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

Authors: Fei Cao; Michael Z. Q. Chen; Yinlong Hu
Journal: Shock and Vibration
 (2020)

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## Abstract

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. The 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. The 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 TMD mass ratio, the isolation performances of the inerter-based isolators are increasingly better than that of TMD.


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## Body

## 1. 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). The 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].The 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). Therefore, 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]. The symbol of inerter is shown in Figure 1. The dynamic equation of inerter is F=bv˙2−v˙1, where the proportional factor b is called the inertance. The physical realization of inerter includes the rack pinion inerter [4, 6], the ball screw inerter [7], and the hydraulic inerter [8–10]. The 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 [11–17], wind turbines [18–20], civil engineering structures [21–25], and other vibration control systems [26–30]. Recently, inerter has also been applied to wave energy converters [31, 32].Figure 1
Symbolic representation of inerter [4].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 et al. proposed the inerter-based isolator, 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 H2 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 inerter-based 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 inerter-based 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. The inerter-based DVA (IDVA) in [36] was illustrated based on six networks containing at most one inerter, one spring, and one damper. The TVMD in [38] is essentially a parallel connection of an inerter and a damper, and then in series with a spring. The 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. This 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. The 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 multi-degree-of-freedom model is employed in this paper which is more practical than the single-degree-of-freedom model in [34, 35].The organization of this paper is as follows. In Section2, the optimal solutions for the H2 performance of the low-complexity networks with one inerter, one spring, and one damper, and the TMD are derived, based on a multi-degree-of-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.


## 2. Analytic Solution of H2 Performance Optimization

### 2.1. Method of H2 Optimization of Inerter-Based Isolator

The 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 Figure2.Figure 2
The low-complexity networks with only one inerter, one spring, and one damper [35]. C4 is TID [39, 40], and C6 is TVMD [38]. (a) C1. (b) C2. (c) C3. (d) C4. (e) C5. (f) C6. (g) C7. (h) C8.
(a)(b)(c)(d)(e)(f)(g)(h)In Section2.1, structure C3 is taken as an example to illustrate the method of H2 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. The 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.Figure 3
Schematic diagram of structure C3 installed in the multi-degree-of-freedom model.For regular uniform multistorey buildings, the low-order mode usually plays a major role under seismic excitation [40]. Therefore, this paper will conduct H2 performance optimization for the first-order mode of a multistorey building and conduct a comparative analysis with the H2 performance of TMD installed at the top floor of the building.Its motion equation is(1)MX¨+KX+FΓ0=−MEx¨g,F=bx¨b=kx1−xc=cx˙c−x˙b,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 X=x1,x2,…,xnT; x¨b, x˙b, and xb are the acceleration, velocity, and displacement of the inerter, respectively; x¨g is the earthquake acceleration; respectively, x˙c and xc represent the velocity and displacement of the damper; Γ0=1,0,…,0T is the indicator vector for installation; E=1,1,…,1T is the unit volume; the mass matrix of the main structure M = diagm1,m2,…,mn; and the stiffness matrix of the main structure K=k1+k2−k2…−k2k2+k3⋮−kn−knkn; the above responses are the responses to the ground.LetX=ϕ1q, where ϕ1 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:(2)Hs=qx¨g=−α2δζw1s2+δw12/λs+2ζw13/λs2+w122δζw1s2+δw12/λs+2ζw13/λ+2γδζw13/λs2,where w1=K1/M1 is the first mode frequency, M1 is the mass of the first mode, K1 is the stiffness of the first mode, and δ=b/M1 is inertance-to-mass ratio; damping ratio is ζ=c/cr, where cr=2w1M1=2M1K1 is the critical damping coefficient of the system without structure C3; stiffness ratio is λ=K1/k; α=ϕ1TME/M1; γ=ϕ1TΓ0ϕ11, where ϕ11 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 H2 norm to evaluate the seismic damping effect of the adopted structure on the main structure. The goal of H2 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 H2 performance index I of the system can be expressed as(3)I=H^s22.For a linear time-invariant closed-loop system,(4)x˙=Aclx+Bclw,z=Cclx+Dclw,where w is the external input and z is the control output. The H2 norm from the input w to the output z of the system can be directly calculated with the following theorem.Theorem 1.
(see [41]). Considering closed-loop system (4), if Acl is unstable or Dcl≠0, then the H2 norm from the input w to the output z of the system is infinite; otherwise,(5)H22=traceCclPCclT=traceBclTSBcl,where P and S are the solutions of the following Lyapunov equation:(6)AclP+PAclT+BclBclT=0,AclTS+SAcl+CclTCcl=0.According to Theorem1, for a given stable transfer function Ts,(7)Ts=bn−1sn−1+⋯+b1s+b0sn+an−1sn−1+⋯+a1s+a0.Its H2 norm can be calculated as follows:(8)Ts22=CsI−A−1B22=CLCT,where matrices A, B, and C are the minimum phase realization of Ts, and matrix L is the solution of the following Lyapunov equation:(9)AL+LAT+BBT=0.Therefore, H2 performance index I of the system can be calculated analytically using the above method, and the specific steps are summarized as follows.The method of H2 optimization of the inerter-based isolator is as follows:(i)
Step 1. The transfer function Hs of the inerter-based isolator is listed, and the analytic expression of the H2 performance index I of the system is obtained by using the method of solving the H2 performance index for the given inertance-to-mass ratio δ.(ii)
Step 2. The analytic expression of the H2 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 λopt and the optimal damping ratio ζopt.(iii)
Step 3. The obtained optimal stiffness ratio λopt and damping ratio ζopt are substituted into the analytic expression of the system H2 performance index I to obtain the optimal H2 performance Iopt.


### 2.2. Analytic Solution of H2 Performance Optimization for Inerter-Based Isolator

On the premise of the given inertance-to-mass ratioδ, the analytic solutions of H2 performance optimization, optimal stiffness ratio λopt, and optimal damping ratio ζopt of all the inerter-based isolator C1–C8 installed in the multi-degree-of-freedom model are given.


#### 2.2.1. Structure C1

The analytic expression of the H2 performance index of structure C1 is(10)IC1=λα24λ+γζγw13.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC1,opt is 0, or the optimal stiffness ratio λopt is 0, and the optimal H2 performance index IC1,opt is 0.


#### 2.2.2. Structure C2

The analytic expression of the H2 performance index of structure C2 is(11)IC2=λα24λγδζ2+4λζ2+λδ2+γδ24γζw13λ+γ2δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is 0, and the optimal H2 performance index IC2,opt is 0.


#### 2.2.3. Structure C3

The analytic expression of the H2 performance index of structure C3 is(12)IC3=α24λ2δ2ζ2−8λδζ2+4γδζ2+δ2+4ζ24γζw13δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(13)λopt=1δ.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is(14)ζopt=44δγ.The optimal H2 performance index IC3,opt is(15)IC3,opt=α2γw13δ/γ.


#### 2.2.4. Structure C4 (TID)

The analytic expression of the H2 performance index of structure C4 is(16)IC4=α24λ2γδζ2+δ2λ2+4λ2ζ2−λγδ2−2δλ+γ2δ2+2γδ+14λ2γζw13δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(17)λopt=2γ2δ2+2γδ+1δγδ+2.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is(18)ζopt=δ3γ2δ2+4γδ/16+16γδ1+γδ.The optimal H2 performance index IC4,opt is(19)IC4,opt=α23γδ+481+γδw133γ2δ2+4γδ/16+16γδ.


#### 2.2.5. Structure C5

The analytic expression of the H2 performance index of structure C5 is(20)IC5=α24λ2γδζ2+δ2λ2+4λ2ζ2−2δλ+14γζw13λ2δ2−2δλ+1.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(21)λopt=0.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC5,opt is 0.


#### 2.2.6. Structure C6 (TVMD)

The analytic expression of the H2 performance index of structure C6 is(22)IC6=α2δ2λ2+4λ2ζ2+λγδ2−2δλ+14γζw13.For a given inertance-to-mass ratioδ, λ has no optimal value when δ=23−22/3γ,23+22/3γ; when δ is equal to anything else, the optimal stiffness ratio λopt is(23)λopt=2−γδ2δ3/8γ2δ2−3/2γδ+3/2−1/89γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.For a given inertance-to-mass ratioδ, ζ has no optimal value when δ=23−22/3γ,23+22/3γ; when δ is equal to anything else, the optimal damping ratio ζopt is(24)ζopt=18δ8+6γ2δ2−24γδ−29γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.When the optimal stiffness ratioλopt and damping ratio ζopt are obtained, the optimal H2 performance index IC6,opt is(25)IC6,opt=α28+6γ2δ2−24γδ−29γ4δ4−72γ3δ3+184γ2δ2−160γδ+164γδw139/8γ2δ2−9/2γδ+5/2−3/89γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.


#### 2.2.7. Structure C7

The analytic expression of the H2 performance index of structure C7 is(26)IC7=α24λ2ζ2+γδ+14γζw13γδ+1.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(27)λopt=0.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC7,opt is 0.


#### 2.2.8. Structure C8

The analytic expression of the H2 performance index of structure C8 is(28)IC8=α24γζw13.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC8,opt is 0.It should be noted that the H2 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. These 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.Figure 4
Simple structure after degradation: (a) TC1. (b) TC2. (c) TC3.
(a)(b)(c)Although the H2 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 H2 performance index under the condition of obtaining the optimal stiffness ratio and damping ratio. In practical applications, the optimal H2 performance can be obtained by selecting the corresponding optimal stiffness ratio and damping ratio.


### 2.3. Analytic Solution of H2 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 Figure5, ignoring the influence of damping of the main structure.Figure 5
Schematic diagram of TMD installed in the multi-degree-of-freedom model.Its motion equation is(29)MX¨+KX+FΓn=−MEx¨g,F=mbx¨b+x¨g=kxn−xb+cx˙n−x˙b,where M and K are, respectively, the mass matrix and stiffness matrix of the main structure; mb, 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 X=x1,x2,…,xnT; x¨b, x˙b, and xb are the acceleration, velocity, and displacement of the TMD, respectively. x¨g is earthquake acceleration; Γn=0,0,…,1T is the indicator vector for installation; E=1,1,…,1T is the unit volume; the mass matrix of the main structure M = diagm1,m2,…,mn; and the stiffness matrix of the main structure K=k1+k2−k2…−k2k2+k3⋮−kn−knkn; the above responses are the response to the ground.LetX=ϕ1q, where ϕ1 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:(30)Hs=qx¨g=−αδTMDs2+2ζw1s+w12/λ−βδTMD2ζw1s+w12/λs2+w12δTMDs2+2ζw1s+w12/λ+γnδTMDs22ζw1s+w12/λ,where w1=K1/M1 is the first mode frequency, M1 is the mass of the first mode, K1 is the stiffness of the first mode, and δTMD=mb/M1 is TMD mass ratio; damping ratio is ζ=c/cr, where cr=2w1M1=2M1K1 is the critical damping coefficient of the system without TMD; stiffness ratio is λ=K1/k; α=ϕ1TME/M1; β=ϕ1TΓn;γn=ϕ1TΓnϕ1n, where ϕ1n is the first mode coordinates of the top layer.TMD structure is optimized by using the above method of the H2 optimization of inerter-based isolator, and the analytic solution of H2 performance optimization of similar structures C1–C8, as well as the analytic expressions of optimal stiffness ratio λopt and optimal damping ratio ζopt, is obtained.The analytic expression of the H2 performance index of TMD structure is(31)ITMD=14γnλ2ζw13δTMD24β2λ2γnδTMD3ζ2+8αβλ2γnδTMD2ζ2+4α2λ2γnδTMDζ2+4β2λ2δTMD2ζ2+β2λγnδTMD4+β2γn2δTMD4+8αβλ2δTMDζ2+2αβγn2δTMD3+α2λ2δTMD2+4α2λ2ζ2−α2λγnδTMD2+α2γn2δTMD2+2β2γnδTMD3−2αβλδTMD2+4αβγnδTMD2−2α2λδTMD+2α2γnδTMD+β2δTMD2+2αβδTMD+α2.For a given TMD mass ratioδTMD, the optimal stiffness ratio λopt is(32)λopt=2βγn2δTMD3+αγn2δTMD2+2βγnδTMD2+2αγnδTMD+βδTMD+αδ−βγnδTMD2+αγnδTMD+2α.For a given TMD mass ratioδTMD, the optimal damping ratio ζopt is(33)ζopt=δ−βγn2δTMD3+3αγn2δTMD2+4αγnδTMD/16βγnδTMD2+16αγnδTMD+16βδTMD+16α1+γnδTMD.The optimal H2 performance index ITMD,opt is(34)ITMD,opt=−β2γnδTMD3+2αβγnδTMD2+3α2γnδTMD+4αβδTMD+4α281+γnδTMDw13−βγn2δTMD3+3αγn2δTMD2+4αγnδTMD/16βγnδTMD2+16αγnδTMD+16βδTMD+16α.


## 2.1. Method of H2 Optimization of Inerter-Based Isolator

The 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 Figure2.Figure 2
The low-complexity networks with only one inerter, one spring, and one damper [35]. C4 is TID [39, 40], and C6 is TVMD [38]. (a) C1. (b) C2. (c) C3. (d) C4. (e) C5. (f) C6. (g) C7. (h) C8.
(a)(b)(c)(d)(e)(f)(g)(h)In Section2.1, structure C3 is taken as an example to illustrate the method of H2 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. The 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.Figure 3
Schematic diagram of structure C3 installed in the multi-degree-of-freedom model.For regular uniform multistorey buildings, the low-order mode usually plays a major role under seismic excitation [40]. Therefore, this paper will conduct H2 performance optimization for the first-order mode of a multistorey building and conduct a comparative analysis with the H2 performance of TMD installed at the top floor of the building.Its motion equation is(1)MX¨+KX+FΓ0=−MEx¨g,F=bx¨b=kx1−xc=cx˙c−x˙b,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 X=x1,x2,…,xnT; x¨b, x˙b, and xb are the acceleration, velocity, and displacement of the inerter, respectively; x¨g is the earthquake acceleration; respectively, x˙c and xc represent the velocity and displacement of the damper; Γ0=1,0,…,0T is the indicator vector for installation; E=1,1,…,1T is the unit volume; the mass matrix of the main structure M = diagm1,m2,…,mn; and the stiffness matrix of the main structure K=k1+k2−k2…−k2k2+k3⋮−kn−knkn; the above responses are the responses to the ground.LetX=ϕ1q, where ϕ1 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:(2)Hs=qx¨g=−α2δζw1s2+δw12/λs+2ζw13/λs2+w122δζw1s2+δw12/λs+2ζw13/λ+2γδζw13/λs2,where w1=K1/M1 is the first mode frequency, M1 is the mass of the first mode, K1 is the stiffness of the first mode, and δ=b/M1 is inertance-to-mass ratio; damping ratio is ζ=c/cr, where cr=2w1M1=2M1K1 is the critical damping coefficient of the system without structure C3; stiffness ratio is λ=K1/k; α=ϕ1TME/M1; γ=ϕ1TΓ0ϕ11, where ϕ11 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 H2 norm to evaluate the seismic damping effect of the adopted structure on the main structure. The goal of H2 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 H2 performance index I of the system can be expressed as(3)I=H^s22.For a linear time-invariant closed-loop system,(4)x˙=Aclx+Bclw,z=Cclx+Dclw,where w is the external input and z is the control output. The H2 norm from the input w to the output z of the system can be directly calculated with the following theorem.Theorem 1.
(see [41]). Considering closed-loop system (4), if Acl is unstable or Dcl≠0, then the H2 norm from the input w to the output z of the system is infinite; otherwise,(5)H22=traceCclPCclT=traceBclTSBcl,where P and S are the solutions of the following Lyapunov equation:(6)AclP+PAclT+BclBclT=0,AclTS+SAcl+CclTCcl=0.According to Theorem1, for a given stable transfer function Ts,(7)Ts=bn−1sn−1+⋯+b1s+b0sn+an−1sn−1+⋯+a1s+a0.Its H2 norm can be calculated as follows:(8)Ts22=CsI−A−1B22=CLCT,where matrices A, B, and C are the minimum phase realization of Ts, and matrix L is the solution of the following Lyapunov equation:(9)AL+LAT+BBT=0.Therefore, H2 performance index I of the system can be calculated analytically using the above method, and the specific steps are summarized as follows.The method of H2 optimization of the inerter-based isolator is as follows:(i)
Step 1. The transfer function Hs of the inerter-based isolator is listed, and the analytic expression of the H2 performance index I of the system is obtained by using the method of solving the H2 performance index for the given inertance-to-mass ratio δ.(ii)
Step 2. The analytic expression of the H2 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 λopt and the optimal damping ratio ζopt.(iii)
Step 3. The obtained optimal stiffness ratio λopt and damping ratio ζopt are substituted into the analytic expression of the system H2 performance index I to obtain the optimal H2 performance Iopt.


## 2.2. Analytic Solution of H2 Performance Optimization for Inerter-Based Isolator

On the premise of the given inertance-to-mass ratioδ, the analytic solutions of H2 performance optimization, optimal stiffness ratio λopt, and optimal damping ratio ζopt of all the inerter-based isolator C1–C8 installed in the multi-degree-of-freedom model are given.


### 2.2.1. Structure C1

The analytic expression of the H2 performance index of structure C1 is(10)IC1=λα24λ+γζγw13.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC1,opt is 0, or the optimal stiffness ratio λopt is 0, and the optimal H2 performance index IC1,opt is 0.


### 2.2.2. Structure C2

The analytic expression of the H2 performance index of structure C2 is(11)IC2=λα24λγδζ2+4λζ2+λδ2+γδ24γζw13λ+γ2δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is 0, and the optimal H2 performance index IC2,opt is 0.


### 2.2.3. Structure C3

The analytic expression of the H2 performance index of structure C3 is(12)IC3=α24λ2δ2ζ2−8λδζ2+4γδζ2+δ2+4ζ24γζw13δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(13)λopt=1δ.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is(14)ζopt=44δγ.The optimal H2 performance index IC3,opt is(15)IC3,opt=α2γw13δ/γ.


### 2.2.4. Structure C4 (TID)

The analytic expression of the H2 performance index of structure C4 is(16)IC4=α24λ2γδζ2+δ2λ2+4λ2ζ2−λγδ2−2δλ+γ2δ2+2γδ+14λ2γζw13δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(17)λopt=2γ2δ2+2γδ+1δγδ+2.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is(18)ζopt=δ3γ2δ2+4γδ/16+16γδ1+γδ.The optimal H2 performance index IC4,opt is(19)IC4,opt=α23γδ+481+γδw133γ2δ2+4γδ/16+16γδ.


### 2.2.5. Structure C5

The analytic expression of the H2 performance index of structure C5 is(20)IC5=α24λ2γδζ2+δ2λ2+4λ2ζ2−2δλ+14γζw13λ2δ2−2δλ+1.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(21)λopt=0.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC5,opt is 0.


### 2.2.6. Structure C6 (TVMD)

The analytic expression of the H2 performance index of structure C6 is(22)IC6=α2δ2λ2+4λ2ζ2+λγδ2−2δλ+14γζw13.For a given inertance-to-mass ratioδ, λ has no optimal value when δ=23−22/3γ,23+22/3γ; when δ is equal to anything else, the optimal stiffness ratio λopt is(23)λopt=2−γδ2δ3/8γ2δ2−3/2γδ+3/2−1/89γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.For a given inertance-to-mass ratioδ, ζ has no optimal value when δ=23−22/3γ,23+22/3γ; when δ is equal to anything else, the optimal damping ratio ζopt is(24)ζopt=18δ8+6γ2δ2−24γδ−29γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.When the optimal stiffness ratioλopt and damping ratio ζopt are obtained, the optimal H2 performance index IC6,opt is(25)IC6,opt=α28+6γ2δ2−24γδ−29γ4δ4−72γ3δ3+184γ2δ2−160γδ+164γδw139/8γ2δ2−9/2γδ+5/2−3/89γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.


### 2.2.7. Structure C7

The analytic expression of the H2 performance index of structure C7 is(26)IC7=α24λ2ζ2+γδ+14γζw13γδ+1.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(27)λopt=0.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC7,opt is 0.


### 2.2.8. Structure C8

The analytic expression of the H2 performance index of structure C8 is(28)IC8=α24γζw13.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC8,opt is 0.It should be noted that the H2 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. These 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.Figure 4
Simple structure after degradation: (a) TC1. (b) TC2. (c) TC3.
(a)(b)(c)Although the H2 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 H2 performance index under the condition of obtaining the optimal stiffness ratio and damping ratio. In practical applications, the optimal H2 performance can be obtained by selecting the corresponding optimal stiffness ratio and damping ratio.


## 2.2.1. Structure C1

The analytic expression of the H2 performance index of structure C1 is(10)IC1=λα24λ+γζγw13.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC1,opt is 0, or the optimal stiffness ratio λopt is 0, and the optimal H2 performance index IC1,opt is 0.


## 2.2.2. Structure C2

The analytic expression of the H2 performance index of structure C2 is(11)IC2=λα24λγδζ2+4λζ2+λδ2+γδ24γζw13λ+γ2δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is 0, and the optimal H2 performance index IC2,opt is 0.


## 2.2.3. Structure C3

The analytic expression of the H2 performance index of structure C3 is(12)IC3=α24λ2δ2ζ2−8λδζ2+4γδζ2+δ2+4ζ24γζw13δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(13)λopt=1δ.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is(14)ζopt=44δγ.The optimal H2 performance index IC3,opt is(15)IC3,opt=α2γw13δ/γ.


## 2.2.4. Structure C4 (TID)

The analytic expression of the H2 performance index of structure C4 is(16)IC4=α24λ2γδζ2+δ2λ2+4λ2ζ2−λγδ2−2δλ+γ2δ2+2γδ+14λ2γζw13δ2.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(17)λopt=2γ2δ2+2γδ+1δγδ+2.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is(18)ζopt=δ3γ2δ2+4γδ/16+16γδ1+γδ.The optimal H2 performance index IC4,opt is(19)IC4,opt=α23γδ+481+γδw133γ2δ2+4γδ/16+16γδ.


## 2.2.5. Structure C5

The analytic expression of the H2 performance index of structure C5 is(20)IC5=α24λ2γδζ2+δ2λ2+4λ2ζ2−2δλ+14γζw13λ2δ2−2δλ+1.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(21)λopt=0.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC5,opt is 0.


## 2.2.6. Structure C6 (TVMD)

The analytic expression of the H2 performance index of structure C6 is(22)IC6=α2δ2λ2+4λ2ζ2+λγδ2−2δλ+14γζw13.For a given inertance-to-mass ratioδ, λ has no optimal value when δ=23−22/3γ,23+22/3γ; when δ is equal to anything else, the optimal stiffness ratio λopt is(23)λopt=2−γδ2δ3/8γ2δ2−3/2γδ+3/2−1/89γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.For a given inertance-to-mass ratioδ, ζ has no optimal value when δ=23−22/3γ,23+22/3γ; when δ is equal to anything else, the optimal damping ratio ζopt is(24)ζopt=18δ8+6γ2δ2−24γδ−29γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.When the optimal stiffness ratioλopt and damping ratio ζopt are obtained, the optimal H2 performance index IC6,opt is(25)IC6,opt=α28+6γ2δ2−24γδ−29γ4δ4−72γ3δ3+184γ2δ2−160γδ+164γδw139/8γ2δ2−9/2γδ+5/2−3/89γ4δ4−72γ3δ3+184γ2δ2−160γδ+16.


## 2.2.7. Structure C7

The analytic expression of the H2 performance index of structure C7 is(26)IC7=α24λ2ζ2+γδ+14γζw13γδ+1.For a given inertance-to-mass ratioδ, the optimal stiffness ratio λopt is(27)λopt=0.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC7,opt is 0.


## 2.2.8. Structure C8

The analytic expression of the H2 performance index of structure C8 is(28)IC8=α24γζw13.For a given inertance-to-mass ratioδ, the optimal damping ratio ζopt is ∞, and the optimal H2 performance index IC8,opt is 0.It should be noted that the H2 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. These 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.Figure 4
Simple structure after degradation: (a) TC1. (b) TC2. (c) TC3.
(a)(b)(c)Although the H2 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 H2 performance index under the condition of obtaining the optimal stiffness ratio and damping ratio. In practical applications, the optimal H2 performance can be obtained by selecting the corresponding optimal stiffness ratio and damping ratio.


## 2.3. Analytic Solution of H2 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 Figure5, ignoring the influence of damping of the main structure.Figure 5
Schematic diagram of TMD installed in the multi-degree-of-freedom model.Its motion equation is(29)MX¨+KX+FΓn=−MEx¨g,F=mbx¨b+x¨g=kxn−xb+cx˙n−x˙b,where M and K are, respectively, the mass matrix and stiffness matrix of the main structure; mb, 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 X=x1,x2,…,xnT; x¨b, x˙b, and xb are the acceleration, velocity, and displacement of the TMD, respectively. x¨g is earthquake acceleration; Γn=0,0,…,1T is the indicator vector for installation; E=1,1,…,1T is the unit volume; the mass matrix of the main structure M = diagm1,m2,…,mn; and the stiffness matrix of the main structure K=k1+k2−k2…−k2k2+k3⋮−kn−knkn; the above responses are the response to the ground.LetX=ϕ1q, where ϕ1 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:(30)Hs=qx¨g=−αδTMDs2+2ζw1s+w12/λ−βδTMD2ζw1s+w12/λs2+w12δTMDs2+2ζw1s+w12/λ+γnδTMDs22ζw1s+w12/λ,where w1=K1/M1 is the first mode frequency, M1 is the mass of the first mode, K1 is the stiffness of the first mode, and δTMD=mb/M1 is TMD mass ratio; damping ratio is ζ=c/cr, where cr=2w1M1=2M1K1 is the critical damping coefficient of the system without TMD; stiffness ratio is λ=K1/k; α=ϕ1TME/M1; β=ϕ1TΓn;γn=ϕ1TΓnϕ1n, where ϕ1n is the first mode coordinates of the top layer.TMD structure is optimized by using the above method of the H2 optimization of inerter-based isolator, and the analytic solution of H2 performance optimization of similar structures C1–C8, as well as the analytic expressions of optimal stiffness ratio λopt and optimal damping ratio ζopt, is obtained.The analytic expression of the H2 performance index of TMD structure is(31)ITMD=14γnλ2ζw13δTMD24β2λ2γnδTMD3ζ2+8αβλ2γnδTMD2ζ2+4α2λ2γnδTMDζ2+4β2λ2δTMD2ζ2+β2λγnδTMD4+β2γn2δTMD4+8αβλ2δTMDζ2+2αβγn2δTMD3+α2λ2δTMD2+4α2λ2ζ2−α2λγnδTMD2+α2γn2δTMD2+2β2γnδTMD3−2αβλδTMD2+4αβγnδTMD2−2α2λδTMD+2α2γnδTMD+β2δTMD2+2αβδTMD+α2.For a given TMD mass ratioδTMD, the optimal stiffness ratio λopt is(32)λopt=2βγn2δTMD3+αγn2δTMD2+2βγnδTMD2+2αγnδTMD+βδTMD+αδ−βγnδTMD2+αγnδTMD+2α.For a given TMD mass ratioδTMD, the optimal damping ratio ζopt is(33)ζopt=δ−βγn2δTMD3+3αγn2δTMD2+4αγnδTMD/16βγnδTMD2+16αγnδTMD+16βδTMD+16α1+γnδTMD.The optimal H2 performance index ITMD,opt is(34)ITMD,opt=−β2γnδTMD3+2αβγnδTMD2+3α2γnδTMD+4αβδTMD+4α281+γnδTMDw13−βγn2δTMD3+3αγn2δTMD2+4αγnδTMD/16βγnδTMD2+16αγnδTMD+16βδTMD+16α.


## 3. Numerical Examples

In this paper, the three-story uniform structure model used by Lazar [39] is employed. The 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 inerter-based 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 IC4 to be less than ITMD in equation (31) is 5. This 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. The 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.Figure 6
El Centro seismic wave.Then, taking C1 as an example, the parameter selection method of C1, C2, C5, C7, and C8 in simulation is illustrated. C1 obtains the optimal H2 performance when the damping ratio ζ=c/cr is infinite, which means that the damping coefficient c is infinite, when the stiffness ratio λ is fixed, and the H2 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.Figure 7
H2 performance index with respect to damping ratio for structure C1 under the excitation of El Centro seismic wave.In the same way, C1 obtains the optimal H2 performance when the stiffness ratio λ=K1/k is 0, which means that the stiffness coefficient k is infinite, when the damping ratio ζ is fixed, and the H2 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 δTMD = 0.07 is shown in Figure 9.Figure 8
H2 performance index with respect to stiffness coefficient for structure C1 under the excitation of El Centro seismic wave.Figure 9
Generalized displacement when TMD mass ratio is 0.07 under the excitation of El Centro seismic wave.As can be seen from Table1, 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.Table 1
Root mean square of generalized displacement with different TMD mass ratios and different structures installed, unit (×10−3 m).
δTMDTMDC3C4C6TC1TC2TC30.016.0796.0256.0195.9990.2750.0850.1120.035.0544.9354.94.8630.054.8684.5694.5634.3650.074.8134.6314.4264.013In order to illustrate the performance of the inerter-based isolator, a total of 11 seismic waves [42] are employed for the simulation experiment, and the power spectral density of various seismic wave accelerations is shown in Figure 10.Figure 10
Power spectral density of various seismic wave accelerations. (a) Chichi. (b) Friuli. (c) Hollister. (d) Imperial Valley. (e) Kobe. (f) Kocaeli. (g) Landers. (h) Loma Prieta. (i) Northridge. (j) Trinidad.The comparative data of generalized displacement are summarized in Table2, where δTMD=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.Table 2
Root mean square of generalized displacement with different earthquakes and different structures installed, unit (×10−3 m).
EarthquakePGA (g)TMDC3C4C6TC1TC2TC3Chichi0.361013.8712.1312.7510.771.4920.1620.519El Centro0.34874.8134.3614.4264.0130.2750.0850.112Friuli0.35135.4784.9714.9994.6840.0740.1570.167Hollister0.194810.759.7319.9059.5410.8590.1560.375Imperial Valley0.31525.75.4085.3814.8740.1030.0630.1Kobe0.34474.9214.6634.6084.290.1360.1540.169Kocaeli0.34904.0263.8653.7713.4830.0690.0910.109Landers0.78031715.5815.6314.740.2722.552.534Loma Prieta0.36742.1061.9091.9551.8160.260.0350.113Northridge0.56838.1477.7277.8016.8530.2950.1270.179Trinidad0.19362.3722.1572.1562.0340.0370.0730.08After 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(35)MX¨+KX+FΓ0=−MEx¨g,F=bx¨b=kx1−xc=cx˙c−x˙b.Let(36)Z=XX˙.The state-space equation can be written as(37)Z˙=AZ+BU,y=CZ,A=0I−M−1K0,B=00−M−1Γ0−E,C=I0,U=Fx¨g.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 Table3, where δTMD=0.07.Table 3
Root mean square of displacement of each floor with different structures installed, unit (×10−3 m).
UncontrolledTMDC3C4C6TC1TC2TC3First floor5.5734.8634.8454.8564.8390.3920.0890.246Second floor9.7228.6968.6638.6868.65826.4811.5515.2Third floor12.1110.8110.7710.810.7742.5518.6624.42As can be seen from Table3, 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 Table3, 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 H2 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. Therefore, 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 [43–46], the nonlinear effects of inerter would make the constitutive behavior different from the linear one, especially at the low frequencies for civil engineering applications. This should be considered in the future.


## 4. 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. The analytical H2 optimization method was adopted based on a multi-degree-of-freedom model to derive the optimal solutions of the considered low-complexity networks. The 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.



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