Matematica | RATIONAL MECHANICS

cod. 0512300015

RATIONAL MECHANICS

	0512300015
	DIPARTIMENTO DI MATEMATICA
	EQF6
	MATHEMATICS
	2020/2021

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2018
	SECONDO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/07	9	72	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

	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 COURSE AIMS AT THE FOLLOWING EDUCATIONAL OBJECTIVES: 1) KNOWLEDGE AND UNDERSTANDING THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF CINEMATICS AND DYNAMICS OF MATERIAL POINT AND MATERIAL SYSTEM, FREE AND WITH CONSTRAINT. 2) APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS: 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). 3) COMMUNICATION SKILLS ABILITY TO APPLY KNOWLEDGE IN DIFFERENT SITUATIONS THAN THOSE PRESENTED IN THE COURSE AND ABILITY TO REFINE OWN KNOWLEDGE. 4) COMMUNICATION SKILLS – TRANSVERSAL SKILLS: ABILITY TO EXPOSE ORALLY, WITH APPROPRIATE TERMINOLOGY, THE TOPICS OF THE COURSE.

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 COURSE AIMS AT THE FOLLOWING EDUCATIONAL OBJECTIVES:

1) KNOWLEDGE AND UNDERSTANDING
THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF CINEMATICS AND DYNAMICS OF MATERIAL POINT AND MATERIAL SYSTEM, FREE AND WITH CONSTRAINT.

2) APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS:
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).
3) COMMUNICATION SKILLS
ABILITY TO APPLY KNOWLEDGE IN DIFFERENT SITUATIONS THAN THOSE PRESENTED IN THE COURSE AND ABILITY TO REFINE OWN KNOWLEDGE.

4) COMMUNICATION SKILLS – TRANSVERSAL SKILLS:
ABILITY TO EXPOSE ORALLY, WITH APPROPRIATE TERMINOLOGY, THE TOPICS OF THE COURSE.

	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. 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 POINT CONSTRAINED TO A CURVE. SIMPLE PENDULUM. GEOMETRY OF THE MASSES: CENTER OF GRAVITY AND PROPERTIES. CENTERS OF GRAVITY OF PLANE SYSTEMS. 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.

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.
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 POINT CONSTRAINED TO A CURVE. SIMPLE PENDULUM.
GEOMETRY OF THE MASSES:
CENTER OF GRAVITY AND PROPERTIES. CENTERS OF GRAVITY OF PLANE SYSTEMS. 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.

	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 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 FORMULATE THE DIFFERENTIAL EQUATIONS DESCRIBING THE DYNAMIC OF MATERIAL SYSTEMS. THE EXAMINATION IS DIVIDED INTO A SELECTIVE WRITTEN TEST AND IN AN ORAL INTERVIEW. WRITTEN PROOF: THIS LASTS 2 HOURS AND CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE. IN THE CASE THAT THIS TEST IS SUFFICIENT, IT WILL BE EVALUATED BY THREE SCALES. ORAL TEST: THIS TEST HAS A DURATION OF APPROXIMATELY 20 MINUTES AND 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%. LAUDE FOLLOWS FROM BRILLIANT WRITTEN PROOF AND ORAL INTERVIEW.

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 FORMULATE THE DIFFERENTIAL EQUATIONS DESCRIBING THE DYNAMIC OF MATERIAL SYSTEMS.
THE EXAMINATION IS DIVIDED INTO A SELECTIVE WRITTEN TEST AND IN AN ORAL INTERVIEW.
WRITTEN PROOF: THIS LASTS 2 HOURS AND CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE. IN THE CASE THAT THIS TEST IS SUFFICIENT, IT WILL BE EVALUATED BY THREE SCALES.
ORAL TEST: THIS TEST HAS A DURATION OF APPROXIMATELY 20 MINUTES AND 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%.
LAUDE FOLLOWS FROM BRILLIANT WRITTEN PROOF AND ORAL INTERVIEW.

	Texts
	M. FABRIZIO, ELEMENTI DI MECCANICA CLASSICA, ED. ZANICHELLI. P. BISCARDI, INTRODUZIONE ALLA MECCANICA RAZIONALE. ELEMENTI DI TEORIA CON ESERCIZI, ED. SPRINGER. S. CHIRITA, M. CIARLETTA, V. TIBULLO, MECCANICA RAZIONALE, ED. LIGUORI.