RATIONAL MECHANICS

Expected Learning Outcomes according to the 2030 Agenda

At the end of the course, the student will be able to:

1. Vector calculus skills and mathematical modeling

   • Apply vector calculus tools (dot product, cross product, mixed product) and kinematics/dynamics to model engineering and architectural problems.

   • 2030 Agenda links:

     • SDG 4 – Quality Education: development of advanced mathematical and analytical skills.

     • SDG 9 – Industry, Innovation, and Infrastructure: use of modeling for sustainable design.

2. Analysis of forces and applied vector systems

   • Evaluate static and dynamic equilibrium conditions for bodies and structures, identify equivalent systems, and reduce the complexity of models.

   • 2030 Agenda links:

     • SDG 11 – Sustainable Cities and Communities: design of safe and resilient buildings.

     • SDG 12 – Responsible Consumption and Production: resource optimization through more efficient static schemes.

3. Kinematics and dynamics of points and systems of points

   • Describe and predict the motions of material points and rigid bodies, distinguishing between inertial and non-inertial references, and recognizing the implications in real systems (inclined planes, mobile structures, foundations).

   • 2030 Agenda links:

     • SDG 7 – Affordable and Clean Energy: improved understanding of mechanical energy transformations.

     • SDG 13 – Climate Action: design of structures capable of responding to dynamic stresses linked to extreme natural events.

4. Geometry of masses and structural strength

   • Calculate centroids, moments of inertia, and fundamental dynamic quantities for structural stability.

   • 2030 Agenda links:

     • SDG 9 – Industry, Innovation, and Infrastructure: analysis of infrastructure safety.

     • SDG 11 – Sustainable Cities and Communities: ensuring reliability and durability of architectural and urban structures.

5. Energy, work, and conservation principles

   • Interpret the mechanical behavior of systems through the use of kinetic energy, potential energy, work, and power of forces.

   • 2030 Agenda links:

     • SDG 7 – Affordable and Clean Energy: promotion of energy efficiency in design.

     • SDG 12 – Responsible Consumption and Production: reduction of energy and material waste.

6. Systemic approach to dynamics and statics of rigid bodies

   • Formulate the cardinal equations of dynamics and statics, recognizing internal and external forces and applying conservation principles.

   • 2030 Agenda links:

     • SDG 9 – Industry, Innovation, and Infrastructure: strengthening skills for safer and more sustainable design.

     • SDG 16 – Peace, Justice, and Strong Institutions: promotion of safety ethics in civil construction and infrastructure.

7. Energy methods and variational principles

   • Apply the principles of virtual work and potential stationarity to analyze complex structures and constraints.

   • 2030 Agenda links:

     • SDG 4 – Quality Education: development of critical and variational thinking.

     • SDG 11 – Sustainable Cities and Communities: use of efficient methods for verifying structural safety and stability.

In-person lectures
Lectures will be conducted in person, in accordance with current regulations.

Information for students with disabilities and/or learning disorders (DSA)
To ensure equal opportunities and compliance with current legislation, students with disabilities or DSA may request an individual meeting to plan any compensatory and/or dispensatory measures, in line with the learning objectives and their specific needs.
It is also possible to contact the CInAP faculty representative for further information and support.

1. Elements of Vector Calculus

• Scalar, vector, mixed, and double vector products

• Vector-valued functions

2. Applied Vectors and Moment Theory

• Polar and axial moments

• Systems of applied vectors

• Couples and central axes

• Equivalent and balanced systems

• Concurrent and parallel vector systems

• Centroid and planar vector systems

3. Point Kinematics

• General concepts of space and time

• Velocity and acceleration of a material point

• Planar, circular, harmonic, and helical motion

4. Kinematics of Systems of Material Points

• Constraints: holonomic, non-holonomic, fixed, movable, unilateral, bilateral

• Degrees of freedom and Lagrangian parameters

• Rigid motion and rigid bodies

• Degrees of freedom of a rigid system

• Euler angles and Poisson’s formula

• Fundamental formula of rigid body kinematics

• Rigid motions: translational, rotational, helical, roto-translational, polar, precessional

• Motion element, Mozzi’s theorem, Mozzi axis

• Relative kinematics: absolute and relative velocities; absolute, relative, transport, and Coriolis accelerations

• Theorem of composition of velocities and accelerations

• Equivalent reference frames

• Planar rigid motion and instantaneous center of rotation

5. Dynamics and Statics of a Point

• Principles of dynamics

• Statics of free and constrained points

• Dynamics and statics in non-inertial frames

• Terrestrial mechanics: weight

• Kinetic energy, work, and power of a force

• Conservative forces

• Work-energy theorem

• Conservation of mechanical energy

6. Mass Geometry

• Center of mass and moment of inertia

• Huygens’ theorem

• Inertia ellipsoid and principal axes of inertia

• Kinetic energy of point systems and motion around the center of mass

• König’s theorem

• Kinetic energy of a rigid system

• Angular momentum

7. Dynamics and Statics of Systems of Material Points

• Internal and external forces

• Cardinal equations of statics and dynamics

• Balance equations and conservation laws

• Examples and applications

• Work-energy theorem

• Work for infinitesimal rigid displacements

• Conservative forces and stresses

• Conservation of mechanical energy

• Statics of constrained rigid bodies, with fixed point or fixed axis

• Statics of a rigid body resting on a plane (e.g., ladder problem)

8. Possible, Virtual, and Elementary Displacements

• Smooth constraints

• Principle of constraint reactions

• Symbolic representation of dynamics

• Principle of virtual work

• Principle of stationary potential energy

• Mauro Fabrizio, " Elementi di Meccanica Classica", Zanichelli
  
• L. Barletti, G. Frosali,  F. Ricci, "Esercizi di Meccanica Razionale per l'Ingegneria",  Società Editrice Esculapio

The exam consists of:

• Written test – prerequisite for the oral exam

• Oral exam

Written Test

The written test is divided into two parts:

1. First part: calculation of centers of mass, moments of inertia, central axes, and principal axes of inertia.

2. Second part: determination of a system’s equilibrium configurations and calculation of constraint reactions.

Oral Exam

The oral exam assesses the theoretical knowledge of the topics covered during the course.

Grading Criteria

Grade 29–30 with honors

• In-depth knowledge of the subject

• Ability to integrate and critically analyze the presented situations

• Independent resolution of complex problems

• Excellent communication skills and command of language

Grade 26–28

• Good knowledge of the subject

• Ability to analyze situations critically and coherently

• Fairly independent resolution of complex problems

• Clear presentation with appropriate language

Grade 22–25

• Satisfactory knowledge, limited to main topics

• Critical analysis not always coherent

• Presentation fairly clear with adequate command of language

Grade 18–21

• Minimal knowledge of the subject

• Limited ability to integrate and critically analyze situations

• Presentation sufficiently clear, but language skills poorly developed

Exam not passed

• Insufficient knowledge of the main course contents

• Very limited or no ability to use specific terminology

• Unable to independently apply acquired knowledge

Compensatory and Dispensatory Measures

To ensure equal opportunities and comply with current regulations:

• Interested students may request a personal meeting to arrange any compensatory and/or dispensatory measures, according to the learning objectives and specific needs.

• Students may also contact the CInAP representative (Centre for Active and Participatory Integration – Services for Disabilities and/or Learning Disorders) of their Department.

Oral Exam Sample Questions

1. Vector Calculus and Theory

• Can you explain the difference between the scalar, vector, and mixed product?

• How is the double vector product calculated, and what properties does it have?

• How would you define an applied vector, and how does it differ from a free vector?

2. Moments and Systems of Vectors

• How do you calculate the polar and axial moment of an applied vector?

• What is a couple, and how is it represented geometrically?

• Can you explain what is meant by equivalent and balanced systems of applied vectors?

3. Point Kinematics

• How are the velocity and acceleration of a material point defined?

• Can you describe the characteristics of harmonic and helical motion?

• How does one generalize from planar motion to three-dimensional motion?

4. Kinematics of Systems of Material Points

• What types of constraints exist, and how do they affect degrees of freedom?

• Can you explain the concept of degrees of freedom for a rigid body?

• How are Euler angles and Poisson’s formula applied to describe rigid body motion?

• Can you distinguish between absolute, relative, transport, and Coriolis velocities?

5. Dynamics and Statics of a Point

• What are the fundamental principles of dynamics?

• How is the work-energy theorem (theorem of live forces) applied to a material point?

• What is the difference between conservative and non-conservative forces?

6. Mass Geometry

• How do you calculate the center of mass of a system of material points?

• What is the moment of inertia about an axis, and how is the inertia ellipsoid used?

• Can you explain König’s theorem and its application in calculating kinetic energy?

7. Dynamics and Statics of Systems of Material Points

• How are the cardinal equations of statics and dynamics written?

• Can you explain how to calculate the constraint reactions for a rigid body resting on a plane?

• How is the principle of virtual work applied to a system of material points?

8. Virtual and Elementary Displacements

• What is the difference between possible, virtual, and elementary displacements?

• How is the principle of stationary potential used to determine equilibrium?

• Can you give a practical example where the principle of virtual work is applied?