ME%20440%20Intermediate%20Vibrations

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ME 440Intermediate Vibrations

• Spring 2009
• Tu, January 20

Dan NegrutUniversity of Wisconsin, Madison
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Before we get started

• Today
• ME440 Logistics
• Syllabus
• Grading scheme
• Start Chapter 1, Fundamentals of Vibrations
• HW Assigned 1.79
• HW due in one week

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ME440

• Course Objective
• 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
• 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

• Students must have the ability to
• 1. Derive the equations of motion of single and
  multi-degree of freedom systems, using Newton's
  Laws and energy methods.
• 2. Determine the natural frequencies and mode
  shapes of single and multi-degree of freedom
  systems.
• 3. Evaluate the dynamic response of single and
  multi-degree of freedom systems under impulse
  loadings, harmonic loadings, and general periodic
  excitation.
• 4. Apply modal analysis and orthogonality
  conditions to establish the dynamic
  characteristics of multi-degree of freedom
  systems.
• 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

• Polytechnic Institute of Bucharest, Romania
• B.S. Aerospace Engineering (1992)
• The University of Iowa, Iowa-City
• Ph.D. Mechanical Engineering (1998)
• MSC.Software, Ann Arbor, MI
• Product Development Engineer 1998-2005
• The University of Michigan
• Adjunct Assistant Professor, Dept. of Mathematics
  (2004)
• Division of Mathematics and Computer Science,
  Argonne National Laboratory
• Visiting Scientist 2004-2005, 2006
• The University of Wisconsin-Madison, Joined in
  Nov. 2005
• Research Computer Aided Engineering (tech. lead,
  Simulation-Based Engineering Lab)
• Focus Computational Dynamics (http//sbel.wisc.ed
  u)

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Good to know

• Time 930-1045 AM
• Location
• 3349EH (through end of Jan)
• 3126ME (after Feb. 1)
• Office 2035ME
• Phone [phone]
• E-Mail negrut_at_engr.wisc.edu
• Grader Naresh Khude, khude_at_wisc.edu

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ME 440 Fall 2009

• Office Hours
• Monday 2 4 PM
• Wednesday 2 4 PM
• Friday 3 4 PM

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Text

• S. S. Rao Mechanical Vibrations

• Pearson Prentice Hall
• Fourth edition (2004)
• Well cover material out of first six chapters
• 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

• Handouts will be printed out and provided before
  each lecture
• Good idea to organize material provided in a
  folder
• Useful for PhD Qualifying exam, useful in
  industry
• Lecture slides will be made available online
• http//sbel.wisc.edu/Courses/ME440/2009/index.htm
  
• Im in the process of reorganizing the class
  material
• Moving from transparency to slide format
• Grades will be maintained online at
  https//LearnUW.wisc.edu
• Schedule will be updated as we go and will
  contain info about
• Topics we cover
• Homework assignments

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Grading

• Homework Projects 40
• Exam 1 (Feb. 24) 20
• Exam 2 (Apr. 7) 20
• Exam 3 (May 7) 20
• Total 100

• 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

• Weekly if not daily homework
• Assigned at the end of each class
• Due at the beginning of the class, one week later
• No late homework accepted
• Two lowest score homeworks will be dropped
• Grading
• 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_at_UW

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Midterm Exams

• Scheduled dates on syllabus
• 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
• One day prior to the exam, at 715PM
• Will run about two hours long
• Room 3126ME

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Final Exam

• There will be no final exam
• The third exam will be a comprehensive exam

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Scores and Grades
Score Grade 94-100 A 87-93 AB 80-86 B 73-79 BC
66-72 C 55-65 D lt54 F

• Grading will not be done on a curve
• Final score will be rounded to the nearest
  integer prior to having a letter assigned
• 86.59 becomes AB
• 86.47 becomes B

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PrerequisiteME340
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MATLAB and Simulink

• Integrated into every chapter in the text
• You are responsible for brushing up on your
  MATLAB skills
• Ill 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
• MATLAB workshop tutorial

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ME440 Major Topics

• Chapter 1 Fundamentals of Vibrations
• Chapter 2 Free Vibrations of Single DOF Systems
• Chapter 3 Harmonically Excited Vibration
• Chapter 4 Vibration Under General Forcing
  Conditions
• Chapter 5 Two Degree of Freedom Systems
• Chapter 6 Multidegree of Freedom Systems

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This Course

• Be active, pay attention, ask questions
• A rather intense class
• 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
• 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

• How has this topic, Mechanical Vibrations, come
  to be?
• Just like many other topics in Engineering

• A physical system is given to you (you have a
  problem to solve)
• You generate an abstraction of that actual system
  (problem)
• 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
• Post-processing might be necessary

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Mechanical Vibrations The Framework (Contd)

• Picture worth all the words on previous slide

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What is the problem here?

• Both the mass-spring-damper system and the string
  system lead to an oscillatory motion
• Vibration, Oscillation
• Any motion that repeats itself after in interval
  of time
• For the mass-spring-damper
• One degree of freedom system
• 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
• An infinite number of degrees of freedom
• 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

• Degree of Freedom
• This concept means different things to different
  people
• In ME440
• 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

• Two different problems lead to two different
  models
• Lumped systems lead to ODEs
• Continuous systems leads to PDEs
• PDEs are significantly more difficult to solve
• In this class, well almost exclusively deal with
  systems that lead to ODE problems (lumped
  systems, discrete systems)
• See next slide

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Typical ME440 Problem

• 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
• Well work in the blue boxes

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Linear or Nonlinear ODE
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How Things Happen

• In a oscillatory motion, one type of energy gets
  converted into a different type of energy time
  and again
• Think of a pendulum
• 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
• Air resistance, heat dissipation due to friction,
  etc.

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Vibration, the Characters in the Play

• One needs elements capable of storing/dissipating
  various forms of energy
• Springs capable of storing potential energy
• Masses capable of acquiring kinetic energy
• Damping elements involved in the energy
  dissipation
• Actuators the elements that apply an external
  forcing or impose a prescribed motion on parts of
  a system
• NOTE The systems (problems) that well analyze
  in 440 lead to models based on a combination of
  these four elements

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Springs

• A component/system that relates a displacement to
  a force that is required to produce that
  displacement
• Physically, its often times a mechanical link
  typically assumed to have negligible mass and
  damping
• Well work most of the time with linear springs

• NOTE After reaching the yield point A, even a
  linear spring stops behaving linearly

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Spring (Stiffness) Element

• F is the force exerted by the spring
• x1, x2 are the displacements of spring end
  points
• Spring deflection ?x x2-x1

x1
x2
F
F

• Linear springs

k stiffness (units N/m or lb/in)
Energy Stored (linear springs)
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Springs Dont Necessarily Look Like
SpringsSpring Constants of Common Elements
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Example (Equivalent Spring)

• Assume that mass of beam is negligible in
  comparison with end mass.
• Denote by Wmg weight of the end mass
• Static deflection of the cantilever beam is given
  by

• The equivalent spring has the stiffness

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Springs Acting in Series
F
Note that two springs are in series when a) They
are experiencing the same tension (or
compression) b) Youd add up the deformations to
get the total deformation x
Exercise Show that the equivalent spring
constant keq is such that
The idea is that you want to determine one
abstract spring that has keq that deforms by the
same amount when its subject to F.
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Springs Acting in Parallel
F
F
Note that two springs are in parallel when a)
They experience the same amount of deformation b)
Youd 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
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Equivalent Spring Stiffness

• Another way to compute keq draws on a total
  potential energy approach

• Example provided in the textbook

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