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
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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)
General equation of motion for a linear system under random excitation
M x ¨ + C x ˙ + K x = F ( t ) Mẍ + Cẋ + Kx = F(t) M x ¨ + C x ˙ + K x = 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 ] x ¨ + [ C ] x ˙ + [ K ] x = F ( t ) [M]ẍ + [C]ẋ + [K]x = F(t) [ M ] x ¨ + [ C ] x ˙ + [ 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 ] = ∫ − ∞ ∞ ∣ H ( ω ) ∣ 2 S ( ω ) d ω E[x^2] = \int_{-\infty}^{\infty} |H(\omega)|^2 S(\omega) d\omega E [ x 2 ] = ∫ − ∞ ∞ ∣ H ( ω ) ∣ 2 S ( ω ) d ω
Root-mean-square (RMS) values derived from mean square response
Displacement RMS: x R M S = E [ x 2 ] x_{RMS} = \sqrt{E[x^2]} x RMS = E [ x 2 ]
Velocity RMS: v R M S = E [ x ˙ 2 ] v_{RMS} = \sqrt{E[ẋ^2]} v RMS = E [ x ˙ 2 ]
Acceleration RMS: a R M S = E [ x ¨ 2 ] a_{RMS} = \sqrt{E[ẍ^2]} a RMS = E [ x ¨ 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 = ∫ 0 ∞ ω n S ( ω ) d ω m_n = \int_{0}^{\infty} \omega^n S(\omega) d\omega m n = ∫ 0 ∞ ω n S ( ω ) d ω
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 x = [Φ]q x = [ Φ ] q
[Φ] represents mode shape matrix
q represents modal coordinates
Decoupled equations in modal space:
x ¨ i + 2 ζ i ω i x ˙ i + ω i 2 x i = f i ( t ) ẍ_i + 2ζ_i ω_i ẋ_i + ω_i^2 x_i = f_i(t) x ¨ i + 2 ζ i ω i x ˙ 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 ( ω ) ] = [ H ( ω ) ] [ S F ( ω ) ] [ H ( ω ) ] H [S_x(\omega)] = [H(\omega)][S_F(\omega)][H(\omega)]^H [ S x ( ω )] = [ H ( ω )] [ S F ( ω )] [ H ( ω ) ] 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 o u t ( ω ) = ∣ H ( ω ) ∣ 2 S i n ( ω ) S_{out}(\omega) = |H(\omega)|^2 S_{in}(\omega) S o u t ( ω ) = ∣ H ( ω ) ∣ 2 S in ( ω )
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