MAE147 L1

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MAE 147 “Vibrations”
Lecture 1
Instructor: 
Professor Andrei Shkel
Email: ashkel@uci.edu
Office: EG4202
 
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
History of Vibrations
n 2000BC first evidences
n 582BC Pythagoras studied music produced by a vibrating 
string
….
n 1564 Galileo studied string musical instruments and 
pendulum
n 1664 Newton established lows of motion
n …Taylor, Euler, Bernoulli, D’Alembert, Fourier, Lagrange, 
Poisson, Coulomb
n 1877 Rayleigh published “The Theory of Sound”
n 20th century: Timoshenko, Stadola, De Laval, Frahm,
Minorsky, Duffing, Von der Pol 
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
History, cont’d
n Before 1930 it was not a part of formal engineering
education
n Why now?
¨ So many devices are powered by engines and motors
¨ Unwanted noise
¨ Uncomfortable motion
¨ Structural failures 
n Engineering disasters
¨ Tacoma Bridge 1940: https://www.youtube.com/watch?v=3mclp9QmCGs
¨ Submarine Coolant pump disaster
¨ Fuel pumps in aircraft and rocket engines 
¨ Helicopter rotors, turbines, elec. generation machines
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-4
Introduction
n Instructor: Professor Andrei M. Shkel
¨ UC Irvine 2000-
n Teaching: Vibrations, Dynamics, MEMS, Inertial Navigation
n Research: Micro-scale gyroscopes and Inertial Measurement Units 
(design, modeling, fabrication, control electronics, advanced T&E)
n Teaching Assistant
¨ Austin Parrish
¨ Research interests: microdevices for biomagnetism
n Micro-electro-mechanical systems
n Nuclei Magnetic Resonance (NMR) for magnetometers and gyroscopes 
Responsibilities:
• Discussion sessions
• Office hours
• Grading
• Class bookkeeping 
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-5
Information
n EEE Canvas
¨ Our main communication hub
¨ Announcements, Assignments, Discussions, Grades, 
Files, Lectures, etc. 
n EEE GrandCentral: https://grandcentral.eee.uci.edu
¨ Gateway to IEEE Canvas
n Yuja: https://uci.yuja.com
¨ Recorded lectures (also available via Canvas)
n Zoom
¨ Replay of lectures with Q&A
¨ Discussions
¨ Office Hours
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
Zoom meeting rooms
n Lectures: MWF 3:00 pm – 3:50 pm in Zoom Room: 
https://uci.zoom.us/j/810747699
n Office hours: MW 4:00-5:00PM via ZoomRoom
https://uci.zoom.us/j/692564711 or by appointment
n Discussions & TA Office hours: 
https://uci.zoom.us/j/4108731809
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
Tentative class logistics
n Lectures 
¨ PDFs available on class website before lecture
¨ All lectures are recorded
¨ Participation extra credit 
n HWs – due Monday nights @ 11:59PM (upload)
¨ Each set ~5 problems
¨ Assigned on rolling basis during the week
¨ Matlab/Simulink problem in every assignment
¨ Check-off grading: 
n “0”, “1”, “2” ó “Nothing”, “Something”, “Everything”
¨ Download/Submit from class website
n Weekly Quizzes
¨ 30 min. Open from 4PM on Friday to 4PM on Sunday 
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-8
Textbook
n Will cover Chapters 1-8 
n ~1 week per Chapter
n ~1 HW per Chapter
n Midterm 1: Ch. 1-4 (April 29th)
n Midterm 2: Ch. 1-6 (May 20th)
n Final: Ch. 1-8 (June 8th)
n Self-study/review MATLAB
n Supplements by instructor
Primary: Mechanical Vibration by William J. Palm III
Secondary: System Dynamics by William J. Palm III
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-9
Grading
n Weekly Quizzes (~8) – 10%
n HWs (~7) - 10%
n Midterm 1 - 20%
n Midterm 2 – 25% 
n Final Exam – 35% 
n Total: 100%
n Absolute Scale
NOTE: Specific score-related questions (homeworks/exams) must be raised 
prior to the next class after receiving the score.
A (90-100)
B (80-89)
C (70-79)
D (50-69)
F (0-49)
2018 Statistics
• Mean: 67.42
• Standard deviation: 12.24
• Maximum: 94.6
• Minimum: 7.4
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-10
Prerequisites 
n MAE 80 “Dynamics”
¨ Kinematics and Dynamics of Particles and Rigid Bodies
¨ The Newton-Euler, Work/Energy, and Impulse/Momentum methods for deriving 
equations 
n MAE 140 “Introduction to Engineering Analysis”
¨ Nonhomogeneous linear ODE
¨ Variable coefficient linear ODE
¨ Eigenfunction expansions
¨ Laplace transforms 
¨ Introduction to Fourier transforms 
n Math 2E or equivalent
¨ Directional derivatives and gradient
¨ Vector fields
¨ Linear Integrals
¨ Curl and Divergence
¨ Parametric surfaces and their areas
¨ Stokes’s Theorem and the Divergence Theorem
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-11
MAE 147 Topics
n 1 – Introduction 
¨ Description of Vibrational motion
¨ Spring and Damper elements
n 2 – Modeling of Dynamics
n 3 – Free Response with 1 DOF
n 4 – Harmonic Response with 1 DOF
n 5 – General Force Response
n 6 – 2 DOF Systems
n 7 – Vibration Suppression and 
Control
n 8 – Matrix Methods for MDF Systems
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-12
MAE 147 Topics
n 1 – Introduction 
n 2 – Modeling of Dynamics
¨ Deriving Equations of motion
n 2nd Newton’s Law
n Work-Energy Method
n Impulse Momentum Method
¨ Equivalent Mass and Inertia
n 3 – Free Response with 1 DOF
n 4 – Harmonic Response with 1 DOF
n 5 – General Force Response
n 6 – 2 DOF Systems
n 7 – Vibration Suppression and 
Control
n 8 – Matrix Methods for MDF Systems
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-13
MAE 147 Topics
n 1 – Introduction 
n 2 – Modeling of Dynamics
n 3 – Free Response with 1 DOF
¨ Free undamped vibrations
¨ Free vibrations with viscous damping
¨ Characteristic roots and stability 
n 4 – Harmonic Response with 1 DOF
n 5 – General Force Response
n 6 – 2 DOF Systems
n 7 – Vibration Suppression and 
Control
n 8 – Matrix Methods for MDF Systems
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-14
MAE 147 Topics
n 1 – Introduction 
¨ Vibrational Motion
¨ Spring and Damping Elements 
n 2 – Modeling of Dynamics
n 3 – Free Response with 1 DOF
n 4 – Harmonic Response with 1 DOF
¨ Solution for the harmonic response
¨ Resonance and bandwidth
¨ Base excitation, rotating unbalance
n 5 – General Force Response
n 6 – 2 DOF Systems
n 7 – Vibration Suppression and 
Control
n 8 – Matrix Methods for MDF Systems
Midterm 1
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-15
MAE 147 Topics
n 1 – Introduction 
¨ Vibrational Motion
¨ Spring and Damping Elements 
n 2 – Modeling of Dynamics
n 3 – Free Response with 1 DOF
n 4 – Harmonic Response with 1 DOF
n 5 – General Force Response
¨ Response to generic periodic inputs
¨ The Laplace Transform
¨ Pulse and Impulse response
n 6 – 2 DOF Systems
n 7 – Vibration Suppression and 
Control
n 8 – Matrix Methods for MDF Systems
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-16
MAE 147 Topics
n 1 – Introduction 
¨ Vibrational Motion
¨ Spring and Damping Elements 
n 2 – Modeling of Dynamics
n 3 – Free Response with 1 DOF
n 4 – Harmonic Response with 1 DOF
n 5 – General Force Response
n 6 – 2 DOF Systems
¨ Models of 2 DOF systems
¨ Lagrange’s equations
¨ Modes and system response
¨ Harmonic and general force response
n 7 – Vibration Suppression and 
Control
n 8 – Matrix Methods for MDF Systems
Midterm 2
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-17
MAE 147 Topics
n 1 – Introduction 
¨ Vibrational Motion
¨ Spring and Damping Elements 
n 2 – Modeling of Dynamics
n 3 – Free Response with 1 DOF
n 4 – Harmonic Response with 1 DOF
n 5 – General Force Response
n 6 – 2 DOF Systems
n 7 – Vibration Suppression and 
Control
¨ Isolator Design for fixed-base system
¨ Isolation with base motion
¨ Dynamic vibration absorber
¨ Active vibration control
n 8 – Matrix Methods for MDF Systems
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-18
MAE 147 Topics
n 1 – Introduction 
¨ Vibrational Motion
¨ Spring and Damping Elements 
n 2 – Modeling of Dynamics
n 3 – Free Response with 1 DOFn 4 – Harmonic Response with 1 DOF
n 5 – General Force Response
n 6 – 2 DOF Systems
n 7 – Vibration Suppression and 
Control
n 8 – Matrix Methods for MDF Systems
¨ Matrix form of the equations of motion
¨ Mode shapes and eigenvalue problem
¨ Modal analysis
¨ Effects of damping 
Final
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-19
Why do I need this stuff, anyway ?
n Mechanical Engineering today is not 
what it used to be yesterday
¨ Micro and Nano Systems
¨ New methods of design and analysis, mostly 
related to the advancement of computational 
science
n Some of you are going be at the forefront 
of research tomorrow
¨ Micro/Nano Electro Mechanical Systems
¨ Prosthetics, implantable systems 
¨ Molecular Dynamics, Protein Folding
¨ Design of materials
¨ Drug design and new methods of drug 
delivery
n Others will do business in a global 
market
¨ Hi-Tech companies, Venture Capital, etc.
¨ Stay competitive
¨ Don’t paint yourself in a corner, always learn 
new things
n You can only build a tall pyramid with a 
large and solid foundation
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-20
This Course … 
n Be active, pay attention, ask questions
n Discussions (during replay and sessions) 
n A key, but intense, class in the ME program
¨ Reading the text is important
¨ Doing your homework is critical (group work is encouraged…)
¨ The class builds on itself – essential to start strong and keep up
n Your feedback is important
¨ Provide feedback (“NO to silent pain”)
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
Review topics
n Given a system - derive equations of motion
¨ Week 2 and discussion sessions 
n Special functions:
¨ Exponential:
¨ Harmonic functions: 
¨ Taylor series
¨ Linearization
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/( ) ty t Ae t-=
( ) sin( ) sin cosy t A t B t C tw f w w= + = +
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
Review topics, cont’d
n Parallel and Series Spring/Damping Elements
n Degrees of freedom
n Kinematics of gears, belts, lever, rolling, etc.
n Modeling
¨ Newton’s 2nd Law
¨ Conservation of Energy
¨ Rayleigh’s Method
¨ Linear impulse-momentum principle
¨ Lagrange’s Equations (may be)
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Functions that 
we will use a lot
MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
Systems with spring elements
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
“E” as coordinate reference 
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
A useful observation 
n A general principle of modeling systems 
containing spring elements
Any constant force or moments will not appear in 
the equations of motion of a system containing a 
linear restoring force or moment if the mass 
displacements are measured from the equilibrium 
positions
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
Illustration 
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020
Conclusion 
n The advantages of choosing the equilibrium 
as the coordinate origin are
¨ We need not specify the geometric dimensions of the 
mass
¨ This choice simplifies the equation of motion by 
eliminating the static force
n This only generally true for a linear spring 
force and a constant gravity or moment 
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MAE 147 “Vibrations”
Professor Andrei M. ShkelSpring 2020 L1-28
For next lecture
n Chapter 1 – self-study
¨ History of Mechanical Vibrations 
¨ Oscillation, Vibration, Sound, Acoustics
¨ Description of Vibrational Motion
¨ Spring and Damping Elements
¨ Linearization Method
¨ Parameter identification
n Chapter 2 – review MAE80
¨ We will talk a lot during Week 2
n Review MATLAB
¨ Start eLab on MATLAB as soon as possible
¨ Install and follow help and links to tutorials
¨ All HWs will use Matlab