.B. Driessen TUE Eindhoven, The Netherlands
More information2 Law of conservation of energy
1 Newtonian viscous Fluid 1 Newtonian fluid For a Newtonian we already have shown that σ ij = pδ ij + λd k,k δ ij + 2µD ij where λ and µ are called viscosity coefficient. For a fluid under rigid body motion
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationModeling of Fiber-Reinforced Membrane Materials Daniel Balzani. (Acknowledgement: Anna Zahn) Tasks Week 2 Winter term 2014
Institute of echanics and Shell Structures Faculty Civil Engineering Chair of echanics odeling of Fiber-Reinforced embrane aterials OOC@TU9 Daniel Balani (Acknowledgement: Anna Zahn Tasks Week 2 Winter
More informationSoft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies
Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed
More informationME 309 Fluid Mechanics Fall 2010 Exam 2 1A. 1B.
Fall 010 Exam 1A. 1B. Fall 010 Exam 1C. Water is flowing through a 180º bend. The inner and outer radii of the bend are 0.75 and 1.5 m, respectively. The velocity profile is approximated as C/r where C
More informationPage 1. Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.)
Page 1 Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (-1 point if not circled, or circled incorrectly): Prof. Vlachos Prof. Ardekani
More informationFluid mechanics and living organisms
Physics of the Human Body 37 Chapter 4: In this chapter we discuss the basic laws of fluid flow as they apply to life processes at various size scales For example, fluid dynamics at low Reynolds number
More informationMathematical Model of Blood Flow in Carotid Bifurcation
Excerpt from the Proceedings of the COMSOL Conference 2009 Milan Mathematical Model of Blood Flow in Carotid Bifurcation E. Muraca *,1, V. Gramigna 1, and G. Fragomeni 1 1 Department of Experimental Medicine
More informationPHYSICS. Course Structure. Unit Topics Marks. Physical World and Measurement. 1 Physical World. 2 Units and Measurements.
PHYSICS Course Structure Unit Topics Marks I Physical World and Measurement 1 Physical World 2 Units and Measurements II Kinematics 3 Motion in a Straight Line 23 4 Motion in a Plane III Laws of Motion
More informationFluid Mechanics. du dy
FLUID MECHANICS Technical English - I 1 th week Fluid Mechanics FLUID STATICS FLUID DYNAMICS Fluid Statics or Hydrostatics is the study of fluids at rest. The main equation required for this is Newton's
More information20. Alfven waves. ([3], p ; [1], p ; Chen, Sec.4.18, p ) We have considered two types of waves in plasma:
Phys780: Plasma Physics Lecture 20. Alfven Waves. 1 20. Alfven waves ([3], p.233-239; [1], p.202-237; Chen, Sec.4.18, p.136-144) We have considered two types of waves in plasma: 1. electrostatic Langmuir
More informationCHAPTER -6- BENDING Part -1-
Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and
More informationLecture 10 Acoustics of Speech & Hearing HST 714J. Lecture 10: Lumped Acoustic Elements and Acoustic Circuits
Lecture 0: Lumped Acoustic Elements and Acoustic Circuits I. A Review of Some Acoustic Elements A. An open-ended tube or Acoustic mass: units of kg/m 4 p linear dimensions l and a
More informationDifferential relations for fluid flow
Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow
More informationBasic Fluid Mechanics
Basic Fluid Mechanics Chapter 6A: Internal Incompressible Viscous Flow 4/16/2018 C6A: Internal Incompressible Viscous Flow 1 6.1 Introduction For the present chapter we will limit our study to incompressible
More informationSTEADY VISCOUS FLOW THROUGH A VENTURI TUBE
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 12, Number 2, Summer 2004 STEADY VISCOUS FLOW THROUGH A VENTURI TUBE K. B. RANGER ABSTRACT. Steady viscous flow through an axisymmetric convergent-divergent
More informationChapter 5 Torsion STRUCTURAL MECHANICS: CE203. Notes are based on Mechanics of Materials: by R. C. Hibbeler, 7th Edition, Pearson
STRUCTURAL MECHANICS: CE203 Chapter 5 Torsion Notes are based on Mechanics of Materials: by R. C. Hibbeler, 7th Edition, Pearson Dr B. Achour & Dr Eng. K. El-kashif Civil Engineering Department, University
More informationPhD Course in Biomechanical Modelling Assignment: Pulse wave propagation in arteries
PhD Course in Biomechanical Modelling Assignment: Pulse wave propagation in arteries Jonas Stålhand Division of Mechanics, Linköping University September 2012 1 The assignment Your first task is to derive
More informationWave Equation in One Dimension: Vibrating Strings and Pressure Waves
BENG 1: Mathematical Methods in Bioengineering Lecture 19 Wave Equation in One Dimension: Vibrating Strings and Pressure Waves References Haberman APDE, Ch. 4 and Ch. 1. http://en.wikipedia.org/wiki/wave_equation
More informationMacroscopic conservation equation based model for surface tension driven flow
Advances in Fluid Mechanics VII 133 Macroscopic conservation equation based model for surface tension driven flow T. M. Adams & A. R. White Department of Mechanical Engineering, Rose-Hulman Institute of
More informationLab Exercise #5: Tension and Bending with Strain Gages
Lab Exercise #5: Tension and Bending with Strain Gages Pre-lab assignment: Yes No Goals: 1. To evaluate tension and bending stress models and Hooke s Law. a. σ = Mc/I and σ = P/A 2. To determine material
More informationDIVIDED SYLLABUS ( ) - CLASS XI PHYSICS (CODE 042) COURSE STRUCTURE APRIL
DIVIDED SYLLABUS (2015-16 ) - CLASS XI PHYSICS (CODE 042) COURSE STRUCTURE APRIL Unit I: Physical World and Measurement Physics Need for measurement: Units of measurement; systems of units; SI units, fundamental
More informationCENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer
CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer You are assigned to design a fallingcylinder viscometer to measure the viscosity of Newtonian liquids. A schematic
More informationWhat s important: viscosity Poiseuille's law Stokes' law Demo: dissipation in flow through a tube
PHYS 101 Lecture 29x - Viscosity 29x - 1 Lecture 29x Viscosity (extended version) What s important: viscosity Poiseuille's law Stokes' law Demo: dissipation in flow through a tube Viscosity We introduced
More informationCONVECTIVE HEAT TRANSFER
CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 4 HEAT TRANSFER IN CHANNEL FLOW BASIC CONCEPTS BASIC CONCEPTS Laminar
More informationPart 7. Nonlinearity
Part 7 Nonlinearity Linear System Superposition, Convolution re ( ) re ( ) = r 1 1 = r re ( 1 + e) = r1 + r e excitation r = r() e response In the time domain: t rt () = et () ht () = e( τ) ht ( τ) dτ
More informationPhysical and Biological Properties of Agricultural Products Acoustic, Electrical and Optical Properties and Biochemical Property
Physical and Biological Properties of Agricultural Products Acoustic, Electrical and Optical Properties and Biochemical Property 1. Acoustic and Vibrational Properties 1.1 Acoustics and Vibration Engineering
More informationLaminar Flow. Chapter ZERO PRESSURE GRADIENT
Chapter 2 Laminar Flow 2.1 ZERO PRESSRE GRADIENT Problem 2.1.1 Consider a uniform flow of velocity over a flat plate of length L of a fluid of kinematic viscosity ν. Assume that the fluid is incompressible
More informationFluid Mechanics II Viscosity and shear stresses
Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small
More informationChapter 5: Torsion. 1. Torsional Deformation of a Circular Shaft 2. The Torsion Formula 3. Power Transmission 4. Angle of Twist CHAPTER OBJECTIVES
CHAPTER OBJECTIVES Chapter 5: Torsion Discuss effects of applying torsional loading to a long straight member (shaft or tube) Determine stress distribution within the member under torsional load Determine
More informationCandidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request.
UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book
More information2.6 Oseen s improvement for slow flow past a cylinder
Lecture Notes on Fluid Dynamics.63J/.J) by Chiang C. Mei, MIT -6oseen.tex [ef] Lamb : Hydrodynamics.6 Oseen s improvement for slow flow past a cylinder.6. Oseen s criticism of Stokes approximation Is Stokes
More informationQ1. Which of the following is the correct combination of dimensions for energy?
Tuesday, June 15, 2010 Page: 1 Q1. Which of the following is the correct combination of dimensions for energy? A) ML 2 /T 2 B) LT 2 /M C) MLT D) M 2 L 3 T E) ML/T 2 Q2. Two cars are initially 150 kilometers
More informationLaminar Boundary Layers. Answers to problem sheet 1: Navier-Stokes equations
Laminar Boundary Layers Answers to problem sheet 1: Navier-Stokes equations The Navier Stokes equations for d, incompressible flow are + v ρ t + u + v v ρ t + u v + v v = 1 = p + µ u + u = p ρg + µ v +
More informationExam 3 Review. Chapter 10: Elasticity and Oscillations A stress will deform a body and that body can be set into periodic oscillations.
Exam 3 Review Chapter 10: Elasticity and Oscillations stress will deform a body and that body can be set into periodic oscillations. Elastic Deformations of Solids Elastic objects return to their original
More informationMotion of a sphere in an oscillatory boundary layer: an optical tweezer based s
Motion of a sphere in an oscillatory boundary layer: an optical tweezer based study November 12, 2006 Tata Institute of Fundamental Research Co-workers : S. Bhattacharya and Prerna Sharma American Journal
More informationCOVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: MECHANICAL ENGINEERING
COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: MECHANICAL ENGINEERING COURSE: GEC 223 DISCLAIMER The contents of this document are intended for practice and leaning purposes at the
More informationCHAPTER 6: Shearing Stresses in Beams
(130) CHAPTER 6: Shearing Stresses in Beams When a beam is in pure bending, the only stress resultants are the bending moments and the only stresses are the normal stresses acting on the cross sections.
More informationOscillatory Motion SHM
Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A
More informationBERNOULLI EQUATION. The motion of a fluid is usually extremely complex.
BERNOULLI EQUATION The motion of a fluid is usually extremely complex. The study of a fluid at rest, or in relative equilibrium, was simplified by the absence of shear stress, but when a fluid flows over
More informationFluid Mechanics II. Newton s second law applied to a control volume
Fluid Mechanics II Stead flow momentum equation Newton s second law applied to a control volume Fluids, either in a static or dnamic motion state, impose forces on immersed bodies and confining boundaries.
More informationLecture 18. In other words, if you double the stress, you double the resulting strain.
Lecture 18 Stress and Strain and Springs Simple Harmonic Motion Cutnell+Johnson: 10.1-10.4,10.7-10.8 Stress and Strain and Springs So far we ve dealt with rigid objects. A rigid object doesn t change shape
More informationcos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015
skiladæmi 10 Due: 11:59pm on Wednesday, November 11, 015 You will receive no credit for items you complete after the assignment is due Grading Policy Alternative Exercise 1115 A bar with cross sectional
More informationIntroduction to Waves in Structures. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil
Introduction to Waves in Structures Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Waves in Structures Characteristics of wave motion Structural waves String Rod Beam Phase speed, group velocity Low
More information6. Bending CHAPTER OBJECTIVES
CHAPTER OBJECTIVES Determine stress in members caused by bending Discuss how to establish shear and moment diagrams for a beam or shaft Determine largest shear and moment in a member, and specify where
More information2 Navier-Stokes Equations
1 Integral analysis 1. Water enters a pipe bend horizontally with a uniform velocity, u 1 = 5 m/s. The pipe is bended at 90 so that the water leaves it vertically downwards. The input diameter d 1 = 0.1
More information[7] Torsion. [7.1] Torsion. [7.2] Statically Indeterminate Torsion. [7] Torsion Page 1 of 21
[7] Torsion Page 1 of 21 [7] Torsion [7.1] Torsion [7.2] Statically Indeterminate Torsion [7] Torsion Page 2 of 21 [7.1] Torsion SHEAR STRAIN DUE TO TORSION 1) A shaft with a circular cross section is
More informationCIRCULAR MOTION, HARMONIC MOTION, ROTATIONAL MOTION
CIRCULAR MOTION, HARMONIC MOTION, ROTATIONAL MOTION 1 UNIFORM CIRCULAR MOTION path circle distance arc Definition: An object which moves on a circle, travels equal arcs in equal times. Periodic motion
More informationCIVL222 STRENGTH OF MATERIALS. Chapter 6. Torsion
CIVL222 STRENGTH OF MATERIALS Chapter 6 Torsion Definition Torque is a moment that tends to twist a member about its longitudinal axis. Slender members subjected to a twisting load are said to be in torsion.
More informationMasters in Mechanical Engineering. Problems of incompressible viscous flow. 2µ dx y(y h)+ U h y 0 < y < h,
Masters in Mechanical Engineering Problems of incompressible viscous flow 1. Consider the laminar Couette flow between two infinite flat plates (lower plate (y = 0) with no velocity and top plate (y =
More informationSection 3.7: Mechanical and Electrical Vibrations
Section 3.7: Mechanical and Electrical Vibrations Second order linear equations with constant coefficients serve as mathematical models for mechanical and electrical oscillations. For example, the motion
More informationMECHANICAL PROPERTIES OF FLUIDS:
Important Definitions: MECHANICAL PROPERTIES OF FLUIDS: Fluid: A substance that can flow is called Fluid Both liquids and gases are fluids Pressure: The normal force acting per unit area of a surface is
More informationnot to be republished NCERT OSCILLATIONS Chapter Fourteen MCQ I π y = 3 cos 2ωt The displacement of a particle is represented by the equation
Chapter Fourteen OSCILLATIONS MCQ I 14.1 The displacement of a particle is represented by the equation π y = 3 cos 2ωt 4. The motion of the particle is (a) simple harmonic with period 2p/w. (b) simple
More informationKINEMATICS & DYNAMICS
KINEMATICS & DYNAMICS BY ADVANCED DIFFERENTIAL EQUATIONS Question (**+) In this question take g = 0 ms. A particle of mass M kg is released from rest from a height H m, and allowed to fall down through
More informationFormation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas )
Formation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas ) Yasutomo ISHII and Andrei SMOLYAKOV 1) Japan Atomic Energy Agency, Ibaraki 311-0102, Japan 1) University
More informationBending of Simply Supported Isotropic and Composite Laminate Plates
Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,
More informationThe... of a particle is defined as its change in position in some time interval.
Distance is the. of a path followed by a particle. Distance is a quantity. The... of a particle is defined as its change in position in some time interval. Displacement is a.. quantity. The... of a particle
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More information7 TRANSVERSE SHEAR transverse shear stress longitudinal shear stresses
7 TRANSVERSE SHEAR Before we develop a relationship that describes the shear-stress distribution over the cross section of a beam, we will make some preliminary remarks regarding the way shear acts within
More informationFlow and Transport. c(s, t)s ds,
Flow and Transport 1. The Transport Equation We shall describe the transport of a dissolved chemical by water that is traveling with uniform velocity ν through a long thin tube G with uniform cross section
More information3. BEAMS: STRAIN, STRESS, DEFLECTIONS
3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets
More informationIntroduction to Aerodynamics. Dr. Guven Aerospace Engineer (P.hD)
Introduction to Aerodynamics Dr. Guven Aerospace Engineer (P.hD) Aerodynamic Forces All aerodynamic forces are generated wither through pressure distribution or a shear stress distribution on a body. The
More informationChemical Engineering 160/260 Polymer Science and Engineering. Lecture 14: Amorphous State February 14, 2001
Chemical Engineering 160/260 Polymer Science and Engineering Lecture 14: Amorphous State February 14, 2001 Objectives! To provide guidance toward understanding why an amorphous polymer glass may be considered
More informationAdvanced Structural Analysis EGF Cylinders Under Pressure
Advanced Structural Analysis EGF316 4. Cylinders Under Pressure 4.1 Introduction When a cylinder is subjected to pressure, three mutually perpendicular principal stresses will be set up within the walls
More informationExam paper: Biomechanics
Exam paper: Biomechanics Tuesday August 10th 2010; 9.00-12.00 AM Code: 8W020 Biomedical Engineering Department, Eindhoven University of Technology The exam comprises 10 problems. Every problem has a maximum
More informationProf. Scalo Prof. Vlachos Prof. Ardekani Prof. Dabiri 08:30 09:20 A.M 10:30 11:20 A.M. 1:30 2:20 P.M. 3:30 4:20 P.M.
Page 1 Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (-1 point if not circled, or circled incorrectly): Prof. Scalo Prof. Vlachos
More informationIntroduction to Aerospace Engineering
Introduction to Aerospace Engineering Lecture slides Challenge the future 3-0-0 Introduction to Aerospace Engineering Aerodynamics 5 & 6 Prof. H. Bijl ir. N. Timmer Delft University of Technology 5. Compressibility
More informationNDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.
CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo
More informationStrain Gages. Approximate Elastic Constants (from University Physics, Sears Zemansky, and Young, Reading, MA, Shear Modulus, (S) N/m 2
When you bend a piece of metal, the Strain Gages Approximate Elastic Constants (from University Physics, Sears Zemansky, and Young, Reading, MA, 1979 Material Young's Modulus, (E) 10 11 N/m 2 Shear Modulus,
More information2, where dp is the constant, R is the radius of
Dynamics of Viscous Flows (Lectures 8 to ) Q. Choose the correct answer (i) The average velocity of a one-dimensional incompressible fully developed viscous flow between two fixed parallel plates is m/s.
More informationME3560 Tentative Schedule Spring 2019
ME3560 Tentative Schedule Spring 2019 Week Number Date Lecture Topics Covered Prior to Lecture Read Section Assignment Prep Problems for Prep Probs. Must be Solved by 1 Monday 1/7/2019 1 Introduction to
More informationUnit - 7 Vibration of Continuous System
Unit - 7 Vibration of Continuous System Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Continuous systems are tore which
More informationNORMAL STRESS. The simplest form of stress is normal stress/direct stress, which is the stress perpendicular to the surface on which it acts.
NORMAL STRESS The simplest form of stress is normal stress/direct stress, which is the stress perpendicular to the surface on which it acts. σ = force/area = P/A where σ = the normal stress P = the centric
More informationIntroduction to Marine Hydrodynamics
1896 1920 1987 2006 Introduction to Marine Hydrodynamics (NA235) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering First Assignment The first
More informationDesign and Modeling of Fluid Power Systems ME 597/ABE Lecture 7
Systems ME 597/ABE 591 - Lecture 7 Dr. Monika Ivantysynova MAHA Professor Fluid Power Systems MAHA Fluid Power Research Center Purdue University Content of 6th lecture The lubricating gap as a basic design
More informationREE Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics
REE 307 - Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics 1. Is the following flows physically possible, that is, satisfy the continuity equation? Substitute the expressions for
More informationFluid dynamics - viscosity and. turbulent flow
Fluid dynamics - viscosity and Fluid statics turbulent flow What is a fluid? Density Pressure Fluid pressure and depth Pascal s principle Buoyancy Archimedes principle Fluid dynamics Reynolds number Equation
More informationCFD with OpenSource software
CFD with OpenSource software The Harmonic Balance Method in foam-extend Gregor Cvijetić Supervisor: Hrvoje Jasak Faculty of Mechanical Engineering and Naval Architecture University of Zagreb gregor.cvijetic@fsb.hr
More information