MEEN 617 - Mechanical Vibrations
Dr.
Luis San Andrés (From 1993-2022
taught this course 17 times, once every other year)
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.
References:
Mechanical and Structural Vibration:
Theory and Applications,
J. H. Ginsberg
Vibrations of Mechanical and
Structural Systems,
L. James, Harper & Row
Structural Dynamics, R. Craig, J. Wiley Pubs, NY, 1981.
Mechanical Vibrations, S.S. Rao, Addison-Wesley Pubs, 2nd Ed., 1990.
Finite Element Procedures in
Engineering Analysis,
K. Bathe, Prentice Hall.
Dynamics in Engineering Practice, Childs. D., Lecture notes for ME363 course
(TAMU)
Lecture Notes
(handouts & worked examples)
The content below is copyrighted by Dr. San Andrés. Hence, you do not have the right to distribute
freely the lecture notes, problems and their solutions, unless the author
expressly grants permission.
warning: Pps
(power point slide shows) may not work on your Mac, Tablet, etc.
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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 |
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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) |
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Deriving
the fundamental EOM Linear
and nonlinear springs. Te choice of coordinate system. Linearization. pps file |
Appendix B. Linearization |
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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 |
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Appendix D. Note on assumed
modes (one DOF and MDOF) |
Appendix
E. Vibration sensors and their
applications
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Vibrations of single degree of freedom (SDOF)
linear systems
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Dynamic
response of second order mechanical systems (1 DOF
systems) |
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Handout 2a Free
response to initial conditions: viscous and coulomb damping systems. Forced
response: impulse and step loads. Examples
Transient response |
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Handout 2b Periodic
forced response and Frequency response function of second order systems. Examples Periodic Forced
Response |
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Handout 2c Interpretation
of forced periodic response. Transmissibility (forces transmitted to base or
foundation). Frequency
Response due to base or foundation motions |
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Handout 2d Fourier
series. Forced response to a periodic forced excitation. Example square wave load |
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Appendix F Uses of the
FRF on the design of mechanical systems |
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A brief
introduction to the Discrete Fourier Transform |
Some interesting
problems for Assessment
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WORKED
EXAMPLES: 1 DOF systems |
Derivation
of Equations of Motion for Simple
Mechanical Systems. |
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Translation |
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Rotation & Rolling w/o slipping |
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How to
measure mass moments of inertia |
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Work and Energy – Single particle.
Constraints – degrees of freedom. Principle of virtual work. D’Alembert Principle. |
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Numerical
solution of EOM for a single degree of freedom (SDOF) system |
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The basics of numerical integration. Issues on
convergence and numerical stability |
Vibrations of multiple degree of freedom (MDOF)
linear systems
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Free & Force Vibrations of undamped
MDOF systems. Orthogonality properties of natural modes. Rayleigh energy
methods. Mode
superposition (displacement and acceleration methods) methods. |
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Application
of knowledge. Design and limitations. |
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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. |
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Methods for calculation of Eigen values and
eigenvectors. (Self-study guide for the inquisitive
student) |
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State-space equations, modal
coordinates and orthogonality properties, relations to undamped model |
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Fundamentals and FEM matrices for
bars and beams. Assembly and solution of global system of equations. |
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Numerical
methods for the dynamic response of MDOF damped systems |
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Stability analysis and minimum time
step. Average acceleration methods
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Free vibrations of elastic bars and
beams. Properties of normal mode functions. Forced response |
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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 spring-mass 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
Send all
comments, corrections or questions to LSanAndres@tamu.edu