# Simplified Vibration PSD Synthesis Method for MIL-STD-810
## 2 Fatigue Damage Spectrum (FDS)
### FDS in MIL-STD-810  
There are various formulas calculating FDS, but here we thoroughly review the derivation process of the DP formula. As a special case, if the input is a random waveform featuring (1) Gaussian distribution and (2) constant PSD, and also (3) the damping coefficient of the SDOF system is low (\(\zeta<0.1\)), then the response of the system is also normally distributed random waveform and the peak distribution shows a Rayleigh one [PERSON] et al. (2014) as depicted in Figure 2.  
This peak distribution corresponds to the histogram \(\mathbf{e}\) in Figure 1, and is represented by the following equation. \[n(s)=f_{n}T\frac{s}{\sigma^{2}}\exp(-\frac{s^{2}}{2\sigma^{2}})=f_{n}T\cdot p(s) \tag{1}\]  
\(p(s)\) is the probability density function of the Rayleigh distribution, where \(\sigma\) is the scale parameter, which is equal to the standard deviation from the response waveform. Assuming an SDOF system with a low damping coefficient, the response waveform oscillates based on its natural frequency \(f_{n}\), so the total number of peaks during time \(T\) can be approximated by the product of \(f_{n}\) and \(T\). Multiplying  
Figure 2: Velocity response of the SDOF system to Gaussian random acceleration input with constant power spectral density (PSD) shows Gaussian distribution. The peak distribution of the response becomes a Rayleigh function. The standard deviation of the response distribution matches the scale parameter of the Rayleigh function. a(t) is input acceleration and s(t) is stress waveform (above). Stress PSD is calculated by multiplying input PSD and squared stress amplitude of system’s frequency response function (below). the Rayleigh function by the total number of peaks yields the distribution function of peaks over stress, \(n(s)\). The standard deviation, \(\sigma\), of the normally distributed output random waveform is squared root of the area of the stress response PSD, \(G_{s}(f)\), which can be obtained by multiplying the input PSD with the squared amplitude of the system's frequency response function (FRF), \(H(f)\), in the frequency domain as shown in Equation (2). \(G\) is set to be a constant to enable the integration and \(k\) is conversion factor from velocity to stress. \(H\) is the FRF of pseudo velocity output to the acceleration input. These FRF relationships are well documented in ISO 18431-4 [12]. \[\sigma^{2}=\int_{0}^{\infty}G_{s}(f)df=\int_{0}^{\infty}k^{2}G\cdot\left|H(f) \right|^{2}df=k^{2}G\int_{0}^{\infty}\left|H(f)\right|^{2}df=\left(\frac{k}{2 \pi f_{n}}\right)^{2}\frac{G\pi f_{n}}{4\varsigma} \tag{2}\]  
\[\left|H(f)\right|^{2}=\left|\frac{pv}{a}\right|^{2}=\frac{\left(1/2\pi f_{n} \right)^{2}}{\left(1-\left(f/f_{n}\right)^{2}\right)^{2}+\left(2\zeta f/f_{n} \right)^{2}} \tag{3}\]  
Since \(n(s)\) has been completely defined, we calculate the damage index by stress and add them all up to get the cumulative damage index \(D\) with Equation (4). This corresponds to step f in Figure 1. Here, the Basquin's Equation (5) is used to represent \(N(s)\) which is the number to failure. \[D(f_{n})=\int_{0}^{\infty}\frac{n(s)}{N(s)}ds=\int_{0}^{\infty}\frac{f_{n}T \cdot p(s)}{Cs^{-b}}ds=\frac{f_{n}T}{C}\int_{0}^{\infty}p(s)s^{b}ds \tag{4}\]  
\[N(s)=Cs^{-b} \tag{5}\]  
The last integral term in Equation (4) is the \(b\)-th moment of the Rayleigh function, \(p(s)\), and can be expressed using the Gamma function [13] as below. \[\int_{0}^{\infty}p(s)s^{b}ds=\sigma^{2}2^{b/2}\Gamma[1+b/2] \tag{6}\]  
Applying the results of Equations (2) and (6) to Equation (4) yields Equation (7). \[D(f_{n})=\frac{f_{n}T}{C}\left[\frac{G(f_{n})k^{2}}{8\pi f_{n}\zeta}\right]^{ b/2}\Gamma[1+b/2] \tag{7}\]  
The equation above is a formula for calculating the fatigue damage index accumulated in a lightly damped SDOF system excited by Gaussian random vibration, and it can be called the FDS equation.