Aim of the course is to provide the students with the skills to deal with the dynamics of machines and vibrating systems and their behavior under different force fields and interaction with other systems. At the beginning, the models used to define the response of dynamic systems will be presented, providing the basics of physics and the governing laws with special attention to vibrations. Vibration of lumped and distributed models will be investigated. Then, models will be used to study the system interaction with force fields investigating problems of vibration induced by dynamic loads (lumped of distributed loads, travelling loads), rotor dynamics (train wheelset dynamic on rails, rotor-foundation interaction, rotor unbalancing and vibration transmissibility) and fluid-structure interaction (flutter instability, aquaplaning) considering specifically the stability features. During the practical lessons, numerical codes will be developed by the students to study the dynamic response of vibrating systems. 

KNOWLEDGE AND UNDERSTANDING

Once passed the exam, the student:

• knows the methodologies to analytically model vibrating systems with N degrees of freedom or string/beam continuous systems moving in a plane.
• knows how to compute the free response of these systems starting from initial conditions
• knows how to compute the steady response of these systems to external forces.
• knows the methodologies to study the stability of vibrating systems excited by external forces that are dependent from the motion of the system itself.
• knows the physical principles of the rotor critical velocities and of the lubricated bearing instability.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING

Once passed the exam, the student:

• is able to model simple vibrating systems with lumped or continuous distribution (limited to string and beam) of stiffness and mass, moving in a plane and write their equation of motion
• is able to solve the equation of motions through analytical solutions
• is able to implement the analytical solution in a Matlab code and compute the free response to initial conditions and the steady response to known time dependent harmonic external forces.
• is able to represent the response of the vibrating systems and its dependence from the most important parameters through plots and graphical representations
• is able to compute the design parameters of a tuned mass damper to control the response of the vibrating systems in resonance.
• is able to analyse the stability of a vibrating system excited by external forces depending from the system motion
• is able to study the response of a string or a beam moving in the plane excited by a travelling load considering dynamic effects
• is able to describe the effects induced by the unbalancing in a rotor and the simple techniques to balance rigid rotors

Vibrations: Review of 2D kinematics and dynamics of rigid bodies and single and multi degree of freedom vibrating systems. (Applications: simple models of a vehicle running on an irregular road, tuned mass damper, foundation insulation)

Strings: Transverse vibrations in stretched strings. Wave equations, wave propagation solution and standing wave solution. Natural frequencies, vibration modes, forced response. Damping systems for cables. (Applications: bridge stays)

Beams: Euler-Bernoulli beam theory. Axial vibration in beams. Torsional vibration in beams. Bending vibrations in slender beams. Natural frequencies, vibration modes, forced response, modal parameter identification. (Applications: rails under travelling loads, beams under lumped and distributed loads)

Rotor dynamics: Lubricated journal bearings. Balancing. Critical bending velocities. Rotor-Foundation interaction.

Stability of mechanical systems in non linear force field: Definition of force field. Conservative and not conservative force fields. Stability analysis of a single degree of freedom system about equilibrium position. Static and dynamic instability. Stability analysis of a two degree of freedom system about equilibrium position. (Applications: Railway wheelset instability, Flutter instability, lubricated bearing instability).

Knowledge of mathematical analysis and geometry are required with specific reference to second order differential equations, complex numbers operations, differential and integral calculus, algebra of matrices, eigenvalue-eigenvector analysis. The basic knowledge of mechanics provided by the course of physics are required with reference to the kinematics of a point and of a rigid body and the fundamentals dynamic laws.

The course is organised in lectures and practical lessons where the topics of the course are presented and applied to practical examples, solving numerical exercises through analytical solutions and numerical solutions obtained using Matlab codes that are partially developed by the students.

During the practical lessons the students will be guided to develop a Matlab code to compute the dynamic response of vibrating systems moving in a plane.

The exam is composed by a written test and an oral test concerning the topics declared in the official program of the Course. The written test is the computation of the dynamic response of a vibrating system excited by an external force having boundary conditions and initial conditions similar to the examples solved during the practical lessons using the methodologies and the Matlab code developed by the students.

The oral part, for those students who have passed the written part, is composed by answering questions on the topics of the course, combining oral explainations with equations and graphs.