Introduction to the fundamental concepts of the random variables, random processes and stochastic differential equations. Introduction to the analysis of the mechanical systems subjected to random excitation. Understanding and application of the software tools used in the process of analysis.

learning outcomes Students gain knowledge about the theory of random vibration of mechanical systems and about software tools needed to the analysis.

Sigma algebra, probability function, probability space, random variables, distribution function, probability density function and normal distribution, random process, stationarity and ergodicity, auto correlation, white noise, Brownian motion and Wiener process, stochastic differential equations (SDE), Ito’s integral, Fokker-Planck equation, linear SDE, Euler method, Monte Carlo Simulation, linear structures with single degree of freedom, system response to random excitation, nonstationary excitation, means and covariance, linear structures with multi degrees of freedom, response to atmospheric turbulence, linear continuous structures, response to boundary layer turbulence, nonlinear structures, nonlinear stress, structural failure resulting from dynamic response, types of structural failure, envelope distribution, fatigue failures.

MatLab, generating random variables, functions rand, randn, generating random process white noise and Wiener, computation of Ito’s integrals, solving SDE, computation distribution function, computation of the probability density function (PDF), evolution of PDF, Euler method, simulation of mechanical systems, simulation of mechanical systems using Euler method, computation of the probability density function of the response of the mechanical systems, computation of the probability of failure, simulation of the probability of fatigue failure.

Without conditions

Written lectures (handouts) MATLAB software Cvetkovic, A., Radojevic, S.: MATLAB I, Faculty of Mechanical Engineering, Belgrade, 2012. Gilat, A., MATLAB: An Introduction with Applications, John Wiley & Sons, 2005

Total assigned hours: 65

New material: 25
Elaboration and examples (recapitulation): 25

Auditory exercises: 0
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 0
Review and grading of seminar papers: 0
Review and grading of the project: 0
Test: 0
Test: 0
Final exam: 15

Activity during lectures: 0
Test/test: 60
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Final exam: 40
Requirement for taking the exam (required number of points): 30

Y.K. Lin Probabilistic Theory of Structural Dynamics, Robert E. Krieger publishing Company, Florida, 1976; J. B. Roberts, P.D. Spanos, Random Vibrations and Statistical Linearization, Dover Publications, New York, 1999.; L.C. Ewans, An Introduction to Stochastic Differential Equations,, American Mathematical Society, 2014; A. Cvetkovic, S. Radojevic, MATLAB I, Faculty of Mechanical Engineering, Belgrade, 2012.