Performance Analysis of a Vibration Absorber with the Inerter Element and Amplifying Mechanism

Based on the good performance of the new control elements, we introduce the viscoelastic model, the amplification mechanism and the inerter element into the dynamic vibration absorber (DVA). Then, we take it as the research object to obtain the DVA parameters when the system achieves the optimal vibration reduction effect based on the fixed-point theory. Finally, it is eventually demonstrated that the DVA discussed in this paper possesses significant benefits in mitigating vibrations, which provides theoretical basis and data support for the parameters design of DVAs in engineering application.


Introduction
A passive damping equipment known as the dynamic vibration absorber (DVA) typically consists of three components: mass block, spring element, and damping mechanism.Its effectiveness and high cost performance of harmful vibration suppression have been widely recognized in various practical tests such as machinery, bridges, and buildings [1].The original intention of the design is to install the parameter-matched DVA on the primary vibration system, and use the resonance principle to change the distribution and transmission of vibration energy.In recent years, scholars have further improved the structure of DVA based on the three classical models [2][3][4].
With the deepening of research, the types of coupling modes between DVA and the primary system are divided into more and more detailed.The primary system can be classified according to whether it has damping [5].It can also be subdivided into continuous [6] and discrete [7] according to the structural composition.The continuous type has different continuous approximation models such as rods, beams, and plates.Stiffness can be divided into linear [8] and nonlinear [9].The connection positions of spring elements and damping elements are also varied.In recent years, new components such as negative stiffness spring [10], amplification mechanism [11], and inerter device [12] have been deeply investigated in structural vibration suppression due to their excellent vibration control performance, which play an important role and bring the possibility for further improvement of DVA performance.In order to pursue better vibration reduction performance and higher cost performance under different actual needs, the collocation and combination of the whole system and the corresponding parameter optimization method have always been the research content of various scholars.
Inspired by this, we pay attention to the multi-parameter optimization and performance analysis of a DVA model.In section 2, the dynamic equation and detailed parameter optimization process are given.In Section 3, the discussed DVA is compared to the three typical DVA models.Finally, we give the conclusions.

Model Establishment and Parameters Analysis
Figure 1 proposes a novel DVA model that considers negative stiffness, inerter, and amplifying mechanism.The distances between the lever support point and the hinge points of the two sliders are denoted as 1 r and 2 r , respectively.The mass and stiffness coefficients of the primary system and subsystem are denoted as 1 m , 1 k , 2 m and 2 k separately.3 k and 4 k stand for the stiffness coefficient of the spring connected to the lever in the subsystem and the grounded negative stiffness spring, respectively.c and b express the damping coefficient and the inerter coefficient of the DVA.
F and ω are used to represent the amplitude and frequency of external excitation.

The Analytical Solution
In terms of the similar triangle theorem, when the system vibrates, the ratio of 2 r to 1 r is constant.The ratio of the lever arm's length is defined as . The system equation is Introducing the dimensionless parameters , , 2 , , , , and inserting the following substitutions Define the primary system amplitude amplification factor where Given the parameters value In Figure 2, we can see three amplitude-frequency curves go through three fixed points

, α v
When ξ tends to zero or infinity, we get By equating the above two expressions, one can get the equation about 2  λ Moreover, according to equation ( 5), the following formula holds. where . Based on the  H optimization criteria, three fixed points λ , , and R λ need to be adjusted to the same height.This process is divided into two steps.Firstly, we adjust Combining equation ( 6) and equation ( 8), the horizontal coordinate of three fixed points can be obtained as Secondly, the heights of point P (or R ) and point Q are adjusted to be equal.We can get Optimization of the negative stiffness term is crucial in order to prevent system instability caused by an improper negative stiffness value.It is found that when the formula  is established, the system is in a stable state.Then, the expressions for the potential optimal negative stiffness ratio are calculated as follows.
We present the three-dimensional surface diagrams of Table 1 gives the optimal parameter values. 2 opt δ represents the mean square response of the primary system, and this parameter will be discussed in a later section.We select N 1000  F . The comparison between the numerical and analytical solutions is illustrated in Figure 4, indicating a high degree of consistency between the two curves.This demonstrates the accuracy of the specific expression of the optimal parameters.Table 1.Optimal parameter values in different situations.

Analysis of Response Characteristics under Random Excitation
We compare the vibration suppression effect of the model with typical models such as Voigt-type DVA [2], Ren-type model [4], and Shen-type DVA [13].As shown in

L
. Figure 5 shows the time history diagrams of the corresponding DVAs.Under the given initial parameters, the peak amplitude of the primary system of the proposed model does not exceed 0.008, which is significantly lower than the other three models.Therefore, our DVA has a good vibration suppression characteristic.Furthermore, we respectively compare the effects of magnification ratios and inerter-mass ratios on vibration suppression in Figure 6, and found that reducing the response amplitude could be achieved through increasing the ratios of magnification and inerter-mass ratios.The vibration energy depends directly on the displacement and velocity response.The corresponding energy change curves can be obtained in Figure 7.Our model has the capability to substantially decrease the vibration energy, yielding a remarkable reduction in vibrations.

Conclusions
A novel DVA model including negative stiffness, inerter device, and amplification mechanism is proposed.The dynamic equations are simplified by two parameter transformations, and four adjustable parameters are optimized.Under random excitation, our model is compared with the three DVA models from the perspectives of the mean square responses, the time history diagrams, and the vibration energy.The DVA proposed in this paper works better.In future investigations, we can combine the dynamic characteristics of DVA, installation location, and other engineering practical factors to consider more, and explore better structural design.

Figure 1 .
Figure 1.Mechanical model of a novel DVA.

Figure 3 . 2 Qλ
Figure 3. Three-dimensional surface diagrams of possible negative stiffness ratio.When the values of the two resonance peaks are the same, it can be observed that the tangent at point Q is almost horizontal, namely

Figure 4 .
Figure 4. Comparison diagrams of numerical solutions and analytical solutions with two groups of magnification ratios L and different inerter-mass ratios β .

Table 1 ,
the model presented in this paper has lower mean square responses value compared to three classical