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.