# Fatigue Assessment of Explosive Bolts Considering Vibration of Fixtures
## 2 Generation of Road Surface Spectrum  
Road roughness function \(x_{m}(L)\) is generally defined as the variation of height \(x_{m}\) of a road surface to a reference plane along road length \(L\)[8]. Stochastic data of road roughness are measured and processed to obtain a displacement PSD \(G_{x}(n)\).  
According to ISO 8608: 1995 [9],  
\[G_{x}(n)=G_{x}(n_{0})\bigg{|}\frac{n}{n_{0}}\bigg{|}^{-\omega}. \tag{1}\]  
\(G_{x}(n)\) is namely road roughness factor. On the basis of displacement PSD, road surfaces are divided into eight classes from A to H. \(n\) is spatial frequency, i.e., the number of wavelengths per meter, \(n_{0}\) a reference value of \(n\), and \(\omega\) an exponent of fitted PSD. \(G_{x}(n_{0})\) is the road roughness value at \(n_{0}\), which is a constant for a given road class. From the national standard [9], \(n_{0}=0.1\) m\({}^{-1}\) and \(\omega=2\).  
If a road is sampled \(N\) times with an interval \(\Delta L\), the total sampling length is \(L=N\Delta L\) and the spatial frequency resolution is \(\Delta n=1/L\). With the discrete Fourier transform, the obtained data \(x_{m}(m=0,\ 1,\ \ldots,N-1)\) is transformed to  
\[X_{k}=\sum_{m=0}^{N-1}x_{m}e^{-\frac{\beta x_{m}}{N}}(k=0,1,\ldots,N-1). \tag{2}\]  
The relationship between \(X_{k}\) and the displacement PSD \(G_{x}(n_{k})\) is [10]  
\[|X_{k}|=\sqrt{\frac{N}{2\Delta l}G_{x}(n_{k})}\bigg{(}k=0,1,\ldots,\frac{N}{2 }\bigg{)}, \tag{3}\]  
where \(n_{k}=k\Delta n\). Equation (3) gives the modulus value of the discrete Fourier transform. Suppose the phase angle is \(\varphi_{k}\), randomly selected in [0,2\(\pi\)], then  
\[X_{k}=|X_{k}|e^{j\varphi_{k}}\bigg{(}k=0,1,\ldots,\frac{N}{2}\bigg{)}. \tag{4}\]  
In Equation (4), only the first \(\frac{N}{2}+1\) terms are transformed by the discrete Fourier transform, in which \(X_{0}=\sum_{n=0}^{N-1}x_{m}\). Through mean normalization, we have \(X_{0}=0\), and then \(X_{1}\) and \(X_{N-1}\), \(X_{2}\) and \(X_{N-2}\), \(\ldots\), \(X_{\frac{N}{2}-1}\) and \(X_{\frac{N}{2}+1}\) are respectively conjugate.  
Performing the discrete inverse Fourier transform for \(X_{k}\), we get [11]  
\[x_{m}=\frac{1}{N}\sum_{k=0}^{N-1}X_{k}e^{\frac{2\pi\mathrm{km}}{N}}(m=0,1,\ldots, N-1). \tag{5}\]  
Statistical data shows that most of the highways in China belong to Class A, B, and C [12]. So, in this paper Class A is taken as an ideal condition, Class C as a common one, and Class E as a severe one. We then simulate the road roughness and correspondingly study the dynamical response of the missile vehicle in maneuver.  
Due to the existence of the vehicle vibration isolation system, the spatial frequency \(n\) is sampled within the range of (0.011, 2.830) [13]. Here, the sampling length is \(L=600\) m with an interval \(\Delta L=0.1\) m. From Ref. [9], \(G_{\mathrm{x}}(n_{0})\) is \(16\times 10^{-6}\) m\({}^{3}\), \(256\times 10^{-6}\) m\({}^{3}\), and \(4096\times 10^{-6}\) m\({}^{3}\) for Class A, C, and E roads, respectively. For each class, the same random sequence is used for phase angle \(\varphi_{k}\) in Equation (3). Hence the road roughness of different classes changes proportionally and the ratio is \(x_{\mathrm{A}}:x_{\mathrm{C}}:x_{E}=1:4:16\), as plotted in Figure 1.