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The exhaustive list of topics in Random Vibrations  in which we provide Help with Homework Assignment and Help with Project is as follows:

  • Random vibrations & Failure Analysis.
  • Random vibration & probabilistic modeling.
  • Axioms of probability theory:
    • Probability space.
    • Random variables.
    • Probability distributions and density functions of random variables.
    • Joint and marginal distribuition.
    • Density functions.
    • Functions of random variables.
    • Expectations and moments of random variables.
    • Baye's theorem.
    • Conditional random variables.
    • Conditional expectations.
    • Characteristic functions.
    • Moment generating functions.
    • Cumulants.
    • Relationship between joint probability density functions and characteristic functions.
    • Numerical issues.
    • Covariance and independence.
    • Sequences of random variables.
    • Stochastic convergence.
    • Limit theorems.
    • Concepts of stochastic processes, probability distributions, moments, correlation and covariance functions.
    • The power spectral density function.
    • Stationarity and non-stationarity of stochastic processes, ergodicity of a stochastic process.
    • Limits of a stochastic process.
    • Continuity & differentiability.
    • Stochastic derivatives and integrals.
    • Special random processes -1: Gaussian random processes.
    • Special random processes -2: Poisson processes and random pulses.
    • Special random processes -3: Random walk & wiener processes.
    • Special random processes -4: Markov processes.
    • The Fokker-Planck-Kolmogorov equation.
    • Stochastic calculus.
    • Numerical simulation of random processes.
    • Deterministic dynamics and impulse response functions of systems.
    • System response to random excitations.
    • Response to stationary & weakly stationary excitations.
    • Delta-correlated excitations.
    • Response to Gaussian excitations.
    • Non-stationary excitations.
    • Joint behavior of the time deriative and its response & Markov vector approach.
    • Linear dynamics and harmonic transfer functions.
    • Generalization to multi degree-of-freedom systems.
    • State space formulation of equations of motion.
    • The Fokker-Planck equation for linear systems.
    • The Fokker-Planck equation for sdof systems.
    • The Fokker-Planck equation for mdof systems.
    • Methods for Numerical solutions for the FPK equation: finite difference.
    • Methods for Numerical solutions for the FPK equation: finite element method.
    • Numerical solutions for the FPK equation: Path integral method.
    • Method of equivalent statistical linearization.
    • State space moment and cumulant equations.
    • Level crossings and the first passage time.
    • Probability distribution of maxima & failure probability.
    • Peak distributions & their applications.
    • Envelope crossings and their distributions.
    • Generralization to non-Gaussian processes
    • Numerical evaluation of Rice's formula.
    • The saddle point approximation.
    • Damage accumulation and random fatigue.
    • Random Fatigue.
    • Rainflow counting algorithm.
    • Probabilistic crack growth.