# Research on the Scope of Spectral Width Parameter of Frequency Domain Methods in Random Fatigue
## 2 Common Frequency Domain Methods
### Spectrum Simulation and the Characteristic Parameters  
The simulated spectrum taken in this article can be divided into four categories: band-limited white noise spectrum, single-peak spectrum, double-peak spectrum and multi-peak spectrum. In order to control the variables, all the simulated spectra are curves with the same root mean square value of stress and varying spectrum width parameters. The root mean square value of the stress here is taken as 147 MPa, and the spectrum width parameter range is between 0 and 1. In order to cover the spectrum width parameter range as far as possible, Matlab, a commercial mathematics software produced by MathWorks in the United States, is used to adjust the spectrum amplitude and frequency domain range and randomly generate signals with different spectrum width parameters.  
1. Single peak spectrum.  
As shown in Figure 1, the single peak spectrum refers to the power spectrum density with only one peak in the whole spectrum line. When the natural frequency of the structure is consistent with or close to the load frequency, a resonance phenomenon will occur, and the response spectrum will generate a peak, so this peak spectrum is also a common response spectrum in engineering. This set of single-peak spectra contains 12 spectral lines, the root mean square stress is 147 MPa, and the spectral width parameters are respectively 0.1875, 0.2657, 0.389, 0.4546, 0.5504, 0.6061, 0.6688, 0.7272, 0.8184, 0.8942, 0.9221, 0.965.  
2 Double peak spectrum.  
As shown in Figure 2, the double-peak spectrum refers to the entire spectral line that contains only two main frequency peaks. This is also one of the main forms of the dynamic response of engineering structures. When the structure generates two-order resonance, a double-peak response spectrum will be generated. This set of bimodal spectra contains 12 spectral lines with spectral width parameters of 0.1951, 0.2445, 0.3568, 0.4821, 0.5543, 0.6534, 0.7386, 0.8129, 0.8852, 0.9483, 0.9746, 0.9829. Here, we also consider the impact of different background noises on life calculation.  
3 Multi-peak spectrum.  
As shown in Figure 3, the multi-peak spectrum refers to more than three peaks in the entire spectrum, which is a typical broadband spectrum. When the frequency range of the external load is wide, it can cover the multi-order natural frequency of the structure, then the result will be multi-order resonance, and the multi-peak spectrum will appear in the response spectrum of this case, which is also a common spectrum in structural, mechanical, marine, and aerospace engineering. This group  
Figure 1: Single peak simulation spectrum.  
Figure 2: Double peak simulation spectrum.  
contains a total of 10 sets of spectral lines, the spectral width parameters are: 0.3863, 0.4858, 0.5844, 0.6415, 0.6972, 0.7511, 0.8396, 0.9099, 0.9298, 0.966.  
4 Band-limited white noise spectrum.  
In structural vibration fatigue analysis, the environmental load and stress response of the structure can usually be approximated as a band-limited white noise process or a combination of band-limited white noise processes [35; 36], as shown in Figure 4. This group takes a total of 10 sets of band-limited white noise analogue spectrum curves, and changes its spectrum width parameters by changing its frequency range and amplitude. The specific spectral width parameters are: 0.115, 0.18812, 0.2733, 0.3498, 0.4167, 0.474, 0.5225, 0.5631, 0.6248, 0.6667.