MEEN 617  Mechanical Vibrations – SPRING 2018
Dr.
Luis San Andrés
Last update:
January 18, 2018
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 5:30 –6:45 pm, ENPH 205
Instructor: Dr. Luis San Andrés, MEOB 117, Phone: 862
4744, LSanAndres@tamu.edu
Office
hours: T,R 4:205:20 pm, or scheduled appointment (email in
advance).
PLEASE see eCampus for
more frequent updates on handouts, worked problems, etc.
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