Study on Optimal Design of Grotto-Eave System with Cable Inerter Viscous Damper for Vibration Control

Abstract

In this paper, the mechanical model of grotto–eave system with cable inerter viscous damper (CIVD) is established, and the vibration control equations are established. Firstly, the stochastic response is carried out, and the optimization design of design parameters of CIVD is carried out for the grotto–eave systems with different connection types. Finally, the vibration mitigation control performance of CIVD under different seismic inputs is analyzed. The research shows that in the optimal design of CIVD, the inerter–mass ratio and damping ratio should be reduced as much as possible to improve the feasibility of the application of CIVD in cultural relics protection engineering under the condition of meeting the target damping ratio. The demand-based optimal method can minimize the cost by enhancing damping element deformation in a small damping ratio, while ensuring that the value of displacement index of grotto–eave system can be reached. Hence, the deformation and damping force of CIVD will increase simultaneously due to the efficient tuning and damping amplification of CIVD. CIVD can enlarge the apparent mass through rotation and damping force through enhancement deformation. Hence, compared with other conventional dampers (such as viscous damper), optimal CIVD has lower damping ratio under the same demand index of grotto–eave system. It can be realized that the lightweight and high efficiency of the damper, and can be applied to the vibration mitigation and reinforcement of the grotto–eave system.

1. Introduction

Historic buildings and cultural relics are valuable and must be preserved carefully. However, such structures, such as grotto–eave system, may be destroyed during earthquakes owing to the deterioration of their structural performance over time [1]. Moreover, the seismic resistance of most historic buildings is inferior to that of modern structures because the mechanical performance of the construction materials deteriorates with time [2,3]. Therefore, it is of significance to protect historic buildings and the cultural relics housed in such buildings. There are various types of immovable heritage structures, including wooden, bricked and rammed structures [4,5,6]. The historical buildings of the most historical and cultural value are mainly wooden structures and bricked structures. In general, non-structural components are often used for decoration, and they may have a certain architectural or utility function. However, in the immovable heritage structures [7], the non-structural components are also regarded as part of the heritage. Their cultural value is higher than the value of structures. It is the main difference between heritage buildings and modern buildings in seismic protection that non-structural components are as important as the main structure.

In order to improve the seismic performance of immovable cultural relics, we can use traditional reinforcement methods that including strengthening the connecting parts of structural members or adding shear walls and supports, and improving the rigidity of the structure. Bento et al. [8] proposed the use of steel or concrete to strengthen the connection of bricked walls, thereby increasing the lateral resistance of the structure. Aty [9] proposed another modification method by using wood support system, and analyzed the seismic performance of X-shaped wood support and K-shaped wood support structure. Witzany et al. [10] conducted experimental research and analysis on the failure mechanism and ultimate compressive bearing capacity of carbon fiber reinforced polymer (CFRP) reinforced bricked walls. Akcay et al. [11] performed reinforcement tests and numerical analyses of historic bricked buildings using different conventional techniques. Although the structural rigidity increases, it also increases the seismic damage to the structure, which heritage buildings may not be able to withstand. These traditional renovation methods all help to improve the seismic performance of heritage buildings. However, the original information of the cultural relics cannot be preserved, and the modern buildings and the cultural relics inside may be damaged during the renovation. Structural control is an efficient approach for suppressing the dynamic response of civil structures under external actions such as those exerted by wind, earthquake, and other hazardous events. Hence, it is necessary to develop a lightweight and high efficiency damper that can control the vibration response of historic buildings and cultural relics simultaneously. An cable inerter viscous damper (CIVD) for seismic response mitigation of typical historic buildings and immovable cultural relics system is proposed in this paper.

The vibration control technologies of inerter-based dampers have been developed based on electromechanical similarity theory [12]. Compared with the TMD, the inerter-based damper can control the inertial force at the two terminals directly. Moreover, the inerter element can effectively enlarge the small apparent mass through converting the translational motion into rotary motion such as ball screw. In 2001, Smith [13] put forward the concept of inerter element and gave the basic forms of ball screw inerter element and rack and pinion inerter element, and designed a hydraulic inerter element in 2013 [14]. Subsequently, shock absorbers were proposed such as tuned viscous mass damper (TVMD) and tuned inerter damper (TID). The design method of the inerter system was also studied. Ikago et al. [15] derived a closed-form formula for TVMD optimization design based on fixed-point theory. Pan et al. [16] considered the natural damping of the original structure and the cost of the inerter-based damper, and make up for the deficiency of fixed-point theory. Then, he proposed the design method of SPIS-Ⅱ inerter-based damper based on stochastic response mitigation ratio [17]. Hwang et al. [18] proposed a rotation inerter system connected with a toggle brace based on ball screw. It is shown that the system can be effectively used in the structure with small drift. Zhang et al. [19,20] applied the inerter damper system to high-rise structures such as chimneys and wind power towers, and proved the effectiveness of the inerter-based damper in high-rise structures. Gao et al. [21] put forward an optimum design method of viscous inerter damper (VID) based on the feedback control theory. De Domenico [22,23,24,25,26,27,28,29,30,31,32] proposed the optimal design methods of inerter-based TMD systems for seismic response mitigation.Although some scholars have used the inerter-based damper in practical engineering [33], most of the research on the inerter-based damper is still in the stage of theoretical analysis, and only a few scholars have proposed the connection mode and design method of the inerter system applied in building structures [34]. Xie et al. [35,36] put forward a cable-bracing inerter system (CBIS), and shows that it is easy to install and can effectively control displacement and acceleration of structure. Wang et al. [37] put forward a new tuned inerter-negative-stiffness damper (TINSD) based on fixed-point method, which is more effective than the TID, TVMD, and INSD in reducing the dynamic response of structures. It is the satisfactory scheme using inerter-based damper to suppress vibrations of some special structure, such as transmission line [38], tall building [39] and transformer-bushing system [40].

At present, the fixed-point method is mainly used in the parameters design of inerter systems and other vibration control systems, such as TMD, TID and TVMD. A better damping ratio or stiffness ratio can be obtained by fixed-point method, but the optimal inerter-mass ratio (or mass ratio) cannot be obtained directly. In this way, the obtained damping ratio is larger and the inerter element cannot amplify the damping effect as far as possible. At the same time, it is necessary to install lighter dampers to meet the higher seismic requirements for immovable historic buildings and cultural relics such as grottoes and eaves.

In this paper, a cable inerter viscous damper (CIVD) system is proposed. The end of the lightweight inerter viscous damper is directly connected with the elastic cable, which can be quickly installed in various historic buildings and cultural relics. The system can not only realize structural reinforcement and improve the integrity of the structure, but also realize the lightweight of the shock absorber. Firstly, Section 2 introduces the basic principle of CIVD. The motion control equations and frequency response functions of grotto–eave system with CIVD are established. In the Section 3, the parameter analysis is carried out to obtain the minimum additional damping ratio of CIVD under different vibration mitigation ratios. Additionally, the demand-based optimal design method of CIVD is proposed; Finally, it is carried out that the dynamic time history analysis of the grotto–eave system installed with CIVD under the ground motions in Section 4 to verify the vibration control effect and parameter optimization results of CIVD. The research in this paper can provide reference for the design of efficient and lightweight vibration mitigation scheme of immovable historic buildings and cultural relics based on inerter damping system.

2. Theoretical Analysis of Grotto-Eave System with CIVD

2.1. Mechanical Model of Inerter Element and CIVD

Equation (1) shows the output force of inerter element and it has different accelerations in two terminals. Hence, the output of inerter element is proportional to the relative acceleration at two terminals, which can be shown as follows:

$$f_I=m_d\left(a_2-a_1\right)$$

(1)

where, f_I is the output force of the inerter element, a₁ and a₂ are the accelerations at terminals, m_d is the inertance, and Figure 1 is the mechanical model of the inerter element.

The inerter element cannot dissipate energy by itself and it is generally used in combination with the damper. Figure 2a shows the inerter viscous damper (IVD), the translation $\mathsf{\Delta }$ of the structure, can be converted into rotation $\phi$ in the IVD through the ball screw. The input energy can be dissipated by the viscous fluid in the damper. IVD can be regarded as an inerter element with a damping element connected in parallel, and the mechanical model of IVD is shown in Figure 2b.

Where m_d and c_d are equivalent mass and equivalent damping coefficient, corresponding to the translation $\mathsf{\Delta }$ at both ends of IVD; J and c_vd are the apparent mass (moment of inertia) and viscous damping constants. The expressions of m_d and c_d can be obtained, where L is the lead of the ball screw:

$$m_d=\frac{4{\pi }^2}{L^2}J,c_d=\frac{4{\pi }^2}{L^2}{c}_{vd},\phi =\frac{2\pi }{L}\mathsf{\Delta }$$

(2)

CIVD is IVD connected with an elastic cable, and the cable in this paper is a short-span cable. The reason is that the prestress relaxation of the short cable can be almost ignored. When the angle of cable is an appropriate large constant, the sag of cable and stress relaxation in the short-span structure can be ignored. Compared with the long cable, the short cable has high efficiency of force transmission. The mechanical model is shown in Figure 3. Where k_d is the equivalent stiffness of the elastic cable.

2.2. Mechanical Model of Grotto-Eave System with CIVD

The grotto–eave system is generally composed of eaves, connection and grotto. Generally, the eaves of the caves are wooden or steel structures. To protect the grottoes, the wooden structures are built in ancient times, while the steel structures are the modern architecture. The height of the structure is generally about 6–20 m, and its overall stiffness is small. The connection is the beam with the eaves lapped on the grotto or inserted into the grotto, so that the two parts are connected into a complete system. The material, type and position of the connection will influence the stiffness of the connection. Similar to the modern frame, the main structure of the grotto is generally an internal hollow structure. The materials of grotto are mostly rock materials such as sandstone, and grotto is generally adjacent to the mountain, the stiffness is relatively large. There is a large difference in stiffness between the eaves and the grottoes. When subjected to external excitation, the incongruity of deformation between the eaves and the grotto body will occur. Many precious cultural heritages are preserved in the grottoes, which have high historical and humanistic value, but are easily damaged by external disturbances; at the same time, since the grottoes are immovable cultural relics, it can only be protected at the original site. It will also lead to the cumulative damage of the cultural relics under frequent external vibration, and resulting in the instantaneous brittle failure of the stone cultural relics.

Under external excitation, it is generally considered to reduce the vibration of the eaves itself to reduce the vibration effect of the eaves on the grottoes. Due to the high demand for vibration mitigation of the system, the bearing capacity of the eave is small and the internal space is limited. Traditional vibration control system (such as TMD) cannot achieve the expected effect. Therefore, it needs to find a lightweight and efficient vibration control system, such as the CIVD, which has the flexible arrangement and the small size damper, and the obvious effect of vibration mitigation. Figure 4 is a layout of CIVD in an eave–grotto system.

For the grotto–eave system, the shock absorber cannot be directly installed in the grotto, so the CIVD can be installed inside the wooden eaves. Figure 5 shows the mechanical model of the grotto-eave system with CIVD. Where m_e, m_c, m_g and m_d are the mass of the eaves, the connection and the grotto, and the inerter of the CIVD; c_e, c_c, c_g and c_d are the damping coefficients of the eaves, connecting sections, grottoes, and the equivalent damping coefficient of CIVD; k_e, k_c, k_g, and k_d are the stiffnesses of the eaves, connection, grottoes and CIVD.

2.3. Motion Control Equation of Grotto-Eave System with CIVD

According to the mechanical model shown in Figure 5, the motion equation of the grotto–eave system with CIVD is established:

$$M\ddot{X}+C\dot{X}+KX=-MI{a}_g$$

(3)

where, $\ddot{X},\dot{X},X$ is the acceleration, velocity and displacement vector of the grotto-eave system, a_g is the acceleration of ground motion; M, C, K are the mass, damping and stiffness matrices, and I is the ground motion excitation vector. Therefore, M, C, K, I and motion vector can be expressed as:

$$M=\left[\begin{array}{ccc}{m}_e+m_c& & \\ & m_d& \\ & & m_g\end{array}\right],C=\left[\begin{array}{ccc}{c}_e+c_c& & -c_c\\ & c_d& \\ -c_c& & c_g+c_c\end{array}\right],K=\left[\begin{array}{ccc}{k}_e+k_c+k_d& -k_d& -k_c\\ -k_d& k_d& \\ -k_c& & k_g+k_c\end{array}\right]$$

(4)

$$X=\left\{\begin{array}{l}{u}_e\\ u_d\\ u_g\end{array}\right\},\dot{X}=\left\{\begin{array}{l}{\dot{u}}_e\\ {\dot{u}}_d\\ {\dot{u}}_g\end{array}\right\},\ddot{X}=\left\{\begin{array}{c}{\ddot{u}}_e\\ {\ddot{u}}_d\\ {\ddot{u}}_g\end{array}\right\},I=\left\{\begin{array}{l}1\\ 0\\ 1\end{array}\right\}$$

(5)

where $u,\dot{u},\ddot{u}$ is the vectors of displacement, velocity and acceleration; the subscripts e, d, g represents the eaves, CIVD and grottoes. For the convenience of parameter analysis, we define the following dimensionless parameters. Where Equation (6) is the natural vibration frequency and damping ratio parameters of the eaves, connection and grottoes; Equation (7) is parameters of CIVD; Equation (8) is the parameters of relative mass.

$${\omega }_e=\sqrt{\frac{k_e}{m_e}}; {\zeta }_e=\frac{c_e}{2m_e{\omega }_e}; {\omega }_c=\sqrt{\frac{k_c}{m_c}}; {\zeta }_c=\frac{c_c}{2m_c{\omega }_c}; {\omega }_g=\sqrt{\frac{k_g}{m_g}}; {\zeta }_g=\frac{c_g}{2m_g{\omega }_g}$$

(6)

$${\zeta }_d=\frac{c_d}{2m_e{\omega }_e}; {\kappa }_d=\frac{k_d}{k_e}$$

(7)

$${\mu }_d=\frac{m_d}{m_e}; {\mu }_{ce}=\frac{m_c}{m_e}; {\mu }_{cg}=\frac{m_c}{m_g}$$

(8)

where ω_e is the natural circular frequency of the eaves, ζe is the damping ratio of the eaves; ω_c is the natural circular frequency of the connection, and ζ_c is the damping ratio of the connection; ω_g is the natural circular frequency of the grotto, and ζ_g is the damping ratio of the grotto; ζ_d is the damping ratio of CIVD, and κ_d is the stiffness ratio of CIVD; μ_d is the inerter–mass ratio, μ_ce is the mass ratio of connection and eave, and μ_cg is the mass ratio of connection and grotto.

From Equations (5)–(8), Laplace transform is applied to Equation (3), so the equation of motion (9) and (10) of the CIVD grotto–eave system in frequency domain can be obtained, where $\widehat{M},\widehat{C},\widehat{K},\widehat{I}$ is the dimensionless mass, damping, stiffness matrix and external excitation vector of the system.

$$\left(\widehat{M}{s}^2+\widehat{C}s+\widehat{K}\right)\widehat{U}=\widehat{I}{A}_g; \widehat{U}=\left\{\begin{array}{c}{U}_e\\ U_d\\ U_g\end{array}\right\}; \widehat{I}=\left\{\begin{array}{c}-1-{\mu }_{ce}\\ 0\\ -1\end{array}\right\}$$

(9)

$$\widehat{M}=\left[\begin{array}{ccc}1+{\mu }_{ce}& & \\ & {\mu }_d& \\ & & 1\end{array}\right]; \widehat{C}=\left[\begin{array}{ccc}2{\omega }_e{\zeta }_e+2{\omega }_c{\zeta }_c{\mu }_{ce}& & -2{\omega }_c{\zeta }_c{\mu }_{ce}\\ & 2{\omega }_e{\zeta }_d& \\ -2{\omega }_c{\zeta }_c{\mu }_{cg}& & 2{\omega }_g{\zeta }_g+2{\omega }_c{\zeta }_c{\mu }_{cg}\end{array}\right];\widehat{K}=\left[\begin{array}{ccc}{\omega }_e^2(1+\kappa )+{\omega }_c^2{\mu }_{ce}& -{\omega }_e^2{\kappa }_d& -{\omega }_c^2{\mu }_{ce}\\ -{\omega }_e^2{\kappa }_d& {\omega }_e^2{\kappa }_d& \\ -{\omega }_c^2{\mu }_{cg}& & {\omega }_g^2+{\omega }_c^2{\mu }_{cg}\end{array}\right]$$

(10)

where s is the Laplace operator, s = iΩ, Ω is the external excitation frequency, U_e, U_d, U_g and A_g are the Laplace transforms of u_e, u_d, u_g and a_g. The linear matrix Equations (9) and (10) can be solved to obtain U_e, U_d and U_g, at the same time, the transfer functions H_Ue(s), H_Ud(s) and H_Ug(s) of the grotto–eave system with CIVD can be obtained:

$$H_{U_e}\left(s\right)=\frac{U_e\left(s\right)}{A_g\left(s\right)},H_{U_d}\left(s\right)=\frac{U_d\left(s\right)}{A_g\left(s\right)},H_{U_g}\left(s\right)=\frac{U_g\left(s\right)}{A_g\left(s\right)}$$

(11)

Meanwhile, the transfer function of output force of CIVD H_Fd(s) can be expressed as follows:

$$H_{F_d}\left(s\right)=\frac{U_d\left(s\right)}{A_g\left(s\right)}\left({\mu }_d{s}^2+2{\zeta }_d{\omega }_{e}s\right)$$

(12)

3. Parameter Analysis

According to Parseval’s theorem, the root mean square (RMS) response $\sigma$ of the system excited by white noise is obtained:

$$\sigma =\sqrt{{{\displaystyle {\int }_{-\infty }^{+\infty }\left|H\left(i\mathsf{\Omega }\right)\right|}}^2{S}_0\mathrm{d}\mathsf{\Omega }}$$

(13)

where S₀ is the power spectrum of white noise. The primary index is the γ_Ur, the relative displacement root mean square response ratio of the eaves and the grottoes before and after vibration mitigation by CIVD. It can be compared to evaluate the interaction between the eaves and the grottoes. At the same time, the γ_U, the displacement root mean square response ratio of the eaves before and after the vibration mitigation by CIVD, can directly measure the effect of the shock absorber, it can be regarded as an additional index. The smaller the γ_U and γ_Ur are, the better the vibration mitigation effect of the grotto–eave system is; that γ_U and γ_Ur are greater than 1 indicate that the CIVD has a displacement amplification effect on the system. The relative displacement ratio γ_Ur and displacement ratio γ_U are expressed as:

$${\gamma }_{U_r}\left({\zeta }_d,{\kappa }_d,{\mu }_d\right)=\frac{\left|{\sigma }_{U_e}-{\sigma }_{U_g}\right|}{\left|{\sigma }_{U_{e,0}}-{\sigma }_{U_{g,0}}\right|}.{\gamma }_U\left({\zeta }_d,{\kappa }_d,{\mu }_d\right)=\frac{{\sigma }_{U_e}}{{\sigma }_{U_{e,0}}}$$

(14)

where σ_Ue,0 and σ_Ug,0 are the displacement root mean square responses of the eaves and grottoes in the uncontrolled state. The CIVD parameter analysis selects the inerter–mass ratio μ_d, the stiffness ratio κ_d and the damping ratio ζ_d; and the parameter analysis index is the relative displacement ratio γ_Ur. Similarly, using Equations (12) and (13), the RMS response of output force of CIVD σ_Fd can be determined. In parameter analysis, the value range of parameters of CIVD is 0.01 to 1.

In addition to the CIVD parameters, it is also necessary to analyze the influence of the stiffness of the connection between grotto and eave. Defining the parameters of the Benchmark model of the grotto–eave system: ω_g/ω_e = 5, ζ_c = 0.015, ζ_e = ζ_g = 0.02, μ_ce = 0.2, μ_cg = 0.04, and subsequent analyses are carried out according to this Benchmark model; then defining the frequency ratio of the connection: β = ω_c/ω_e, β can reflect the stiffness of the connection, and the smaller the β, the smaller the stiffness of the connection.

Table 1. Frequency ratio of different types of connection.

Types of Connection	Soft Connection	Equal-Stiffness	Hard Connection
Symbol	SC	EC	HC
β	0.2	1	5

shows the frequency ratio β of different types of connection.

3.1. Parameter Analysis of CIVD

Based on the above analysis indexes, the parameters of the grotto–eave Benchmark model and different types of connection, the influence study of design parameters of CIVD, including inerter–mass ratio μ_d, stiffness ratio κ_d and damping ratio ζ_d.

Firstly, the influence of types of connection on grotto-eave system under different inerter-mass ratio μ_d is analyzed, and Figure 6, Figure 7 and Figure 8 show three-dimensional contour plot of γ_Ur. When the stiffness ratio is 0.01 to 0.3 and the damping ratio is 0.1 to 1, it can be seen from figures that as the stiffness ratio and damping ratio increase, the relative displacement between the grotto and the eaves will decrease; when the damping ratio is 0.01 to 0.1, the value of the minimum γ_Ur changes with the increase of the inerter–mass ratio, the larger the inerter-mass ratio, the greater requirement of additional stiffness of CIVD. However, in the case of low damping ratio and low stiffness ratio, the hard connection system has displacement amplification, which should be avoided in parameters selection. The maximum γ_Ur of three types of connection is 0.2, but the stiffness ratio and damping ratio required by CIVD should be as large as possible. Figure 6, Figure 7 and Figure 8 show the analytical solutions of the equivalent mathematical problems, which cannot be realized in engineering applications. Generally, it is more appropriate to control the damping ratio within 0.2. Hence, further analysis of other parameters and indexes should be carried out.

Taking γ_Ur as the index and fixing the damping ratio ζ_d to be 0.05, 0.1, and 0.15, from contour plot of γ_Ur with κ_d-μ_d space, we can get the variation trend of γ_Ur with ζ_d. Figure 9, Figure 10 and Figure 11 show the 2D contour plot of γ_Ur in κ_d-μ_d space under different types of connection, in which the numbers in the figure are contour values, representing the value of γ_Ur, the variation range is 0 to 1, and the interval value is 0.1. When the damping ratio ζ_d varies from 0.05 to 0.15, the value of the contour line of the soft connection system is 0.2 to 1, the equal stiffness connection system is 0.4 to 1, and the hard connection system is 0.5 to 1. Furthermore the smaller the damping ratio, the higher the contour line value, and the worse effect of vibration mitigation of relative displacement. At the same time, as the stiffness of the connection between the grotto and the eaves increases, CIVD needs greater the damping ratio to obtain the same relative displacement ratio γ_Ur. We can also draw a conclusion from the number of contour lines intuitively, the more “hard” the connection (means the stiffness of connection is large), the less the number of contour lines. CIVD has a better control effect on the relative displacement index γ_Ur, and it can indicate that the arrangement of CIVD can reduce the impact of the eaves on the grotto. However, at the same time, considering the vibration mitigation effect of the eaves themselves, if the displacement of the eaves themselves are too large, the CIVD cannot be used as the best vibration control device. To this end, we should continue to study another index γ_U with CIVD parameters.

The γ_U is also controlled by three parameters. For the convenience of research, a three-dimensional μ_d-ζ_d-κ_d space is established. The γ_U is any point in the parameter space, and it is represented by a specified color and a value corresponding to the color. We make slice plots on some specific parameter planes, and the two-dimensional spaces under the specified parameters are displayed. In this paper, we fixed μ_d to 0.01, 0.03, 0.1, 0.3 and 1, and it is used to study the response trend of the eaves under different types of connection based on the Benchmark model. Figure 12 is the γ_U slice plot of different connection systems. It can be seen from the figure that the displacement ratio of the SC system is the smallest, and the minimum value is 0.2. As the stiffness of the connection increasing, the minimum value of γ_U is larger, and the displacement mitigation of single eaves is worse. Figure 12a shows the displacement ratio of the eaves without connection, and it is similar to EC system. If the target displacement ratio of the eaves is specified as 0.5 (γ_U = 0.5), it can be seen that when μ_d is in the range of 0.03 to 0.3, the enclosed area of target contour γ_U is larger (the blue area), and the enclosed area tends to increase firstly and then decrease. When μ_d is in the range of 0.01 to 0.03, the γ_U is almost constant, so the range of μ_d can be determined between 0.03 and 0.3, and further discussions of influence of stiffness are continued.

Since the large stiffness of the cable is easier to achieve in practical, it can be assumed that the stiffness of the cable can be taken as any value. The γ_U of SC system under different inerter–mass ratios are shown in Figure 13. It can be seen from the figure that a large inerter–mass ratio requires a large stiffness to achieve the same displacement ratio of single eaves. At the same time, when ζ_d is less than 0.05, γ_U decreases firstly and then increases under a small damping ratio. Additionally, as the inerter–mass ratio increasing, this phenomenon becomes more obvious. Hence, the robustness of CIVD with small damping ratio is low. When ζ_d is greater than 0.1, increasing the stiffness of the cable cannot reduce the displacement ratio. In addition, a larger damping ratio (ζ_d is greater than 0.3) does not continue to reduce γ_U, but the stability of the curve and the trend of γ_U are better. However, in practical engineering, increasing damping ratio will lead to a higher cost and a larger volume of the damper. Therefore, while ensuring the index, further parameter optimization design is required to reduce the damping ratio.

3.2. Demand-Based Optimal Design of CIVD

The parameters of CIVD and other inerter-based shock absorbers can be determined by the fixed-point method [41]. Ikago et al. [16] has proposed corresponding inerter system design methods for single degree of freedom (SDOF) structures and multi-degrees of freedom structures. For the SDOF system of the eaves structure with CIVD, if the arrangement angle of the cables is not considered, the stiffness ratio and damping ratio can be obtained by the following formulas after the inerter–mass ratio μ_d is determined:

$${\kappa }_d=\frac{{\mu }_d}{1-{\mu }_d},{\zeta }_d=\frac{{\mu }_d}{2}\sqrt{\frac{3{\mu }_d}{\left(1-{\mu }_d\right)\left(2-{\mu }_d\right)}}$$

(15)

However, the fixed-point theory does not consider the inherent damping ratio, external excitation characteristics and performance demands of main structure. For example, it cannot reflect the key index such as the relative displacement response of grottoes and eaves. Zhang et al. [42] consider the main performance of the structure based on closed-form solution and use extreme conditions to determine parameters of CIVD. Compared with fixed-point method, better structural performance index can be obtained by this methods. In addition to performance index, output force of CIVD can represent the cost index of shock absorber. The contour plot of σ_Ue and σ_Fd of SC system are shown in Figure 13.

It can be seen from Figure 14 that the minimum value of σ_Ue is located at the upper right of the contour plot (white dot), but at this time, it corresponds to a large stiffness ratio and damping ratio. That means the output value of CIVD is also large, which is difficult to achieve in practical. However, compared with Figure 14a,b, it can be seen that there exists a parameter combination of a small stiffness ratio and a damping ratio when the inerter–mass ratio is determined, which can make the displacement and output smaller simultaneously (dark yellow dot). It means that the parameter combination can meet both the performance index (displacement ratio) and the cost index (output force of CIVD). If the main performance index of grottoes and eaves is determined, the optimization problem can be transformed from multi-objectives optimization problem to single-objective optimization problem, as shown in the following formula 16. It can be seen that the objective function of single-objective optimization problem is σ_Fd.

$$\begin{array}{l}\mathrm{minimize}{\sigma }_{Fd}\left({\zeta }_d,{\kappa }_d,{\mu }_d\right)\\ subjectto\{\begin{array}{c}{\gamma }_U{}_r\left({\zeta }_d,{\kappa }_d,{\mu }_d\right)={\gamma }_U{}_{r,t}\\ {\zeta }_d\in \left(0,0.2\right)\\ {\kappa }_d\in \left(0,1\right)\\ {\mu }_d\in \left(0,1\right)\end{array}\end{array}$$

(16)

where, γ_Ur,t is the relative displacement ratio of the grotto and the eaves. It is difficult to realize in practical engineering with large damping ratio. For this reason, the damping ratio is set within 0.2.

Moreover, compared with the conventional viscous damper (VD), the different topological connection forms of the inerter element can enhance the deformation of the damping element and improve the capacity of energy dissipation [43]. Therefore, a new index of damping effects of CIVD is proposed by comparing VD, namely, inerter-enhanced energy dissipation coefficient η:

$$\eta =\frac{{\sigma }_U{}_0\left({\zeta }_0\right)-{\sigma }_{U_e}\left({\zeta }_0,{\zeta }_d,{\kappa }_d,{\mu }_d\right)}{{\sigma }_U{}_0\left({\zeta }_0\right)-{\sigma }_{U_{eVD}}\left({\zeta }_0,{\zeta }_d\right)}$$

(17)

where σ_UeVD are the RMS displacement responses of the eave with VD. Additionally, η means the damping element with inerter element has a higher deformation effect under the same additional damping ratio. When η is greater than 1, it shows that the inerter can strengthen the energy dissipation of the structure, and it can be used as an index to evaluate the robustness of the CIVD. According to massive numerical case studies, the recommended range of η is [1,2] and the η can be also a constraint condition of formula 16. In this paper, η is fixed as 1.5 and γ_Ur,t is fixed as 0.5, 0.6 and 0.7.

Table 2. Optimal parameters of CIVD.

γUr	Parameters	β = 0	SC (β = 0.2)	EC (β = 1)	HC (β = 5)
0.7	μd	0.0121	0.0163	0.0277	0.0391
	κd	0.0213	0.0254	0.0442	0.0782
	ζd	0.0054	0.0061	0.0114	0.0183
0.6	μd	0.0221	0.0225	0.0485	0.0584
	κd	0.0332	0.0455	0.0782	0.1327
	ζd	0.0091	0.0111	0.0216	0.0341
0.5	μd	0.0344	0.0483	0.0692	0.0994
	κd	0.0671	0.0755	0.1301	0.2031
	ζd	0.0164	0.0192	0.0366	0.0485

shows the results of parameters of CIVD after optimization. It can be seen from Table 2 that the value of parameters’ combination of HC system are larger, but the values of CIVD are generally small, especially the damping ratio (less than 0.05).

Figure 15 shows the transfer function of relative displacement of the grotto–eaves system under different types of connection. Where, the modulus of transfer function of the relative displacement H_Ur of the grotto–eaves is:

$$\left|H_{U_r}\right|=\left|H_{U_e}-H_{U_g}\right|$$

(18)

It can be seen from Figure 15 that peak relative displacements of the grotto–eaves system are reduced under harmonic excitation, which controls the narrow-band resonance response of the system significantly. However, the HC system needs a smaller relative displacement ratio to control the resonance response. In order to better illustrate the advantages of this optimal method, the response of eaves and the deformation of damping element by using the classic fixed-point method should be compared under the same target relative displacement ratio.

Figure 16 and Figure 17 show the displacement amplification factor of eaves and transfer function of deformation of damping element of CIVD in SC system with different optimal design methods. It can be seen from Figure 16 and Figure 17 that the maximum displacement of demand-based optimal method is less than it of the fixed-point method. Meanwhile, the damping element deformation enhancement of the inerter system can be brought into full play by using demand-based optimal method. Hence, the demand-based optimal method can minimize the cost by enhancing damping element deformation in a small damping ratio.

4. Dynamic Response

In the previous study, parameter analyses in the frequency domain are carried out based on the performance index. In order to further verify the vibration mitigation effect of CIVD and the optimal parameters. Dynamic response analyses of the grotto–eave system with CIVD are carried out under the non-stationary ground motions in this section. The structural parameters of the grotto–eaves system refer to the Benchmark model in Section 2.1, and the parameters of CIVD are shown in Table 2. The ground motion records EL Centro record, Taft record, Chi-chi record and Kobe record are selected. The predominant frequencies of the four records are all different. Figure 18 shows the acceleration response spectrum of the four ground motion records.

Figure 19 shows the relative displacement time history response of SC eave–grotto system. It can be seen that CIVD has better vibration control performance intuitively. Additionally, the continuous vibration mitigation of CIVD is good on the grotto–eave system, and the controlled system can be stable and have small displacements quickly. As the seismic response progresses, the vibration control effects of CIVD in the later stage are better, indicating that a stable vibration state of eave–grotto system can be quickly got under control of CIVD.

Figure 20 shows that the decline of RMS displacement of the eave is between 25% and 60%. At the same time, it can be seen that the deformation of damping element in CIVD is always greater than displacement of main structure (eave). Compared with the hysteretic loops of CIVD and VD under the same damping coefficient as shown in Figure 21, it can be seen that the deformation and damping force of CIVD will increase simultaneously due to the efficient tuning and damping amplification of CIVD.

Hence, if the damping ratio ζ_d is a fixed, the damping force will be greater as the deformation of the damping element of CIVD is greater. Additionally, the damping effect of CIVD will be improved. Then, the deformation enhancement coefficient of damping element γ_Ud can be defined, as shown in Equation (19):

$${\gamma }_{U_d}\left({\zeta }_d,{\kappa }_d,{\mu }_d\right)=\frac{{\sigma }_{U_d}}{{\sigma }_{U_e}}$$

(19)

where, σ_Ud is the displacement root mean square responses of damping element of CIVD. The larger γ_Ud means the better deformation amplification effect of the inerter on the damping element, corresponding the smaller displacement mitigation ratio γ_U. Additionally, the deformation enhancement coefficient of damping element of VD γ_Ud,VD is equal to 1, indicating there is no deformation enhancement effect of VD. A large damping ratio is required in VD system in order to achieve the same displacement mitigation ratio γ_U as CIVD. That means the VD will produce large damping force even under small displacement the eave, which will have a great impact on historical buildings. Due to a small optimal damping ratio, CIVD have a smaller damping force under small displacement the eave, but can still control the eave through better tuning capacity.

Figure 22 shows the contour plot of γ_U and γ_Ud of eave with optimal μ_d of CIVD, it can be seen that under the same damping ratio, the deformation of damping element of CIVD is about 2 times of VD, which can obtain better energy dissipation effect and reduce the energy transmitted to the historical buildings’ underground motions. Meanwhile, about five times the damping coefficient of CIVD are required for VD to achieve the same vibration mitigation effect as CIVD. Thus, compared with VD, the damping force of CIVD transmitted to the historical buildings will be much lower.

5. Conclusions

In this paper, the motion equation of grotto–eave systems with cable inerter viscous damper (CIVD) is established. Then, the stochastic analysis and demand-based parameters optimization of CIVD are carried out. Finally, the vibration control performance of CIVD is analyzed under different seismic inputs. The main conclusions are as follows:

• The CIVD is easy to install, and can quickly improve the seismic performance of the structure. Therefore, it can be used in immovable cultural relics such as eaves and grottoes where the damper installation conditions are harsh.
• In the parameters optimization design of CIVD, damping ratio should be reduced as much as possible under the condition of satisfying the target vibration mitigation ratio based on performance demand. The optimal design of a CIVD in grotto–eave system should be a balance process between response of different types of grotto–eave system and the cost of CIVD. It can improve the feasibility of the application of CIVD in cultural relics protection projects.
• The proposed demand-based optimal method can minimize the cost by enhancing damping element deformation in a small damping ratio, while ensuring that the value of displacement index of grotto–eave system can be reached. Moreover, the inerter–mass ratio of the CIVD can be also determined by displacement index of grotto–eave system when using fixed-point method.
• Considering the lower damping ratio and inerter–mass ratio of CIVD, applications of the CIVD designed by demand-based optimal method can be extended and its installation made more flexible in specific structures like eaves. The corresponding verification experiment should be conducted in the near future.