Random vibrations shake up our understanding of mechanical systems. This section dives into how linear systems respond to unpredictable forces, using statistical methods to make sense of the chaos. It's like predicting the weather for your machine!

We'll explore equations of motion, transfer functions, and analysis techniques for both single and multi-degree-of-freedom systems. By the end, you'll have tools to tackle real-world random vibration problems in engineering design and analysis.

Equations of Motion for Random Excitation

Characterizing Random Excitation

• Random excitation in vibration analysis varies unpredictably over time, requiring statistical methods for characterization
• Stochastic processes model random excitation
  • White noise (constant power spectral density across all frequencies)
  • Colored noise (power concentrated in specific frequency ranges)
  • Band-limited noise (power confined to a finite frequency band)
• Key statistical measures characterize random excitation
  • Autocorrelation function R(τ) describes temporal correlation
  • Power spectral density S(ω) represents frequency content distribution
• Stationary random processes have time-invariant properties, simplifying analysis
• Non-stationary processes require complex techniques (evolutionary power spectral density methods)

Formulating Equations of Motion

• General equation of motion for a linear system under random excitation $Mẍ + Cẋ + Kx = F(t)$
  • M represents mass matrix
  • C represents damping matrix
  • K represents stiffness matrix
  • F(t) represents random forcing function
• For single degree-of-freedom (SDOF) systems, matrices reduce to scalar values
• Multi-degree-of-freedom (MDOF) systems use matrix form $[M]ẍ + [C]ẋ + [K]x = F(t)$
• Equations account for system properties and external random forces

Response of Single-DOF Systems to Random Excitation

Statistical Characterization of Response

• Response characterized by probability density function (PDF) and statistical moments
• Mean square response calculated using frequency response function H(ω) and input power spectral density S(ω) $E[x^2] = \int_{-\infty}^{\infty} |H(\omega)|^2 S(\omega) d\omega$
• Root-mean-square (RMS) values derived from mean square response
  • Displacement RMS: $x_{RMS} = \sqrt{E[x^2]}$
  • Velocity RMS: $v_{RMS} = \sqrt{E[ẋ^2]}$
  • Acceleration RMS: $a_{RMS} = \sqrt{E[ẍ^2]}$
• Probability of exceeding response level determined using cumulative distribution function (CDF)
• Gaussian random excitation results in Gaussian response for linear SDOF systems

Advanced Analysis Techniques

• Spectral moments characterize frequency content of response $m_n = \int_{0}^{\infty} \omega^n S(\omega) d\omega$
• Peak factors estimate maximum response amplitude
• Time domain analysis (Fokker-Planck equation) used for non-linear SDOF systems
• Monte Carlo simulations generate multiple response realizations for statistical analysis

Analysis of Multi-DOF Systems with Random Excitation

Modal Analysis and Decoupling

• Modal analysis decouples equations of motion into independent SDOF systems
• Transformation to modal coordinates: $x = [Φ]q$
  • [Φ] represents mode shape matrix
  • q represents modal coordinates
• Decoupled equations in modal space: $ẍ_i + 2ζ_i ω_i ẋ_i + ω_i^2 x_i = f_i(t)$
• Each mode analyzed independently, results combined for total response

Response Computation and Spatial Correlation

• Frequency response matrix [H(ω)] relates input excitation to output response
• Cross-correlation and cross-spectral density functions account for spatial correlations
• Response statistics computed using matrix operations $[S_x(\omega)] = [H(\omega)][S_F(\omega)][H(\omega)]^H$
  • [S_x(ω)] represents response power spectral density matrix
  • [S_F(ω)] represents input power spectral density matrix
  • [H(ω)]^H represents conjugate transpose of frequency response matrix
• Principal coordinate analysis simplifies MDOF system analysis
• Monte Carlo simulations employed for complex systems or intractable analytical solutions

Transfer Functions in Random Vibration Analysis

Transfer Function Fundamentals

• Transfer function H(s) defined as ratio of output to input Laplace transforms
• Frequency response function H(ω) relates steady-state response to harmonic excitation
• Magnitude |H(ω)| represents system amplification factor at different frequencies
• Phase angle of H(ω) indicates phase shift between input and output
• Transfer function used to compute output power spectral density $S_{out}(\omega) = |H(\omega)|^2 S_{in}(\omega)$

Applications and Experimental Techniques

• Transmissibility related to transfer function, analyzes vibration isolation effectiveness
• Experimental methods determine transfer functions for complex systems
  • Sine sweep tests apply sinusoidal excitation at varying frequencies
  • Impact hammer tests use impulse excitation to excite system
• Transfer functions enable prediction of system response to various input types
• Used in design optimization, structural health monitoring, and vibration control systems