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,

Office hours: T,R 4:20-5:20 pm, or scheduled appointment  (e-mail 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 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.

                                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)



Problem numbers



5, 8, 13, 14, 15, 20, 43, 44, 56






3, 7, 19, 20, 22, 25, 50, 52



Derive EOMs MODF systems: 25, 27, 30, 33, 36, 38, 39, 44, 49



10,19, 21, 30, 36, 39,34, 55



14, 18, 29



3, 11, 43, 49



3, 9, 11, 14, 15, 28, 38, 54



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.


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


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

Springs in cart (rel motion)  pps file      (more examples below)


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



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
                         Convolution integral and response to arbitrary loading  Example using Convolution


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




A brief introduction to the Discrete Fourier Transform
A little about Discrete Fourier Transforms (DFT) and their application to vibrations. Examples (zip)

       Some interesting problems for Exam I



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


USEFUL FORMULAS for prediction of 1DOF responses






Spring & masses 1   

Spring pulls a block    springs in series & block


Spring & mass & pulleys

Sensor in a rocket


Rotation & Rolling w/o slipping



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

Gondola swinging 


Elevator System start up

Pendulum & base motion








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
Lecture notes, Prof. Dara Childs ME 613 L13 and L14  (reproduced with permission from the author)



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



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



The dynamic vibration absorber


Application of knowledge. Design and limitations.



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. Mode displacement & mode acceleration methods.



 Numerical evaluation of natural modes and frequencies


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



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



Finite element modeling of mechanical systems


Fundamentals and FEM matrices for bars and beams. Assembly and solution of global system of equations.



Numerical methods for the dynamic response of MDOF damped systems


Stability analysis and minimum time step. Average acceleration methods



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



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 


Brown University  UG dynamics and vibrations


MIT Open Courseware   

Dynamics and Vibration


Structural mechanics


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


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.

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