Identifying Data 2019/20
Subject (*) Theory of Vibration Code 730G03040
Study programme
Grao en Enxeñaría Mecánica
Descriptors Cycle Period Year Type Credits
Graduate 1st four-month period
Fourth Optional 6
Teaching method Face-to-face
Department Enxeñaría Naval e Industrial
Gutierrez Fernandez, Ruth Maria
Gutierrez Fernandez, Ruth Maria
Web http://
General description Esta materia persegue a adquisición de competencias específicas para analizar o comportamento de estruturas e elementos mecánicos sometidos a vibración e para deseñar estruturas e elementos mecánicos baixo cargas dinámicas

Study programme competencies
Code Study programme competences / results
B5 CB05 - Que os estudantes desenvolvan aquelas habilidades de aprendizaxe necesarias para emprenderen estudos posteriores cun alto grao de autonomía
B7 B5 - Ser capaz de realizar unha análise crítica, avaliación e síntese de ideas novas e complexas
B9 B8 - Adquirir unha formación metodolóxica que garanta o desenvolvemento de proxectos de investigación (de carácter cuantitativo e/ou cualitativo) cunha finalidade estratéxica e que contribúan a situarnos na vangarda do coñecemento

Learning aims
Learning outcomes Study programme competences / results
Handle the principles of vibration theory to analyze dynamic systems: response under free and forced vibration to single degrees of freedom SDOF and multiple degrees of freedom MDOF systems, harmonic load, and general type excitations. B5
Apply properly theoretical concepts not laboratory. Understand and apply some technical computing solution: numerical methods for the analysis of vibrating systems. B5
Use a rigorous language in the engineering structural dynamics in order to show and to explain information and results B5

Topic Sub-topic
Chapter 0. The following topics develop the contents set up in the verification memory. Dynamic equations. Modelling. Vibration of systems of 1 and N degrees of freedom. Buffer. Vibration of continuous systems
Chapter 1. Introduction to structural dynamics:dynamic equations and modeling. Basic concepts. Classification of vibrations. Modelling systems: stiffness, inertia, and damping elements. Mathematical models of Single Degree Of Freedom (SDOF) systems. Application of Newton's laws. Application of the principle of virtual displacements. Hamilton principle. Application of the Lagrange equations.
Chapter 2. Free vibration of SDOF system. Damping. Free vibration of undamped SDOF systems. Free vibration of viscous damped SDOF systems. Other types of damping.
Chapter 3. Response of SDOF to harmonic excitation. Damping. Response of undamped SDOF to harmonic excitation. Response of viscous damped SDOF to harmonic excitation. Complex frequency response. Vibration isolation. Force Transmissibility. Base motion. Response of SDOF due to unbalance in rotating machines.
Chapter 4. Analytical methods of solution. Response of SDOF to a general dynamic excitation Response of SDOF to special forms of excitation. Ideal step input, rectangular pulse and ramp loadings. Short-duration impulse. Unit impulse response. Classification of methods. Duhamel Integral Method.
Chapter 5. Numerical methods of solution. Response of SDOF to a general excitation. Numerical evaluation of the integral of convolution. Method of linear forces. Step by step methods. The average acceleration method. Methods of Newmark family.
Chapter 6. Continuous systems. Mathematical models of Multiple Degrees Of Freedom (MDOF) systems Continuous systems. Discrete systems: application of Newton's laws, application of the Lagrange equations. Equations of motion.
Chapter 7. Free vibration response of MDOF systems Natural frequencies and modes of vibration of MDOF systems. Free vibration response of MDOF systems. Rigid body modes of vibration. Some properties of the natural frequencies and natural modes. Scaling or normalizing. Orthogonality. Expansion theorem. Free vibration response of MDOF systems. Mode-superposition method.
Chapter 8. Forced vibration response of MDOF systems. Mode-superposition method response of undamped MDOF systems. Truncation. Damped MDOF systems. Orthogonal, modal, classic or proportional damping. Rayleigh damping. Non-proportional damping.

Methodologies / tests Competencies / Results Teaching hours (in-person & virtual) Student’s personal work hours Total hours
Laboratory practice B1 B2 B4 B5 B6 B7 B9 C4 C6 10 35 45
Supervised projects B1 B2 B4 B5 B6 B7 B9 C6 C4 12 25 37
Problem solving B1 B2 B4 B5 B6 B7 B9 C4 C6 4 14 18
Guest lecture / keynote speech B1 B2 B4 B5 B6 B7 B9 C4 C6 16 32 48
Personalized attention 2 0 2
(*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students.

Methodologies Description
Laboratory practice Methodology that allows the realization of activities of practical character, with computer, such as modelization, analysis and dynamic simulation of mechanical and structural elements.
Supervised projects Methodology designed to promote autonomous learning of students, solving a problem that involves the contents of the course and involves specific skills, under teacher supervision.
Problem solving Técnica a través da cal hai que resolver unha situación problemática específica, a partir da
coñecemento que se traballou e que pode ter máis dunha solución.
Guest lecture / keynote speech Oral lecture supplemented with the use of audiovisual means, aiming transmit knowledge and facilitate the learning within the scope of vibration analysis

Personalized attention
Laboratory practice
Supervised projects
Guidance and revision about specific problems posed at the development of the different activities proposed in the course. Revision and help when making supervised projects.

Methodologies Competencies / Results Description Qualification
Laboratory practice B1 B2 B4 B5 B6 B7 B9 C4 C6 Students must systematically attend practices. The proposed activities have to be done along the practical sessions, in order to be revised and evaluated by the teacher. The practices that aren’t developed during the practical classes, and periodically revised by the teacher will not be considered in the qualification.

The evaluation process of the laboratory lessons includes a two hour practice session, where the student solves with the computer the problems proposed by the teacher, individually.
Supervised projects B1 B2 B4 B5 B6 B7 B9 C6 C4 The projects include the theoretical and practical contents of the course. They are to be done individually. The projects will be developed during the practical sessions along the course and completed at home on the student personal work hours. The tasks will be followed and revised during the practical lessons. If the projects aren’t matured during the practical classes, nor periodically revised by the teacher, will not be considered in the qualification. 60
Assessment comments

Students, whose presence throughout the
semester where insufficient to track their work, by academic waiver or other
causes, must also develop and present practices and tutored work for their
evaluation. The follow-up of this work shall be carried out in tutoring
sessions. In this case, the process of evaluation may include in addition to
the presentation of practices and tutored work, a practice session,
individually or in group, in which the student addresses manually or with the
computer the problems raised by the teacher.


For the second chance you can present or improve
practices and tutored work. The tracking is done in tutorial sessions. The
assessment is done through presentation of practices and tutored work pending
and/or improved. The process of evaluation may include, in addition to the
presentation of practices and tutored work, a practical session, individually
or in group, in which the student addresses manually or with the computer the
problems posed by the teacher.

Sources of information
Basic Dassault Systèmes Simulia Corp. (2011). Abaqus Analysis User’s Manual. Providence, RI, USA. (1998)
R. R. Craig (1981). Structural Dynamics. John Wiley and Sons, Inc
R. Gutiérrez, E. Bayo, A. Loureiro y L.E. Romera (2009). Teoría de Estructuras III. Servicio de publicaciones de la Universidade da Coruña
S.S. Rao (2012). Vibraciones Mecánicas.Quinta Edición. Pearson Education, México.


Subjects that it is recommended to have taken before
Diferential Equations/730G03011
Theory of Structures /730G03021
FEM of Structures/730G03069

Subjects that are recommended to be taken simultaneously
Structural Typologies/730G03070

Subjects that continue the syllabus
Simulation of Mechanic and Structural Systems/730497224

Other comments

To help achieve a sustained immediate environment and meet the objective of the action number 5: "Teaching and healthy and sustainable environmental and social research" of the "Plan of action Green Campus Ferrol":


Documentary work presented in this matter:

* Should be requested in virtual format or computer support

* Will take place through Moodle, in digital format without having to print them

* Should be required on paper:

   -Not be  they used plastic

   -There will be double-side printing.                 

   -Will use recycled paper.                 

   -Prevent printing drafts.


You should make a sustainable use of resources and the prevention of negative impacts on the natural environment


(*)The teaching guide is the document in which the URV publishes the information about all its courses. It is a public document and cannot be modified. Only in exceptional cases can it be revised by the competent agent or duly revised so that it is in line with current legislation.