Basics of Spectral Analysis, Structural Dynamics and Statistical Inference

This chapter brings together key ideas from spectral analysis, structural dynamics, and statistical inference, tailored to equip readers with practical tools for understanding subsequent system identification and structural health monitoring methodologies. It begins with stationary stochastic processes, modeling vibration test data through time-domain correlation functions and frequency-domain power spectral density. Key estimators for finite-duration, discretely sampled data are discussed, highlighting their asymptotic forms under long data durations, a prerequisite for operational modal analysis and damage detection. Next, structural dynamics fundamentals are covered for single- and multi-degree-of-freedom systems, including free vibration, forced responses (harmonic and arbitrary excitations), and modal superposition. Stochastic excitation is introduced via power spectral density matrices, connecting deterministic and random vibration frameworks in the modal domain. Finally, statistical inference essentials are outlined for stochastic system identification. Complex-valued probability distributions are introduced to handle frequency-domain quantities such as Fourier transform coefficients and transmissibility functions. Bayesian inference principles, highlighting Laplace approximation, stochastic sampling, and variational inference, are simplified to prepare readers for uncertainty-quantified system identification in later chapters. This chapter prioritizes clarity over theoretical rigor, ensuring accessibility to foundational concepts critical for algorithmic developments and advanced applications.