# Ingegneria Civile per l'Ambiente ed il Territorio | Rational Mechanics

## Ingegneria Civile per l'Ambiente ed il Territorio Rational Mechanics

 0612500007 DIPARTIMENTO DI INGEGNERIA CIVILE CIVIL AND ENVIRONMENTAL ENGINEERING 2015/2016

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2012 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/07 12 120 LESSONS BASIC COMPULSORY SUBJECTS
 FRANCESCA PASSARELLA T
Objectives
THE COURSE AIMS TO PROVIDE AND DEVELOP USEFUL TOOLS FOR A MATHEMATICAL TREATMENT OF THE PROBLEMS AND PHYSICAL PHENOMENA IN THE CONTEXT OF CLASSICAL MECHANICS.
THE AIM OF THE COURSE IS THE ACQUISITION OF GOOD CAPACITY OF FORMULATION AND SOLUTION OF DIFFERENTIAL EQUATIONS DESCRIBING THE DYNAMICS OF MATERIAL SYSTEMS (MATERIAL SYSTEMS PROPERLY MODELED AS: THE MATERIAL POINT, THE RIGID BODY WITH A FIXED AXIS, RIGID BODY WITH A FIXED POINT, THE FREE RIGID BODY, HOLONOMIC SYSTEMS), ALSO USING THE METHODS OF ANALYTICAL MECHANICS.
Prerequisites
FOR THE SUCCESSFUL ACHIEVEMENT OF OBJECTIVES, STUDENTS ARE REQUIRED BASIC MATHEMATICAL KNOWLEDGE, WITH PARTICULAR REFERENCE TO THE CONCEPTS AND TECHNIQUES FOR SOLUTIONS RELATED TO THE THEORY OF INTEGRATION AND RESOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS. KNOWLEDGE OF ALGEBRA AND VECTOR MATRIX THEORY IS ALSO REQUIRED.
Contents
VECTOR CALCULUS:
CARTESIAN REPRESENTATION OF VECTORS AND OPERATIONS. VECTOR-VALUED FUNCTIONS. APPLICATIONS TO DIFFERENTIAL-GEOMETRIC CURVES. FERNET FORMULAS.
APPLIED VECTORS:
RESULTANT AND RESULTANT MOMENT OF A SYSTEM OF APPLIED VECTORS. CENTRAL AXIS. SYSTEM OF APPLIED VECTORS EQUIVALENT. VECTOR SYSTEM PLANS AND PARALLEL.
KINEMATICS OF A POINT:
SPEED. ACCELERATION. MOTION IN A PLANE. PLANE MOTIONS. HARMONIC MOTION.
KINEMATICS OF MATERIALS:
DEGREES OF FREEDOM AND LAGRANGIAN COORDINATES. HOLONOMIC SYSTEMS. KINEMATICS OF RIGID BODIES. EULER ANGLES. RIGID MOTIONS: TRANSLATIONAL MOTION, ROTATIONAL MOTION AND ROTARY AND TRANSLATORY MOTION. POISSON'S FORMULAS. MOZZI'S THEOREM. INSTANTANEOUS AXIS OF ROTATION AND TRANSLATION.
KINEMATICS OF RELATIVE MOTION.
RIGID MOTION PLANS AND THE THEOREM OF CHASLES.
STATICS AND DYNAMICS OF A FREE MATERIAL POINT.
WORK OF A FORCE. CONSERVATIVE FORCES. ENERGY THEOREM FOR A FREE MATERIAL SYSTEM AND ENERGY CONSERVATION MECHANICS. DIFFERENTIAL EQUATIONS OF MOTION OF A FREE MATERIAL POINT. DIFFERENTIAL EQUATIONS OF MOTION OF A POINT WITH RESPECT TO TWO NON-INERTIAL REFERENCES (APPARENT FORCES, GRAVITATIONAL FORCE). STATIC FREE MATERIAL POINT. HARMONIC OSCILLATOR, DAMPED HARMONIC MOTION, RESONANCE.
STATICS AND DYNAMICS OF A CONSTRAINED POINT:
EQUATIONS OF MOTION OF A POINT CONSTRAINED. STATICS OF A POINT CONSTRAINED. FRICTION AND EQUILIBRIUM POSITIONS. DYNAMICS OF A MATERIAL POINT CONSTRAINED TO A SURFACE, SPONTANEOUS MOTION OF A POINT ON A SURFACE. DYNAMICS OF POINT CONSTRAINED TO A CURVE. SIMPLE PENDULUM.
GEOMETRY OF THE MASSES:
CENTER OF GRAVITY AND PROPERTIES. PLANE SYSTEMS: CENTERS OF GRAVITY AND STATIC MOMENTS. RADIUS OF INERTIA. MOMENTUM AND MOMENT OF MOMENTUM. THEOREM OF KOENIG. KINETIC ENERGY AND MOMENT OF INERTIA. WAY TO VARY THE MOMENT OF INERTIA TO CHANGE THE AXIS: HUYGENS THEOREM AND ELLIPSOID OF INERTIA. APPLICATIONS.
GENERAL THEOREMS OF MECHANICS OF MATERIAL SYSTEMS: CARDINAL EQUATIONS OF DYNAMICS. THEOREM OF MOTION OF THE CENTER OF GRAVITY. WORK OF THE INTERNAL FORCES FOR A RIGID SYSTEM. ENERGY THEOREM AND CONSERVATION OF MECHANICAL ENERGY FOR A CONSTRAINED MATERIAL.
STATICS OF RIGID BODIES:
CARDINAL EQUATIONS OF STATICS. GENERAL CONDITIONS OF EQUILIBRIUM OF A RIGID BODY. APPLICATIONS FOR A FREE RIGID BODY, RIGID BODY WITH A FIXED POINT AND RIGID BODY WITH A FIXED AXIS. REACTION FORCES ON A RIGID BODY IN EQUILIBRIUM. FRICTION AND EQUILIBRIUM POSITIONS. CALCULATION OF REACTION FORCES IN EQUILIBRIUM CONDITION.
RIGID BODY DYNAMICS:
MOTION OF A RIGID BODY WITH A FIXED AXIS FRICTIONLESS. MOTION OF A RIGID BODY WITH A FIXED POINT. MOTION OF A RIGID BODY FREE. POINSOT MOTION.
ELEMENTS OF ANALYTICAL MECHANICS:
VIRTUAL DISPLACEMENTS OF A HOLONOMIC SYSTEM. VIRTUAL WORK. SYMBOLIC EQUATION OF THE DYNAMICS AND THE PRINCIPLE OF D'ALEMBERT. SYMBOLIC EQUATION OF STATICS AND THE PRINCIPLE OF VIRTUAL WORK. EQUILIBRIUM CONDITIONS FOR A HOLONOMIC SYSTEM. CALCULATION OF REACTION FORCES THROUGH THE PRINCIPLE OF VIRTUAL WORK. HOLONOMIC SYSTEMS STRESSED BY CONSERVATIVE FORCES. LAGRANGE EQUATIONS AND APPLICATIONS. KINETIC ENERGY OF A HOLONOMIC SYSTEM AND STUDY OF LAGRANGE'S EQUATIONS. ENERGY THEOREM FOR A HOLONOMIC SYSTEM CONSTRAINT INDEPENDENT OF TIME. LAGRANGE EQUATIONS FOR A CONSERVATIVE SYSTEM. LAGRANGIAN SYSTEMS AND THEIR FIRST INTEGRALS.
STABILITY AND SMALL OSCILLATIONS:
STABILITY, DEFINITION OF STABILITY FOR A HOLONOMIC SYSTEM, SMALL FLUCTUATIONS AROUND A STABLE EQUILIBRIUM POSITION.
Teaching Methods
THE COURSE COVERS THEORETICAL LESSONS, DURING WHICH WILL BE PRESENTED DURING THE COURSE TOPICS THROUGH LECTURES AND CLASSROOM EXERCISES, DURING WHICH PROVIDE THE MAIN TOOLS NEEDED FOR SOLVING EXERCISES RELATED TO THE CONTENT OF THE THEORETICAL ASPECTS.
Verification of learning
THE ASSESSMENT OF THE ACHIEVEMENT OF THE OBJECTIVES WILL BE DONE THROUGH A WRITTEN TEST AND AN ORAL INTERVIEW.
Texts
S. CHIRITA, M. CIARLETTA, V. TIBULLO, MECCANICA RAZIONALE, ED. LIGUORI.
M. FABRIZIO, ELEMENTI DI MECCANICA CLASSICA, ED. ZANICHELLI.