# Simplified Vibration PSD Synthesis Method for MIL-STD-810
## 3 Vibration PSD Synthesis
### Vibration PSD Synthesis Made Simpler  
Originally, the concept of damage potential (DP) was proposed to describe the potential damage delivered to the virtual test items, which are the SDOF systems with different natural frequencies, by random vibration test [7]. So, DP itself is meaningful for comparison purpose. In the combination process, however, DP is an intermediate process, not a final output. Therefore, the Equation (9) that produce the individual DPs and the Equation (10) that combines them is substituted into the Equation (11) that calculates the test PSD, which can be summarized as follows. \[G_{test}(f_{n})=f_{n}\zeta\bigg{[}\frac{DP_{total}(f_{n})}{f_{n}T_{test}}\bigg{]} ^{\frac{2}{b}}=f_{n}\zeta\bigg{[}\sum_{i=1}^{N}\frac{T_{i}}{T_{test}}\bigg{(} \frac{G_{i}(f_{n})}{f_{n}\zeta}\bigg{)}^{b/2}\bigg{]}^{2/b}=\bigg{(}\sum_{i=1}^ {N}\frac{T_{i}(G_{i}(f_{n}))^{b/2}}{T_{test}}\bigg{)}^{2/b} \tag{13}\]  
This formula can be rewritten as follows to look simple,  
\[\text{for each }f_{n},G_{test}=\bigg{(}\sum_{i=1}^{N}\frac{T_{i}G_{i}^{b/2}}{T_{ test}}\bigg{)}^{2/b}=\bigg{(}\frac{T_{1}G_{1}^{b/2}+T_{2}G_{2}^{b/2}+\ldots+T_{N}G_{N}^{b /2}}{T_{test}}\bigg{)}^{2/b} \tag{14}\]This can be rearranged as follows in a symmetric form which shows the ultimately simple relationship between field conditions and test condition in fatigue damage perspective when input PSDs are Gaussian waveform. \[T_{test}G_{test}^{b/2}=T_{1}G_{1}^{b/2}+T_{2}G_{2}^{b/2}+\ldots+T_{N}G_{N}^{b/2} \tag{15}\]  
Equations (15) shows that we do not need natural frequency and damping ratio in combining the PSDs. Now we can remove the unnecessary parameters and define a new index for only synthesizing process as follows and named it as Gaussian random synthesis (GRS) to emphasize its inherent constraints and usage at the same time,  
\[GRS(f_{n})=T\cdot G(f_{n})^{b/2} \tag{16}\]  
The inverse form is  
\[G(f_{n})=\left(\frac{GRS(f_{n})}{T}\right)^{2/b} \tag{17}\]  
The same results as Equations (14) and (15) are obtained from other FDS equation. The nCode, fatigue and durability analysis software, uses the FDS formula under the assumption that the relative displacement response instead of the velocity response of the virtual specimen is proportional to stress [4]. The equation below shows the process that the constant terms are collected in the FDS as in Equation (8) when DP is obtained. \[D(f_{n})=f_{n}T\cdot\frac{k^{b}}{C}\Bigg{[}\frac{Q\cdot G(f_{n})}{2(2\pi f_{n} )^{3}}\Bigg{]}^{b/2}\Gamma\Bigg{[}1+\frac{b}{2}\Bigg{]}=\textit{const}\cdot f _{n}T\Bigg{[}\frac{Q\cdot G(f_{n})}{f_{n}{}^{3}}\Bigg{]}^{b/2} \tag{18}\]  
Here, \(Q=1/2\zeta\), The inverse of Equation (18) calculating the test PSD is  
\[G_{test}(f_{n})=\frac{f_{n}^{3}}{Q}\Bigg{[}\frac{DP_{total}(f_{n})}{\textit{ const}\cdot f_{n}T_{test}}\Bigg{]}^{\frac{2}{b}}=\frac{f_{n}^{3}}{Q}\Bigg{[} \sum_{i=1}^{N}\frac{T_{i}}{T_{test}}\Bigg{(}\frac{Q\cdot G_{i}(f_{n})}{f_{n}{} ^{3}}\Bigg{)}^{\frac{2}{b}}\Bigg{]}^{\frac{2}{b}}=\left(\sum_{i=1}^{N}\frac{T _{i}(G_{i}(f_{n}))^{\frac{2}{b}}}{T_{test}}\right)^{\frac{2}{b}} \tag{19}\]  
The last expression is same as Equation (14) again which means we would get the same synthesized PSD using both FDS formula based on different FRF. Table 1 is a simple example synthesizing two PSDs by using FDS based on pseudo velocity, DP, and GRS. The frequency is fixed at 300 Hz, and \(b=4\), \(\zeta=0.05\), \(C=1\), and \(k=1\) is given then the intermediate calculation results are different but the final test PSDs to generate the equivalent fatigue damage index for 10 min test time are exactly the same, 30 (m/s\({}^{2}\))\({}^{2}\)/Hz.