Introduction to random vibrations

Lecturer(s): Alain LE BOT, Joël PERRET LIAUDET, Julien HUILLERY
Course ⋅ 12 hTC ⋅ 8 hPW ⋅ 8 hStudy ⋅ 4 h


Many mechanical engineering problems concern the vibratory response of mechanical structures subjected to random forces. Examples include slender structures excited by wind, offshore platforms excited by wave action, road-excited vehicles, buildings subjected to earthquakes, frictional noise induced by roughness, aircraft fuselages excited by a turbulent boundary layer, etc. The objective of this course is to provide the student with elements of random vibrations. We will introduce the concepts of random signals and their spectral properties and provide methods for estimating the probabilistic response of systems excited by random forces. Many concrete examples will illustrate the concepts and results discussed.


Vibration, linear systems, stochastic process, spectral properties


Chapter 1 Stochastic processes Generalities of probability, random variable, random vector, stochastic process, spectral analysis, continuity, derivation, integration Chapter 2 Vibrations in small deformations 1 degree of freedom oscillator, n degree of freedom systems, deformable solids Chapter 3 Spectral Response of Linear Systems Presentation of the problem, average, correlation and spectrum of the response, correlation and spectrum between excitation and response Chapter 4 Threshold and Maximum Probability Probability density of the response, threshold crossing, occurrence of maxima

Learning Outcomes

  • Know how to calculate and interpret the power spectral densities of random signals
  • Know how to calculate the frequency response functions of simple mechanical systems
  • Know how to estimate the spectral quantities of the responses of linear systems excited by random signals


How to do (practical activities) : 50% Initial knowledge (final test) : 50 %