A Study on the Amplification Effect and Optimum Control of the Intermediate Column–Lever Negative Stiffness Viscous Damper

Abstract

Currently, the energy dissipation efficiency of intermediate column dampers is extremely low, and traditional lever amplification damping systems occupy a large space in buildings. Aiming at solving these problems, this paper puts forward a new intermediate column–lever negative stiffness viscous damper (CLNVD), which has the characteristics of small impact on building space and significant amplification of the damper displacement. The CLNVD consists of the following four parts: the viscous damper, the negative stiffness device, the lever, and the intermediate column. This paper introduces the displacement amplification coefficient (f_d) to assess the CLNVD’s displacement amplification effect and introduces the energy dissipation coefficient (f_E) to assess the CLNVD’s energy dissipation effect. The expressions for f_d and f_E are derived according to the geometric magnification coefficient and effective displacement factor. Moreover, the impacts of multiple factors including the CLNVD’s position, the lever’s amplification coefficient, the bending line stiffness of beam, the negative stiffness, the damping coefficient, the damping index, and the inter-story displacement on the CLNVD’s f_d and f_E are elaborated. The analysis results reveal the following: when the CLNVD is located in the middle of the span, the f_d and f_E of the CLNVD will be maximized, and f_E will increase first and then decrease as the beam’s bending line stiffness increases. Meanwhile, the amplification capability of the CLNVD increases as the lever’s amplification coefficient χ rises. When the negative stiffness does not exist, there exists an optimum lever’s amplification coefficient χ that maximizes f_E. When the combination of damping coefficient c and index α satisfies a specific relationship, f_E of the CLNVD reaches its largest value. When the negative stiffness and the loss stiffness of VD are within the region proposed in this paper, the CLNVD will achieve a higher f_d and avoid providing significant negative stiffness. Subsequently, this paper proposes an optimization design method of the CLNVD. Finally, the amplification effect of CLNVD as well as the effectiveness of its optimization design method are verified through examples. In the case study, the CLNVD offers a larger damping ratio under the circumstance of fortification earthquakes. Under fortification and rare earthquakes, the inter-story displacement of Scheme 1 has been decreased by half roughly. According to the above-mentioned results, the CLNVD provides a brand-new approach for designers in the seismic design of buildings. Furthermore, this paper will provide beneficial reference for the damping design of other amplification devices equipped with negative stiffness dampers.

1. Introduction

Ensuring that buildings in earthquake areas have good seismic performance has become one of the key tasks for building designers. Currently, the commonly used technical methods include seismic isolation strategy [1,2,3], energy dissipation and vibration reduction passive control strategy [4,5,6], semi-active control strategy [7], and active control strategy [8]. In comparison with seismic isolation strategy, the passive energy dissipation control strategy possesses the characteristics of having less impact on the construction process and construction time. Moreover, the related energy dissipation devices can be installed after the completion of the main structure. Meanwhile, in contrast to the other two kinds of control strategies, the passive control strategy has the following characteristics: it does not need external energy; relevant technical means are simple and reliable; the subsequent maintenance is extremely simple; and the overall cost is extremely low. In the passive control strategy, in comparison with metal dampers [9] and friction dampers [10,11], viscous dampers are favored by engineers due to their lack of stiffness and strong energy dissipation capabilities. However, when the structure’s deformation is extremely small, or the damper’s linking component deformation is extremely large, the damper’s deformation will become very small, which cannot efficiently reduce the structure’s seismic response [12,13,14]. Therefore, it is essential to put forward a damper which not only has a large displacement amplification capability but also has a low stiffness requirement for the brace.

The intermediate column damper [15,16] takes the upper and lower intermediate columns as connecting components for dampers and then installs them within the span of the beam. Therefore, its impact on building functionality is minimal. Lan et al. [17,18] studied the impact of the position of the column on the damper’s displacement and the structure’s vibration response as per vibration table experiments. The experimental results were as follows: when the damper system lies at the mid of the beam span, the damping ratio will be the highest, and the structure’s inter-story deformation will be the smallest. Other related researchers studied the optimal installation position of dampers in structures. For instance, Sonmez [19,20] adopted the artificial bee colony algorithm; Lan [21] applied response surface methodology, and Bishop et al. [22,23,24] employed the genetic algorithm to seek the dampers’ optimal installation location. However, in contrast to the amplification device, the improvement effect on those dampers’ displacement produced by the above-mentioned optimization method is poor. Therefore, it is essential to improve the traditional intermediate column damping system to enhance its displacement amplification effect.

In order to enhance the damper’s displacement, scholars put forward a variety of devices. Ribakov et al. [25,26,27] utilized the slant support lever damping system to control the structure’s dynamic response and developed a structural optimization design program to realize the structural response close to the optimal control theory. Numerical simulation showed that the seismic response of structures that are equipped with lever damping systems was significantly improved. Yang [28] proposed a symmetrical lever amplification-style oil damper and tested the vibration-reduction effect of the amplification-style oil damper through finite element simulation. He et al. [29,30] designed a lever amplification-style viscous damping wall and conducted a vibration table experiment on a steel frame installed with enlarged viscous damping walls. The results indicated that this amplification device could significantly improve the seismic safety of the structure. Meanwhile, Yang [31] proposed a kind of joint damper with an amplification effect and studied the performance of this enlarged soft steel joint damper through numerical simulation. In addition to the above-mentioned lever amplification devices, toggle-style amplification devices are also widely applied in engineering. Constantinou et al. [32,33] proposed a toggle damping system utilizing crank connecting rod technology. The theoretical analysis results indicated that the toggle significantly increased the damper’s displacement. Moreover, the stronger the amplification capability of the toggle device is, the more sensitive it is to changes in the length of the connecting rod. Subsequently, Taylor [34,35,36] and Sigaher [37] proposed the improved toggle brace devices and scissor-style devices, respectively. On this basis, in order to reduce the impact on building functionality, Polat [38] proposed an open damping amplification device, and Lan [39] proposed a local toggle brace damper. Meanwhile, Kang et al. [40,41] proposed a seesaw amplification mechanism composed with a bracket, a seesaw, and a viscous damper. Through a dynamic analysis of the 6-floor framework attached to the mechanism, it was found that the seesaw mechanism had strong energy dissipation capacity. Meanwhile, it was pointed out that the focus of the mechanism design was to balance the damping coefficient and brace stiffness. However, the above-mentioned amplification devices either have an enormous impact on building function or are susceptible to small changes in the connecting rod’s length of the toggle or need a high stiffness of the brace component. As a result, these problems increase the difficulties of their extensive applications.

To significantly enhance the damper’s energy dissipation, the brace components or amplification devices connected to the damper themselves need to have a high stiffness. However, Javanbakh [42] found that when the support is flexible, the negative stiffness device will aggravate the damper’s displacement. Currently, various forms of negative stiffness devices have been proposed successively. Nagarajaiah et al. [43,44,45] proposed realizing negative stiffness through preloading springs or gap springs. Meanwhile, Chen et al. [46,47] applied a preloading spring to the inclined surface to realize negative stiffness. Zhu [48] launched a negative stiffness damper on the basis of turbine damping and permanent magnets. Sun [49] put forward a double triangular damper and applied it to replace traditional lead rubber isolation bearings in the foundation isolation system. Numerical simulations show that the damping system has a significant vibration isolation effect. Zhou [50] analyzed the multimodal seismic control effect of the negative stiffness damping system on cables, and the results indicated that a small negative stiffness damper equipped with flexible braces can achieve high damping effects. Wang [51] analyzed the negative stiffness damper’s impact on the dynamic responses of the structures under pulse excitation, random excitation, and earthquake excitation. The results revealed the following: adding the negative stiffness device to the flexible brace damper significantly increased the damper’s energy dissipation while effectively reducing the structure’s displacement and acceleration responses. The energy dissipation capacity of the damper was improved by 5 times to 16 times, even if the ratio of the damper’s damping to the structure’s damping was lowered to 2.8%.The above studies indicate the following: negative stiffness devices are capable of improving the performance of dampers significantly while decreasing the stiffness demands on the damper’s connecting components. However, there is currently limited research on the combination of amplification devices and negative stiffness damping systems, which will be a new and effective passive control technology.

Aimed at improving the damper’s displacement, decreasing the stiffness requirements on the damper connecting components, and minimizing the obstruction to building space, this paper introduces the intermediate column–lever negative stiffness viscous damper (CLNVD). The intermediate column–lever negative stiffness viscous damper (CLNVD) consists of an intermediate column, a lever, a negative stiffness device, and a viscous damper. In contrast to the displacement amplification effect of the toggle device, which is highly sensitive to changes in the length of the toggle’s connecting rod, the amplification capability of the CLNVD is relatively stable. In comparison to the traditional lever amplification devices, the CLNVD has less impact on building space because it can be flexibly installed within the span of the beam. In contrast to the traditional intermediate column damper, the CLNVD has a greater displacement amplification effect. This makes the cost of the CLNVD become lower under the same control effect on the seismic response of the structure. Meanwhile, due to the negative stiffness device, the CLNVD further reduces its stiffness requirements for the intermediate column and lever and further increases the damper’s deformation and energy dissipation.

Meanwhile, this paper introduces the displacement amplification coefficient f_d and energy dissipation coefficient f_E to assess the function of the CLNVD. Specifically, in this paper, the geometric magnification coefficient is employed to describe impacts of the CLNVD’s position on the CLNVD’s amplification ability; the effective displacement coefficient is utilized to present weakening of equivalent stiffness on the CLNVD’s amplification ability, and the relationship between geometric magnification coefficient and effective displacement factor with f_d and f_E is established. Subsequently, the geometric magnification coefficient is analyzed and derived. An equivalent simplified model for CLNVD is established, and the effective displacement factor is derived. The variation in CLNVD’s f_d and f_E with different factors, such as the position of the CLNVD, lever’s amplification coefficient, the beam’s bending line stiffness, damping index, damping coefficient, and inter-story displacement, is expounded. Subsequently, a damping design method considering CLNVD optimization is proposed. Ultimately, CLNVD’s strong displacement amplification capability is verified through engineering examples. Meanwhile, the effectiveness of the optimum design method is verified through the comparison among the three schemes.

2. Composition of the CLNVD and Solution of Its Amplification Coefficient

2.1. Composition and Features of the CLNVD

Figure 1 illustrates the negative stiffness device’s composition, the CLNVD’s structure form, and the hysteresis curve. In the multi-story structure, CLNVD is installed between the upper beam and the lower beam, which is shown in Figure 1a. The intermediate column’s position can be selected as per the building space. Hence, CLNVD can be flexibly arranged in the span. The viscous damper (VD) and negative stiffness device (NSD) are linked in parallel to constitute an NVD.

When the inter-story deformation occurs, the beam drives the intermediate column to move, and the intermediate column moves the upper node of the lever arm through the connecting rod. The lever arm rotates through the node of the lever, causing significant deformation of the viscous damper and negative stiffness device. Due to the preloading effect of the internal spring, the deformation of NSD further increases, thereby increasing the deformation of VD. When the structure undergoes inter-story deformation, the node connecting the upper intermediate column and beam will rotate, causing the intermediate column to rotate. This means that the lower node of the upper intermediate column has undergone a rotational displacement, which is different from the structural inter-story deformation. When the intermediate column is located in the middle of the span, this rotational displacement is greater than the inter-story deformation of the structure. The lever arm causes significant deformation of NVD by rotating the node. When NSD deforms, the preload of the preload spring is released and acts on the piston rod of NSD through the convex surface, with the direction of this force being the same as the direction of the piston rod’s movement. Therefore, when the NSD moves, the preloading spring will increase the displacement of the NSD. The viscous damper undergoes significant deformation under the combined action of the lever, lower intermediate column, and NSD, thereby resulting in significant energy dissipation to significantly control the seismic response of the structure.

In Figure 1b, the convex surface shape of the piston rod is f(x); the stiffness and displacement of the preloaded spring are k_SP and △_SP, respectively; the number of springs is N. Because the friction generated by the pulley rolling on the convex surface is much smaller than the viscous force generated by the viscous element VD, and in order to simplify the analysis, the friction force is ignored in this paper. The NSD’s negative stiffness force f(x) is calculated through Equation (1). The specific derivation process can be found in reference [52].

$$F(x)=N{\displaystyle \frac{f^{\prime }(x)}{{[1+f^{\prime }(x)]}^2}}{k}_{SP}[{\Delta }_{SP}+f(x)-f(0)]$$

(1)

When NVD generates sinusoidal deformation, NVD and NSD’s hysteresis curves are presented in Figure 1c. This sinusoidal deformation has an amplitude of 100 mm and a circular frequency of 1 rad/s. The spring stiffness is 1.2 kN/mm, and the preload displacement is 60 mm; the shape of the convex surface is f(x) = 50cos(2π/600x) − 7cos(2π/300x); the VD’s damping coefficient c = 2 kN/(mm/s). In NSD, there are a total of 12 preload springs acting on the convex surface. The simplified curve in Figure 1c represents the hysteresis curve with a negative stiffness of −2 kN/mm and a damping coefficient of c = 2 kN/(mm/s). According to Figure 1c, the simplified curve is in good accordance with the theoretical curve. Therefore, NVD can be composed of linear negative stiffness elements and viscous elements in parallel.

2.2. The CLNVD’s Displacement Amplification Coefficient and Energy Dissipation Coefficient

In Figure 1a, the structural damping is Rayleigh damping. The mass proportional damping of the i-th floor is C_mi, and the stiffness proportional damping is C_ki. The constitutive relationship of NVD formed by VD and NSD in parallel is $F=c\mathrm{sgn}({\dot{u}}_d){\left|{\dot{u}}_d\right|}^{\alpha }+k_N{u}_d$, where c, α, and k_N are damping coefficients, damping index, and negative stiffness, respectively.$u_d$ and ${\dot{u}}_d$ are VD’s deformation and velocity, respectively. Under earthquake action, the maximum displacement and velocity between the i-th floor of the structure are ${{\displaystyle U}}_i$ and ${{\displaystyle \dot{U}}}_i$, respectively; VD’s maximum displacement on the i-th floor is $u_{do}$. When ${{\displaystyle U}}_i$ is kept constant and c = 0 and k_N = 0, the VD’s displacement is $u_{ao}$. We can define it as follows:

$${\gamma }_i=u_{ao}/U_i, {\eta }_{di}=u_{do}/u_{ao}$$

(2)

${\gamma }_i$ is referred to as the geometric amplification coefficient in this paper. ${\eta }_{di}$ represents the effect of the equivalent stiffness of the intermediate column–lever on the VD’s displacement. In this paper, it is called effective displacement factor.

The displacement amplification coefficient $f_{di}$ of CLNVD is defined as the ratio of VD displacement $u_{do}$ on the i-th story to the inter-story displacement ${{\displaystyle U}}_i$, which is illustrated in Equation (3). The energy dissipation coefficient $f_{Ei}$ of CLNVD is defined as the ratio of the maximum energy dissipation of VD on the i-th floor in a single cycle to the largest energy dissipation of the structural stiffness proportional damping in a single cycle, which is shown in Equation (4).

$$f_{di}={\displaystyle \frac{u_{do}}{U_i}}={\gamma }_i{\eta }_{di}$$

(3)

$$f_{Ei}={\displaystyle \frac{\lambda c{u}_{do}{\dot{u}}_{do}^{\alpha }}{\mathsf{\pi }{C}_{ki}{U}_i{\dot{U}}_i}}={\displaystyle \frac{\lambda c{f}_{di}^{1+\alpha }{\dot{U}}_i^{\alpha -1}}{\mathsf{\pi }{C}_{ki}}}$$

(4)

where the value of $\lambda$ is referenced in the Chinese Code [50].

The amplification ability of CLNVD is related to coefficient ${\gamma }_i$ and coefficient ${\eta }_{di}$. $f_{di}$ describes the displacement amplification capability of the CLNVD, and $f_{Ei}$ describes the energy consumption capacity of the CLNVD. Therefore, $f_{di}$ and $f_{Ei}$ can be applied to describe the comprehensive performance of CLNVD.

2.2.1. The CLNVD’s Geometric Amplification Coefficient

The force analysis of the i-th story in the structure shown in Figure 1a is conducted (See Figure 2a). F is the horizontal force of the i-th story; P₁–P₃ is the column axial force of the i + 1st story; M₁–M₃ is the bending moment at the column bottom on the i + 1st story; △₁ − △₃ is the cumulative axial deformation of the column from the first story to the i − 1th story; and θ is the corner angle at the column bottom on the i-th story. ${{\displaystyle U}}_i$ is the inter-story displacement on the i-th story. ${{\displaystyle u}}_{ao}$ is the displacement of VD on the i-th story when c = 0 and k_N = 0. The isolation body illustrated in Figure 2a has the following relationship:

$$U_i=U_{Fi}+U_{Mi}+U_{\Delta i}+U_{\theta i}$$

(5)

$$u_{ao}=u_{Fao}+u_{Mao}+u_{\theta ao}+u_{\Delta ao}$$

(6)

In Equations (5) and (6), $U_{Fi}$, $U_{Mi}$, $U_{\Delta i}$, and $U_{\theta i}$ are the inter-story displacements under the action of F, M, △, and θ, respectively. $u_{Fao}$, $u_{Mao}$, $u_{\Delta ao}$, and $u_{\theta ao}$ are displacements of VD under the action of F, M, △, and θ, which is shown from Figure 2b–e. They are defined as follows:

$$\rho =i_b/i_c,\hspace{1em}\beta =a /l,\hspace{1em}\chi =l_2/l_{ 1}$$

(7)

$${\gamma }_F=u_{Fao}/U_{Fi} ,\hspace{1em}{\gamma }_M=u_{Mao}/U_{Mi} ,\hspace{1em}{\gamma }_{\theta }=u_{\theta ao}/U_{\theta i} ,\hspace{1em}{\gamma }_{\Delta }=u_{\Delta ao}/U_{\Delta i}$$

(8)

In Equations (7) and (8), as shown in Figure 1, a is the distance between the intermediate column and the left column. l is the span length, and h is the floor height. l₁ and l₂ are the lengths of the lever arm at the upper and lower ends; i_b and i_c are the beam and column’s bending line stiffness, respectively. β represents the intermediate column’s position, ρ represents the stiffness ratio of the beam to column, and χ represents the lever’s amplification coefficient.

The coefficients in Equation (8) are analyzed. ${\gamma }_F$, ${\gamma }_M$, ${\gamma }_{\theta }$, and ${\gamma }_{\Delta }$ is expressed as follows:

$$\left\{\begin{array}{l}{\gamma }_F=\chi -3\chi \left[6({\beta }^2-\beta )+1\right]/\left[2(3\rho +2)\right]\hfill \\ {\gamma }_M={\gamma }_{\theta }=6\chi (\beta -{\beta }^2)\hfill \\ {\gamma }_{\Delta }=\chi (6+2/\rho )({\beta }^2-\beta )\hfill \end{array}\right.$$

(9)

Equation (9) can be found in reference [15]. And the derivation of ${\gamma }_F$ can be found in Appendix A. By combining Equations (2), (5), (6), and (8), we obtain the geometric amplification coefficient as follows:

$${\gamma }_i={\displaystyle \frac{u_{ao}}{U_i}}={\displaystyle \frac{{\gamma }_F{U}_{Fi}+{\gamma }_M{U}_{Mi}+{\gamma }_{\theta }{U}_{\theta i}+{\gamma }_{\Delta }{U}_{\Delta i}}{U_{Fi}+U_{Mi}+U_{\theta i}+U_{\Delta i}}}$$

(10)

According to Equation (10), ${\gamma }_i$ is related to ${\gamma }_F$,${\gamma }_M$,${\gamma }_{\theta }$, and ${\gamma }_{\Delta }$. Figure 3 reveals the variation pattern of ${\gamma }_F$, ${\gamma }_M$, ${\gamma }_{\theta }$, and ${\gamma }_{\Delta }$ with ρ and β when χ = 4. As β increases, ${\gamma }_F$ increases at first, and then decreases, reaching its maximum value when β = 0.5. There are two commonalities in ${\gamma }_F$ under different ρ. ${\beta }_{\mathrm{A}}=1/2-1/\sqrt{12}$, ${\beta }_{\mathrm{B}}=1/2+1/\sqrt{12}$. When ${\beta }_{\mathrm{A}}<\beta <{\beta }_{\mathrm{B}}$, ${\gamma }_F$ decreases as ρ increases. As is shown in Figure 3b, we can see that just like the ${\gamma }_F$, ${\gamma }_M$ also reaches its maximum value when β = 0.5. From Figure 3c, it can be seen that when $\beta \in (0,1)$, ${\gamma }_{\Delta }$ decreases as ρ decreases, and all values are negative. Therefore, the cumulative axial deformation of the column reduces the geometric amplification coefficient ${\gamma }_i$. Combining Figure 3a–c, we can see that when β = 0.5, ${\gamma }_F$ and ${\gamma }_M$ obtain the positive maximum value, and ${\gamma }_{\Delta }$ obtains the negative maximum value. For frame structures, due to the fact that the proportion of $U_{\Delta i}$ is very small in ${{\displaystyle U}}_i$, it is recommended that CLNVD lies at the middle of the span to obtain a larger geometric amplification coefficient.

2.2.2. The CLNVD’s Effective Displacement Factor

The derivation of the effective displacement factor ${\eta }_{di}$ requires transforming the CLNVD’s simplified mechanical model into the Kelvin model, which is displayed in Figure 4a. $k$ is the equivalent stiffness of the lower intermediate column, the upper intermediate column, and the lever in series, which is calculated by Equation (11). In Equation (11), $k_{le}$ is calculated by Equation (12), and $k_{mc-up}$ is calculated by Equation (13). Equation (11) can refer to reference [18]. $k_{le}$ is the stiffness of the lever amplification device, which is illustrated in Figure 4b. $k_{mc-up}$ is the stiffness of the upper middle column after it is connected with the lever (See Figure 4c). $k_{mc-low}$ is the stiffness of the lower middle column after it is connected with the lever (See Figure 4d). $i_{mc}$ and $i_{vmc}$ are the intermediate column’s flexural line stiffness and shear line stiffness, respectively. $i_{le}$ is the lever arm’s bending line stiffness; $l_b$ is the brace length; $A_b$ is the brace’s cross-sectional area; and $\theta$ is the brace’s horizontal included angle.

$$k={\displaystyle \frac{k_{le}{k}_{mc-up}}{2k_{le}+k_{mc-up}}}$$

(11)

$$k_{le}={\displaystyle \frac{1}{\frac{{(l_1+l_2)}^2{\chi }^2}{3i_{le}{(\chi +1)}^2}+\frac{{(\chi +1)}^2{l}_b}{2E{A}_b{\cos }^2\theta }}}$$

(12)

$$k_{mc-up}={\displaystyle \frac{1}{\frac{{\chi }^2[\beta f_1+(1-\beta )f_2]h^2}{2f_3{i}_b}+\frac{{\chi }^2(12i_{mc}+i_{vmc}{h}^2)}{12i_{mc}{i}_{vmc}}}}$$

(13)

In Equation (5), $f_1=(1+{\displaystyle \frac{\beta }{\rho }})[\beta (2-3\beta )-2(1+{\displaystyle \frac{1}{\rho }})(1-\beta )(3\beta -1)]$, $f_2=[1+(1-\beta )/\rho ][(1-\beta )(3\beta -1)-2\beta (1+1/\rho )(2-3\beta )]$, $f_3=2-8{(1+1/\rho )}^2$.

In order to achieve the transformation between the mechanical model of CLNVD and the Kelvin model, equal force, equal deformation, and equal energy dissipation are utilized here. The Kelvin model’s damping coefficient is $c_a^{\prime }$, and the spring stiffness is $k_a^{\prime }$, which is shown in Figure 4a. Assuming the sinusoidal deformation generated by CLNVD, according to the definition of $u_{ao}$, we obtain $u_{ao}={\gamma }_i{U}_i$. According to equal force and equal deformation, $u_d(t)$, ${\dot{u}}_d(t)$, and $F(t)$ of VD are revealed as follows:

$$\left\{\begin{array}{l}{u}_d(t)=u_{ao}\sin (\omega t)-\left[k_a^{\prime }{u}_{ao}\sin (\omega t)+c_a^{\prime }\omega u_{ao}\cos (\omega t)\right]/({\gamma }_i^{2}k) \hfill \\ {\dot{u}}_d(t)=\omega u_{ao}\cos (\omega t)-\left[k_a^{\prime }\omega u_{ao}\cos (\omega t)-c_a^{\prime }{\omega }^2{u}_{ao}\sin (\omega t)\right]/({\gamma }_i^{2}k)\hfill \\ F(t)=c{\dot{u}}_d(t)+k_N{u}_d(t)=\left[k_a^{\prime }{u}_{ao}\sin (\omega t)+c_a^{\prime }\omega u_{ao}\cos (\omega t)\right]/{\gamma }_i^2\hfill \end{array}\right.$$

(14)

Equation (14) is utilized to solve the stiffness $k_a^{\prime }$ and damping coefficients $c_a^{\prime }$ in the Kelvin model. Details are as follows:

$$\left\{\begin{array}{l}{k}_a^{\prime }={\gamma }_i^{2}k\left[{(k_N/k)}^2+k_N/k+{(c\omega /k)}^2\right]/\left[{(1+k_N/k)}^2+{(c\omega /k)}^2\right]\hfill \\ c_a^{\prime }={\gamma }_i^{2}c/\left[{(1+k_N/k)}^2+{(c\omega /k)}^2\right]\hfill \end{array}\right.$$

(15)

The loss stiffness of VD is defined as $k_d^{″}=c\omega$. ${\eta }_d$ is solved based on equal energy consumption. Details are as follows:

$${\eta }_d=u_{do}/u_{ao}=1/{\left[{(1+k_N/k)}^2+{(k_d^{″}/k)}^2\right]}^{0.5}$$

(16)

When damping index $\alpha =0$, $k_d^{″}=c/u_{do}$, $k_a^{\prime }={\gamma }_i^{2}k(k_N/k+k_d^{″}/k)/(1+k_N/k+k_d^{″}/k)$, $c_a^{\prime }=f_d c$, and ${\eta }_d=1/(1+k_N/k+k_d^{″}/k)$. According to the reference [53], when $0<\alpha \le 1$, $k_d^{″}$ and ${\eta }_d$ are approximately expressed by interpolation as follows:

$$k_d^{″}=c{\omega }^{\alpha }/{u_{do}}^{1-\alpha } , 0<\alpha \le 1$$

(17)

$${\eta }_d={1/\left[{(1+k_N/k)}^{1+\alpha }+{(k_d^{″}/k)}^{1+\alpha }\right]}^{1-0.5\alpha } , 0<\alpha \le 1$$

(18)

When $0<\alpha \le 1$, $k_a^{\prime }$ is approximately expressed by interpolation. $c_a^{\prime }$ is Solved through Equation (18). $k_a^{\prime }$ and $c_a^{\prime }$ are as follows:

$$\left\{\begin{array}{ll}{k}_a^{\prime }={\gamma }_i^{2}k\left[{(1+k_N/k)}^{\alpha }{k}_N/k+{(k_d^{″}/k)}^{1+\alpha }\right]/\left[{(1+k_N/k)}^{1+\alpha }+{(k_d^{″}/k)}^{1+\alpha }\right]\hfill & , 0<\alpha \le 1\hfill \\ c_a^{\prime }={({\gamma }_i{\eta }_d)}^{1+\alpha }c\hfill & , 0<\alpha \le 1\hfill \end{array}\right.$$

(19)

Figure 5 reveals the variation in the ${\eta }_d$ and the $k_a^{\prime }/({\gamma }_i^{2}k)$ with $k_d^{″}/k$. As $k_d^{″}/k$ increases, ${\eta }_d$ decreases while $k_a^{\prime }/({\gamma }_i^{2}k)$ increases. In comparison with the traditional viscous dampers, NVD’s ${\eta }_d$ is improved due to the negative stiffness effect, while the degree of improvement weakens as $k_d^{″}/k$ increases. Meanwhile, when $k_d^{″}/k$ is smaller, $k_a^{\prime }/({\gamma }_i^{2}k)<0$. Therefore, the CLNVD provides a negative stiffness to the structure. On the one hand, the negative stiffness property of the CLNVD reduces the shear force demand of the structure; on the other hand, it raises the displacement demand. However, higher ${\eta }_d$ of NVD will suppress the displacement demand of the structure.

3. Parameter Analysis and Optimization Design

The coefficients $f_d$ and $f_E$ of CLNVD are jointly affected by the intermediate column’s position $\beta$, the lever’s amplification factor $\chi$, the bending line stiffness $i_b$ of the beam, negative stiffness $k_N$, and damping coefficient $c$. For the convenience of parameter analysis, the single-story structure shown in Figure 6 is taken as the analysis object.

For single-story structures, ${\gamma }_M={\gamma }_{\theta }={\gamma }_{\Delta }=0$. Combining Equations (2)–(4), (9), (10), and (18), we obtain the following:

$$f_d={\displaystyle \frac{\chi -3\chi \left[6({\beta }^2-\beta )+1\right]/\left[2(3\rho +2)\right]}{{\left[{(1+k_N/k)}^{1+\alpha }+{(k_d^{″}/k)}^{1+\alpha }\right]}^{1-0.5\alpha }}}$$

(20)

$$f_E={\displaystyle \frac{\lambda c{f}_d^{1+\alpha }{(\omega U_1)}^{\alpha -1}}{\mathsf{\pi }{C}_{k1}}}$$

(21)

where $k$ is calculated by Equation (11), and $k_d^{″}$ is calculated by Equation (17); $U_1$ is the inter-story displacement, and $C_{k1}$ is the structural stiffness proportional damping coefficient. For $\omega$, we take the first modal frequency of the structure.

When the damping index $\alpha <1$, Equation (17) cannot be directly employed since $u_{do}$ is unknown. However, the following iterative method is applied. This iterative method is as follows:

(1)

The damper displacement $u_{do}$ is assumed to be equal to the inter-story displacement $U_1$.

(2)

$u_{do}$ is substituted into Equation (17) for calculation, and Equations (20) and (21) are utilized to calculate $f_d$ and $f_E$.

(3)

The error $\epsilon =u_{do}/(f_d{U}_1)-1$ is calculated. When $\epsilon >1\%$, $u_{do}$ is made to be equal to $f_d{U}_1$, and steps from 2 to 3 are repeated. When $\epsilon <1\%$, the iteration ends.

For the convenience of discussion, an analysis is conducted on the single-story structure illustrated in

Table 1. Parameters of the improved TNVD components.

$\mathbf{Column} \mathbf{Section}$	$\mathbf{Beam} \mathbf{Section}$	$\begin{array}{l}\mathbf{Intermediate} \mathbf{Column} \\ \mathbf{Section}\end{array}$	$\mathit{\beta }$	$\mathit{\chi }$	$\mathit{c}$	${\mathit{k}}_{\mathit{N}}$
$(\mathbf{mm}\times \mathbf{mm})$	$(\mathbf{mm}\times \mathbf{mm})$	$(\mathbf{mm}\times \mathbf{mm})$			${(\mathbf{kN}\cdot (\mathbf{s}/\mathbf{mm})}^{0.5})$	$(\mathbf{kN}/\mathbf{mm})$
600 × 600	400 × 800	1600 × 250	0.5	4	5	−2

. The material of the lever amplification device is Q235, and the concrete material grade of the structure is C35. The cross-section of the lever box is 230 × 140 × 25 × 25 (unit: mm), with the length of 1000 mm; the box section of slant support is 200 × 200 × 25 × 25 (unit: mm). The structure maintains elasticity, and the modal frequency is $\omega =8.1 \mathrm{rad}/\mathrm{s}$; the structure’s stiffness proportional damping is $C_{k1}=1.56\mathrm{kN}/(\mathrm{mm}/\mathrm{s})$, and the inter-story displacement under the circumstance of fortification earthquakes is $U_1=h/400$. The method adopted for parameter analysis in this section is to keep other parameters unchanged and observe the impact of changing a certain parameter on the efficiency of CLNVD.

3.1. Parameter Analysis

3.1.1. Impact of the Intermediate Column’s Position and Its Bending Line Stiffness

The intermediate column’s position $\beta$ affects CLNVD’s $f_d$ and $f_E$. Figure 7 reveals the curves of $\gamma$, ${\eta }_d$, $f_d$, and $f_E$ with $\beta$. As is described in Figure 7a, when the intermediate column moves from the ends of the span to the middle of the span, $\gamma$ increases, while ${\eta }_d$ increases at first and decreases subsequently. However, the increase in $\gamma$ is even greater. Therefore, when $\beta =0.5$, $f_d$ and $f_E$ reach its maximum value. SAP2000 (Version 23.1.0) is used for numerical simulation, and seven seismic waves from Section 4.2 are selected. The analysis results of $f_d$ in this section are taken as the average of the seven seismic wave calculation results. The results indicate that the theoretical calculation results of $f_d$ are highly consistent with the numerical simulation results. In Figure 7b, $i_{mc1}$ refers to the intermediate column’s bending stiffness with the cross-section of 1600 mm × 250 mm. As is displayed in Figure 7b, $f_d$ and $f_E$ rapidly increase at first, and then slowly decrease with the increase in $i_{mc}$. A larger cross-section of the intermediate column is not conducive for CLNVD to obtain the peak value of $f_d$ and $f_E$. In order to ensure that CLNVD has good energy consumption capacity, the CLVD is required to lie at the middle span under the condition that the building space allows this.

3.1.2. Impact of Beam’s Bending Line Stiffness

This section expounds the law of $f_E$ and $f_d$ as the beam’s bending line stiffness changes. In Figure 8, $i_{b1}$ refers to the beam’s bending stiffness with the cross-section of 400 mm × 800 mm. When the intermediate column’s position $\beta$ changes, $f_d$ and $f_E$ do not change in the same way with $i_b$. As is shown in Figure 8a, when $\beta =0.1$, $\gamma$ and ${\eta }_d$ increase with the increase in $i_b$. Hence, $f_d$ and $f_E$ increase as $i_b$ increases. As is revealed in Figure 8b, when $\beta =0.5$, with the increase in $i_b$, $\gamma$ slowly decreases, and ${\eta }_d$ first increases and then decreases. Therefore, as $i_b$ increases, $f_d$ and $f_E$ first rapidly increase and then decrease. For different $\beta$, the impact law of $i_b$ on $f_E$ is shown in Figure 8c. From Figure 8a–c, it is concluded that CLNVD located at the mid span can achieve higher $f_E$ with a smaller beam cross-section.

3.1.3. Impact of Lever’s Bending Line Stiffness and Leverage Amplification Coefficient

This section expounds the variation in $f_d$ and $f_E$ with the lever’s bending line stiffness $i_{le}$ as well as the lever’s amplification coefficient $\chi$. In Figure 9, $i_{le1}$ refers to the bending line stiffness with the box section of 230 × 140 × 25 × 25 (unit: mm). The impact of the stiffness and amplification coefficient is not the same. As is shown in Figure 9a, like $i_{mc}$, $f_d$ and $f_E$ rapidly increase at first and then decrease slowly with the increase in $i_{le}$. As is shown in Figure 9b, when $k_N=0$, there exists an optimal leverage amplification coefficient $\chi$ which enables $f_E$ to reach the maximum. When $k_N\ne 0$, $f_d$ and $f_E$ increase with the increase in the lever’s amplification coefficient $\chi$. However, this does not mean that larger options can be selected without limitations. The larger $\chi$ causes the lever and beam to bear a greater force, which requires a larger cross-section to maintain elasticity. SAP2000 (Version 23.1.0) is used for numerical simulation. The results indicate that the theoretical calculation results and numerical simulation results of $f_d$ are very consistent under different χ conditions.

3.1.4. Impact of the Damping Coefficient and Damping Exponent

This section expounds the variation law of $f_d$ and $f_E$ with the damping index $\alpha$ as well as the damping coefficient $c$. From Figure 10a,b, we obtained a conclusion that ${\eta }_d$ gradually decreases when increasing the damping coefficient $c$ or index $\alpha$. Therefore, $f_d$ gradually decreases, while $f_E$ first increases and then reduces. Therefore, it is necessary to select appropriate index $\alpha$ and damping coefficient $c$ to enable $f_E$ to reach the largest value. Under different exponent $\alpha$, the variation law of $f_E$ with the change in the damping coefficient $c$ is shown in Figure 10c. $f_{PE}$ in Figure 10c refers to the peak value of the energy dissipation coefficient $f_E$. Figure 10d presents the curve of $k_d^{″}/k$ and ${\eta }_d$ corresponding to $f_{PE}$ in Figure 1c with the index $\alpha$. According to Figure 10d, it is indicated that $f_E$ reaches the maximum value when the $c$ and $\alpha$ make $k_d^{″}/k=1+k_N/k$ or ${\eta }_d=1/{[2{(1+k_N/k)}^{1+\alpha }]}^{1-0.5\alpha }$.

3.1.5. Impact of NSD’s Negative Stiffness

This section analyzes the variation in $f_d$ and $f_E$ with the changes in NSD’s $k_N$. As is shown in Figure 11a, when $k_N$ decreases, ${\eta }_d$ gradually increases. Therefore, $f_d$ and $f_E$ gradually increase. From Figure 11b, it is concluded that when $k_N$ decreases, $k_a^{\prime }$ decreases at first and then increases. Moreover, a smaller $c$ will give CLNVD larger negative stiffness. Therefore, the size of $k_N$ should be controlled to make CLNVD achieve a higher $f_d$ while avoiding an influential decrease in the structure’s overall stiffness. SAP2000 (Version 23.1.0) is used for numerical simulation. The results indicate that the theoretical calculation results and numerical simulation results of $f_d$ are very consistent under different $k_N$ conditions. The CLNVD can be guaranteed to achieve a high $f_d$ by controlling the size association between $k_N/k$ and $k_d^{″}/k$. The reference [54] requires that the ratio of the viscous damper’s loss stiffness to the brace stiffness is less than 1/3. When $f_d=1/{(1+1/3^{1+\alpha })}^{1-0.5\alpha }\gamma$, $f_d=\gamma$, and $f_d=2\gamma$, the size relationship between $k_N/k$ and $k_d^{″}/k$ can be calculated from Equation (22) as follows:

$$\left\{\begin{array}{ll}{k}_N/k={\left[1+1/3^{1+\alpha }-{(k_d^{″}/k)}^{1+\alpha }\right]}^{1/(1+\alpha )}-1\hfill & ,f_d=1/{(1+1/3^{1+\alpha })}^{1-0.5\alpha }\gamma \hfill \\ k_N/k={\left[1-{(k_d^{″}/k)}^{1+\alpha }\right]}^{1/(1+\alpha )}-1\hfill & ,f_d=\gamma \hfill \\ k_N/k={\left[{(1/2)}^{1/(1-0.5\alpha )}-{(k_d^{″}/k)}^{1+\alpha }\right]}^{1/(1+\alpha )}-1\hfill & ,f_d=2\gamma \hfill \end{array}\right.$$

(22)

The size of CLNVD’s $k_a^{\prime }$ can also be controlled by controlling the size association between $k_N/k$ and $k_d^{″}/k$. When $k_a^{\prime }={\gamma }^2{k}_N$ or $k_a^{\prime }=-{\gamma }^{2}k$, the size association between $k_N/k$ and $k_d^{″}/k$ are showed in Equation (23).

$$\left\{\begin{array}{ll}{(1+k_N/k)}^{\alpha }(k_N/k{)}^2/(1-k_N/k)={(k_d^{″}/k)}^{1+\alpha }\hfill & ,k_a^{\prime }={\gamma }^2{k}_N\hfill \\ {(1+k_N/k)}^{\alpha }(1+2k_N/k)=-2{(k_d^{″}/k)}^{1+\alpha }\hfill & ,k_a^{\prime }=-{\gamma }^{2}k\hfill \end{array}\right.$$

(23)

Combining Equation (22) as well as Figure 11c, we can see that when $k_N/k\le {\left[1+1/3^{1+\alpha }-{(k_d^{″}/k)}^{1+\alpha }\right]}^{1/(1+\alpha )}-1$, $f_d\ge 1/{(1+1/3^{1+\alpha })}^{1-0.5\alpha }\gamma$. Combining Equation (23) and Figure 11d, we can see that when ${(1+k_N/k)}^{\alpha }(k_N/k{)}^2/(1-k_N/k)\ge {(k_d^{″}/k)}^{1+\alpha }$, $k_a^{\prime }\ge {\gamma }^2{k}_N$. When $(1+k_N/k{)}^{\alpha }(1+2k_N/k)\le -2{(k_d^{″}/k)}^{1+\alpha }$, $k_a^{\prime }\le -{\gamma }^{2}k$. Through Equation (19), it can be obtained that when $k_N/k\ge -0.5$, $k_a^{\prime }\ge -{\gamma }^{2}k$. We can see from Section 3.1.4 that when $\left(k_d^{″}-k_N\right)/k\le 1$, $f_E$ increases as the damping coefficient $c$ increases. Thus, according to the above analysis, it is recommended that the target area in Figure 11d should be taken as the range of values for $k_N/k$ and $k_d^{″}/k$. Within this target area, we can see the following: $f_d\ge 1/{(1+1/3^{1+\alpha })}^{1-0.5\alpha }\gamma$; ${\gamma }^2{k}_N\le k_a^{\prime }$; $f_E$ rises when increasing the damping coefficient $c$.

3.1.6. Impact of Inter-Story Displacement

This section explores the variation law of $f_d$ and $f_E$ with the inter-story displacement $U_1$. $U_1$ affects $f_d$ and $f_E$ by means of changing the loss stiffness $k_d^{″}$ of VD. As is shown in Figure 12a, when increasing $U_1$, $f_d$ gradually increases, and $f_E$ increases first and later decreases. Increasing the lever’s amplification coefficient $\chi$ can increase $f_E$, which has been discussed in Section 3.1.3. As is shown in Figure 12b, when $U_1$ is larger, increasing $\chi$ has a significant improvement on $f_E$. However, when $U_1$ is small, increasing $\chi$ does not have a good effect on improving $f_E$. In Figure 12c, $i_{b1}$ refers to the beam’s bending line stiffness with the section of 400 mm × 800 mm. As is shown in Figure 12c, when $U_1$ is small, increasing $i_b$ has a significant improvement on $f_E$. However, when $U_1$ is larger, increasing $i_b$ will lead to a decrease in $f_E$. Meanwhile, increasing $i_b$ causes the peak energy-consumption efficiency $f_{PE}$ to decrease first and then increase. In Figure 12d, $c_1=5{(\mathrm{kN}/(\mathrm{mm}/\mathrm{s})}^{0.5})$. As is revealed in Figure 12d, when $U_1$ is larger, increasing the damping coefficient $c$ has a significant improvement on $f_E$. However, when $U_1$ is smaller, a higher damping coefficient $c$ leads to a decrease in $f_E$. Meanwhile, increasing the coefficient $c$ does not change the magnitude of the peak energy consumption coefficient $f_{PE}$. Moreover, it will cause $f_{PE}$ to move towards larger displacements.

3.2. CLNVD’s Optimum Design

3.2.1. Optimum Strategies for the CLNVD

In order to ensure that the CLNVD has a large coefficient $f_d$ and coefficient $f_E$ under the circumstance of frequent earthquakes (or fortification earthquakes), on the basis of the analysis in Section 3.1, the following steps are adopted to optimize the CLNVD:

(1)

The position of the intermediate column $\beta =0.5$ is selected, as well as the beam section and lever amplification factor $\chi$; the line stiffness ratio $\rho$ of beam to column is calculated. According to Equations (9) and (10), the geometric amplification coefficient is calculated. $1/{(1+1/3^{1+\alpha })}^{1-0.5\alpha }\gamma$ is judged to estimate whether it is higher in comparison with the pre-set target of $f_d$. If it is not higher, the following methods are applied: to increase $\chi$, or to decrease $\rho$.

(2)

CLNVD’s parameters (such as intermediate column section, lever section, negative stiffness, damping coefficient, and damping index) are selected so that the values of $k_d^{″}/k$ and $k_N/k$ are within the target area (See Figure 11d). The inter-story displacement target value and the iterative method in Section 3.1 can be used to calculate $k_d^{″}$; the ${\eta }_d$ is calculated through Equation (18).

(3)

$\gamma$ and ${\eta }_d$ are substituted into Equation (3) to calculate the displacement amplification coefficient $f_d$, and Equation (4) is utilized to calculate the energy dissipation coefficient $f_E$.

(4)

$f_E$ is judged if the predetermined goals are met. If not, the following methods are applied for optimization: to enhance the stiffness $i_b$ and damping coefficient $c$; to enhance the stiffness $i_b$ and the lever’s coefficient $\chi$; to increase the lever’s coefficient $\chi$ and reduce the damping coefficient $c$; or to reduce the NSD’s negative stiffness $k_N$. It is necessary to verify again if $k_N/k$ and $k_d^{″}/k$ are within the region proposed in Section 3.1.5, and Step (3) and Step (4) should be repeated. If $f_E$ cannot meet the set target when $k_d^{″}/k$ is on the upper limit of the target region in Figure 11d, we should increase the beam section and return to Step (3).

3.2.2. Optimum Design for Structures with the CLNVD

With the aim to evaluate the damping contribution of dampers to the overall structure and control the structure‘s deformation, the current vibration-reduction design methods mostly utilize additional damping ratios as well as the maximum inter-story displacement angle as control objectives. For additional CLNVD structures, this paper proposes a seismic design method with coefficient $f_d$, coefficient $f_E$, maximum inter-story displacement angle, and additional damping ratio as control objectives. In contrast to the structure without the CLNVD, the stiffness and damping properties of the multi-story structures with the CLNVD vary greatly, and such changes are difficult to predict before the actual calculation. Therefore, for the vibration-reduction design of the additional multi-story CLNVD structures, the current design method is to conduct trial calculations, while ensuring that CLNVD has a high energy consumption, until the proposed design goals are met. During the trial calculation process, it is necessary to utilize the optimization strategy in Section 3.2.1 to optimize CLNVD. Subsequently, the design is conducted according to the vibration-reduction design process in Section 3.2.2. Figure 13 illustrates the vibration-reduction design flowchart. The specific design steps are as follows:

(1)

The target values are set for $f_E$, $f_d$, maximum inter-story displacement angle, and additional damping ratio.

(2)

The stiffness ratio damping $C_k$ of the structure, the seismic response of structures without CLNVD, and the seismic response of structures with additional expected damping ratios are calculated. The floors with the CLNVD layout based on the seismic response of the structure without the CLNVD are selected, and the number of CLNVDs based on the building area (a ratio of building area to number of dampers between 100 and 200 is more reasonable) is selected. The number of CLNVDs for each floor based on the total number of CLNVDs and the layout floor is set. The parameters of CLNVD are set, and CLNVD is optimized according to Section 3.2.1.

(3)

The time history analysis is conducted under the circumstance of frequent or fortification earthquakes, and it is judged whether $f_d$, $f_E$, damping ratio, and displacement angle satisfy the goals. If not, we should increase the quantity of the CLNVDs or return to Step (2) for optimum strategies.

(4)

The time history analysis is conducted under the circumstance of rare earthquakes, and it is judged if the displacement angle satisfies the goals. If the goals are not satisfied, we should raise the quantity of the CLNVDs and return to Step (3).

When the viscous damper is linear, the loss stiffness of the damper does not change with the displacement of the damper, thus avoiding the usage of iterative programs. When the viscous damper is nonlinear, if no iteration is carried out, the approximate formula for the damper loss stiffness provided in the reference [53] can be used for relevant calculations, while the accuracy of the approximate formula is not ideal. The relevant parameters of CLNVD are selected in the target region in Figure 11d, which will enable CLNVD to have a large displacement amplification factor and avoid providing a large negative stiffness to the structure. Hence, for single-story structures and linear viscous dampers, relevant parameters can be directly selected in the target area of Figure 11d for ideal CLNVD design. For nonlinear viscous dampers and multi-story structures, the damping design of additional CLNVD structures needs to follow the design process in Section 3.2.2.

4. Engineering Example Analysis

4.1. Project Overview

With the aim to test CLNVD’s vibration-reduction efficiency and optimization strategy, this section takes a 9-floor concrete frame as an example to analyze. This concrete frame building has a height of 36 m and a self-weight of 9784 tons. The seismic fortification intensity is 8 degrees (0.2 g), and the site’s characteristic period is 0.45 s. The structural parameters are revealed in

Table 2. Parameters of the structure.

Story	Concrete Material	Column Section	Main Beam Section	Secondary Beam Section
		(mm × mm)	(mm × mm)	(mm × mm)
1	C40	700 × 700	300 × 600	200 × 400
2	C40	600 × 600	300 × 600	200 × 400
3–4	C40	550 × 550	300 × 600	200 × 400
5–9	C35	500 × 500	300 × 600	200 × 400

, and the dynamic properties are shown in

Table 3. Dynamic properties of the structure.

Mode	1	2	3	4	5	6
Period (s)	1.60	1.58	1.38	0.56	0.54	0.48
Direction	X (E-W)	Y (N-S)	Rotation	X (E-W)	Y (N-S)	Rotation
Effective Mass (%)	77	76	76	11	12	12

. The structure utilizes Rayleigh damping, with a stiffness ratio coefficient of 2.3 × 10⁻³ and a mass ratio coefficient of 0.35. The additional CLNVD structural optimization design objective is as follows: under the circumstance of fortification earthquakes (PGA is 0.2 g), the damping ratio provided by CLNVD is 18%, and the peak inter-story displacement angle is smaller than 1/400; under rare earthquakes (PGA is 0.4 g),the peak inter-story displacement angle is smaller than 1/200, and $f_d>3.5$ and $f_E>30$.

4.2. Scheme Design

In this section, three CLNVD schemes are designed. Based on the optimization design method, Scheme 1 is obtained. In comparison with Scheme 1, the position of the CLNVD in Scheme 2 is $\beta =0.2$. In contrast to Scheme 2, Scheme 3 does not have the NSD. The layout of Scheme 1 to Scheme 3 in the floors is described in Figure 14. The CLNVD’s parameters for each scheme are shown in

Table 4. Parameters of the CLNVD.

Scheme	Story	$\mathit{\beta }$	$\mathbf{Beam} \mathbf{Sec}\mathbf{tion}$	$\mathbf{Column} \mathbf{Sec}\mathbf{tion}$	$\mathit{\chi }$	$\mathit{c}$	${\mathit{k}}_{\mathit{N}}$
			$(\mathbf{mm}\times \mathbf{mm})$	$(\mathbf{mm}\times \mathbf{mm})$		${(\mathbf{kN}\cdot (\mathbf{s}/\mathbf{mm})}^{0.5})$	$(\mathbf{kN}/\mathbf{mm})$
Scheme 1	5–8	0.5	400 × 800	1700 × 350	4	8	−2
	2–4	0.5	400 × 800	1700 × 350	4	12	−2
Scheme 2	5–8	0.2	400 × 800	1700 × 350	4	8	−2
	2–4	0.2	400 × 800	1700 × 350	4	12	−2
Scheme 3	5–8	0.2	400 × 800	1700 × 350	4	8	0
	2–4	0.2	400 × 800	1700 × 350	4	12	0

. The lever is made of Q235 steel. The box-shaped cross-sections corresponding to the support and lever arm are 300 mm × 140 mm × 20 mm × 20 mm, and 200 mm × 200 mm × 25 mm × 25 mm, respectively, with a lever arm’s length of 1000 mm.

The structure’s section design with the CLNVDs is on the basis of the internal forces of the response spectrum design under the circumstance of fortification earthquakes. As per Chinese code [55], 7 seismic waves are selected, and 7 seismic waves’ response spectra are presented in Figure 15. The average of all the waves’ analysis results is taken as the relevant analysis result. As for fortification earthquakes, elastic dynamic analysis is carried out through adopting SAP2000 software (Version 23.1.0); as for rare earthquakes, elastic-plastic dynamics is implemented by means of utilizing PERFORM-3d software (Version 8.0). In elastoplastic analysis, fluid damper elements are used for VD, and a nonlinear elastic gap hook bar is used for NSD. For steel bars, a double-line model is used; for unconstrained concrete and restrained concrete, the trilinear model and Mander model are applied, respectively; for the beam and column components, a nonlinear fiber section model is adopted. Structures without CLNVD are forced to maintain elasticity. In this way, only the CLNVD’s control effect on the structure’s seismic response can be considered, and the changes caused by the structure’s plastic deformation are not taken into account additionally. Meanwhile, this section only analyzes the CLNVD’s damping effect in the structure’s north–south direction.

4.3. Vibration-Reduction Effect

Figure 16 shows $f_d$ and $f_E$ of the CLNVD under the circumstance of fortification earthquakes and rare earthquakes. From Figure 17, it can be seen as follows: Scheme 1 is the most optimal, Scheme 2 is the second optimal one, and Scheme 3 is the poorest. Meanwhile, for Scheme 1, $f_d>3.5$ and $f_E>30$. Therefore, Scheme 1 satisfies the goals. By comparing Scheme 2 and Scheme 1, we can see that the CLNVD has the highest efficiency when the CLNVD is designed at the middle span. Through comparing Scheme 3 and Scheme 2, we can see that the CLNVD’s stiffness k_N greatly increases the energy dissipation capacity of the damper.

Figure 17a reveals the NVD’s hysteresis curve of B (1–2) axis on the 5th floor under the rare earthquake’s Big Bear waves. As is shown in Figure 17a, NSD causes the hysteresis curve of NVD to become inclined, and the damper’s hysteretic curve area in scheme 1 is larger. Scheme 1 is obtained after optimizing and designing in Section 3.2. Therefore, $k_N/k$ and $k_d^{″}/k$ are located within the target region presented in Figure 11d (See Figure 17b).

The damping ratio provided by CLNVDs is determined according to the strain energy method specified in Chinese code [50]. According to

Table 5. Damping ratios provided by CLNVDs under fortification earthquakes.

Earthquake Wave	Scheme 1	Scheme 2	Scheme 3
Manjil	19.1%	13.5%	7.9%
Coalinga	20.6%	13.8%	8.7%
Imperial Valley	21.3%	15.8%	9.2%
Chi-Chi	23.3%	15.5%	8.8%
Big Bear	19.6%	13.8%	8.5%
Artificial wave 1	19.7%	13.8%	8.0%
Artificial wave 2	20.1%	13.5%	8.4%
Average of multiple waves	20.5%	14.2%	8.5%

, the damping ratios of Scheme 1, Scheme 2, and Scheme 3 are 20.5%, 14.2%, and 8.5%, respectively. Scheme 1 can satisfy the set goals. In addition to the additional damping ratio, the ratio of the cumulative energy dissipation of the damper to the seismic input energy can also be employed as an indicator to measure the performance of CLNVD. As per

Table 6. Energy composition of rare earthquakes.

Energy Composition	Scheme 1	Scheme 2	Scheme 3
Internal model damping/Input	21.3%	27.1%	36.5%
Damper/Input	77.9%	70.8%	57.4%
Dissipated inelastic/Input	0.8%	2.1%	6.1%

, under the circumstance of rare earthquakes, the cumulative energy ratios of dampers from Scheme 1, Scheme 2, and Scheme 3 are 77.9%, 70.8%, and 57.4%, respectively. The internal plastic cumulative damage energy ratios from Scheme 1 to Scheme 3 are 0.8%, 2.1%, and 6.1%, respectively. From this, it can be seen that all three proposed schemes in this paper can significantly reduce structural damage, while Scheme 1’s CLNVD system has the strongest energy dissipation capacity and the optimal control effect on structural damage.

Figure 18 and Figure 19 reveal the structure’s inter-story displacement as well as story shear force under three different schemes. As shown in Figure 18, the CLNVD of Scheme 1 reduces the maximum of the structure’s inter-story displacement angle from 1/200 to 1/400 under fortification earthquakes, and from 1/100 to 1/200 under rare earthquakes, meeting the set goals. The inter-story displacement of all floors except for the ground floor has been reduced by almost 50%. From Figure 19, we can draw a conclusion that the shear force of each floor falls to a certain extent, while the magnitude of the decrease is different from the magnitude of the decrease in inter-story displacement. As is shown in Figure 17b, this is due to the positive stiffness effect and large damping effect of the CLNVD under fortification earthquakes and the negative stiffness effect and large damping effect under rare earthquakes.

In summary, by utilizing the current optimization method, the CLNVDs can achieve high displacement amplification and energy dissipation capabilities, thereby effectively controlling the structure‘s seismic response. It should be noted that to guarantee the normal work of the CLNVD, the beams connected to the CLNVD and the columns directly associated with the beams should be reinforced. These reinforcing components are called substructures. This paper uses the internal forces under rare earthquakes to strengthen the substructure design.

The research on the amplification efficiency of the CLNVD in this study mainly focuses on the framework structure. For frame shear wall structures and shear wall structures, the inter-story bending deformation accounts for a large proportion of the total inter-story deformation. Therefore, in such structures, the displacement amplification coefficient and energy dissipation coefficient of the CLNVD will deteriorate. This paper only discusses the displacement amplification effect and the energy dissipation of CLNVD and does not discuss the control effect of the CLNVD on the inter-story displacement and floor shear force of the structure under earthquake action. Meanwhile, the out–of–plane deformation of the CLNVD will have adverse effects on the displacement of the viscous damper. The above points are not covered in this paper, and we will further improve our research on the amplification and damping effects of CLNVD in future work. CLNVD is a combination of the intermediate column–lever and negative stiffness damping system, helping to achieve large displacement amplification capacity and energy dissipation capacity. The mentality and conclusions of this study also have certain reference significance for the combination of other types of amplification devices and negative stiffness dampers (such as scissor-style negative stiffness damper, toggle-scissor negative stiffness damper, etc.). The CLNVD possesses good energy dissipation capacity and can be used not only in buildings but also to control the deformation of transmission towers under earthquake or wind loads. Marine drilling platforms can also use the CLNVD to increase their ability to withstand wave loads, wind loads, or earthquakes.

5. Conclusions

The CLNVD consists of the viscous damper, the negative stiffness device, the lever, and the intermediate column, which has a low demand for stiffness of the connecting components and enhances the damper’s displacement. In this paper, energy dissipation coefficient f_E and displacement amplification coefficient f_d are proposed to intensively assess the CLNVD’s performance. The coefficient γ is introduced to describe the impact of the intermediate column’s position on the CLNVD’s amplification capability, and the factor η_d is introduced to describe the weakening of the equivalent stiffness on the CLNVD’s amplification capability. Moreover, the impacts, such as column’s position, beam’s stiffness, lever’s amplification coefficient, negative stiffness, damping coefficient, and inter-story displacement on CLNVD’s f_d and f_E, are studied. On this basis, this paper proposes a damping design method to optimize the CLNVD’s amplification capability. The main conclusions are as follows:

(1) The CLNVD can employ f_d and f_E to, respectively, describe its displacement amplification capability and energy dissipation capability. The coefficient f_d is related to the geometric amplification coefficient γ as well as the effective displacement factor η_d.

(2) CLNVD’s f_d and f_E will be maximized, and f_E will increase first and then decrease as the beam’s stiffness increases when the column‘s position is β = 0.5.

(3) The amplification capability of CLNVD will increase when increasing the lever’s amplification coefficient χ. When the negative stiffness is 0, there exists an optimum lever amplification coefficient that maximizes CLNVD’s f_E.

(4) There exists an optimal combination of damping coefficient c and index α that maximizes CLNVD’s f_E. Under this optimal combination, c and α need to satisfy $k_d^{″}/k$ = 1 + k_N/k or η_d = 1/[2(1 + k_N/k) ^1+α] ^1−0.5α.

(5) In contrast to the intermediate column damping system or the intermediate column–lever damper, the negative stiffness device enhances the displacement of CLNVD. In order to enable CLNVD to obtain greater displacement while avoiding providing too much negative stiffness, it is very necessary to make k_N/k and k_d″/k within the target area in Section 3.1.5. Within this area, f_d ≥ 1/(1 + 1/3^1+α)^1−0.5αγ, γ²k_N ≤ k_a′, and f_E increases when increasing damping coefficient c.

(6) In order to enable CLNVD to have high energy dissipation capacity in both small and large displacements, the following methods can be utilized: to increase χ and i_b simultaneously; to increase i_b and c; or to increase χ while decreasing c.

(7) In vibration-reduction design, applying the optimization design method in Section 3.2 can enable CLNVD to effectively control structure’s seismic response. The CLNVD provides a damping ratio of 20.5% in Scheme 1 under fortification earthquakes. Under the circumstance of rare earthquakes and fortification earthquakes, the inter-story displacement has a 50% reduction effect. Meanwhile, this optimization design method can provide a reference for vibration-reduction design in other types of negative stiffness dampers with amplification function.