## 4 Solving the Vibration Response of Circuit Unit
### Solving for Optimal Stiffness and Damping Coefficient of the Damper  
Through the decomposition of the vibration response of the above circuit unit, two main parameters, the logarithmic attenuation rate of the amplitude and the displacement amplification factor, are obtained. They are both functions of damping and stiffness, andthe two parameters will affect each other when seeking their optimum values. Therefore, the two functions are used as the objective functions, which are transformed into a multi-objective optimization problem. The optimal stiffness and damping parameters of the vibration system can be obtained by seeking the optimal solution through a suitable algorithm. The multi-objective optimization problem is more than a single-objective optimization problem, multiple objective functions may conflict. Therefore, the result is an optimal solution set. In this study, NSGA-II was chosen to solve the optimal stiffness damping parameters of the vibration system, which has the advantages of fast convergence, high computational efficiency, and good stability compared with other multi-objective optimization algorithms. Based on the previous section, Equations (9) and (16) are used as the objective functions of this multi-objective optimization problem, and Equations (10), (17) and (18) are used as the boundary conditions of this function. This is sorted out as follows:  
\[\left\{ \begin{aligned} f(k,c)\left\{\begin{aligned} & min\beta(k,c)=\frac{k}{\sqrt{(k-m \omega)^{2}+c^{2}\omega^{2}}}\\ & max\delta(k,c)=\frac{2\pi c}{\sqrt{4 mk-c^{2}}}\\ & s.t.\left\{\begin{aligned} & 0<\beta<1\\ & 0<\frac{c}{2\sqrt{mk}}<0.2\\ &\sqrt{2}<\frac{\omega c}{\omega_{n}}=\omega\sqrt{\frac{m}{k}}<5\\ \end{aligned}\right.\end{aligned}\right.\end{aligned} \right. \tag{19}\]  
A non-dominated sorting genetic algorithm was developed based on the above objective function and boundary conditions. To optimize programming and calculation, take the reciprocal of the displacement amplification factor and find the maximum of both objective functions simultaneously using the algorithm. An initial set of populations is first generated randomly when the algorithm is executed. Each spring-damping parameter represents a set of solutions and ensures that these initial values satisfy the boundary conditions in Equation (19). Then, a hierarchy is performed based on the dominance of the spring-damped parameter over the set of multi-objective functions. This can ensure that each individual parameter solved maintains the property of being a non-inferior solution on at least one objective. Next, a crowding calculation is performed for each spring-damped parameter to quantify the intensity of the solution. Thus, this promotes the diversity and equilibrium of the solution set. After that, the spring-damped parameter with a higher grade is preferred. According to the congestion index, the spring-damped parameter with a sparse distribution is selected in each layer. Then, it is added to the population of the subsequent evolution. Then, new populations are obtained through the use of two genetic operations, crossover, and mutation, to increase the diversity of solutions within the population. After the newly generated individuals are merged with the existing population through non-dominated sorting and crowding evaluation, the individuals with higher priority are again filtered to form the new generation of the population. This is carried out until the maximum number of iterations is reached. Eventually, the algorithm will converge to a set of high-quality optimal solution sets, resulting in the optimal stiffness and damping coefficients that are applicable to the model. The optimization process is shown in Figure 4.  
The known circuit unit prototype is 8 kg, and the DTH hammer (China University of Geosciences, Beijing, China) used in the mining of geothermal resources has an operating frequency range of 5-15 Hz. Let the impact frequency be 10 Hz, and then substitute it into Equation (19).