This subject, called Applied Mechanics, is devoted to establish the basic knowledge on the Dynamics of rigid bodies. Every concept developed in this subject is in the frame of Classical Mechaics, i.e. without any relativistic consideration, so as to be a reference for Mechanical Engineering topics. From this starting point, the subject is presented in two clearly different parts: first, a geometrical and vector approach to Dynamics derivated directly from Newton postulates; and a second one, with a more mathematical focus, following the principles of Analytical Mechanics.



Applied Mechanics concepts will be developed in the frame of vector calculus and matrix algebra over the geometrical sketches of mechanical systems. Moreover, the solving of mechanical problems with a large number of variables will be structured algorithmically for a computational solution.

The subject matters will be, first, a base for Applied Mechanics (second semester), and second, a bunch of basic tools for other subjects in the third year (Solid Mechanics, Mechanism Theory and Vibrations) and fourth year (Machine Elements).

- Knowledge and competences for the calculus, design and testing of machines.



Learning Outcomes of the Subject Matter:

- C.1 To be able to analyze with precision and efficiency mechanical phenomena in the field of Dynamics.

- C.2 To be able to choose the most adequate solving tools for the resolution of mechanical problems in the previous fields o the frame of rigid bodies.

- C.3 To be able to evaluate the need for simplifications in modelling a real system and the adequacy of mathematical models in such mechanical systems.

- C.4 To be able to give a correct interpretation of the results.

- C.5 To be able to distribute, interact and explain a mechanical problem, its resolution and results to a group of people in written and oral form.

Topic 1 MOMENTS OF INERTIA

Before getting into depth in the Dynamics, a topic is dedicated to the geometry of mass distribution. Moments of Inertia are defined with respect to points, lines and planes. Inertia Tensors and ellipsoids are obtained as a way to obtain any moment in a rigid body. Steiner theorems are presented along with many examples.



Topic 2 MAGNITUDES AND FUNDAMENTAL AXIOMS OF DYNAMICS

This second topic presents the basics of Newtonian Dynamics. Every dynamic magnitude is defined over the motion of a mechanical system in order to establish the relationships among them that are present in the theorems for dynamic analysis. In this way we will get the correspondence between actions and motion.



Topic 3 PARTICLE DYNAMICS

In this topic, the theorems postulated in topic 2 are applied to the analysis of the motion of a particle, covering from rectilinear motion to central forces.



Topic 4 RIGID BODY DYNAMICS

Expressions of the formulas to find the fundamental magnitudes for rigid bodies are presented, i.e. linear and angular momentum, and kinetic energy. The several types of dynamic problems are explained, and constraint forces are revisited.



Topic 5 DYNAMICS OF PLANAR RIGID BODIES Y PLANE MOTION

We start in this topic the analysis of several specific dynamic problems on mechanical systems conforming mechanisms, by application of the formulas introduced above.



Topic 6 DYNAMICS OF RIGID BODIES WITH A FIXED AXIS

Rigid bodies with a fixed line about which the body rotates have a special interest. In this topic, we will also go over the equilibrium conditions for such mechanical systems.



Topic 7 DYNAMICS OF RIGID BODIES WITH A FIXED POINT

General equations are found for such systems using Euler angles. A classification of gyroscopes is given, and two types are treated in detail. Euler-Poinsot gyroscope is analysed with special attention to the symmetric case. Also, Euler-Lagrange gyroscope is studied. Gyroscopic effects are treated in a general case to end.



Topic 8 IMPACT DYNAMICS

The dynamic theorems are deduced for the case of impact actions and collision between rigid bodies.



Topic 9 ANALYTICAL MECHANICS

This first topic is devoted to the introduction of the fundamentals about constraints, freedom, and generalized coordinates and so on in the frame of Analytical Mechanics.



Topic 10 LAGRANGIAN DYNAMICS

Starting with the approach of D’Alembert, and by application of the principle of virtual work to the dynamic equilibrium, we will arrive at the Lagrange equations. These will be used to solve rigid body dynamic problems with holonomic constraints.



Topic 11 SMALL OSCILLATIONS

To end the subject, we will apply the Lagragian dynamics to the analysis of conservative mechanical systems in order to solve the resonance frequencies and mode shapes of the oscillations about the equilibrium positions of the system.

TEACHING METHODOLOGY

For every week, the subject is scheduled as follows:

- Lecture 1. Theory and practice. 1.5 hours. Theory and examples.

- Lecture 2. Practice. 1 hour. Problem solving.

- Coursework. Recommended duration 2.5 hours. Solution of problems proposed in Moodle.

- Lecture 3. Theory and Practice. 1 hour. Theory and examples.

- Coursework. Recommended duration 3.5 hours. Solution of problems proposed in Moodle.



For the Seminars (three sessions of 1.5 hours every semester): Discussion on difficulties and problem solving.



For the Labs (one session of 3 hours per semester): Practical analysis and writing deliverable.

• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 20
  • Realization of Practical Work (exercises, cases or problems) (%): 80

Evaluation is based on a continuous frame as follows:



1. Theoretical questions to be solved individually in writing:

- Week 19. Mass Geometry. 0.5 hours. Value (10/100)

- Week 30. Dynamics 2. 0.5 hours. Value (10/100)



2. Dynamics 1 Problem Solving. 2 problems to be solved individually in writing:

- Week 23. Dynamics 1. 2 hours. Value (35/100)



3. Dynamics 2 Problem Solving.2 problems to be solved individually in writing:

- Date Final Exam. Dynamics 2. 2 hours. Value (40/100)



4. Labs evaluation.

- Week 30. Deliverable into Moodle. Value (2,5/100)

Questions in Moodel. Value (2,5/100)



Every exercise will be published solved in Moodle.

A day will be dedicated to the revision of qualifications, individually.

For those who want to increase their marks in the first two evaluations, they can do the corresponding exercises of the final exam.



5. Final Exam.

- 5 exercises, 1 theory and 4 problems. 4 hours. Value 100/100.

Final Exam.

- 5 exercises, 1 theory and 4 problems. 4 hours. Value 100/100.

- Mecánica Aplicada: Dinámica, Armando Bilbao, Enrique Amezua y Óscar Altuzarra. Editorial Síntesis. 2008

Basic bibliography

- Mecánica Aplicada: Dinámica, Armando Bilbao, Enrique Amezua y Óscar Altuzarra. Editorial Síntesis. 2008

- Mecánica Aplicada: Estática y Cinemática, Armando Bilbao y Enrique Amezua. Editorial Síntesis. 2006

- Curso de Mecánica, J. M. Bastero y J. Casellas, Ediciones Universidad de Navarra, S.A.

- Mecánica para Ingenieros (Estática y Dinámica), J.L. Meriam y L.G. Kraige, Editorial Reverté, S.A. 1998.

- Mecánica Vectorial para Ingenieros (Estática y Dinámica), F. P. Beer y E. R. Johnston, Editorial Mc-Graw Hill.

- Ingeniería Mecánica (Estática y Dinámica), W. F. Riley y L. D. Sturges, Editorial Reverté, S.A. 1996.

- Mecánica para Ingeniería (Estática y Dinámica), A. Bedford y W. Fowler, Addison-Wesley Iberoamericana. 1996.

- Ingeniería Mecánica (Estática y Dinámica), R. C. Hibbeler, Prentice Hall 1995

- Engineering Mechanics (Statics and Dynamics), E.W. Nelson, Mc Graw Hill 1997

- Fundamental Engineering Mechanics. P.J. Ogrodnik. Addison-Wesley Longman. 1997.

- Mecánica Clásica. H. Goldstein. Editorial Reverté, S.A. 1994.

In-depth bibliography

- KANE, T. , LEVINSON, D. ¿Dynamics: Theory and Applications. McGraw-Hill, New York, 1985.
- GREENWOOD, D. Classical Dynamics. Prentice-Hall. London, 1977.

Journals

- American Society of Mechanical Engineers. Transactions: Journal of Applied Mechanics.
- Anales de Ingeniería Mecánica.
- Applied Mechanics Reviews.
- Computer Methods in Applied Mechanics and Engineering.
- Computer & Structures.
- Engineering Structures.
- Experimental Mechanics.
- International Journal of Mechanical Engineering Education.
- International Journal for Numerical Methods in Engineering.
- Journal of Applied Mechanics.
- Journal of Mechanical Design.
- Journal of Mechanisms and Robotics
- Mechanism and Machine Theory.

Web addresses

http://www.ehu.es/compmech
http://www.thinkmotion.eu
http://www.dmg-lib.org
http://www.europeana.eu
http://www.biblioteka.ehu.es
http://kmoddl.library.cornell.edu
http://www.technologystudent.com
http://www.howstuffworks.com
http://www.physics-online.com/
http://ocw.mit.edu/OcwWeb/web/home/home/index.htm
http://mit.ocw.universia.net/Mechanical-Engineering/index.htm
http://imechanica.org/
http://www7.nationalacademies.org/usnctam/
http://www.mip.berkeley.edu/physics/bookadx.html