## 2 Damage Counting with Acceleration Data Only
### Physical Damage Counting with Energy Isocline Function  
The fatigue damage of a linear system can be derived from acceleration data using an energy isocline function, and the calculated fatigue damage has the physical value of a hot spot, which is an important difference compared to the previous pseudo-damage. The energy isocline function recalls the stress information of the location of interest by using the response acceleration [12; 13; 14; 15]. The energy isocline function can be explained using the frequency response function between the stress and the response acceleration, as shown in Equation (6):  
\[\sigma(\omega_{1})=\|H_{m}(\omega_{1})\|\cdot a(\omega_{1}). \tag{6}\]  
In Equation (6), the stress and acceleration are variables from Equations (1) and (2), respectively, and \(\|H_{m}(\omega_{1})\|\) is the magnitude of the frequency response function \(H_{m}(\omega_{1})\) between \(\sigma(\omega_{1})\) and \(a(\omega_{1})\). The direct calculation of the fatigue damage from the measured acceleration in Equation (2) can be formulated using both the damage counting in Equation (4) and the spectral relationship in Equation (6) during \(T_{1}\), as shown in Equation (7):  
\[D(\omega_{1},T_{1})=\frac{\omega_{1}T_{1}}{\pi\cdot S_{0}^{-1/ \epsilon}}\cdot\|H_{m}(\omega_{1})\|^{\frac{-1}{\epsilon}}\cdot\overline{a}( \omega_{1})^{\frac{-1}{\epsilon}}. \tag{7}\]  
Conversely, the measured acceleration response can be converted into the spectral energy using the power spectral density (PSD) [24; 25; 26]. Given the acceleration signal in Equation (2), the energy of excitation can be expressed in a frequency domain as shown in Equation (8):  
\[\begin{array}{l}E(\omega_{1},T_{1})=\overline{a}(\omega_{1})^{2}\xi_{s}^{2} \\ \xi_{s}^{2}=\int_{0}^{T_{1}}|\sin(\omega_{1}t+\varphi)|^{2}\mathrm{d}t=\frac{ \omega_{1}T_{1}}{2}\end{array}. \tag{8}\]  
From Equations (7) and (8), both the spectral damage and the vibration energy can be derived using only the acceleration where the variable \(\overline{a}(\omega_{1})\) plays a key role in these equations. If the spectral damage is assumed to be proportionally related to the excitation energy with a certain gain \(\kappa(\omega_{1})\), the spectral damage in Equation (7) can be reformulated as Equations (9) and (10) below [12; 13; 20]:  
\[D(\omega_{1},T_{1})=\kappa(\omega_{1},T_{1})E(\omega_{1},T_{1})^{\frac{-1}{2 \epsilon}}, \tag{9}\]  
\[\kappa(\omega_{1},T_{1})=\frac{2^{\frac{-1}{2\epsilon}}}{\pi}\cdot\omega_{1}^ {(1+\frac{1}{2\epsilon})}\cdot T_{1}^{(1+\frac{1}{2\epsilon})}\cdot S_{0}^{ \frac{1}{\epsilon}}\cdot\|H_{m}(\omega_{1})\|^{\frac{-1}{\epsilon}}. \tag{10}\]  
The proportional gain \(\kappa(\omega_{1})\) is the normalized spectral damage over the scaled excitation energy \(E(\omega_{vibration test is defined as \(D\). The corresponding formulation is shown in Equation (11), where the test duration is fixed as \(T\)[12; 13; 20]:  
\[D(T)=\sum_{i=i}^{N}\frac{2^{\frac{1}{2b}}}{\pi}\cdot\omega_{i}^{(1-\frac{1}{2b})} \cdot T^{(1-\frac{1}{2b})}\cdot S_{0}^{\frac{1}{b}}\cdot\|H_{m}(\omega_{i})\| ^{\frac{-1}{b}}\cdot a(\omega_{i})^{\frac{-1}{b}}, \tag{11}\]  
where \(N\) is the discretization number of the input spectrum. The summation of the fatigue damage in Equation (11) contains the material property of the test specimen with coefficients \(b\) and \(S_{0}\), such that it is a different physical value from the previous pseudo-damage, which is useful only for comparing the severity values of test events.