# Simplified Vibration PSD Synthesis Method for MIL-STD-810
## 1 Introduction  
Products must perform adequately under all environmental conditions associated with its service life. Laboratory environmental tests are used extensively instead of more time-consuming and less cost effective field exposure tests. The MIL-STD-810, Environmental Engineering Considerations and Laboratory Tests, establishes laboratory test methods that replicate the effects of environments on the equipment rather than imitating the environments themselves. Vibration test among dozens of environmental test methods are also performed to verify that materiel will function in and withstand the vibration exposures of a life cycle. The most dynamically accurate method to reproduce the full exposure would be to sequentially vibrate the product to all the individual, uncompressed events representing its full lifecycle. However, such an approach is generally not feasible from both schedule and economic perspectives, and some compromises must be made to realize the benefits of testing in the laboratory. Time compression techniques based on fatigue equivalency are typically employed such that vibration testing can be performed in a timely and economic manner. The objective of the laboratory vibration test is to generate the equivalent fatigue damage accumulated in the products by vibration throughout their lifetime [1].  
This fatigue damage by vibration can be quantified by the fatigue damage spectrum (FDS) methods which is a spectral representation of fatigue damage as a function of the SDOF system's natural frequency [2]. The FDS can be computed in time domain [3] and in frequency domain [4; 5; 6]. In frequency domain, the formula calculating FDS can be obtained based on some assumptions. In case of a low damping SDOF system and Gaussian random input, in which the peak distribution of theresponse becomes a Rayleigh one, the FDS equation gets much simpler [7]. By simplifying this further, [PERSON] and [PERSON] proposed a descriptor for comparing random test conditions and named it as damage potential (DP) [7]. MIL-STD-810 uses this damage potential formula for synthesizing random vibration conditions. The purpose of DP is not to calculate the accurate fatigue damage, but to compare the relative fatigue damage between test specifications. However, as the synthesized vibration condition using DP is the same as using FDS, it is more efficient to use DP which is a much simpler formula. However, DP is not an optimized index for vibration condition synthesis. In this paper, we propose the slimmest formula in which unnecessary factors are removed from the DP to suit the purpose of vibration condition synthesis. This simplified equation, called GRS (Gaussian Random Synthesis), produce the same results as DP in vibration synthesis application only.  
To introduce GRS, we first examine the concept of FDS thoroughly in time domain and how the FDS formula, which is the basis of DP, was derived in frequency domain, and then look at how to synthesize random vibration conditions under [PERSON]'s cumulative damage law in MIL-STD-810. Then the more simplified formula, GRS that is newly presented in this paper is derived. Actually, it reveals a concise expression in the process of composing the formula of itself. Finally, we present an example of synthesis using FDS, DP, and the new GRS equations and discuss the results.  
GRS formula aims just to present a more convenient method for those who wish to create vibration test conditions by the method introduced in MIL-STD-810. Therefore, the constraints and problems that the method has inherited must be checked as well.