Random Vibrations Course
This course covers the theory and applications of random vibrations in mechanical systems. It is taught as part of the engineering curriculum at École centrale de Lyon, France by Alain Le Bot and Joel Perret-Liaudet. It comprises lectures, exercises, projects, practical work, and final test, totaling 34 hours.
The reference book is Introduction aux vibrations aléatoires, A. Le Bot, Dunod, 2019.
Further developments may be found in Foundation of statistical energy analysis in vibroacoustics, A. Le Bot, OUP, 2015.
Use the menu on the left to navigate through the chapters.
Introduction
Vibrations of mechanical systems can arise from time-varying forces, acoustical pressure, or imposed displacement applied to the system. But these external loads may be deterministic or random. This course focuses on random vibrations, of a linear mechanical system where the excitations are stochastic. The case where the system parameters are random, with a deterministic or stochastic source, as well as propagation of waves in random media are different subjects not tackled in this course.
Pedagogical Objectives
The main objective of this course is to enable students to:
- Understand the probabilistic characterization of random processes.
- Analyze the spectral response of linear systems to random excitations.
- Calculate the root-mean-square (RMS) and maximum of vibration of a mechanical system subjected to random forces.
Examples
Real-world examples of random vibrations include:
- Buildings vibrating during an earthquake.
- Boats or off-shore platforms excited by waves.
- Bridges or sky-crapers excited by wind.
- Vehicles, aircraft, or other mechanical structures excited by a turbulent flow.
- Cars rolling on a rough asphalt or trains with the wheel/rail contact.
Course Chapters
The course is organized into four chapters:
Chapter 1: Stochastic Processes
This chapter introduces the fundamental concepts of probability theory and stochastic processes, which are essential for analyzing random vibrations.
Go to Chapter 1Chapter 2: Vibration of Mechanical Systems
This chapter focuses on calculating the impulse response and frequency response function (FRF) of vibrating systems, including single-degree-of-freedom (SDOF), multi-degree-of-freedom (MDOF), and continuous systems.
Go to Chapter 2Chapter 3: Spectral Response of Linear Systems
This chapter covers the spectral response of linear systems, including problem presentation, mean of response, correlation of response, spectral density of response, and root-mean-square of response.
Go to Chapter 3Chapter 4: Probability of Thresholds and Maxima
This chapter focuses on the probability of thresholds and maxima, including Gaussian processes, threshold crossings, and rate of maxima.
Go to Chapter 4