


Politecnico di Torino  
Academic Year 2016/17  
06BPTMC, 06BPTMO, 06BPTMQ Analytical Mechanics 

1st degree and Bachelorlevel of the Bologna process in Civil Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Environmental And Land Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
The main goal of the course is to provide knowledge of the mathematical methods and models for the study of mechanical systems. The main topics treated in the course are the mechanics of the rigid body and of the articulated systems and the Lagrangian mechanics.

Expected learning outcomes
Students will learn how to deal with mathematical models of mechanical systems and with the related methods of qualitative analysis.

Prerequisites / Assumed knowledge
Concepts and methodologies from the courses of Calculus I, II and Geometry.

Contents
 Introduction to the mathematical modelling
 Kinematics of the rigid body Kinematics of a point particle. Kinematics of the rigid body. Euler’s angles, angular velocity. Fundamental formula of the velocities. Act of rigid motion. Rigid motion in the plane. Relative kinematics.  Constrained systems Classification of the constraints, positional constraints and rigidity constraints. Virtual displacements and virtual velocities. Holonomic systems, Lagrangian coordinates, degrees of freedom.  Geometry of the masses Centre of mass. Moments of inertia, matrix of inertia, principal axes of inertia. Computation of the moments of inertia of particular planar sections: Tsection, double Tsection, Usection.  Mechanics Classification of active forces. Systems of forces, resultant, moment, equivalent systems of applied vectors and their reduction. Work, potential and conservative forces. Laws of mechanics. Principle of virtual work of reactions.  Statics Elementary and virtual work. Principle of virtual work. Statics of holonomic systems and theorem of stationary potential. Cardinal equations of statics. Introduction to graphic statics.  Dynamics of systems and of rigid bodies Dynamics of a point particle. Linear momentum, angular momentum, kinetic energy and their expressions for rigid systems. König’s Theorem. Cardinal equations of dynamics. Theorems of linear momentum and angular momentum and related first integrals. Euler equations. Motion of a solid with a fixed axis. First integral of energy.  Lagrangian mechanics D'Alembert principle. Kinetic energy of holonomic systems. Lagrange equations. First integrals. Stability and small oscillations. 
Delivery modes
Lectures are complemented by classroom exercises mainly focussed on the following topics:
 kinematics of free and constrained point particles and rigid bodies  relative kinematics  cardinal equations of dynamics  principle of virtual work, stationary potential  cardinal equations of statics  calculation of the motion and of the equilibrium configurations of a mechanical system, calculation of dynamical and statical reactions  Lagrange equations  linearisation of the equations of motion about stable equilibrium configurations 
Texts, readings, handouts and other learning resources
P. Biscari, T. Ruggeri, G. Saccomandi, M. Vianello, Meccanica Razionale (3a edizione), Springer, 2016
Further reading material will be suggested at the beginning of the course. 
Assessment and grading criteria
The examination is written and consists in an exercise with several questions (related to the topics developed in classroom lectures and exercises) aimed at testing the ability of the student to qualitatively deal with mechanical systems. The examination may include also a theoreticalconceptual question (without proofs).
The maximum time allowed is 120 minutes. During the written examination students may consult a form (A4 paper size  see the template available on the official website of the course) that they can fill in personally in advance with the major formulas learnt during the course. An oral examination, integrating the written one, is possible upon request of the student (if satisfactory in the written part) or of the teacher. The oral examination consists in two or three questions, which may include both the proof of theorems and the application of solution techniques developed during the course. 
