The course is an advanced module on vibrations of mechanical systems, with emphasis on distributed parameter systems. Techniques for the discretisation of distributed parameter systems, with special reference to the finite element method, are also covered by the course. The course also contains a short introduction to rotordynamics.
Dynamics and vibrations of discrete systems with more than one degree of freedom (dof)
Methods for writing the equations of motion of multi-dof systems. Equations of motion using the set of principal coordinates end representation of system motion in terms of modal superposition. Techniques for the identification of modal parameters from measurements of the system’s dynamic behaviour. Introduction to the spatial dynamics of rigid bodies.
Dynamics of mono-dimensional continuous systems- Analytical solutions.Axial and torsional vibrations of beams: propagative and stationary solutions, natural frequencies and modes of vibration for different boundary conditions. Transversal vibrations of stretched strings. Bending vibrations of slender beams (Euler-Bernoulli formulation): stationary solutions, natural frequencies and modes of vibration for different boundary conditions. Free vibration of systems of beams, analytical solutions. Forced vibration for systems of beams: the modal superposition principle. Orthogonality of the modes of vibration for mono-dimensional continua.
Dynamics of mono-dimensional continuous systems- The finite element method. Discretisation methods for continuous systems. The finite element method. Beam type finite elements: shape functions, local and global coordinates, stiffness and damping matrices, effect of static axial tension. Expression of the nodal forces corresponding to typical concentrated and distributed loads. Assembling finite elements into a model, boundary conditions. Structural damping model. Calculations of the natural frequencies and modes of vibration, calculation of frequency response function and of the response to arbitrary loads. Outline of the application of the Finite Element method with 2D and 3D discretisation.
Rotordynamicsdynamic problems in rotating machines. Rotor schematisation. Numerical model of the hydrodynamic bearings and linearisation: equivalent stiffness and damping matrices. Instability produced by oil film forces. Critical speeds, methods for rotor balancing.
 Diana G., Cheli F.: Dinamica e Vibrazioni dei Sistemi meccanici, Polipress, 2008.
 Meirovitch L.: Fundamentals of Vibrations, Mc Graw-Hill International Edition.
 Handouts available on the course site