## 1 Introduction
### SDOF and Dual Dynamic Vibration Absorbers  
Figure 4 shows an SDOF dynamic vibration absorber and the system (target structure) for which vibration reduction was sought. The target structure is connected to the ground by stiffness, and the mass and stiffness on it constitute the SDOF dynamic vibration absorber. In the case of variable-frequency excitation, the equations of motion of the SDOF dynamic vibration absorber and the target system for vibration reduction are expressed by Equations (2) and (3), respectively.  
\[m\ddot{x}+kx+k_{a}(x-x_{a})=F(t) \tag{2}\]  
\[m_{a}\ddot{x_{a}}+k_{a}(x_{a}-x)=0 \tag{3}\]  
[PERSON] proposed that the mass ratio \(\mu\) between the target structure and the SDOF dynamic vibration absorber should be designed to be \(0.05\leq\mu=m_{a}\mu m\leq 0.25\)[2]. Further, he suggested that the mass ratio should be set to \(\mu=0.25\) for vibration reduction in wider frequency ranges. In this section, therefore, a simulation of vibration reduction by the SDOF dynamic vibration absorber was carried out with a mass ratio of \(\mu=0.25\). The target frequency selected for vibration reduction was 20 Hz.  
Figure 3: Frequency response function of the target system.  
Figure 2: Experimental setup for modal test.  
The simulation numerically computed the acceleration and frequency responses from the target object for vibration reduction (Figure 5).  
The simulation results revealed that the SDOF dynamic vibration absorber reduced vibrations at the target frequency during variable-frequency excitation but with two resonance frequencies occurring near the target frequency (Figure 5a). One method proposed as a solution to the resonance problem is to install a number of dynamic vibration absorbers [7; 8].  
Figure 6 shows the dual dynamic vibration absorber and target system for vibration reduction, proposed by [PERSON] et al. [7]. The target system for vibration reduction is connected to the ground by stiffness, and two masses on it constitute the dual dynamic vibration absorber. In the case of variable-frequency excitation, the equations of motion of the dual dynamic vibration absorber and target system are given by Equations (4)-(6).  
\[m\ddot{x}+kx+k_{1}(x-x_{1})+k_{2}(x-x_{2})=F(t) \tag{4}\]  
Figure 4: Modeling of single degrees of freedom vibration absorber and target structure.  
Figure 5: Numerical simulation results of single degrees of freedom vibration absorber. (**a**) Frequency response, (**b**) Time response (acceleration).  
\[m_{1}\ddot{x}_{1}+k_{1}(x_{1}-x) = 0 \tag{5}\] \[m_{2}\ddot{x}_{2}+k_{2}(x_{2}-x) = 0 \tag{6}\]  
A numerical simulation of vibration reduction was performed using the dual dynamic vibration absorber's design factors--i.e., mass ratio and damping ratio as proposed by [PERSON] et al., who used a mass ratio of 0.15 and damping ratios of 0.166 and 0.299. A frequency range of 15-23 Hz was selected as the target range for vibration reduction. The simulation numerically computed the acceleration and frequency responses from the target object for vibration reduction.  
The simulation results revealed that the dual dynamic vibration absorber reduced vibrations at target frequencies during variable-frequency excitation. In addition, it was found to overcome the drawbacks of SDOF dynamic vibration absorbers without producing two resonance frequencies outside the target frequency range. However, high damping ratios caused a degradation of 20 Hz in vibration-reducing performance, compared to SDOF dynamic vibration absorbers (Figure 7).  
Figure 6: Modeling of dual dynamic absorber and target structure.  
Figure 7: Numerical simulation results of dual dynamic absorber. (**a**) Frequency response, (**b**) Time response (acceleration).