Functions Derivation Basic derivation theorems. Multivariable functions Introduction to differential equations. Particle and Rigid Bodies Kinematics. Particle and Rigid Bodies Dynamics.

By the end of this subject, students will be able to have: 1. Knowledge and understanding of linear differential equations which are applicable in mechanical vibration problems 2. Knowledge and understanding of key aspects of mechanical vibrations fundamentals; 3. The ability to apply their knowledge and understanding to identify, formulate and solve problems of mechanical vibrations using established methods; 4. The ability to combine theory and practice to solve problems of mechanical vibrations; 5. An understanding of applicable techniques and methods in mechanical vibrations, and of their limitations;

RA1.1 Knowledge and understanding of the scientific and mathematical principles underlying their branch of engineering. RA1.2 An systematic understanding of the key aspects and concepts of their branch of engineering. RA2.1 The ability to apply their knowledge and understanding to identify, formulate and solve engineering problems using established methods. RA4.2 The ability to design and conduct appropriate experiments, interpret the data and draw conclusions. RA4.3 Workshop and laboratory skills. RA5.1 The ability to select and use appropriate equipment, tools and methods. RA5.2 The ability to combine theory and practice to solve engineering problems. RA5.3 An understanding of applicable techniques and methods, and of their limitations. CB1 Students have demonstrated possession and understanding of knowledge in an area of study that builds on the foundation of general secondary education, and is usually at a level that, while relying on advanced textbooks, also includes some aspects that involve knowledge from the cutting edge of their field of study. CB2 Students are able to apply their knowledge to their work or vocation in a professional manner and possess the competences usually demonstrated through the development and defence of arguments and problem solving within their field of study. CG1 Ability to resolve problems with initiative, creativity decision-making and critical reasoning skills, and to communicate and transmit knowledge, skills and abilities in the Industrial Engineering area. CG9 Knowledge and capacity to apply computational and experimental tools for analysis and quantification of mechanical engineering problems. CG20 Knowledge of the fundamentals of mechanism and machine theory.

1.- Introduction to differential calculus 1.1.- Functions of a Variable 1.2.- Mathematical Models 1.3.- Classification of Differential Equations 1.4.- Linear Ordinary Differential Equations 1.5.- Solution of Differential Equations 1.6.- Solution of Homogeneous First Order Diff Eq. 1.7.- Solution of Non homogeneous Linear First Order Diff Eq. 1.8.- Solution of Second Order Diff. Eq. 1.9.- Integration methods 2.- Numerical methods for solving differential equations 2.1.- Introduction 2.2.- Euler's method 2.3.- Taylor's method 2.4.- Runge-Kutta methods 3.- Introduction to MATLAB 3.1.- Introduction 3.2.- General Rules 3.3.- Basic Operations 3.4.- Output formats 3.5.- Matrices and Vectors 3.6.- Graphics in MATLAB 3.7.- Programming in MATLAB 3.8.- SIMULINK 4.- Solution of Differential Equations using MATLAB 4.1.- Introduction 4.2.- ODE function 4.3.- Solution of a first order Differential Equations 4.4.- Solution of second order Differential Equations 4.5.- Solution of Differential Equations by SIMULINK 5.- Introduction to mechanical vibrations: 5.1.- Introduction 5.2.- Classification of vibrations 5.3.- Components of an oscillatory system 5.4.- Simple Harmonic Motion (SHM) 5.5.- Energy of a Simple Harmonic Movement 5.6.- Nonlinear vibrations 6.- Damped mechanical vibrations and forced vibrations 6.1.- Damped harmonic oscillator 6.2.- Differential equation of Damped Movement 6.3.- Damping coefficient 6.4.- Solution to the Differential Equation of Damped Motion 6.5.- Logarithmic Decrease 6.6.- Forced vibrations 6.7.- Transmissibility coefficient 6.8.- Resonance 7.- Systems of 2 Degrees of Freedom and N Degrees of Freedom 7.1.- Introduction 7.2.- Undamped free vibrations for 2 DOF 7.3.- Forced undamped vibrations for 2 DOF 7.4.- Damped forced vibrations for 2 DOF 7.5.- N DOF Vibrations systems

Master class Classroom exercises Laboratories exercices Personal work. Team Work