Seismic Isolation Performance Evaluation for a Class of Inerter-Based Low-Complexity Isolators

In this paper, the seismic base isolation problem for all low-complexity networks containing one inerter, one spring, and one damper is studied based on a multi-degree-of-freedom model. ,e analytical solutions for the H2 performance optimization are derived, and the traditional tuned mass damper (TMD) is employed for comparison. Extensive numerical simulations are performed to verify the effectiveness of the obtained results. ,e results show that for different seismic wave excitations, some isolators are better than TMD in controlling the displacement of the main structure. Moreover, with the increase of the TMDmass ratio, the isolation performances of the inerter-based isolators are increasingly better than that of TMD.


Introduction
With the rapid development of the economy and the continuous progress of urbanization, the safety of buildings is particularly important. According to statistics, more than 5 million earthquakes occur on earth every year, most of which are ignored because of their small magnitudes, but a strong earthquake will cause great damage to buildings.
Structural vibration control is an effective way to reduce a structure's dynamic response under the excitation of earthquake and wind, which can be classified as two basic forms: base isolation and dynamic vibration absorption (DVA). e base isolation is to install a vibration isolator between the source and the controlled object [1], while the dynamic vibration absorption is to attach an auxiliary system to the controlled object, and such a device is also known as tuned mass damper (TMD) [2,3]. e traditional passive structural vibration control mainly utilizes three classical mechanical components, i.e., mass, spring, and damper. A mass has only one endpoint that can move freely, namely, the center of mass, and the other endpoint is the inertial reference frame (usually the ground). erefore, mass is not a real two-endpoint component. At the beginning of this century, Smith first proposed the concept of the inerter [4]. Inerter is defined as a device where the applied force at its two endpoints is proportional to the relative acceleration between the two endpoints [5]. e symbol of inerter is shown in Figure 1. e dynamic equation of inerter is F � b( _ v 2 − _ v 1 ), where the proportional factor b is called the inertance. e physical realization of inerter includes the rack pinion inerter [4,6], the ball screw inerter [7], and the hydraulic inerter [8][9][10]. e unique feature of inerter is that large inertance can be achieved through a small physical mass. Inerter was first applied in the suspension system of Formula One racing cars [6] and was called "J-damper" at that time. Afterwards, inerter has been broadly studied in the mechanical engineering field, such as vehicles [11][12][13][14][15][16][17], wind turbines [18][19][20], civil engineering structures [21][22][23][24][25], and other vibration control systems [26][27][28][29][30]. Recently, inerter has also been applied to wave energy converters [31,32].
In structural vibration control, the base isolation device and the dynamic vibration absorption device see new advances after introducing inerter. In terms of base isolation, Wang et al. proposed two basic inerter-based configurations and verified that appropriate inerter-based configurations can effectively reduce structural vibration under earthquake action [33]. Hu  where the displacement transmissibility problem and the force transmissibility problem were considered, and the damping effect of the proposed inerter-based isolator in terms of the performance of H 2 measurement and the performance of H ∞ measurement was studied based on a single-degree-of-freedom system [34,35]. In terms of dynamic vibration absorption, Hu et al. proposed an inerterbased dynamic vibration absorber (IDVA), where parameter optimization was conducted in the case of single degree of freedom, and made a comparison with the traditional dynamic vibration absorber [36]. In order to provide further explanation to the mass amplification characteristics of inerter, Marian et al. first proposed a tuning mass damper inerter (TMDI) combining the traditional tuning mass damper and inerter, and studied the optimal parameters [37]. Other inerter-based structural control configurations installed at the base or interstorey of a building have also been proposed, such as tuned viscous mass damper (TVMD) [38] and tuned inerter damper (TID) [39,40].
Note that for the existing results on inerter-based structural vibration control, a (or some) specific inerterbased network is commonly employed. For example, the inerter-based isolator in [34,35] is concerned with five networks containing at most one inerter, one spring, and one damper. e inerter-based DVA (IDVA) in [36] was illustrated based on six networks containing at most one inerter, one spring, and one damper. e TVMD in [38] is essentially a parallel connection of an inerter and a damper, and then in series with a spring. e TID in [39,40] is a parallel connection of a spring and a damper, and then in series with an inerter. Since numerous networks can be constructed by using inerters, dampers, and springs, then comprehensive study for an entire subclass of inerter-based networks (such as all the networks containing one inerter, one damper, and one spring in this paper) is essential, such that general conclusions including the performance limitations, amongst others, can be obtained. is motivates the investigation presented in this paper.
In this paper, all the networks containing one spring, one damper, and one inerter proposed in [35] are considered for the seismic vibration problem. e main difference from [34,35] is that in [34,35], the displacement transmissibility problem and force transmissibility problem were considered; while in this paper, the impact of seismic acceleration on the displacement of buildings is studied, which is the main consideration in seismic research. Moreover, a multidegree-of-freedom model is employed in this paper which is more practical than the single-degree-of-freedom model in [34,35]. e organization of this paper is as follows. In Section 2, the optimal solutions for the H 2 performance of the lowcomplexity networks with one inerter, one spring, and one damper, and the TMD are derived, based on a multi-degreeof-freedom model. In Section 3, a numerical example is used for performance comparison and analysis, and eleven seismic waves are used for verification. In Section 4, a brief conclusion is drawn.

Method of H 2 Optimization of Inerter-Based
Isolator. e schematic diagram of the low-complexity networks with only one inerter, one spring, and one damper (hereinafter referred to as inerter-based isolators) is shown in Figure 2.
In Section 2.1, structure C3 is taken as an example to illustrate the method of H 2 optimization of the inerter-based isolator, and the schematic diagram of structure C3 installed in the multi-degree-of-freedom model is shown in Figure 3. e damping of certain structures is generally small in practice and would have little influence on the analysis of these structures. In order to obtain the analytic solution, the influence of damping of the main structure is ignored.
For regular uniform multistorey buildings, the low-order mode usually plays a major role under seismic excitation [40]. erefore, this paper will conduct H 2 performance optimization for the first-order mode of a multistorey building and conduct a comparative analysis with the H 2 performance of TMD installed at the top floor of the building.
Its motion equation is where M and K are, respectively, the mass matrix and stiffness matrix of the main structure; b, c, and k are the inertance, damping coefficient, and stiffness, respectively; € X and X are the acceleration matrix and displacement matrix of the main structure, respectively, where , and x b are the acceleration, velocity, and displacement of the inerter, respectively; € x g is the earthquake acceleration; respectively, _ x c and x c represent the velocity and displacement of the damper; ; the above responses are the responses to the ground. Let X � ϕ 1 q, where ϕ 1 is the first mode and q is the generalized displacement, by substituting it into equation (1) and carrying out the Laplace transform of equation (1), the transfer function of seismic acceleration to the generalized displacement of the main structure can be obtained: where w 1 � ������ K 1 /M 1 is the first mode frequency, M 1 is the mass of the first mode, K 1 is the stiffness of the first mode, is the first mode coordinates of the bottom layer.
Since seismic excitation contains multiple frequency bands and is a random disturbance, it is more appropriate to use H 2 norm to evaluate the seismic damping effect of the adopted structure on the main structure. e goal of H 2 performance optimization is to minimize the total vibration energy of the main structure when it is disturbed by white noise. Using the method in Reference [41], the H 2 performance index I of the system can be expressed as (3) For a linear time-invariant closed-loop system, where w is the external input and z is the control output. e H 2 norm from the input w to the output z of the system can be directly calculated with the following theorem. Figure 2: e low-complexity networks with only one inerter, one spring, and one damper [35]. C4 is TID [39,40], and C6 is TVMD [38].
x g ·· Figure 3: Schematic diagram of structure C3 installed in the multi-degree-of-freedom model.

Shock and Vibration
Theorem 1 (see [41]). Considering closed-loop system (4), if A cl is unstable or D cl ≠ 0, then the H 2 norm from the input w to the output z of the system is infinite; otherwise, where P and S are the solutions of the following Lyapunov equation: According to eorem 1, for a given stable transfer function T(s), Its H 2 norm can be calculated as follows: where matrices A, B, and C are the minimum phase realization of T(s), and matrix L is the solution of the following Lyapunov equation: erefore, H 2 performance index I of the system can be calculated analytically using the above method, and the specific steps are summarized as follows.
e method of H 2 optimization of the inerter-based isolator is as follows: (i) Step 1. e transfer function H(s) of the inerterbased isolator is listed, and the analytic expression of the H 2 performance index I of the system is obtained by using the method of solving the H 2 performance index for the given inertance-to-mass ratio δ. (ii) Step 2. e analytic expression of the H 2 performance index I is obtained, and the derivatives of stiffness ratio λ and damping ratio ζ are obtained, respectively, and set each as 0 to obtain the simplified optimal stiffness ratio λ opt and the optimal damping ratio ζ opt . (iii) Step 3. e obtained optimal stiffness ratio λ opt and damping ratio ζ opt are substituted into the analytic expression of the system H 2 performance index I to obtain the optimal H 2 performance I opt .

Analytic Solution of H 2 Performance Optimization for
Inerter-Based Isolator. On the premise of the given inertance-to-mass ratio δ, the analytic solutions of H 2 performance optimization, optimal stiffness ratio λ opt , and optimal damping ratio ζ opt of all the inerter-based isolator C1-C8 installed in the multi-degree-of-freedom model are given.

Structure C1.
e analytic expression of the H 2 performance index of structure C1 is (10) For a given inertance-to-mass ratio δ, the optimal damping ratio ζ opt is ∞, and the optimal H 2 performance index I C1,opt is 0, or the optimal stiffness ratio λ opt is 0, and the optimal H 2 performance index I C1,opt is 0.

Structure C2.
e analytic expression of the H 2 performance index of structure C2 is For a given inertance-to-mass ratio δ, the optimal stiffness ratio λ opt is 0, and the optimal H 2 performance index I C2,opt is 0.

Structure C3.
e analytic expression of the H 2 performance index of structure C3 is For a given inertance-to-mass ratio δ, the optimal stiffness ratio λ opt is For a given inertance-to-mass ratio δ, the optimal damping ratio ζ opt is e optimal H 2 performance index I C3,opt is

Structure C4 (TID).
e analytic expression of the H 2 performance index of structure C4 is For a given inertance-to-mass ratio δ, the optimal stiffness ratio λ opt is

Shock and Vibration
For a given inertance-to-mass ratio δ, the optimal damping ratio ζ opt is e optimal H 2 performance index I C4,opt is For a given inertance-to-mass ratio δ, the optimal stiffness ratio λ opt is For a given inertance-to-mass ratio δ, the optimal damping ratio ζ opt is ∞, and the optimal H 2 performance index I C5,opt is 0.

Structure C6 (TVMD).
e analytic expression of the H 2 performance index of structure C6 is For a given inertance-to-mass ratio δ, λ has no optimal value when δ � [(2(3 − 2 ))/(3c)]; when δ is equal to anything else, the optimal stiffness ratio λ opt is For a given inertance-to-mass ratio δ, ζ has no optimal value when δ � [(2(3 − 2 when δ is equal to anything else, the optimal damping ratio ζ opt is When the optimal stiffness ratio λ opt and damping ratio ζ opt are obtained, the optimal H 2 performance index I C6,opt is 2.2.7. Structure C7. e analytic expression of the H 2 performance index of structure C7 is For a given inertance-to-mass ratio δ, the optimal stiffness ratio λ opt is For a given inertance-to-mass ratio δ, the optimal damping ratio ζ opt is ∞, and the optimal H 2 performance index I C7,opt is 0.

Structure C8.
e analytic expression of the H 2 performance index of structure C8 is

Shock and Vibration 5
For a given inertance-to-mass ratio δ, the optimal damping ratio ζ opt is ∞, and the optimal H 2 performance index I C8,opt is 0.
It should be noted that the H 2 performance indexes of structures C1, C2, C5, C7, and C8 are optimized when the stiffness ratio λ is zero or the damping ratio ζ is infinite. ese structures will degenerate into simple structures as shown in Figure 4, in which C1 degenerates into TC1 or TC2, C2 degenerates into TC1, C5 and C7 degenerate into TC3, and C8 degenerates into TC2. Although the H 2 performance indexes of C1, C2, C5, C7, and C8 can be equal to zero in the optimal case, they need to have a damping ratio equal to infinity and a stiffness ratio equal to zero, which is impossible in practice. In addition, C1, C2, C5, C7, and C8 after degradation will have other disadvantages, which will be pointed out in the subsequent simulation experiments. C3, C4, and C6 can obtain the optimal H 2 performance index under the condition of obtaining the optimal stiffness ratio and damping ratio. In practical applications, the optimal H 2 performance can be obtained by selecting the corresponding optimal stiffness ratio and damping ratio.

Analytic Solution of H 2 Performance Optimization for TMD.
Similarly, the traditional TMD was installed in the multi-degree-of-freedom model and its damping performance was studied, and the schematic diagram of TMD installed in the multi-degree-of-freedom model is shown in Figure 5, ignoring the influence of damping of the main structure.
Its motion equation is where M and K are, respectively, the mass matrix and stiffness matrix of the main structure; m b , c, and k are the TMD mass, damping coefficient, and stiffness, respectively; € X and X are the acceleration matrix and displacement matrix of the main structure, respectively, where , and x b are the acceleration, velocity, and displacement of the TMD, respectively. € x g is earthquake acceleration; Γ n � [0, 0, . . .
; the above responses are the response to the ground. Let X � ϕ 1 q, where ϕ 1 is the first mode and q is the generalized displacement; by substituting it into equation (29) and carrying out the Laplace transform of equation (29), the transfer function of seismic acceleration to the displacement of the main structure can be obtained: where w 1 � ������ K 1 /M 1 is the first mode frequency, M 1 is the mass of the first mode, K 1 is the stiffness of the first mode, and δ TMD � (m b /M 1 ) is TMD mass ratio; damping ratio is ζ � (c/c r ), where c r � 2w 1 M 1 � 2 ����� � M 1 K 1 is the critical damping coefficient of the system without TMD; stiffness ratio is where ϕ 1(n) is the first mode coordinates of the top layer. TMD structure is optimized by using the above method of the H 2 optimization of inerter-based isolator, and the analytic solution of H 2 performance optimization of similar structures C1-C8, as well as the analytic expressions of optimal stiffness ratio λ opt and optimal damping ratio ζ opt , is obtained. e analytic expression of the H 2 performance index of TMD structure is For a given TMD mass ratio δ TMD , the optimal stiffness ratio λ opt is λ opt � 2 βc 2 n δ 3 TMD + αc 2 n δ 2 TMD + 2βc n δ 2 TMD + 2αc n δ TMD + βδ TMD + α δ − βc n δ 2 TMD + αc n δ TMD + 2α .

Numerical Examples
In this paper, the three-story uniform structure model used by Lazar [39] is employed. e mass of each floor is 1000 kg, and the stiffness of each layer is 1500 kN m − 1 . Ignoring the damping of the main structure, El Centro north-south seismic wave is adopted for simulation, and the damping performance of TMD and inerter-based isolator is analyzed and compared. TMD mass ratios are 0.01, 0.03, 0.05, and 0.07. First of all, only the first mode is considered in the simulation experiment. Inertance-to-mass ratio is five times that of TMD mass ratio [39]. TMD mass ratio is generally selected as (0, 0.1]. Within this range, when the performance of the inerterbased isolators is superior to TMD, there will be a minimum multiple relationship between inertance-to-mass ratio and TMD mass ratio. Taking the configuration C4 as an example, the minimum multiplicity of δ in equation (16) for I C4 to be less than I TMD in equation (31) is 5. is means that the inertance-to-mass ratio of C4 should be 5 times larger than the TMD mass ratio. In practice, the higher amplifications than 5 can be easily realized due to the mass amplification effect of inerter.
El Centro seismic wave is the first fully recorded seismic wave in human history, with a magnitude of 6.7 and an epicenter distance of 9.3 km. e waveform with a duration of 30 s in the north-south direction is selected with a time interval of 0.02 s and a peak acceleration of 3.417 m s − 2 , and the El Centro seismic wave is shown in Figure 6. en, taking C1 as an example, the parameter selection method of C1, C2, C5, C7, and C8 in simulation is illustrated. C1 obtains the optimal H 2 performance when the damping ratio ζ � (c/c r ) is infinite, which means that the damping coefficient c is infinite, when the stiffness ratio λ is fixed, and the H 2 performance index with respect to damping ratio is shown in Figure 7. It can be seen that when the damping ratio is 100, the performance index flattens and is approximately 0, and then C1 degenerates to TC2.
In the same way, C1 obtains the optimal H 2 performance when the stiffness ratio λ � (K 1 /k) is 0, which means that the stiffness coefficient k is infinite, when the damping ratio ζ is fixed, and the H 2 performance index with respect to stiffness coefficient is shown in Figure 8. It can be seen that when the stiffness coefficient is 100000 kN m − 1 , the performance index flattens and is approximately 0, and then C1 degenerates to TC1. Under the excitation of El Centro seismic waves, the generalized displacement when δ TMD � 0.07 is shown in Figure 9.
As can be seen from Table 1, in terms of generalized displacement, structures C3, C4, and C6 have better performance than TMD. With the increase of TMD mass ratio, C3, C4, and C6 can obtain a greater performance improvement than TMD. When TMD mass ratio is 0.07, structures C3, C4, and C6 can provide 9.39%, 8.02%, and 16.62% performance improvement, respectively. Structures TC1, TC2, and TC3 are better than other structures.
In order to illustrate the performance of the inerterbased isolator, a total of 11 seismic waves [42] are employed for the simulation experiment, and the power spectral  Shock and Vibration density of various seismic wave accelerations is shown in Figure 10. e comparative data of generalized displacement are summarized in Table 2, where δ TM D � 0.07. It can be found that structures C3, C4, and C6 are much better than TMD in controlling the generalized displacement. Structures TC1, TC2, and TC3 are better than other structures in controlling the generalized displacement.
After the simulation experiment of the simplified model is completed, the simulation experiment of the three-storey  structure mentioned above is carried out to verify the effectiveness of the optimal parameters. Taking structure C3 as an example, the state-space model is given.

Its motion equation is
e state-space equation can be written as El Centro seismic wave is still used as seismic input, and the optimal parameters obtained from the simplified model are used for simulation; the displacement of each floor with different structures installed is summarized in Table 3, where δ TMD � 0.07.
As can be seen from Table 3, compared with uncontrolled structure, TMD, C3, C4, and C6 have a good control effect on the displacement of each floor, and C3, C4, and C6 have better performance than TMD. Although TC1, TC2, and TC3 can control the displacement of the first floor, they cannot control the displacement of the upper floor, which is the biggest disadvantage of TC1, TC2, and TC3.
Meanwhile, according to Table 3, the interstorey drifts of the two adjacent storeys can be calculated. Compared with an uncontrolled structure, TMD, C3, C4, and C6 reduce the interstorey drifts, and the interstorey drifts of C3, C4, and C6 are smaller than TMD, which can better avoid the destruction of buildings, while TC1, TC2, and TC3 increase the interstorey drifts.
It should be noted that although TC1, TC2, and TC3 have good H 2 performance, in the case of multiple degrees of freedom, only the displacement of the first floor can be controlled, and the displacement of the upper floor will be large, which is not desirable. erefore, the application of structures C3, C4, and C6 is more reasonable.
Note that the nonlinearities of the employed devices are not considered in this paper. However, as shown in the existing results such as [43][44][45][46], the nonlinear effects of inerter would make the constitutive behavior different from the linear one, especially at the low frequencies for civil engineering applications. is should be considered in the future.

Conclusions
In this paper, all the low-complexity networks containing one inerter, one spring, and one damper were employed for the base isolation problem in seismic applications. e analytical H 2 optimization method was adopted based on a multi-degree-of-freedom model to derive the optimal solutions of the considered low-complexity networks. e performance comparison between these low-complexity networks as well as the TMD was conducted. Numerical simulations based on eleven seismic waves were performed to verify the results in this paper. It was found that some inerter-based isolators (such as C3, C4, and C6) are much  better than TMD in controlling the displacement of the main structure, with the increase of TMD mass ratio, and some inerter-based isolators (such as C3, C4, and C6) can obtain a greater performance improvement than TMD.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.