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

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

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

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

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 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.edu) 5

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: 608 890-0914 E-Mail: negrut@engr.wisc.edu Grader: Naresh Khude, khude@wisc.edu 6

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

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

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 http://sbel.wisc.edu/Courses/ME440/2009/index.htm I’m 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 9

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 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 10

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 Learn@UW 11

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

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

Scores and Grades Score Grade 94-100 A April 21, 2017 Scores and Grades Score Grade 94-100 A 87-93 AB 80-86 B 73-79 BC 66-72 C 55-65 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

April 21, 2017 Prerequisite: ME340 15

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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