## 3 Basic Theory
### Structural Response Analysis
#### 3.4.2 Shock Response Spectrum (SRS)  
Shock response spectrum is a vibration analysis technique that characterizes the frequency content of an acceleration-time history, and thereby estimates the dynamic response of structures [29]. This algorithm shows how single degree of freedom (SDOF) systems of varying natural frequencies (\(\omega\)) would respond to an input shock. The SRS is represented in the form of a graph as shown in Figure 5, with the \(x\)-axis showing the natural frequency of varying hypothetical SDOF, and the \(y\)-axis showing the peak acceleration which these SDOF systems undergo when actuated by the shock input. This technique has been shown to be very effective when used to estimate damage induced by transient and pyrotechnic pulses such as earthquakes and transportation shocks [30]. An in-depth mathematical derivation and an open source MATLAB(r) code for the absolute acceleration of a spring mass system was provided by [PERSON] [31]. The shock response spectrum can then be derived by plotting the maximum acceleration obtained from Equation (5), against its corresponding natural frequency:  
\[\begin{array}{ll}\ddot{x}_{k}=+2e^{-\xi\omega_{N}\Delta t}&\cos(\omega_{d} \Delta t)\ddot{x}_{k-1}-e^{-2\xi\omega_{N}\Delta t}\ddot{x}_{k-2}+2\xi\omega_{ N}\Delta t\ddot{y}_{k}\\ &+\omega_{N}\Delta te^{-\xi\omega_{N}\Delta t}\Big{(}\Big{(}\frac{\omega_{N}}{ \omega_{d}}\big{(}1-2\xi^{2}_{\circ}\big{)}\Big{]}sin[\omega_{d}\Delta t]\\ &-2\xi cos[\omega_{d}\Delta t]\Big{)}\ddot{y}_{k-1}\end{array} \tag{5}\]  
where,  
\(\begin{array}{ll}k\\ \ddot{x}(t)&\text{Spring constant of each SDOF system}\\ \ddot{y}(t)&\text{Input shock}\\ \xi&\text{Damping ratio of each SDOF system}\\ N&\text{Number of SDOF systems}\\ \omega&\text{Oscillation frequency}\end{array}\)In order to obtain a more descriptive indicator capable of monitoring the lubricating condition of the toggle clamping system, a relationship between acceleration (\(\ddot{x}_{k}\)) and velocity (\(\dot{x}_{k}\)) must first be established as shown in Equation (6):  
\[\left|\dot{x}_{k}\right|=\frac{1}{2\pi f}\left|\dot{x}_{k}\right| \tag{6}\]  
Given that each SDOF system exhibits a simple harmonic motion, its kinetic energy (\(KE\)) can be defined as shown if Equation (7):  
\[KE=\frac{1}{2}mv^{2}=\frac{1}{2}m\omega^{2}A^{2}sin^{2}(\omega t+\varphi) \tag{7}\]  
The systems elastic potential energy (\(U\)) is stored in the spring as it is displaced a distance \(x\) from its equilibrium position. It is hence calculated using Equation (8):  
\[U=\frac{1}{2}kx^{2}=\frac{1}{2}m\omega^{2}x^{2}=\frac{1}{2}m\omega^{2}A^{2}cos ^{2}(\omega t+\varphi) \tag{8}\]  
The total mechanical energy of the system is expressed as shown in Equation (9):  
\[E=KE+U=\frac{1}{2}m\omega^{2}A^{2}\left(sin^{2}(\omega t+\varphi)+cos^{2}( \omega t+\varphi)\right)=\frac{1}{2}m\omega^{2}A^{2} \tag{9}\]  
The maximum velocity \(\dot{x}_{max}\) is expressed as shown in Equation (10):  
\[\dot{x}_{max}=\omega A \tag{10}\]  
Finally, an index representative of the lubricating condition of the toggle clamping system can be expressed as the sum of the maximum energy of each SDOF system of different natural frequencies as shown by Equation (11).  
\[E=HI_{SRS}=\sum_{m=1}^{m=n}\frac{1}{2}m\left(\dot{x}_{max}\right)^{2} \tag{11}\]  
Figure 5: Shock response spectrum.  
Simplifying the above expression by omitting the constants yields the final health indicator expression as shown by Equation (12):  
\[E=HI_{SRS}=\sum_{m=1}^{m=n}\frac{1}{2}m\big{(}\dot{x}_{max}\big{)}^{2}=\sum_{m=1} ^{m=n}\big{(}\dot{x}_{max}\big{)}^{2} \tag{12}\]