MEEN 617 - Mechanical Vibrations – SPRING 2013
Dr. Luis San Andrés

Last update: March 26, 2013 (updated Handout 12: FE in vibrations)

 

Course Description:    Linear Theory of vibrations of Single and Multiple degree of freedom (DOF) systems via Newtonian and Lagrangian formulations, and with emphasis on analytical methods and computer applications. 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 204

 

Instructor: Dr. Luis San Andrés, MEOB 118, Phone: 862 4744, LSanAndres@tamu.edu

Office hours: T, 12:30 pm-1:30 pm, R 10:00 am-11:00 pm, or by scheduled appointment  (phone call or e-mail in advance).

 

References:       Mechanical and Structural Vibration: Theory and Applications, J. H. Ginsberg

Other:                 Vibrations of Mechanical and Structural Systems, L. James, Harper & Row Pubs., N.Y., 1989.

                        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.

 

SYLLABUS  learn about grading policies, exam schedules, and more

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

Class Notes (handouts & worked examples)

recommendation: Download HDs as needed. Lecturer may update the notes as the semester progresses.
check syllabus to realize which handout(s) will be used in a given week

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 the relevant issues to resolve.

 

The technical memorandum
How to present/report your work in engineering (and this class).

(new) Appendix A. Equivalence of principles of conservation of mechanical energy and conservation of linear momentum (Power Balance)

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).

 


(new) Appendix B. Linearization
  Example – linear and nonlinear springs & derive EOM

Appendix C. Derivation of equations of motion for a multiple degree of freedom system

 

Appendix D. Note on assumed modes (one DOF and MDOF)

 

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.

 

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

 

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. Response to a unit impulse. Convolution integral and response to arbitrary loading

 

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

 

 

WORKED EXAMPLES: 1 DOF systems

Derivation of Equations of Motion for Simple Mechanical Systems. TAKEN from ME363 course

 

Translation

 

 

Spring & masses 1    Spring & mass & pulleys

Spring pulls a block    springs in series & block

 

Springs in cart (rel motion)  Springs & blocks & pulley

Sensor in a rocket

 

Rotation & Rolling w/o slipping

 

 

How to measure mass moments of inertia
Cylinder rolling inside a pipe

Gondola swinging  Pendulum & base motion

 

Elevator SYSTEM start up

 

NEW

USEFUL FORMULAS for prediction of 1DOF responses

Examples Transient response

Examples Periodic Forced Response

 

 

4

Elements of analytical dynamics

 

Work and Energy – Single particle. Constraints – degrees of freedom. Principle of virtual work. D’Alembert Principle. Hamilton Principle. Lagrange’s equations of motion.
Examples: 2DOF- cargo on wheels, 2DOF: rolling-pendulum

 

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

Undamped Modal Analysis of MDOF systems

 

Free & Force Vibrations of undamped MDOF systems. Orthogonality properties of natural modes. Rayleigh energy methods. Mode superposition (displacement and acceleration methods) methods.
Examples:  ANALYSIS-two elastic modes, ANALYSIS-one rigid body mode, ANALYSIS: BEATING

 

10

The dynamic vibration absorber

 

Application of knowledge. Design and limitations.

 

8

Modal Analysis of MDOF systems with Proportional Damping

 

The concept of proportional damping and its use in MDOF systems. Damped natural frequencies and damping ratios. Limitations of proportional damping. Appendix: Mode acceleration

 

 

9

 Numerical evaluation of natural modes and frequencies

 

Methods for calculation of Eigen values and eigenvectors. (Self-study guide for the inquisitive student)

 

11

Modal Analysis of MDOF Systems with Viscous Damping

 

State-space equations, modal coordinates and orthogonality properties, relations to undamped model
Addendum: Direct periodic force response of MDOF systems

 

12

Finite element modeling of mechanical systems new March 2013)

 

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

Dynamic response of continuum systems

 

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

 

OLD: Rayleigh-Ritz energy Methods (assumed modes method) – Handwritten notes

 

15

Identification of parameters in mechanical systems

 

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 

 

Brown University  UG dynamics and vibrations

 

TAMU-ME Mechanical Vibrations    (Luis San Andrés)

 

MIT Open Courseware   

Dynamics and Vibration

Dynamics

Structural mechanics

Tribology

Advanced Fluid Mechanics

 

U South Florida Numerical methods

 

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

University of Colorado Interactive Simulators
Resonance – Harmonic motion 

Pendulum lab

Wave on a string

 

Kettering Univ Acoustics and Vibration Animation  (vibration absorber)

 

Videos – Youtube and others
The Tacoma bridge collapse  (a classic)

Resonance 1DOF mechanical system

Storage-flutter

resonance - destruction

resonance – breaking a glass
Pendulum waves

Wet dogs – an application of FRF
resonance - membranes (mode shapes)
Vibration absorber
Tunable vibration absorber

Rube Goldberg machine

 

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, in-class 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