Vibration Fatigue By Spectral Methods Pdf -

Empirical closed-form solution using Monte Carlo on 70+ PSD shapes:

[ p_\textDK(s) = \frac\fracD_1Q e^-Z/Q + \fracD_2 ZR^2 e^-Z^2/(2R^2) + D_3 Z e^-Z^2/2\sqrtm_0 ]

Where ( Z = s / \sqrtm_0 ), and coefficients ( D_1, D_2, D_3, Q, R ) are functions of ( \alpha_1, \alpha_2, m_0, m_1, m_2, m_4 ).
Accuracy: Within ±5% of rainflow for most stationary Gaussian processes.

Once the PDF of stress ranges $p(S)$ is obtained, damage is calculated using the Palmgren-Miner linear damage rule combined with the material S-N curve (Basquin’s equation: $N S^k = C$).

The expected fatigue life $T$ is calculated as: vibration fatigue by spectral methods pdf

$$E[D] = T \int_0^\infty \fracp(S) \cdot v_pN(S) ds$$

Where $v_p$ is the rate of peaks and $N(S)$ is the number of cycles to failure at stress range $S$.

Vibration fatigue refers to the failure of structures subjected to dynamic loads where the stress history is a random process rather than a deterministic cycle. Traditional fatigue analysis (e.g., Rainflow Counting on time-domain signals) is accurate but computationally expensive, requiring long time-history simulations.

Spectral Methods offer an alternative by operating in the frequency domain. Instead of analyzing a time-history stress signal, these methods utilize the Power Spectral Density (PSD) function of the stress response. The primary advantage is computational efficiency: a frequency-domain analysis takes seconds compared to the hours required for transient dynamic simulations. Empirical closed-form solution using Monte Carlo on 70+

Using the pdsa (Power Density Spectral Analysis) toolbox:

psd = [0 0.01; 10 0.01; 10 0.0001]; % Define PSD
[S, freq] = pdsa_psd(psd);
damage = pdsa_dirlik(S, freq, sn_curve);

An empirical correction applied to the narrowband result:

[ D_WL = \rho(k, \gamma) \cdot D_NB ]

with correction factor:

[ \rho(k, \gamma) = a(k) + [1 - a(k)] (1 - \gamma)^b(k) ]

where ( a(k) = 0.926 - 0.033k ), ( b(k) = 1.587k - 2.323 ). Valid for ( 3 \le k \le 6 ).

If input PSD is ( S_in(f) ) and FRF is ( H(f) ): [ S_\sigma(f) = |H(f)|^2 \cdot S_in(f) ]