Experimental Verication of Analysis Model of Seismic Isolation With High-Static-Low-Dynamic Stiffness

This paper veriﬁes the model of high-static-low-dynamic stiﬀness (HSLDS) for seismic isolation based on an experiment. Seismic isolation is widely used in several countries. Moreover, the number of seismically isolated buildings has rapidly increased in these few decades. Seismic isolation extends a natural period of a building and decreases the absolute acceleration to reduce a seismic force. However, as there is a trade-oﬀ between displacement and absolute acceleration, it might result in the maximum displacement beyond an allowable range. HSLDS is nonlinear, and its restoring force can be approximated cube of a displacement. Thus, HSLDS applies a large restoring force for signiﬁcant displacement, and the force is small for small perturbation around an equilibrium position. To improve the control performance of seismic isolation for displacement, we apply HSLDS for seismic isolation. This paper conducts an experiment and compares the results with a time-history analysis to verify a numerical model of HSLDS.

compares the results with a time-history analysis to verify a numerical model of HSLDS.
Passive isolation has been applied to several fields [1]. In civil engineering, seismic isolation has been used to protect buildings from large earthquakes. To date, several buildings employ seismic isolation. In particular, in Japan, the number of seismically isolated buildings has rapidly increased after the Great Hanshin earthquake (Kobe earthquake) [2].
Seismic isolation introduces a soft layer into a building to extend a natural period to suppress absolute acceleration. However, as there is a trade-off between displacement and absolute acceleration, the displacement of a seismically isolated layer might exceed an allowable range. To overcome this disadvantage, nonlinear devices are used and invented such as lead damper, steel damper, and high-damping rubber bearing [3]. In recent times, a high-staticlow-dynamic stiffness (HSLDS) or quasi-zero stiffness (QZS), which is a Duffing equation-based spring, has been applied to isolation systems. The restoring force of HSLDS is derived from the cube of displacement. Thus, the effect is small caused by small perturbations, and it becomes large caused by significant displacement. The combination of seismic isolation and HSLDS suppresses absolute acceleration for small displacement, and it suppresses the displacement for large earthquakes. To improve the control performance of isolation systems, HSLDS has been applied for vibration isolation, e.g., [4]- [12].
The dynamic behavior of a linear system with HSLDS (HSLDS model) is complicated because the jump phenomenon, which causes bifurcation, occurs. Several studies have estimated the performance of the HSLDS model.
Tamas et al. considered a forced and free response of the HSLDS model and compared the dynamic behavior of it and a linear system [4]. While a linear single-degree-of-freedom (SDOF) system possesses one resonance frequency, an HSLDS model with weak damping exhibits a secondary resonance frequency, which is sub-harmonic resonances. Carella et al. defined a simple equation that estimates the peak gain of transmissibility of an external force to displacement [5]. The peak transmissibility of a linear system depends on a damping ratio. In contrast, the transmissibility of an HSLDS model depends on not only the damping ratio but also the coefficient of HSLDS and maximum steady-state response.
In recent times, HSLDS has been applied to vibration isolation. HSLDS can be achieved using magnets, electrical devices, or multiple springs [5][6][7]. The advantages of using electronic or magnetronic devices are a long stroke, long life span. Zhang   vibration [8]. Wang et al. presented a dual quasi-zero stiffness-based vibration isolator that uses additional springs to improve the control performance of the HSLDS model [9]. HSLDS is used for seismic isolation to protect a building. Watanabe et al. applied HSLDS for seismic isolation with inertia. They showed that HSLDS is effective to reduce displacement, and inertia reduces the absolute acceleration [10]. Zhou et al. used HSLDS to suppress a vertical vibration of a building and showed that HSLDS is effective to reduce transmissibility [11]. Liu et al. analyzed the control performance of an isolation system with HSLDS for nearfault ground motion and demonstrated that HSLDS exhibits good control with regard to the displacement of earthquake waves [12].
Previous studies demonstrated that the combination of HSLDS and seismic isolation yields good control performance. However, most studies are based on analytical and theoretical ways. Unlike these studies, this paper conducts experiments to verify a numerical model of HSLDS. The experimental results are compared with time history analysis (THA) and theoretical results. This paper is organized as follows: Section 2 derives the maximum displacement of an HSLDS model for a sinusoidal wave and defines an equation that estimates the maximum absolute acceleration of an HSLDS model. Section 3 describes the experimental setup and specimen. Section 4 compares the experimental results and THA. Section 5 concludes the paper.
2 Dynamics and maximum response of isolation with HSLDS.
First, a steady-state response and a resonance frequency of HSLDS model are derived. Second, a deformation response factor, R d , i.e., the ratio of a resonant response and static response is derived. Then, an equation that estimates absolute acceleration that is based on [5] is presented, and its validity is verified.

Steady state response of HSLDS model.
HSLDS is achieved using orthogonal two springs (Fig. 1), and the equation of motion of an HSLDS model is as follows: where m is a mass, c is a damping coefficient, k is stiffness, x(t) is a displacement, F n (x(t)) is a restoring force of HSLDS, andẍ g (t) is ground acceleration. The restoring force of HSLDS, F n (x(t)), can be approximated by third Taylor polynomial of the function [5,15]: where Λ N 1 and Λ N 3 are the coefficients of HSLDS, and they are derived from a free length of a spring, d 0 : In this study, we used unstretched springs, i.e. d 0 = d. Thus, the restoring force of HSLDS, F n (x(t)) can be approximated using the following equation: Eq. (4) showed that the restoring force of HSLDS is yielded based on the cube of displacement. In other words, stiffness depends on displacement and is given by the square of displacement. Thus, the stiffness of HSLDS is small for small perturbation and large for big displacement. This point is one of the big differences to linear stiffness. The derivations of Eqs. (2) and (3) are explained in Appendix A. Rewriting Eq. (1) yields the following equation: where To consider the steady-state response of the system, we assume that the ground motionẍ g (t) = −a e cos Ωt. Moreover, we assume that the response of system Eq. (1) can be approximated by following equations:    x(t) = r cos(Ωt + φ), x(t) = −rΩ sin(Ωt + φ), x(t) = −rΩ 2 cos(Ωt + φ).
Substituting Eq. (10) into Eq. (8) and considering only the excitation frequency, Ω, the following equation has been derived: Regrouping Eq. (11) using sin 2 φ + cos 2 φ = 1 gives Thus, the amplitude of a steady-state response r is where 2.2 Estimation of maximum response of HSLDS.
Suppose that the damping ratio is much smaller than 1, ξ ≪ 1; then Eq. (13) considers peak value if Ω satisfies ω 2 s + 3 4 λr 2 − Ω 2 = 0. In this case, Ω is defined as a resonance frequency, Ω p : The resonance frequency of an undamped linear system, Ω p,lin , is given as follows: ω s is the same as Eq. (15) with λ = 0. Since a stiffness of the HSLDS model is a function of a displacement, Eq. (15) reflects the characteristic of it. Substituting Eq. (15) into Eq. (13) yields the peak amplitude r p : Let us consider a deformation response factor, R d , that is a ratio of r p to static displacement of a linear system, x st : Substituting Eqs. (15) and (17) into Eq. (18) yields Note that the deformation response factor of a linear system R d,lin is as follows [14]: This value is identical to R d with λ = 0. Equation Eq. (20) shows that the deformation response factor of a linear system only depends on the damping ratio, ξ. In contrast, the factor of an HSLDS model depends not only on the damping ratio but also the displacement because the stiffness of HSLDS depends on the square of displacement. Substituting Eq. (15) into Eq. (17) gives the following equation: r 2 p is defined using the quadratic formula for Eq. (21): Therefore, r p is given by This equation is identical to [5]. The stiffness of HSLDS increases for large displacement. We assume that the absolute acceleration of an HSLDS model takes maximum value when the displacement is maximum, x max . The equation that estimates the maximum absolute acceleration of an HSLDS model: Since the velocity, which is the derivative of displacement, is 0 if the displacement takes a maximum value; the maximum absolute acceleration can be estimated based on the displacement. Note that the maximum absolute acceleration of the HSLDS model for sinusoidal wave is given by substituting The stiffness, k, of the HSLDS model is The HSLDS coefficient, Λ N 3 , is designed in a way so that nonlinear restoring force, F n (x) = Λ N 3 x 3 (t), is the same as the linear restoring fore at the selected displacement, x sel . Thus, Λ N 3 is given by In this subsection, x sel is selected to be 0.3 m. We performed THA considering 44 earthquakes for numerical models. The waves are recommended to evaluate vibration-suppression performance by the Federal Emergency Management Agency (FEMA) P695 [16]. The detail of the waves are listed in Tabs. 1 and 2. Figure 2 compares the absolute acceleration obtained using THA and Eq. (24). In Fig. 2, the maximum displacement yielded by THA is used in Eq. (24).   The results of THA (Fig. 2) shows that Eq. (24) accurately estimates the maximum absolute acceleration of the HSLDS model. The results demonstrate that it is reasonable to suppose that the absolute acceleration can be maximum value when the displacement is maximum. That is, the maximum absolute acceleration can be estimated considering the maximum displacement.

Experimental setup.
This section illustrates the experimental setup. The specimen was developed at the Institute of Technology of Shimizu Corporation.
The specimen comprises a mass, vertical four linear springs, and HSLDS, which is achieved by two wires and disc springs (Figs. 3 and 4 (a)-(d)). The mass is 4038 kg, the stiffness of the model is 13597 N/m, and the damping ratio, which is caused by friction, is estimated to be 0.04. x sel is selected to be 0.3 m in Eq. (26), and Λ N 3 is 158530 N/m 3 . Λ N 3 can be adjusted by the stiffness of disc springs (Fig. 4(d)).
The mass is guided by two linear-motion guides. The mass oscillates along the horizontal direction. In this experiment, the following responses and forces are measured: the displacement and absolute acceleration of the specimen, the restoring force of HSLDS and linear springs, and the acceleration of the shaking table.

vib. dir.
Linear spring disc. spr.     (Fig. 9): An artificial wave and the pseud velocity spectrum, p S V is approximately 0.8 m/s for a structure whose natural period is longer than 0.64 s and the damping ratio is 5%.
In this study, the sweep wave is described as where A p is the amplitude of the wave, f s is the initial frequency, f e is the target frequency, and T is the length of the time. The frequency of the signal f sw (t) is given by As Eq. (28) showed that the frequency of the signal is a function of time, and it is varied continuously between f s and f e . This paper uses a scaled Hachinohe wave to asses the effectiveness of HSLDS, and the scaling factor is 1.47. Figure 7 shows the scaled wave. The pseudo velocity response spectra, p S V , for these waves show that the dominant period of the Hachinohe wave is about 2.5 s and that of the Kobe wave is about 1.0 s. On the other hand, the spectrum of the Random wave is about 0.8 m/s for a building with a damping ratio of 5%, and the natural period is longer than 0.64 s. This section shows the results of the two sweep waves. The results of three earthquake waves are shown in Appendix B.
An HSLDS model, which is one of the Duffing's oscillator, depends on the "direction" of an excitation frequency. This implies that if an excitation frequency varies from low level to a high level, the response amplitude increases until r p , which is known as resonant branch. However, if an excitation frequency decreases, the maximum amplitude is lower than r p (Fig. 14). Figure  14 shows the envelope curves of maximum displacement and f sw for Sweep Up and Sweep Down. Equation (23) estimates the maximum displacement for resonant branch, and the estimated value is close to the result for Sweep Up. These figures also show that Eq. (24) accurately estimates the maximum absolute acceleration of the system for Sweep Up. Note that in this section, r p , which is obtained from Eq. (23), is used in (24). Since the restoring force with regard to HSLDS is approximated by the cube of the displacement, the restoring force sharply increases as the displacement increases to suppress the maximum displacement (Figs. 11 (b) and 13 (b)). On the other hand, for the small perturbation, the absolute acceleration of HSLDS model reduces because the restoring force of HSLDS decreases. This is one of the major differences between HSLDS and large linear stiffness.

Conclusion.
This paper dealt with seismic isolation with high-static-low-dynamic stiffness (HSLDS). Seismic isolation uses a low stiffness component to decrease the influence of high-frequency waves. However, it might result in significant displacement and causes beyond the allowable range. In contrast, since stiffness Data availability statement The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.  Carella et al showed that the restoring force of an HSLDS can be described by Taylor polynomial of approximation [15], and usually a third Taylor polynomial of it is used (e.g. [4,5,15]). The third Taylor polynomial of restoring force, Fn(x), is

Declarations
Fn(x(t)) = Fn(0) + F (1) is the nth derivative of Fn(x(t)) based on x(t). Since the first to third derivative of Fn(x) are n (0), which are the coefficients of x(t), x 2 (t), and x 3 (t), respectively are defined as In particular, if d 0 = d, which means the springs are unstretched at the initial position, these are given by Appendix A.2 Influence of high-order terms.
Next, we consider the polynomial approximation for HSLDS. Regrouping Eq. (A.1:) yields the following equation: Fn(x(t)) = 2kort . (A.10:) can be approximated by using the binomial series: , (A.11:) + · · · (A.14:) In particular, if d = d 0 , Fn(x(t)) can be approximated as This is worth mentioning that if |x(t)/d 0 | > 1, the value of approximation will not converge. In other words, the restoring force of HSLDS can be approximated if the displacement, x(t), is smaller than d 0 . Figures 11 and 13 demonstrated that the third-order polynomial equation appropriately approximates the restoring force of HSLDS in this paper.
Appendix B Comparison of results for THA and experiment for earthquake waves.
The THA and experimental results for three earthquake waves are compared. The results show that the analytical model is in close agreement with the experimental results for response and the restoring force. Thus, the behavior of an HSLDS model can be estimated using, Eq. (1).