MEEN 617  Mechanical Vibrations – SPRING
2014
Dr. Luis San
Andrés
Last update: February 611,
2014 (added TA:
grader email)
Course Description: Theory of linear vibrations of finite &
infinite number of degree of freedom systems via Lagrange, Newtonian and Energy
approaches. Engineering applications and tools for frequency
domain analyses.
Prerequisites: MEEN 363, MATH 308.
OBJECTIVES: To provide the
fundamental analytical and numerical tools for analysis and modeling of
vibration phenomena in discrete and continuum SDOF and MDOF linear systems. Learning of advanced analytical tools and methods for experimental
identification of system parameters using recorded data, i.e., frequency domain
parameter identification methods.
Class Time: T, R 11:10 am – 12:25 pm, ENPH 205
Instructor: Dr. Luis
San Andrés, MEOB 117, Phone: 862 4744, LSanAndres@tamu.edu
Office
hours: T,R 12:45 pm1:45 pm, or by scheduled
appointment (phone
call or email in advance).
TA (Grader) ChienFan Chen (cfchen@tamu.edu)
References: Mechanical
and Structural Vibration: Theory and Applications, J. H. Ginsberg
Other: Vibrations of Mechanical and Structural Systems, L. James, Harper & Row
Structural
Dynamics, R. Craig, J. Wiley Pubs, NY, 1981.
Mechanical
Vibrations, S.S. Rao, AddisonWesley Pubs, 2^{nd}
Ed., 1990.
Finite
Element Procedures in Engineering Analysis, K. Bathe, Prentice Hall.
Dynamics in Engineering Practice, Childs. D., Lecture notes for ME363 course
(TAMU)
SYLLABUS learn about grading policies, exam schedules,
and more.
…. à Skip the syllabus and just get
the
tips to succeed.
Read/understand and practice: The lecturer when grading
your work will make an effort to READ
your solution to problems in an exam or homework. Correct usage of English
language with explanations of procedures using full sentences will make a large portion of your partial grade.
Worked problems that show a jargon of numbers without definitions and
procedures will MERIT a LOW grade, even if the (numerical) answer should be
correct.
Recommended problems from your textbook
(Be
ahead of the game. These problems may be part of your graded homework)
week 
chapter 
Problem numbers 
1 
1 
5, 8, 13, 14, 15, 20, 43, 44, 56 
2 
2 

3&4 
3 
3, 7, 19, 20, 22, 25, 50, 52 
5 
1 
Derive EOMs MODF systems: 25, 27, 30, 33, 36, 38, 39, 44, 49 
6 
4 
10,19, 21, 30, 36, 39,34, 55 
7 
5 
14, 18, 29 
8 
7 
3, 11, 43, 49 
10 
6 
3, 9, 11, 14, 15, 28, 38, 54 
1114 

More to follow 
Class Notes (handouts
& worked examples)
recommendation: Download handouts as
needed. The lecturer may update the notes as the semester progresses.
Check syllabus to realize which handout(s) will be
used in a given week.
warning:
Pps (power point slide shows) may not work on your Mac,
Tablet, etc.
0 
Introduction to the analysis of vibrations in mechanical
systems. Introduction
to motion in mechanical systems. Definition of design, analysis, and testing.
Steps in Modeling. Continuous and lumped parameter systems. Second Order
Systems and differential equations of motion. Definitions of Free and Forced
Responses. The purpose of analysis and relevant issues to resolve. 
The
technical memorandum 
1 
Modeling of mechanical (lumped
parameter) elements Fundamental elements in mechanical
systems: inertias, stiffness and damping elements. Equivalent spring
coefficients and associated potential energy. Equivalent mass or inertia
coefficients and associated kinetic energy. Equations of motion of a rigid
body in a plane. Equivalent damping coefficients and associated dissipation
energy. Types of damping models (linear or viscous and nonlinear). 
Appendix A. Equivalence of principles of conservation
of mechanical energy and conservation of linear momentum (Power Balance) 

Deriving
the fundamental EOM Linear and nonlinear springs.
Te choice of coordinate system. Linearization. pps file 
Appendix B. Linearization 

Examples derivation of
EOM (1 DOF) E1. Springs
& blocks & pulley pps file 
Appendix C. Derivation of equations of motion for a multiple
degree of freedom system 

Appendix D. Note on assumed modes (one DOF and MDOF) 
Appendix E. Vibration sensors and their
applications 
Vibrations of single degree of freedom (SDOF) linear systems
2 
Dynamic
response of second order mechanical systems (1 DOF
systems) 

Handout 2a Free response to
initial conditions: viscous and coulomb damping systems. Forced response:
impulse and step loads. Examples
Transient response 

Handout
2b Periodic forced response and
Frequency response function of second order systems. Examples Periodic Forced
Response 

Handout
2c Interpretation of forced
periodic response. Transmissibility (forces transmitted to base or foundation). Frequency Response due to base
or foundation motions 

Handout
2d Fourier series. Forced response
to a periodic forced excitation. Example square wave load 

Appendix F Uses of
the FRF on the design of mechanical systems 


3 
A brief introduction
to the Discrete Fourier Transform 
Some
interesting problems for Exam I

WORKED EXAMPLES: 1 DOF systems 
Derivation
of Equations of Motion for Simple Mechanical Systems. TAKEN from ME363
course 




Translation 






Rotation & Rolling w/o slipping 


How to
measure mass moments of inertia 






4 


Work and Energy – Single particle. Constraints – degrees
of freedom. Principle of virtual work. D’Alembert
Principle. 
6 
Numerical
solution of EOM for a single degree of freedom (SDOF) system 

The basics of numerical integration. Issues on convergence
and numerical stability 
Vibrations of multiple degree of freedom (MDOF) linear systems
7 


Free & Force Vibrations of undamped MDOF
systems. Orthogonality properties of natural modes. Rayleigh energy methods. Mode superposition
(displacement and acceleration methods) methods. 
10 


Application
of knowledge. Design and limitations. 
8 


The concept of proportional damping and its use in MDOF systems. Damped
natural frequencies and damping ratios. Limitations of proportional damping. Mode displacement & mode
acceleration methods. 
9 


Methods for calculation of Eigen values and
eigenvectors. (Selfstudy guide for the inquisitive
student) 
11 


Statespace equations, modal coordinates and orthogonality properties, relations to undamped model 
12 


Fundamentals and FEM matrices for bars and beams.
Assembly and solution of global system of equations. 
13 
Numerical
methods for the dynamic response of MDOF damped systems 

Stability analysis and minimum time step. Average
acceleration methods 
14 


Free vibrations of elastic bars and beams. Properties
of normal mode functions. Forced response 

OLD: RayleighRitz energy Methods (assumed modes method) – Handwritten notes 
15 


Time
and frequency domain methods. Curve fits to impedances and the instrumental
variable filter method. 
RESOURCES on the web (there are a million more, below is just a minuscule
sample)
FREE resources: Real Life Engineering Examples in Dynamics
www.EngageEngineering.org www.engineeringexamples.org
MIT Open
Courseware
U
Vibration Institute and Sound and Vibrations magazine (free)
S&V
article: 48 cases of
intriguing machinery problems
Simulators: simple mechanical springmass Double
Spring Oscillation
Resonance –
Harmonic motion
Videos – Youtube and
others
The
Tacoma bridge collapse (a classic)
Resonance 1DOF mechanical
system
resonance – breaking a glass
Pendulum
waves
Wet
dogs – an application of FRF
resonance  membranes
(mode shapes)
Vibration absorber
Tunable vibration
absorber
The handouts and textbook
used in this course are copyrighted. By "handouts," I mean all
materials generated for this class, which include but are not limited to
syllabi, quizzes, exams, lab problems, inclass materials, review sheets, and
additional problem sets. Because these materials are copyrighted, you do not
have the right to distribute freely the handouts, unless the author expressly
grants permission.
Send all comments,
corrections or questions to LSanAndres@tamu.edu