Vibrations and Dynamics of Machines

The course is involved with the mathematical description and study of systems that can vibrate. Systems with one degree of freedom are studied, where the equations of motion are developed, and their vibrational behavior is investigated. Ordinary differential equations and laws of Dynamics are used (Newtons Laws). The analysis always includes both free and forced vibration, subject to various excitations (impact, step, harmonic), with or without the presence of damping mechanisms. Next the study is generalized to systems of multiple degrees of freedom where matrices are used along with elements from Linear Algebra, in order to calculate the modal frequencies and mode shapes by solving the eigenproblem. The method of Modal Analysis is used to calculate the response of the system to various excitations. The analysis is transfered to the frequency domain using the Fourier Transform, and the transfer function is introduced. Next, continuous systems are investigated where partial differential equations are used, and their modal frequencies and mode shapes are calculated in order to understand their vibrational behavior. Also, alternative methods of writing the equations of motion are studied (Principle of Virtual Work and Lagrange Equations) and experimental modal analysis is introduced.

Objectives

The objective is the presentation of the important methodologies for the prediction of the dynamics and vibrations of mechanical systems with linear characteristics. The student must be able to develop simplified models that describe the behavior of mechanical systems, to predict based on the analysis of these models the dynamics and vibrations of mechanical systems, to understand the important system characteristics that affect the dynamics of structural and mechanical systems, and finally to apply the methodologies for the design of vibration isolation systems.

Prerequisites

Dynamics, linear algebra, ordinary differential equations, partial differential equations

Syllabus

Vibration of single-degree-of-freedom systems, mathematical models, equation of motion using Newton’s law, free vibration, vibration to various forms of excitation (impulse, pulse, harmonic, periodic, non-periodic), Fourier analysis, transfer function – Applications (vibration suppression, measuring instruments in vibration analysis) – Vibration of multi-degree-of-freedom discrete systems, mathematical models, equations of motion, modal frequencies and modeshapes, free and forced vibrations, modal analysis method – Vibration of continuous systems (vibration of strings, longitudinal and torsional vibrations of rods, vibration of beams) – Experimental dynamics and modal analysis – Approximate methods of analysis (Rayleigh method, Rayleigh-Ritz method, Galerkin method) –Machine dynamics (rotor dynamics, critical speed, balancing).