Transcript ME 440 Intermediate Vibrations

ME 440
Intermediate Vibrations
Spring 2009
Tu, January 20
Dan Negrut
University of Wisconsin, Madison
Before we get started…
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Today:
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ME440 Logistics
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Syllabus
Grading scheme
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Start Chapter 1, “Fundamentals of Vibrations”
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HW Assigned: 1.79
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HW due in one week
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ME440
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Course Objective
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The purpose of the course is to develop the skills needed to design and
analyze mechanical systems in which vibration problems are typically
encountered. These skills include analytical and numerical techniques that
allow the student to model the system, analyze the system performance
and employ the necessary design changes. Emphasis is placed on
developing a thorough understanding of how the changes in system
parameters affect the system response.
Catalog Description:
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Analytical methods for solution of typical vibratory and balancing problems
encountered in engines and other mechanical systems. Special emphasis
on dampers and absorbers.
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Course Outcomes
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Students must have the ability to:
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1. Derive the equations of motion of single and multi-degree of freedom
systems, using Newton's Laws and energy methods.
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2. Determine the natural frequencies and mode shapes of single and
multi-degree of freedom systems.
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3. Evaluate the dynamic response of single and multi-degree of freedom
systems under impulse loadings, harmonic loadings, and general periodic
excitation.
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4. Apply modal analysis and orthogonality conditions to establish the
dynamic characteristics of multi-degree of freedom systems.
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5. Generate finite element models of discrete systems to simulate the
dynamic response to initial conditions and external excitations.
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Instructor: Dan Negrut
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Polytechnic Institute of Bucharest, Romania
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The University of Iowa, Iowa-City
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Adjunct Assistant Professor, Dept. of Mathematics (2004)
Division of Mathematics and Computer Science, Argonne National Laboratory
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Product Development Engineer 1998-2005
The University of Michigan
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Ph.D. – Mechanical Engineering (1998)
MSC.Software, Ann Arbor, MI
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B.S. – Aerospace Engineering (1992)
Visiting Scientist 2004-2005, 2006
The University of Wisconsin-Madison, Joined in Nov. 2005
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Research: Computer Aided Engineering (tech. lead, Simulation-Based Engineering Lab)
Focus: Computational Dynamics (http://sbel.wisc.edu)
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Good to know…
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Time:
9:30-10:45 AM
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Location:
3349EH (through end of Jan)
 3126ME (after Feb. 1)
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Office:
2035ME
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Phone:
608 890-0914
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E-Mail:
[email protected]
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Grader:
Naresh Khude, [email protected]
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ME 440 Fall 2009
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Office Hours
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Monday
Wednesday
Friday
2 – 4 PM
2 – 4 PM
3 – 4 PM
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Text
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S. S. Rao – Mechanical Vibrations
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Pearson Prentice Hall
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Fourth edition (2004)
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We’ll cover material out of first six chapters
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On a couple of occasions, the material in the book will be
supplemented with notes
Available at Wendt Library (on reserve)
Paperback international edition available for $35 ($150 for
hardcover)
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Other Tidbits
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Handouts will be printed out and provided before each lecture
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Lecture slides will be made available online
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Good idea to organize material provided in a folder
Useful for PhD Qualifying exam, useful in industry
http://sbel.wisc.edu/Courses/ME440/2009/index.htm
I’m in the process of reorganizing the class material
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Moving from transparency to slide format
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Grades will be maintained online at https://LearnUW.wisc.edu
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Schedule will be updated as we go and will contain info about
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Topics we cover
Homework assignments
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Grading
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Homework + Projects
Exam 1 (Feb. 24)
Exam 2 (Apr. 7)
Exam 3 (May 7)
40%
20%
20%
20%
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Total
100%
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NOTE:
• Score related questions (homeworks/exams/projects) must be raised
prior to next class after the homeworks/exams/project is returned.
• Exam 3 will serve as the final exam and it will be comprehensive
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Homework
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Weekly if not daily homework
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Assigned at the end of each class
Due at the beginning of the class, one week later
No late homework accepted
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Two lowest score homeworks will be dropped
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Grading
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Each problem scored on a 1-10 scale (10 – best)
For each HW an average will be computed on a 1-10 scale
Solutions to select problems will be posted at Learn@UW
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Midterm Exams
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Scheduled dates on syllabus
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Tu, 02/24 – covers chapters 1 through 3
Tu, 04/07 – covers chapter 4 through 5
Th, 05/07 – comprehensive, chapters 1 through 6
A review session will be offered prior to each exam
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One day prior to the exam, at 7:15PM
Will run about two hours long
Room: 3126ME
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Final Exam
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There will be no final exam
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The third exam will be a comprehensive exam
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Scores and Grades
Score
94-100
87-93
80-86
73-79
66-72
55-65
<54
Grade
A
AB
B
BC
C
D
F
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Grading will not be done on
a curve
Final score will be rounded to
the nearest integer prior to
having a letter assigned
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86.59 becomes AB
86.47 becomes B
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Prerequisite:
ME340
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MATLAB and Simulink
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Integrated into every chapter in the text
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You are responsible for brushing up on your MATLAB skills
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I’ll offer a MATLAB Workshop (outside class)
 Friday, January 30 1 to 4 PM (room 1051ECB)
 Topics covered: working in MATLAB, working with matrices, m-file:
functions and scripts, for loops/while loops, if statements, 2-D plots
 Actually it covers more than you need to know for ME440
 Offered to ME students, seating is limited, register if you plan to
attend
Resources posted on course website
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MATLAB workshop tutorial
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ME440 Major Topics
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Chapter 1 – Fundamentals of Vibrations
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Chapter 2 – Free Vibrations of Single DOF Systems
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Chapter 3 – Harmonically Excited Vibration
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Chapter 4 – Vibration Under General Forcing Conditions
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Chapter 5 – Two Degree of Freedom Systems
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Chapter 6 – Multidegree of Freedom Systems
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This Course…
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Be active, pay attention, ask questions
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A rather intense class
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The most important thing is taking care of homework
Reading the text is important
The class builds on itself – essential to start strong and keep up
Your feedback is important
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Provide feedback – both during and at end of the semester
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End: ME440 Logistics, Syllabus Discussion
Begin: Chapter 1 - Fundamentals of Vibration
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Mechanical Vibrations: The Framework
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How has this topic, Mechanical Vibrations, come to be?
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Just like many other topics in Engineering:
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A physical system is given to you (you have a problem to solve)
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You generate an abstraction of that actual system (problem)
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In other words, you generate a model of the system
You apply the laws of physics to get the equations that govern the time
evolution of the model
You solve the differential equations to find the solution of interest
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Post-processing might be necessary…
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Mechanical Vibrations: The Framework (Contd)
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Picture worth all the words on previous slide:
Physics System
Discrete Parameter systems
(Lumped Systems)
Mathematical
Model
ODE’s
Distributed Parameter Systems
(Continuous Systems)
PDE’s
x(t)
k
y(x,t)
F(t)
m
c
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What is the problem here?
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Both the mass-spring-damper system and the string system lead to an
oscillatory motion
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Vibration, Oscillation:
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Any motion that repeats itself after in interval of time
For the mass-spring-damper:
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One degree of freedom system
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Everything is settled once you get the solution x(t)
You get x(t) as the solution of an Initial Value Problem (IVP)
For the string:
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An infinite number of degrees of freedom
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You need the string deflection at each location x between 0 and L
You get the string deflection as a function of time and location based on both
initial conditions and boundary conditions – solution of a set of Partial Differential
Equations
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The Concept of Degree of Freedom
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Degree of Freedom
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This concept means different things to different people
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In ME440:
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The minimum number of coordinates (“states”, “unknowns”, etc.)
that you need to have in your model to uniquely specify the
position/orientation of each component in your model
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Type of Math Problems in Vibrations
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Two different problems lead to two different models
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Lumped systems – lead to ODEs
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Continuous systems – leads to PDEs
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PDEs are significantly more difficult to solve
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In this class, we’ll almost exclusively deal with systems that
lead to ODE problems (lumped systems, discrete systems)
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See next slide…
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Typical ME440 Problem
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Not only that we are going to mostly deal with ODEs, but they are
typically linear
Nonlinear ODEs are most of the time impossible to solve in close form
You end up using a numerical algorithm to find an approximate solution
We’ll work in the blue boxes
Linear
Systems
Nonlinear
Free
Response
Forced
Initial
Excitations
Externally applied
forces
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Linear or Nonlinear ODE
y  4y  0
y  5 y  y2  t
y  sin y  5
y  5y  y  t
2
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How Things Happen…
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In a oscillatory motion, one type of energy gets converted
into a different type of energy time and again…
Think of a pendulum
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Potential energy gets converted into kinetic energy which gets
connected back into potential energy, etc.
Note that energy dissipation almost always occurs, so the
oscillatory motion is damped
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Air resistance, heat dissipation due to friction, etc.
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Vibration, the Characters in the Play
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One needs elements capable of storing/dissipating various forms of energy:
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Springs – capable of storing potential energy
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Masses – capable of acquiring kinetic energy
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Damping elements –involved in the energy dissipation
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Actuators – the elements that apply an external forcing or impose a prescribed
motion on parts of a system
NOTE: The systems (problems) that we’ll analyze in 440 lead to models
based on a combination of these four elements
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Springs…
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A component/system that relates a displacement to a force that is
required to produce that displacement
Physically, it’s often times a mechanical link typically assumed to have
negligible mass and damping
We’ll work most of the time with linear springs
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NOTE: After reaching the yield point A,
even a linear spring stops behaving linearly
Spring (Stiffness) Element
x1
x2
F
F
Hardening
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F is the force exerted by the spring
x1, x2 are the displacements of
spring end points
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Spring deflection x= x2-x1
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Linear springs:
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Linear
Force
Softening
F  k x  k  x 2  x1 
k = stiffness (units = N/m or lb/in)
Deflection
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Energy Stored
(linear springs)
1
E  k x 2
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‘Springs Don’t Necessarily Look Like Springs’
Spring Constants of Common Elements
Ewh3
k 3
L
Gd 4
k
64nR 3
k
k
16Ewh3
k
L3
EA
L
4Ewh
L3
3
2Ewh3
k
L3
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Example (Equivalent Spring)
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Assume that mass of beam is negligible in comparison with end mass.
Denote by W=mg weight of the end mass
Static deflection of the cantilever beam is given by
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The equivalent spring has the stiffness:
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Springs Acting in Series
x
x
k1
k2
keq
M
F
M
F
Note that two springs are in series when:
a) They are experiencing the same tension (or compression)
b) You’d add up the deformations to get the total deformation x
Exercise: Show that the equivalent spring constant keq is such that:
1
1 1
 
keq k1 k2
The idea is that you want to determine one abstract spring that has keq
that deforms by the same amount when it’s subject to F.
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Springs Acting in Parallel
x
x
k1
k2
keq
M
F
M
F
Note that two springs are in parallel when:
a) They experience the same amount of deformation
b) You’d add up the force experienced by each spring to come up
with the total force F
Exercise: Show that the equivalent spring constant keq is such that:
keq  k1  k2
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Equivalent Spring Stiffness
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Another way to compute keq draws on a total potential
energy approach:
Example provided in the textbook
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