MEEN 363/502 – Dynamics and Vibration FALL 2007
NOTE: THIS URL SITE IS TEMPORARY UNTIL WEBCT is READY. Created
August 27, 2007
Course Description:
Application of Newtonian methods to model dynamic systems
with ordinary differential equations; system of rigid bodies; solutions of
models; interpreting solutions/performance measures; vibrations; energy
methods.
Prerequisites: ENGR 221, MATH 308, MEEN 357 or CVEN 302 or
registration therein, CVEN 305 or registration therein
Course Objectives:
To introduce fundaments for modeling mechanical systems, to derive differential
equations of motion (kinetics and kinematics), find (predict) the dynamic
response of systems using mathematical analysis, and to provide knowledge for
practice in understanding mechanical systems behavior.
Class
Time: TR 12:45 1:35 pm FERM 303, T 5:307:20 pm RICH
101
Instructor: Dr. Luis San Andrés,
ENPH 118, Phone  8450160, 8624744, LsanAndres@mengr.tamu.edu
Office
hours: T, R: 11:30 am 12:30 pm, T: 4:005:00 pm, or
by appointment (phone call or email in advance).
TA: Benjamin Gronemeyer:
References: Dynamics in Engineering
Practice, 7^{th}
Edition (only), D.W. Childs, TAMU Bookstore
MEEN 363 Lecture Notes, D. Childs, download
from http://elearning.tamu.edu
Engineering
Mechanics, Vol. II: Dynamics,
Meriam, J.L., and L. Kraige, 1992, J. Wiley Pubs., III.
Advanced
Engineering Dynamics, Ginsberg, J.H., 1995,
Theory
of Vibration with Applications, Thompson, W.T., and
Dahleh, M.D., 1993, Prentice Hall.
In addition, Dr. San
Andrés distributes via email and through the class web site (webCT) additional notes (appendices)
and solution to problems & past exams in pdf format as they become
available.
Class description 
DOWNLOAD material 
Comments 
Read this document. What you don’t learn in
school and industry demands of you! The importance of communication skills,
team work, longlife learning, and the RIGHT ATTITUDE to excel in your
professional career, 

Lectures to be covered in Exam 1 

Problem statements SET 1 
Full list of problem statements for lectures
14 (also available in your text book) 
Partial Solutions SET 1 
Partial solutions for problem set 1 
Recommended homework problems (SET 1) 
Lecture 1: Cartesian and polar coordinates:
2.4ac, 2.5ac,2.10ab, 2.11ab Lecture 2: More polar coordinates: 2.12ab,
2.14ab, 2.15ab Lecture 3: Normaltangential (path) coordinates:
2.17ab, 2.18ab, 2.19ab Lecture 4: 2.5ac, 2.13ac, 2.20ac 
Cheat
Sheet for exam 1 – A must to have and bring to exam. No substitutes allowed 


EXAMPLES
of PROBLEMS FROM PAST EXAMS (*) 
EX:
Kinematics of rectilinear motion 

EX:
Analysis of motion and coordinate system (right handed) 

EX:
Same as above but ask for velocities at a location where coordinate system is
LEFT HANDED 

EX:
A simple problem for kinematics of motion and coordinate system 



More elaborate problem – AN ENGINEERING APPROACH 
A
comprehensive problem describing fully the kinematics of roller skating. Learn
and discuss motion over a given time, not merely at an instant. 
(*) These are representative problems asked in
previous classes. It does not mean that you will have an identical problem in
your exam. Please NOTE solution strategy as well as descriptions (written
statements in full sentences) of solution procedure. The lecturer stresses comprehensive
solutions, not merely equations and numbers.
EXAM SCHEDULE:

Topics 
Exam covers Lecture Sets 
Date 
Time 
Place 
Planar kinematics for particle motion 
14 
09/06 

SCOATES 208 

2 
Planar kinetics of 1DOF lumped mechanical systems 
512 
09/27 

RICH 114 
3 
Planar kinetics of 2DOF lumped mechanical systems 
1317 
10/11 

RICH 114 
4 
Planar kinematics for rigid links (mechanisms) 
1922 
10/25 

SCOATES 208 
5 
Planar kinetics of rigid bodies and elastic structures 
2331 
11/15 

RICH 114 
6 
Final
Exam 
517, 2335 
12/12 
810
am 
FERM 303 
Grading:
Five partial exams and a comprehensive final exam. Exams cover material
specified on the SYLLABUS and PERFORMANCE OBJECTIVES. Practice problems
(recommended homework) assigned but not
graded. No makeup exams will be given unless the student has an acceptable and
verifiable excuse (see http://studentrules.tamu.edu/rule7.htm)
and notified the lecture instructor in advance. (If instructor is not in
his office leave a [phone or email] message and return address or phone number).
Exam
1 10%
Exams
2,3,4,5 60
% (15 % each)
Quizzes(7%)+
Assignments (8%)= 15% (MATLAB/MATHCAD/MS Excel Spreadsheets)
Final
Exam 15% (Final is neither optional nor will be
waived)
100%
The
course letter grade is assigned from your numerical grade based on TAMU policy;
i.e., A = 10090, B = 8980, C 7970, D
= 6960, F = 590. There is no “curve” for the exams, assignments or the class
final grades.
Notes:
November
2: Last day for all students to drop course with
no penalty (Qdrop).
All background material on prerequisites is the
responsibility of each student (See page
8 and webCT).
MEEN 363 Class Syllabus FALL 2007
Acronyms
below: M: mass, K: stiffness, C: damping, ODE:
ordinary differential equation, EOM: equation of motion, SEP: static
equilibrium position, DOF: degree of freedom, FBD: free body diagram, CE:
characteristic equation, CME: Principle of Conservation of Mechanical Energy
wk 
dates 
Lecture
Material (subject to revision) 
Topic/
Reading Assignment 


PART 1. PLANAR KINEMATICS OF PARTICLES 
PART
1 
1 
08/28 Tuesday 
Class description: Syllabus, Policies, Grading, Exam, etc. L1. Particle kinematics in a plane using Cartesian coordinates. Review
of vector and matrix algebra. Coordinate transformations: relationships
between components of a vector in two coordinate systems. Example: projectile
motion. L2. Particle motion in a plane using polar coordinates. Radial and
tangential unit vectors and their time derivatives. Components of velocity
and acceleration vectors in polar coordinates. Examples of coordinate
transformations: polar to Cartesian or vice versa. 
CH
2.1 thru 2.7 Lecture Set L1L4 (39 pages) 
2 
09/04 
L3. Particle motion in a plane using path coordinates. Path radius of
curvature, normal and tangential unit vectors and their time derivatives.
Components of velocity and acceleration vectors in path coordinates. Examples
of coordinate transformations: path to Cartesian or polar or vice versa. L4. Expressing motion in different coordinate systems: moving between
Cartesian, polar and path coordinates. Physical units: conversions &
understanding. 
Exam 1: 09/06 


PART 2. PLANAR KINETICS OF LUMPED MASS SYSTEMS 
PART
2 
2 
09/06 
L5. 
(1 DOF systems)
CH
3.1 thru 3.2d Lecture Set L5L8 (41 pages) 
3 
09/11 
L6. Derivation of EOM from CME: kinetic + potential energy = Work.
Definition of static equilibrium position (SEP) and finding small amplitude
motions (perturbation) about SEP. EOM including external (nonconservative)
forces. Necessary condition for static stability (K>0). L7. Examples of simple MK systems
and interpretation of physical responses. Nonlinear EOM for simple pendulum
and linearization for small amplitude motions: evaluation of natural
frequency. Mechanical viscous damping element (C): constitutive relationship
for energy dissipation element. Derivation of fundamental EOM for
massspringdamper (MKC) system. Solution of ODE for arbitrary initial
conditions: system free and constant force response. The concept of damped
natural frequency and viscous damping ratio. Review of physical units.
Necessary condition for dynamic stability (C>0). Types of system
responses: underdamped, overdamped and critically damped. L8. Transient (1) response of
simple MKC systems. Forced response
including effect of initial conditions. Superposition of solutions:
homogeneous + particular: formulas for forcing functions of the forrn = c + at + b t^{2}. 
CH
3.1 thru 3.2d Lecture Set L9L12 (35 pages) 
4 
09/18 
L9. More Transient Response (2)
Solutions  Base excitation L10. Transient Response (3)
Solutions  Base excitation: Ramp loading L11. Harmonic Excitation: solution
to equation of motion = transient + permanent or steadystate components.
Definition of system frequency response. Effect of excitation frequency and damping
ratio on amplitude (amplification factor) and phase lag of steadystate
response. Regimes of operation, the concept of system resonance. 
CH 3.3a 
5 
09/25 
L12. EOM for MKC system due to rotating imbalance. Physical nature of
imbalance, magnitude of excitation force and its importance in rotating
system dynamics. System response to mass imbalance. Effect of excitation
frequency and damping ratio on amplitude (amplification factor) and phase lag
of steadystate response. Interpretation of regimes of operation. Example of
application to reduce vibration amplitude in a mechanical structure. Pulleys:
EOMs and equations of constraint. Applications to MKC systems. App G. Important design and
operation issues/questions applying frequency response functions in simple
MKC mechanical systems. 
Exam 2: 09/27 RICH 114 (L5L12) 


Syllabus  continued 

w 
Dates 
Lecture
Material (subject to revision) 
Topic/ Reading
Assignment 


PART 2. PLANAR KINETICS OF LUMPED 2DOF SYSTEMS 
PART
2 
5 
09/27 
L13 EOMs for 2DOF mechanical MKC systems including base excitation. Free
body diagrams (FBDs), selection of coordinates, and establishment of forces
for mechanical elements connecting masses. Application of 
(2 DOF systems) CH
3.5 Lecture Set L13L17 (74 pages) 
6 
10/02 
L14.Eigenanalysis of free response of 2DOF undamped mechanical systems.
Fundamental response function and derivation of system characteristic
equation (CE). Solving CE: Eigenvalues (natural frequencies) and eigenvectors
(mode shapes) of system, physical interpretation of natural frequencies and
mode shapes. Transformation of coordinates to modal space via modal matrix of
eigenvectors and uncoupling of system EOMs. L15. Transient response of 2DOF undamped mechanical systems. Prediction
of system transient (free) response using modal (natural) coordinates.
Transformation to physical space. Examples of analysis and interpreting
solutions (system responses). Examples of systems with rigid body motions:
interpretation of null or zero natural frequency. Insights into the analysis
of systems with viscous damping: lightly damped systems and systems with
proportional damping. The concept of modal damping ratio. Application to the
analysis of a 2DOF system response after collision. L16. Transient response of 2 DOF MKC system with proportional damping.
Example of usage of modal coordinates – Collision problem 
CH
3.5 
7 
10/09 
L17 Forced periodic response of 2DOF undamped mechanical systems. Steady
state solution to harmonic force excitation using (a) modal coordinates and
(b) direct substitution. Interpretation of regimes of operation: below, around
and above natural frequencies. Effect of excitation frequency on amplitude
(amplification factor) and phase lag of steadystate response. Insights into
the forced periodic response of 2DOF systems with viscous damping. Examples: vibration absorber 
Exam 3: 10/11 RICH 114 L13L17 


PART 3. PLANAR KINEMATICS OF RIGID BODIES 
PART
3 
7 
10/11 
L19 Introduction to rigid body motion in a plane. Concept of rigid body.
Rotation about a fixed axis and angular velocity vector. Vectors expressed
relative to fixed coordinate system (X,
Y) and coordinate system (x, y)
attached to rigid body. Transformation of coordinates. Time derivatives of
unit vectors in rotating coordinate system (x, y). Derivation of velocity and acceleration vectors in two
coordinate systems. Derivation of equations for velocity and acceleration
vectors for two material points in rigid body. Geometrical interpretation of
vector terms forming velocity and acceleration vectors. Example of
application to pulley system. 
CH
4.1 thru 4.6 Lecture Set L19L23 (44 pages) 
8 
10/16 
L20 Rolling without slipping. Physical description of motion for wheels
and gears. Fundamental constraint: geometrical development and visual demo.
Derivation of equations for velocity and acceleration of material points in a
rolling wheel: geometricdifferential approach and vector analysis approach.
Examples or rolling inside and outside of curved surfaces.L21. Planar kinematics of mechanisms:
geometricdifferential approach and vector analysis to determine velocity and
acceleration of desired material points. Identifying constraints and DOF in
linkage problems. 
CH
4.1 thru 4.6 
9 
10/23 
L2223. Examples of threebar mechanism, slidercrank mechanism, etc. Dr. San Andres away 
Exam 4: 10/25 Scoates 208 L19L22 


Syllabus
 continued 

w 
Dates 
Lecture Material
(subject to revision) 
Topic/
Reading Assignment 


PART 4. PLANAR KINETICS OF RIGID BODIES (1 DOF) 
PART 4 
10 
10/30 
L24. Rigid body: inertia properties (center of mass and mass moments of
inertia): development of equations. Parallel axis formula and restrictions to
its application. Example: determining mass center and moment of inertia of a
mass assembly. Steps in procedure. Use of published Tables containing useful
mass properties L25. Motion of a rigid body on a plane: derivation of force and moment
EOMs in Cartesian coordinates. Vector analysis. Reduced forms for the moment
EOM: about mass center, about fixed point in inertial space. 
CH 5.1 thru 5.5 Lecture Set 2427 (35 pages) 
11 
11/06 
L26. Kinetic energy of rigid body in planar motion (translation and
rotation). Examples of fixed axis rotation: derivation of equations from CME:
rotor on bearings, torsional vibrations, pulleys connected by belts. L27.Nonlinear EOMs for compound pendulum, including damping: FBDs,
application of force and moment equations. EOM derived from CME. Equilibrium
and small amplitude motions about SEP, stable and unstable configurations.
Example: Nonlinear EOMs for a swinging plate. L28.Nonlinear EOMs for compound pendulum connected to (linear) spring and
viscous damper: FBDs, application of force and moment equations, geometric
nonlinearities at linear element (K, C) connections, linearization of EOM
about SEP. EOM derived from CME. Preload in spring elements. Finding natural
frequencies and motions for bars and plates connected to springs and dampers. 
CH
5.5 ac 
12 
11/13 
L29 Rigid body motion with prescribed acceleration of pivot support
point: FBDs, application of force and moment equations. Equations for
reaction forces, interpretation of results.
L30. Motion of cylinders rolling w/o slipping. FBDs, identification of
forces and rolling constraint, derivation of EOM. Definition of Coulomb (dry
friction) forces. When will the cylinder slip and not roll? EOM derived from
CME. EOM for imbalanced cylinder rolling down an inclined plane, oscillations
of a half cylinder on a flat plane: prediction and measurement of natural
frequency. L31.More examples of rolling motion: cylinder restrained by spring,
cylinder rolling inside a concave surface: FBDs, identification of forces and
rolling constraint, derivation of EOM, linearization and identification of
natural frequency. EOM derived from CME. Example: pulley assembly connected
to spring element. 
CH
5.6 ad Lecture Set 2831 (25 pages) Exam 5: 11/15 RICH 114 L23L31 


PART 5. PLANAR KINETICS FOR MULTIBODY SYSTEMS 
PART
5 
13 
11/20 
L32.Torsional vibrations of rotating assemblies. Methods to estimate mass
moments of inertia from oscillating assemblies. Motion of disks connected
with flexible shafts: FBDs, identification of elastic moments, derivation of
multiple DOF EOMs, eigenvalue analysis and determination of natural
frequencies and mode shapes. Interpretation of natural modes of motion. Thanksgiving 11/2223 
5.6c Lecture Set 3235 (28 pages) 
14 
11/27 
L33. Lateral vibrations of mass connected to an elastic beam. Brief review
of lateral deflections of elastic beams. Definition of lumped stiffness (K)
for cantilever beam. Derive EOM for mass supported at beam end:
identification of system natural frequency. Analysis for development of beam
stiffness matrix from force/moment relationships to beam
displacement/rotation. Applications to building and bridge frames – 2DOF
problems  eigensolutions L34. 2DOF examples: vehicle suspension system, rotorbearing
system, rolling w/o slipping. FBDs,
identification of constraints and reaction forces, geometric approach to
derive mechanism kinematics, derivation of EOMs from rigid body force and
moment equations. Eigenanalysis – natural frequencies and interpretation of
mode shapes. 
CH
5.6 bd

15 
12/04 
L35. Nonlinear 2DOF systems: A swinging bar supported by cord, a double compound pendulum. FBD,
identification of constraints and reaction forces, derivation of nonlinear
EOMs from rigid body force and moment equations, matrix form for numerical
evaluation, linearization for small amplitude motions about SEP. Closure. 
CH
5.6 d, CH 5.8 Last day of class, Tuesday 12/04 
16 
12/12 
Wed,
DECEMBER 12, 810 am FINAL Exam Content: L5L17, L23L35 
FINAL EXAM FERM 303 