Advanced Dynamics and Vibration
MET 4124- Vibrations and Advanced Dynamics (3-0-3)
Professor: Simin Nasseri
Office: Q231 (office hours)
Web Page: http://met.spsu.edu/snasseri
If you wish to see me at times outside my office hours, you can send me an e-mail to arrange an appointment.
Any use of my course materials on any other website or networked computer environment
for any purpose is prohibited. The materials on this website are copyrighted and any
unauthorized use of any materials may violate copyright and other laws (P&P 603). Other
instructors can only use the lecture notes by permission and they should not add their
names to them.
Course introduction/ Objectives:
Theory of mechanical vibrations with applications to machinery and the kinematics and kinetics of three dimensional motion of rigid bodies are covered. Conventional and computer methods are used.
Prerequisites: MATH 2306 (ODE), ENGR 3122 (Dynamics)
Course Text/ Software:
Mechanical Vibrations by William J. Palm, Published: Hoboken, NJ : John Wiley, c2007. ISBN: 9780471345558
Book review: This is a very good book on Mechanical Vibrations. Palm takes the reader into a systematic exposition of the theory of mechanical vibrations. Showing how this can be understood in terms of the basic physics. He walks through progressively more intricate cases, starting with the simplest of systems with 1 degree of freedom. The book is positioned as a text for an undergrad course, with numerous problem sets and chapter summaries.
Also, he chooses MATLAB in order to give numerical methods that can be applied to
various problems. Separate MATLAB sections at the end of most chapters show how to
use the most recent features of this standard engineering tool, in the context of
solving vibration problems. Students are advised to buy the student version of the
software which can be bought online. This is OPTIONAL. I recommend that you buy the
DVD and don’t download the materials.
Other suggested books:
“An Introduction to Mechanical Vibrations”, Steidel, R.F., 3rd edition, John Wiley & Sons, 1989.
“Mechanical Vibrations”, Kelly, S.G., Schaum’s Outline Series in Engineering, McGraw-Hill, 1996.
Course grade determination:
Your grade in this course will be determined from your performance on quizzes, projects, and tests. The main emphasis of the course is on gaining practical skills so it can be used for solving real engineering problems.
o Mini-tests 30%
o 3 Tests 50%
o Project 20%
[90 - 100% = A, 80 - 89% = B 70 - 79% = C 60 - 69% = D Below 60% = F]
Course content- Topic coverage:
Table of Contents
Chapter 1- Introduction to mechanical vibration (page 1)
Chapter 2- Models with one degree of freedom (p 59)
Chapter 3- Free response with a single degree of freedom (p 115)
Chapter 4- Harmonic response with a single degree of freedom (p 203)
Chapter 6- Two-degrees-of-freedom systems (p 362)
Chapter 8- Matrix methods for multi-degree of freedom systems (p 486)
By the end of this course, students will be able to:
(1) Write and solve the differential equations of motion of a mechanical system to determine the natural frequencies and response to free vibrations and to external periodic forces.
(2) Understand the various damping models and their effects on system behavior.
(3) Use energy methods to solve for the system vibration behavior.
(4) Solve for the transient vibration response and the effect of initial conditions.
(3) Understand the matrix methods and other numerical approaches to solve for the vibration characteristics.
Chapter 1- Introduction to Mechanical Vibration Excluding: Pages 22-26 and 36 to the end
·Simple harmonic motion:
o Spring elements
o The solution for simple harmonic motion
o Displacement- velocity and acceleration
o Shock absorber
o Wave on a string (understanding the components of a wave)
Chapter 2- Models with one degree of freedom Review pages 67 to 87 and recall Dynamics concepts
Chapter 3- Free response with a single degree of freedom Start from page 12. Also omitted sections are: page 127 to 131 (linearization) and graphical interpretation
o Damped free oscillation
o Solution for free undamped oscillation (simple harmonic motion)
o Solution for free damped oscillation , Check the viscously damped equations here (Play here!)
o Example of Highway Crash Barrier
o Car crash test 1
o Car crash test 2
o Understand Rayleigh’s method by reading this document on a simple pendulum
· Summary of the equations
Chapter 4- Harmonic response with a single degree of freedom (Forced Harmonic Responses)
o Internet source
o Aircraft Engine Vibration (check the noise it produces and how it changes at higher frequencies: Transient and Steady State solutions)
o Beating (Examples: Condensate pumps , Pumping station, Microwave beat frequency)
o Free & Harmonic Motions
o Chinook Helicopter ground resonance test
o Top ten Crosswind and Scary Aircraft Landings (also: Boeing 767 Windy Approach)
o Flutter Tests of a Lockheed Electra Model (Explanation (Video))
o SolidWorks/COSMOS Vibration/Resonance/Frequency Simulation
o Base Excitation (My Method)
o Rotating Unbalance (Rotor Excitation)
Chapter 6 and 8 combined: Coupled Oscillators
· Linear Algebra: Matrix operation (Gaussian Elimination)
· Two-degrees-of-freedom Examples:
o A cool double pendulum (good to make and play with!)
· Two-Degrees-of-Freedom Systems- Part 1
· Two-Degrees-of-Freedom Systems- Part 2 (Last page is added and page 8 is modified à omega1) Check the HW solution here
o Coupled Oscillators (consider case 1: equal positive displacements, case 2: equal positive and negative displacements for both masses and case 3: just one positive displacement for one mass)
· Two-Degrees-of-Freedom Systems- Part 3 (Example 6.1-7)
· Two-Degrees-of-Freedom Systems- Part 4- Forced Response
· Multiple Degree-of-Freedom Example (from Efunda)
Video clip 1
Video clip 2
More video clips:
o Simple Harmonic Motion I, Demonstrating that one component of uniform circular motion is simple harmonic motion. View
o Simple Harmonic Motion II, Illustrating and comparing Simple Harmonic Motion for a spring-mass system and for a oscillating hollow cylinder. View
o Damped Simple Harmonic Motion, The damping factor may be controlled with a slider. The maximum available damping factor of 100 corresponds to critical damping. View
o Driven Simple Harmonic Motion, A harmonic oscillator driven by a harmonic force. The frequency and damping factor of the oscillator may be varied. View
o Coupled Harmonic Oscillators, Two simple pendulums connected by a spring. The mass of one of the pendulums may be varied.