# Ingegneria Civile per l'Ambiente ed il Territorio | RATIONAL MECHANICS

## Ingegneria Civile per l'Ambiente ed il Territorio RATIONAL MECHANICS

 0612500007 DEPARTMENT OF CIVIL ENGINEERING EQF6 CIVIL AND ENVIRONMENTAL ENGINEERING 2020/2021

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2018 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/07 12 120 LESSONS BASIC COMPULSORY SUBJECTS
 ADA AMENDOLA T IVANA BOCHICCHIO
Objectives
THE COURSE HAS THE FOLLOWING EDUCATIONAL OBJECTIVES:

• EXPECTED LEARNING OUTCOMES AND SKILLS TO BE ACQUIRED:
KNOWLEDGE OF THE BASIC ELEMENTS OF MECHANICS ADDRESSED FROM A RATIONAL POINT OF VIEW, THAT IS THE KINEMATICS AND DYNAMICS OF THE MATERIAL POINT AND OF THE SYSTEMS OF POINTS, IN PARTICULAR THE RIGID ONES. KNOWLEDGE OF THE FORMULATION OF MECHANICAL PROBLEMS IN LAGRANGIAN FORMALISM. ABILITY TO ACQUIRE METHODS FOR SOLVING PROBLEMS OF RATIONAL MECHANICS BUT ABOVE ALL THE RELATIONSHIP BETWEEN THE PROPERTIES OF A PHYSICAL SYSTEM AND THE MATHEMATICAL STRUCTURES FOR THEIR REPRESENTATION.

• KNOWLEDGE AND UNDERSTANDING.
ACQUIRE THE NECESSARY METHODS TO SET UP AND DEAL WITH MECHANICAL PROBLEMS OF FREE AND CONSTRAINED SYSTEMS. IN PARTICULAR, TO KNOW THE BASIC ELEMENTS OF MASS GEOMETRY AND VECTOR CALCULUS FOR THE STUDY OF KINEMATICS AND POINT DYNAMICS, POINT SYSTEMS AND RIGID BODIES, ALSO IN LAGRANGIAN FORMALISM. UNDERSTAND AND SCHEMATIZE MECHANICAL SYSTEMS WITH MATHEMATICAL MODELS WITH A FINITE DEGREE OF FREEDOM, CONSISTING OF TWO OR MORE MATERIAL ELEMENTS AND / OR RIGID BODIES.

• ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING.
KNOWING HOW TO SOLVE SIMPLE, BUT CONCRETE, MECHANICAL PROBLEMS CONCERNING THE GEOMETRY OF THE MASSES (IN THE CASE OF FLAT SYSTEMS), THE DYNAMICS (FLAT SYSTEMS WITH ONE OR MORE DEGREES OF FREEDOM) AND THE STATIC OF MECHANICAL SYSTEMS (FOR FLAT SYSTEMS, WITH THE EQUATIONS CARDINALS OF STATICS AND WITH THE PRINCIPLE OF VIRTUAL WORKS). IN PARTICULAR, ABILITY TO MODEL CONSTRAINED AND NON-CONSTRAINED MECHANICAL SYSTEMS, THROUGH AN APPROPRIATE CHOICE OF COORDINATES IN THE SPACE OF CONFIGURATIONS FOR WRITING THE EQUATIONS OF MOTION.

• AUTONOMY OF JUDGMENT.
TO AUTONOMOUSLY DEEPEN WHAT HAS BEEN LEARNED, IN ORDER TO USE THE KNOWLEDGE ACQUIRED AS A STARTING POINT THAT ALLOWS TO FACE NEW PROBLEMS AND TO REACH FURTHER RESULTS THROUGH AN EVER GREATER MATURITY AND AN EVER WIDER AUTONOMY OF JUDGMENT.

• COMMUNICATION SKILLS.
KNOWING HOW TO EXPLAIN IN A CONSCIOUS AND RIGOROUS WAY WHAT ARE THE METHODS AND TECHNIQUES ADOPTED TO SOLVE A MECHANICAL PROBLEM AND WHAT ARE THE PROCEDURES USED TO REACH THE RESULTS OBTAINED. ABILITY TO EXPLAIN, EVEN TO NON-EXPERT PEOPLE, TO WHAT PRACTICAL EXPERIENCES ALL THAT HAS BEEN LEARNED CAN BE APPLIED.

•LEARNING ABILITY.
KNOWING HOW TO APPLY THE KNOWLEDGE ACQUIRED ALSO TO CONTEXTS OTHER THAN THOSE PRESENTED DURING THE COURSE, AS WELL AS DEEPEN THE TOPICS COVERED USING ALTERNATIVE APPROACHES AND / OR PROCEDURES.
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.
MATEMATICA II IS PREPARATORY FOR Rational Mechanics.
Contents
VECTOR CALCULUS (3/1/-):
CARTESIAN REPRESENTATION OF VECTORS AND OPERATIONS. VECTOR-VALUED FUNCTIONS.

APPLICATIONS TO DIFFERENTIAL-GEOMETRIC CURVES (3/-/-):
FERNET FORMULAS.

APPLIED VECTORS (8/3/-):
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:(4/2/-):
SPEED. ACCELERATION. MOTION IN A PLANE. PLANE MOTIONS. HARMONIC MOTION.

KINEMATICS OF MATERIALS (8/1/-):
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 (4/1/-).

RIGID MOTION PLANS AND THE THEOREM OF CHASLES (1/-/-).

STATICS AND DYNAMICS OF A FREE MATERIAL POINT(6/5/-):
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(4/2/-):
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 (5/7/-):
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(8/2/-):
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 (4/4/-):
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 (5/4/-):
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(13/6/-):
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 (4/2/-):
STABILITY, DEFINITION OF STABILITY FOR A HOLONOMIC SYSTEM, SMALL FLUCTUATIONS AROUND A STABLE EQUILIBRIUM POSITION.
Teaching Methods
THE COURSE IS OF 12 CFU AND COVERS THEORETICAL LESSONS, 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.
IS AN OBLIGATION AT LEAST 70% FREQUENCY OF HOURS OT TEACHING EXPERIENCE
Verification of learning
THE EXAM IS AIMED AT EVALUATING THE KNOWLEDGE AND THE ABILITY TO UNDERSTAND THE CONCEPTS EXPOSED DURING LESSONS AND THE ABILITY TO APPLY SUCH KNOWLEDGE AND TO FORMULATE THE DIFFERENTIAL EQUATIONS DESCRIBING THE DYNAMIC OF MATERIAL SYSTEMS.
THE EXAMINATION IS CARRIED OUT AT THE END OF THE COURSE AND IS DIVIDED INTO A SELECTIVE WRITTEN TEST AND IN AN ORAL INTERVIEW. THE WRITTEN TEST IS CARRIED OUT IN 3 HOURS AND PROPOSES EXERCISES AND QUESTIONS WITH OPEN ANSWERS. THE ORAL INTERVIEW EVALUATES THE ACQUIRED KNOWLEDGE.
IN THE FINAL EVALUATION, EXPRESSED IN THIRTIETHS, THE ASSESSMENT OF WRITTEN TEST WEIGHS FOR 40%, WHILE THE ORAL INTERVIEW WEIGHS FOR THE REMAINING 60%.
TO ACHIEVE THE LEVEL OF EXCELLENCE IT
WILL BE CONSIDERED: 1.THE USE OF APPROPRIATE SCIENTIFIC LANGUAGE; 2. THE TRANSVERSE CORRELATION BETWEEN THE DIFFERENT ARGUMENTS OF THE COURSE AND WITH OTHER DISCIPLINES; 3. THE AUTONOMY OF JUDGMENT.
Texts
S. CHIRITA, M. CIARLETTA, V. TIBULLO, MECCANICA RAZIONALE, ED. LIGUORI.
M. FABRIZIO, ELEMENTI DI MECCANICA CLASSICA, ED. ZANICHELLI.
F. STOPPELLI, APPUNTI DI MECCANICA RAZIONALE, LIGUORI ED