This subject, called Mechanics, is devoted to establish the basic knowledge on statics and kinematics of rigid bodies. It is a subject closely related to Physics and mathematics. Mechanics provide alumni with an enhanced practical understanding of physical phenomena, a competence in the analysis of problems by disassembling them into simpler parts to relate them once solved. Also, this is the first subject in which alumni will come across the mechanical engineering application of the fundamental knowledge that they are acquiring.



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 on principles of mechanism and machine theory.

- Knowledge and usage of principles of Solid Mechanics.



Learning Outcomes of the Subject Matter:

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

- 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 FUNDAMENTALS OF VECTORIAL CALCULUS

This first topic starts with a simple introduction with the aim to show the content matters of this subject and the previous fundamental issues whose knowledge is required. Mechanics is inside the historical frame of Physics and here its milestones are highlighted. Then, basic operations of vector calculus will be revisited (e.g. scalar, cross and combined products). Also, vector functions of scalar variables, Frenet-Serret frame, and derivation of vectors on moving frames. A special focus is placed on sliding vectors, vector fields, moment operation on vector fields and equivalence of vector systems.



Topic 2 CENTER OF MASS AND CENTER OF GRAVITY

In this topic, a series of concepts used throughout Statics are introduced. Thus, the concept of center of gravity is explained on the basis of the central point of a system of bound vectors. This approach requires the explanation of a series of concepts on bound vectors and their systems, such as virial, central plane and point. Afterwards, static moment and its properties are defined in order to end with Pappus and Guldin theorems.



Topic 3 STATICS OF POINT MASSES

Statics considers systems of mass that cannot move (or have inertial motion) with respect to an inertial frame whatever the actions applied. The objective of its analysis is to find the forces that keep in equilibrium each of the elements of such a system. We will start with the analysis of the equilibrium of a point of mass. The fundamental axiom of equilibrium is proposed, and concepts such as constraint and reaction force are introduced prior to the analysis of rigid bodies.



Topic 4 STATICS OF A RIGID BODY

The Theorem of transmissibility of forces is explained as a first step towards the definition of the rigid body equations of equilibrium. The diverse types of constraint in the absence of friction are defined with an explanation of the constraint forces generated.



4.1 TRUSSES

This topic will go over the equilibrium of a system of rigid bodies whose characteristics are to have a dimension much bigger than the other two, being linked with pinned joints or spherical joints, and loaded at such joints called nodes. These systems are referred to as trusses. The objective is to solve the axial forces on such elements using two methods capable to be computationally effective.



4.2 INTERNAL EFFORTS IN STRUCTURAL ELEMENTS

Internal efforts are defined for a section of a rigid body. Simple isostatic systems will be analyzed on structural structures in order to get axial and shear forces, as well as bending moments. Plots of these values will be obtained for beams.



Topic 5 FLEXIBLE CABLES

This topic is dedicated to the analysis of the equilibrium of flexible slender systems that under load support tension actions but not bending moments. Such systems acquire the shape of certain algebraic curves. We will start with the simple analysis of a mass less cable loaded at discrete points, then move to the analysis of continuous loading, to end with the study of catenary systems.



Topic 6 FRICTION

Subject matters in the previous topics are extended to systems with friction under Coulomb approach.



Topic 7 RIGID BODY KINEMATICS

Kinematics is the analysis of the motion of bodies with no regards to the causes. The student is introduced into the concepts of motion and the mathematical treatment using rotation and transformation matrices.



Topic 8 EQUATIONS OF MOTION

On the basis of the rigid body, simple motions are treated from translation to rotation, to end with the equation of velocity for screw motion. Angular velocity is explained along with its properties. The instantaneous screw axis and the axodes are defined. To follow with the acceleration analysis. Relative motion is defined and Cosiolis and Resal components obtained. A brief description of kinematics pairs is done to end with the analysis of the motion between bodies in contact.



Topic 9 PLANE MOTION

Kinematics of the planar motion is approach as a sequel of the matters in previous topics. Concepts such as Instantaneous Center of Rotation and Centrodes are defined. Accelerations are dealt with in the plane. Acceleration pole and circles are defined. A deep analysis of the accelerations at contact points is performed to end.

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. Three theoretical questions to be solved individually in writing:

- Week 9. Statics. 0.5 hours. Value (10/100)

- Week 15. Spatial and Planar Kinematics. 0.5 hours. Value (10/100)



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

- Week 8. Statics with friction. 2 hours. Value (35/100)



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

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



4. Labs evaluation.

- Week 15. 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.

At least a 40% of the Value is compulsory in the evaluation of the third point.



5. Final Exam.

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

At least a 40% of the Value is compulsory in the evaluation of the Kinematics.

Final Exam.

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

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

Basic bibliography

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



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

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

BOTTEMA, O., ROTH, B. "Theoretical Kinematics". Nort-Holland, Amsterdam, 1979.

Web addresses

http://www.ehu.es/compmech
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