1. ME 440 Intermediate Vibrations
April 21, 2017
ME 440 Intermediate Vibrations
Spring 2009
Tu, January 20
M1051 ECB 2, 2hr EES seminars:
2-4 pm on Jan 20
4-6 pm on Jan 20
Dan Negrut University of Wisconsin, Madison

2. Before we get started… Today: ME440 Logistics
April 21, 2017
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 2

3. ME440 Course Objective Catalog Description:
April 21, 2017
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. 3

4. Course Outcomes Students must have the ability to:
April 21, 2017
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. 4

5. Instructor: Dan Negrut
April 21, 2017
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
The University of Michigan
Adjunct Assistant Professor, Dept. of Mathematics (2004)
Division of Mathematics and Computer Science, Argonne National Laboratory
Visiting Scientist , 2006
The University of Wisconsin-Madison, Joined in Nov. 2005
Research: Computer Aided Engineering (tech. lead, Simulation-Based Engineering Lab)
Focus: Computational Dynamics ( 5

6. Good to know… Time: 9:30-10:45 AM Location: Office: 2035ME
April 21, 2017
Good to know…
Time: 9:30-10:45 AM
Location:
3349EH (through end of Jan)
3126ME (after Feb. 1)
Office: 2035ME
Phone:
Grader: Naresh Khude, 6

7. ME 440 Fall 2009 Office Hours Monday 2 – 4 PM Wednesday 2 – 4 PM
April 21, 2017
ME 440 Fall 2009
Office Hours
Monday 2 – 4 PM
Wednesday 2 – 4 PM
Friday 3 – 4 PM 7

8. Text S. S. Rao – Mechanical Vibrations 8 Pearson Prentice Hall
April 21, 2017
Text
S. S. Rao – Mechanical Vibrations
Pearson Prentice Hall
Fourth edition (2004)
We’ll 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) 8

9. April 21, 2017
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
I’m in the process of reorganizing the class material
Moving from transparency to slide format
Grades will be maintained online at
Schedule will be updated as we go and will contain info about
Topics we cover
Homework assignments 9

10. Grading Homework + Projects 40% Exam 1 (Feb. 24) 20%
April 21, 2017
Grading
Homework + Projects 40%
Exam 1 (Feb. 24) 20%
Exam 2 (Apr. 7) 20%
Exam 3 (May 7) 20%
Total %
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 10

11. Homework Weekly if not daily homework
April 21, 2017
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 11

12. Midterm Exams Scheduled dates on syllabus
April 21, 2017
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 7:15PM
Will run about two hours long
Room: 3126ME 12

13. Final Exam There will be no final exam
April 21, 2017
Final Exam
There will be no final exam
The third exam will be a comprehensive exam 13

14. Scores and Grades Score Grade 94-100 A
April 21, 2017
Scores and Grades
Score Grade A AB B BC C D
<54 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 14

15. April 21, 2017
Prerequisite: ME340 15

16. MATLAB and Simulink Integrated into every chapter in the text
April 21, 2017
MATLAB and Simulink
Integrated into every chapter in the text
You are responsible for brushing up on your MATLAB skills
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
MATLAB workshop tutorial 16

17. ME440 Major Topics Chapter 1 – Fundamentals of Vibrations
April 21, 2017
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 17

18. This Course… Be active, pay attention, ask questions
April 21, 2017
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 18

19. End: ME440 Logistics, Syllabus Discussion
April 21, 2017
End: ME440 Logistics, Syllabus Discussion
Begin: Chapter 1 - Fundamentals of Vibration 19

20. Mechanical Vibrations: The Framework
April 21, 2017
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… 20

21. Mechanical Vibrations: The Framework (Contd)
April 21, 2017
Mechanical Vibrations: The Framework (Contd)
Picture worth all the words on previous slide: 21

22. What is the problem here?
April 21, 2017
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 22

23. The Concept of Degree of Freedom
April 21, 2017
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 23

24. Type of Math Problems in Vibrations
April 21, 2017
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, we’ll almost exclusively deal with systems that lead to ODE problems (lumped systems, discrete systems)
See next slide… 24

25. April 21, 2017
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
We’ll work in the blue boxes 25

26. Linear or Nonlinear ODE
April 21, 2017
Linear or Nonlinear ODE 26

27. April 21, 2017
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. 27

28. Vibration, the Characters in the Play
April 21, 2017
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 we’ll analyze in 440 lead to models based on a combination of these four elements 28

29. April 21, 2017
Springs…
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
NOTE: After reaching the yield point A, even a linear spring stops behaving linearly 29

30. Spring (Stiffness) Element
April 21, 2017
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
Hardening
Linear springs:
Linear
Deflection
Force
Softening
k = stiffness (units = N/m or lb/in)
Energy Stored
(linear springs) 30

31. ‘Springs Don’t Necessarily Look Like Springs’ Spring Constants of Common Elements
April 21, 2017 31

32. Example (Equivalent Spring)
April 21, 2017
Example (Equivalent Spring)
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
The equivalent spring has the stiffness: 32

33. Springs Acting in Series
April 21, 2017 k 1 2 M x
keq F 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:
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. 33

34. Springs Acting in Parallel
April 21, 2017 k 1 2 M x
keq M x F 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: 34

35. Equivalent Spring Stiffness
April 21, 2017
Equivalent Spring Stiffness
Another way to compute keq draws on a total potential energy approach:
Example provided in the textbook 35