Experimental evaluation of eccentric rotational channel-type damping system for vibration control of building structures

Abstract

A theoretical and experimental investigation of an eccentric rotational channel-type damping system is presented. The proposed damping system can incorporate any type of existing passive dampers and provide a wider field of view to residents compared with existing damping systems. Furthermore, the efficiency of the proposed damping system can be magnified by modifying the geometry of the channel-type secondary system. Cyclic loading and free vibration tests of a full-scale test model with steel dampers were conducted to investigate the validity of the suggested simple behavior prediction model and the vibration characteristics of the proposed damping system. The experimental results were in good agreement with the numerical analysis. The results of numerical prediction studies on a single degree of freedom system with the proposed damping system also showed the effect of the eccentric rotational channel-type damping system on seismic response reduction.

Keywords

cyclic loading test eccentric rotational channel-type damping system effective deformation ratio free vibration test magnification factor seismic response reduction simple behavior prediction model vibration characteristics

Introduction

For more than two decades, structural engineers have worked with innovative structural systems and damping devices as well as smart materials to develop structural damping technologies that can reduce structural damage caused by seismic events. It is well known that passive dampers absorb a considerable amount of earthquake-induced energy using mechanical mechanisms such as yielding of steel materials, friction, fluid inflow, and viscoelastic deformation of solid bodies (Aiken et al., 1993; Kim et al., 2012a, 2012b; Kori and Jangid, 2008; Mazza and Vulcano, 2011; Michael and Constantin, 2014; Mirzabagheri et al., 2015; Mohammad and Hosein, 2015; Neflze and Alp, 2016; Patil and Jangid, 2009; Phocas and Pocanschi, 2003; Saidi et al., 2011; Tani et al., 2009; Teruna et al., 2015; Zhang and Zhu, 2007). Existing passive damping systems are classified into vibration isolation systems, tuned mass damper systems, and story-installation-type damping systems (Soong and Dargush, 1997). The efficiency of damping systems significantly depends on the geometry and stiffness of the elements that support an energy dissipation device (hereafter called a secondary system). Thus, it is important for the secondary system to be effectively designed to control deformation in a damper.

Several damping systems with amplification mechanisms such as toggle brace dampers (Constantinou et al., 1997; Constantinou et al., 2001; Hwang et al., 2005; Hwang et al., 2006), scissor-jack systems (Sigaher and Constantinou, 2003), seesaw-damper systems (Kang and Hiroshi, 2013; Kang and Tagawa, 2013, 2014), and eccentric lever-arm systems (Sebastián et al., 2016) have been proposed and investigated to mitigate the disadvantages of existing diagonal and chevron systems, for example, insufficient deformation in stiff structural systems. However, these damping systems obstruct residents’ views because their diagonal elements or wires are connected to the main structure.

A theoretical and experimental investigation of an eccentric rotational channel-type damping system (hereafter referred to as an ERC damping system) in which any type of passive damper (i.e. metallic, friction, viscous, or viscoelastic damper) can be incorporated is herein presented. Compared with existing damping systems, two pairs of channel-type secondary systems positioned near upper and lower floors can provide a wider field of view to residents (see Figure 1). Moreover, because each secondary system needs to support only half of the load transferred from the dampers, an efficient cross-section of the secondary systems can be designed. If required, the length of the secondary systems can be modified to magnify the deformation of the dampers.

Figure 1.

Eccentric rotational channel-type damping system.

Objectives of this study included (a) clarification by analytical and experimental investigations of the mechanical behavior of an ERC damping system affecting the restoring force characteristics of dampers and (b) evaluation by cyclic loading and free vibration tests of the appropriateness of the design of the channel-type secondary system and the employed damper. The experimental results were compared with analytical results to verify the validity of an analytical model considering the displacement mechanism of the ERC damping system. A numerical analysis was additionally conducted to analytically verify the effect of the ERC damping system on the response reduction of a single degree of freedom (SDOF) system.

Eccentric rotational channel-type damping system

The proposed ERC damping system consists of a pair of channel-type secondary systems assembled with prefabricated H-beams, a pair of damper units including steel dampers, and four ball bearing hinges, as shown in Figure 1. A channel-type secondary system of the ERC damping system comprises three parts: a pinned–pinned column frame (a vertical frame) and a pair of fixed–fixed beam frames (upper and lower horizontal frames). However, the upper and lower fixed–pinned supports presented in Figure 1 can be altered or omitted depending on the type of building in which the ERC damping system is installed. The channel-type secondary systems are rotated by the hinge joints to transmit deformation from the main frame to the damper. Although viscoelastic or viscous dampers can be incorporated, for convenience in predicting the hysteretic behavior of the ERC damping system, in this study, steel hysteretic dampers were employed as energy dissipation devices.

Simple behavior prediction model

The configuration of the ERC damping system before and after deformation is schematically presented in Figure 2(a). A pair of dampers is located between two analytical elements that simplify the channel-type secondary systems (A₁-C₁-C₂-A₂ and B₁-D₁-D₂-B₂). All analytical elements are symmetrical with respect to the central axis of the analytical model. The inter-story drift, $δ_{F}$ , generates eccentric rotation of the channel-type secondary system connected at the lower hinge joints; then, the rotated upper and lower beam elements of the channel-type secondary systems (hereafter called horizontal frames) deform the dampers. In addition, the column elements (hereafter called vertical frames) of the analytical model were assumed to be rigid to simplify the analytical process.

Figure 2.

Simple behavior prediction model of ERC damping system: (a) configuration before and after deformation and (b) lateral forces on damping system corresponding to damper deformation.

Figure 2(b) shows a simple behavior prediction model that visualizes the deformation of a steel damper and both sides of the horizontal frames connected in series. If it is assumed that the relative displacement between the upper and lower hinge joints (C₁-C₂ or D₁-D₂) is equal to the inter-story drift, $δ_{F}$ , the rigid rotation angle of the vertical frame of the channel-type secondary systems (hereafter called vertical frames), $θ_{CV}$ , can be expressed as

θ_{C V} = \sin^{- 1} [\frac{δ_{F}}{h_{1} + h_{2} + h_{3}}] ≃ \frac{δ_{F}}{h_{1} + h_{2} + h_{3}}

(1)

where $h_{1}$ $[= h_{3}]$ denotes the distance from the center of the upper hinge joint to the center of the upper horizontal frame and $h_{2}$ denotes the center-to-center distance between the upper and lower horizontal frames. In addition, the vertical displacement of C₁ or D₁ due to the rigid rotation of the vertical frame, $δ_{CV}$ , is expressed as

δ_{C V} = (h_{1} + h_{2} + h_{3}) \cdot (\cos θ_{C V} - 1)

(2)

However, $δ_{CV}$ can be neglected in the analytical model because $θ_{CV}$ may be less than 5% rad even for large deformations. Considering that the ratio of $δ_{CV}$ to the total displacement of the ERC damping system, $δ_{C}$ , is very small, $δ_{C}$ can be approximately estimated as

\begin{matrix} δ_{C} = (l_{1} + l_{3}) \cdot \sin θ_{CH} + 2 δ_{CV} \approx 2 l_{1} \cdot θ_{CV} \\ = 2 l_{1} \cdot δ_{F} / (h_{1} + h_{2} + h_{3}) \end{matrix}

(3)

where $l_{1}$ $[= l_{3}]$ , $θ_{CH}$ $[= θ_{CV}]$ , and $δ_{D}$ denote the initial length of the left-side horizontal frames, the rigid rotation angle of the horizontal frame, and the displacement of a steel damper, respectively. In this study, a concept called magnification factor, $α_{C}$ , is introduced to evaluate how much the inter-story drift of a main frame is transferred to the displacement of the damping system by the configuration of the ERC damping system. $α_{C}$ denotes the ratio of $δ_{C}$ to $δ_{F}$ , which is proportional to $l_{1}$ and inversely proportional to the total length of the vertical frames, as

α_{C} = \frac{δ_{C}}{δ_{F}} = \frac{2 l_{1}}{h_{1} + h_{2} + h_{3}}

(4)

Here, $δ_{C}$ can be represented by the sum of the displacement of a steel damper, $δ_{D}$ , and the displacement of the horizontal frames, $δ_{CH}$ , as

δ_{C} = δ_{D} + 2 δ_{CH}

(5)

The rotation of the horizontal frame, however, causes not only vertical displacement but also axial elongation of the damper, as shown in Figure 2(b). Considering the use of steel dampers in the ERC damping system, the length of a steel damper after deformation, ${l_{2}}^{'}$ , the strain, $ε_{D}$ , the rotation angle, $θ_{D}$ , and the elongations in the longitudinal direction of a steel damper, $Δ l$ , can be calculated as

δ_{D} = 2 l_{1} \sin θ_{CH}^{D}

(6)

Δ l = 2 l_{1} (1 - \cos θ_{CH}^{D})

(7)

{l^{'}}_{2} = ​ \sqrt{{(l_{2} + Δ l)}^{2} + δ_{D}^{2}} = \sqrt{{[l_{2} + 2 l_{1} (1 - \cos θ_{C H}^{D})]}^{2} + δ_{D}^{2}}

(8)

ε_{D} = \frac{{l'}_{2} - l_{2}}{l_{2}}

(9)

θ_{D} = \sin^{- 1} [\frac{δ_{D}}{\sqrt{{(l_{2} + Δ l)}^{2} + {(δ_{D})}^{2}}}]

(10)

Here, $θ_{CH}^{D}$ denotes the rotation angle of the horizontal frames after deformation. Particularly in the case of using steel dampers, an additional lateral force acts on a steel damper due to $Δ l$ . Thus, the total lateral force on a steel damper, $Q_{D}$ , can be expressed as

Q_{D} = Q_{DS} + Q_{DP}

(11)

where $Q_{DS}$ and $Q_{DP}$ are the lateral force and the normal component of the axial tension force on a steel damper, respectively. $Q_{DS}$ and $Q_{DP}$ can be expressed as

Q_{DS} = k_{D} \cdot δ_{D} [if, δ_{D} \leq δ_{Dy}]

(12a)

Q_{DS} = Q_{Dy} = k_{D} \cdot δ_{Dy} [if, δ_{D} > δ_{Dy}]

(12b)

Q_{DP} = E \cdot ε_{D} \cdot A_{D} \cdot \sin θ_{D} [if, ε_{D} \leq ε_{Dy}]

(13a)

Q_{DP} = σ_{Dy} \cdot A_{D} \cdot \sin θ_{D} [if, ε_{D} > ε_{Dy}]

(13b)

where $k_{D}$ , $Q_{Dy}$ , $δ_{Dy}$ , E, $A_{D}$ , $σ_{Dy}$ , and $ε_{Dy}$ denote the lateral stiffness, the yield strength, the yield displacement, the modulus of elasticity, the cross-sectional area, the yield stress, and the yield strain of the steel damper, respectively.

As presented by the analytical model, the steel damper elongates under loading on either direction due to the behavior of the ERC damping system. Compared with existing metallic dampers with buckling issues, this is an attractive feature of the ERC damping system in addition to the displacement mechanism and improved view, as shown in Figure 2.

Effective deformation ratio of ERC damping system

The ERC damping system consists of a pair of channel-type secondary systems and a pair of steel dampers, as shown in Figure 2(b). In the case of connecting the dampers to secondary systems in series, deformations in the secondary systems affect the global efficiency of the damping system. It is important to clarify how much deformation of the structural frame is transferred to the steel damper by the ERC damping system because damper deformation is directly related to the seismic performance of damping systems. An existing concept called effective deformation ratio, that is, the ratio of actual damper deformation to main frame deformation (Kim et al., 2012a), was employed in this study to evaluate the efficiency of the ERC damping system. In this section, the effective deformation ratio of the ERC damping system, $R_{D}$ , is analytically examined as

R_{D} = \frac{δ_{D}}{δ_{F}} = α_{C} \cdot \frac{δ_{D}}{δ_{C}}

(14)

In general, considering the inelastic behavior of materials, iterative calculations should be implemented to examine the total deformation in any analytical model. To evaluate the efficiency of an ERC damping system without iterative calculations, an analytical procedure based on the inverse-problem approach is presented in this section. The spring model for a structure with an ERC damping system is shown in Figure 3(a), and a flowchart for estimating $R_{D}$ is presented in Figure 3(b). In the analytical procedure, the given is a maximum displacement of a steel damper, $δ_{D}$ . Then, the total lateral force on a steel damper, $Q_{D}$ , corresponding to a given $δ_{D}$ can be determined using equations (6) to (13). Since the lateral force acting on horizontal frames connected to a steel damper in series is the same as $Q_{D}$ , the displacement of the horizontal frames, $δ_{CH}$ , is

δ_{CH} = \frac{Q_{D}}{k_{CH}}

(15)

where $k_{CH}$ is the lateral stiffness of the horizontal frames. Then, $δ_{C}$ is calculated from equation (5). Finally, $δ_{F}$ and $R_{D}$ are obtained from equations (4), (5), and (14), with magnification factor, $α_{C}$ , determined by the dimensions of the vertical and horizontal frames at the design stage. For the simplicity of the proposed spring model, it was assumed that the horizontal frames are relatively stiff compared with the steel damper, and displacements in the longitudinal direction of the horizontal frame due to $Q_{D}$ do not have a significant effect on the seismic performance of the ERC damping system. Based on this assumption, only $δ_{CH}$ was considered in equation (15).

Figure 3.

Analytical model based on inverse-problem solution: (a) spring model for ERC damping system and (b) flowchart of analytical procedure for determination of $R_{D}$ .

A parametric analysis was conducted to evaluate the effect of the lateral stiffness of the steel damper, the main frame, and the horizontal frame on the efficiency of the ERC damping system. The main variables employed in the analysis were the relative stiffness of a steel damper and a main frame, $κ$ , and the relative stiffness of horizontal frames and a steel damper, $ρ$ , and $α_{C}$ . It was assumed in the analysis that the main frame behaves elastically under a given maximum deformation. Table 1 presents the values of structural variables given in the parametric analysis.

Table 1.

Structural variables in parametric analysis.

Variable	$k_{F}$ $(N / mm)$	$h_{1} + h_{2} + h_{3}$ $(mm)$	E $(N / m m^{2})$	$σ_{Dy}$ $(N / m m^{2})$	$δ_{Dy}$ $(mm)$	$α_{C}$ (–)	$κ$ (–)	$ρ$ (–)
Value	10,310.0	1800.0	2,00,000.0	235.0	2.0	1.0	0.5	10
							1.0	25
							2.0	50
							3.0	100
						1.5	0.5	10
							1.0	25
							2.0	50
							3.0	100

The ratio of $2 δ_{CH}$ to $δ_{C}$ and the ratio of $δ_{D}$ to $δ_{C}$ with respect to $δ_{C}$ are shown in Figure 4(a) and (b), respectively. The sum of the two ratios for a single value of $δ_{C}$ is 1.0. While the ratio of $2 δ_{CH}$ to $δ_{C}$ converges to zero, the ratio of $δ_{D}$ to $δ_{C}$ converges to 1.0 with the increase of $ρ$ . Moreover, as the lateral stiffness of the horizontal frames increases, the range in which the two ratios remain constant becomes narrower. This shows that if $ρ$ becomes larger, most of the deformation of the ERC damping system is concentrated on the steel dampers.

Figure 4.

Deformation ratios with respect to total deformation of ERC damping system: (a) $2 δ_{CH} / δ_{C}$ and (b) $δ_{D} / δ_{C}$ .

The relationship between $R_{D}$ and $δ_{F}$ for various values of $ρ$ is shown in Figure 5. The $R_{D}$ for assumed values of $ρ$ remains constant but has lower values than the designed magnification factors in the elastic region; however, $R_{D}$ converges on the designed $α_{C}$ with the increase of $δ_{F}$ . Another set of analytical results show that a minimum magnification factor of about 1.25 at $ρ = 10$ is required to ensure an $R_{D}$ of more than 0.9 in the elastic region. From the analytical results, designing the magnification factor, $α_{C}$ , to be 1.25 or more can increase the efficiency of the ERC damping system and reduce the amount of steel used for channel-type secondary systems.

Figure 5.

Relationship between effective deformation ratio $(R_{D})$ and inter-story drift $(δ_{F})$ for various relative stiffness, ρ: (a) $α_{C} = 1.00$ and (b) $α_{C} = 1.50$ .

Experimental investigation

Description of test model

A full-scale test model was constructed to evaluate the behavior of the proposed ERC damping system. Cyclic loading and free vibration tests with a full-scale test model (Kim et al., 2012a) were undertaken to meet the objectives of this study. The test model, which consisted of a loading frame and the ERC damping system, is shown in Figure 6. The loading frame consisted of two H-beams connected to a concrete block covered with steel plates by high-strength bolts. The mass of the loading frame including the concrete block, m, was about 1583 kg. The loading frame was suspended from a steel reaction beam by high-strength bolts.

Figure 6.

Test model with ERC damping system.

Table 2 summarizes the dimensions and property of the loading frame and the ERC damping system. All properties of the loading frames were determined so that these frames could behave elastically within an amplitude range of 8.0 mm. The designed free vibration period of the loading frame was about 0.46 s. The vertical and the horizontal frames of the test model were designed so that $α_{C}$ would be 1.0. Thus, the designed total lengths of the vertical frames including the upper and lower ball bearing hinges, $h_{1} + h_{2} + h_{3}$ , and the horizontal frame including the damper unit elements, $l_{1}$ $(= l_{2})$ , shown in Figure 2(a), were 1800 and 900 mm, respectively.

Table 2.

Specifications of loading frame and ERC damping system.

Element name	Material	Dimension	Mass (kg)
Loading frame	$SS 400$	$H - 200 \times 100 \times 5.5 \times 8$ , $L = 2760 mm \times 2$	117.7
Gusset plate	$SS 400$	$PL - 200 \times 20$ , $L = 420 mm \times 4$	53.8
Steel plate on concrete block	$SS 400$	$PL - 420 \times 10$ , $L = 2600 mm \times 2$	174.9
Concrete block	$SS 400$	$Concrete - 300 \times 420$ , $L = 2600 mm \times 1$	753.5
Upper/lower support	$SS 400$	$H - 400 \times 400 \times 13 \times 21$ , $L = 2100 mm \times 2$	724.2
Vertical frame	$SS 400$	$H - 200 \times 200 \times 8 \times 12$ , $L = 1580 mm \times 2$	157.7
Horizontal frame	$SS 400$	$H - 200 \times 200 \times 8 \times 12$ , $L = 640 mm \times 4$	127.7
Stiffener	$SS 400$	$PL - 10 \times 95.5$ , $L = 176 mm \times 10$	26.9
Hinge plate	$SS 400$	$PL - 15 \times 130$ , $L = 82 mm \times 12$	15.4
Steel plates of vertical frame	$SS 400$	$PL - 10 \times 200$ , $L = 200 mm \times 4$	12.8
Steel plates of horizontal frame	$SS 400$	$PL - 10 \times 150$ , $L = 200 mm \times 4$	9.6
Steel plates of hinge joint	$SS 400$	$PL - 20 \times 200$ , $L = 200 mm \times 8$	51.3.
Damper unit element 1	$SS 400$	$PL - 10 \times 140$ , $L = 200 mm \times 8$	9.0
Damper unit element 2	$SS 400$	$PL - 8 \times 140$ , $L = 100 mm \times 4$	3.6

ERC damping system: eccentric rotational channel-type damping system.

The steel dampers used in this study were designed using structural variables such as the relative stiffness, $κ$ $[= k_{D} / k_{F}]$ , the damper’s strength ratio, $β$ $[= Q_{D} / Q]$ , and the ductility ratio of steel dampers, $μ_{D}$ $[= δ_{D} / δ_{Dy}]$ . Considering the capacity of the prepared loading device in the laboratory, the cross-sectional area of the steel damper was appropriately determined so that the equivalent viscous damping ratios, $h_{eq}$ , were about 0.044 $[κ = 0.33]$ , 0.077 $[κ = 0.66]$ , and 0.106 $[κ = 0.99]$ when $μ_{D}$ was 3.0. The yield displacement of the steel damper was designed to be 2.0 mm. Figure 7 shows details of the designed steel dampers.

Figure 7.

Detail of designed steel damper.

Test setup

The measurement plan is presented in Figure 8. In this figure, relative displacement of both sides of the steel damper, $d 4$ and $d 5$ (or $d 6$ and $d 7$ ), represents the displacement of the steel dampers, $δ_{D}$ . $d 1$ and $d 2$ (or $d 3$ ) represent the inter-story drift of the test model, $δ_{F}$ , and the displacement of the horizontal frames, $δ_{CH}$ , respectively. According to the analytical procedure, $δ_{C}$ and $R_{D}$ during the cyclic loading tests, therefore, could be estimated as

δ_{C} = δ_{D} + 2 δ_{CH} = (d 4 + d 5) + 2 \cdot d 2

(16)

R_{D} = \frac{δ_{D}}{δ_{F}} = \frac{d 4 + d 5}{d 1}

(17)

Figure 8.

Measurement plan for full-scale test model.

In addition, the total lateral force on the test model, Q, during the free vibration test was estimated as the inertia force obtained by multiplying the mass by the absolute acceleration, $a 1$ . In the free vibration test, a screw-type actuator was used to gradually increase $d 1$ until the inter-story drift reached about 8.0 mm; then, the loading was discontinued to permit free vibration of the test model. The sampling frequency of data collection during the free vibration test was 100 Hz. The loading displacement history of the cyclic loading test is shown in Figure 9, and the loading cases and their objectives are shown in Table 3.

Figure 9.

Cyclic loading history.

Table 3.

Loading program.

Test model	Cyclicloading	Free vibration	The number ofsteel dampers	Objectives of tests
A	○	○	Withoutdampingsystem	Lateral stiffness and free vibration period of loading frame
B	○	○	$κ = 0.00$	Mass, structural damping, and deformation ratio
C-2	○	○	2 $(κ = 0.33)$	Damper’s strength ratio, effective deformation ratio, equivalent viscousdamping ratio, force–displacement relationship
C-4	○	○	4 $(κ = 0.66)$
C-6	○	○	6 $(κ = 0.99)$

A: loading frame only; B: loading frame + damping system without steel damper; C: loading frame + damping system with steel damper; $κ$ : relative stiffness of steel damper and loading frame; ○: whether the test is performed.

Experimental results and discussion

Restoring force characteristics

The hysteresis loop of the loading frame without the ERC damping system under a cyclic loading test is represented by the dashed line in Figure 10(a). The lateral stiffness of the loading frame, $k_{F}$ , is about 216 N/mm, and this value was used in the numerical analysis.

Figure 10.

Comparison of experimental measurements with analytical hysteresis loops: (a) test model B, (b) test model C-2, (c) test model C-4, and (d) test model C-6.

The hysteresis loops of the test models are compared with those of the analytical models for cyclic loading in Figure 10(b) to (d). The experimental results showed good agreement between the experimental measurements and the numerical predictions. The results also demonstrated that the increase in the area surrounded by the hysteresis loop in each cycle was proportional to the relative stiffness, $κ$ . In addition, the analytical results showed that nonlinearity in the restoring force due to the mechanical behavior of the ERC damping system decreased with the increase of the yield strength of the steel damper. It can be understood that steel dampers have sufficient strength, and their restoring force characteristics can be approximated as elastic-perfectly plastic curves.

Comparison of experimental and analytical effective deformation ratios

The $R_{DA}$ obtained from the analysis with $R_{DT}$ obtained from monotonic loading tests is compared in Figure 11. $R_{DT}$ in the amplitude range between 0 and 2.0 mm was smaller than the design value of $α_{C} = 1.0$ . However, after the behavior of test models became stable, $R_{DT}$ in the amplitude range between 2.0 and 8.0 mm was found to be about 0.96. This shows that the proposed mechanism converted about 96% of the inter-story drift to deformation of the steel dampers at $α_{C} = 1.0$ . It was found that $R_{DA}$ converged to the design value of $α_{C}$ even at small deformations, whereas $R_{DT}$ showed a difference from the design value of $α_{C}$ at displacements less than 2.0 mm. The difference between the experimental and analytical results seems to have been caused by the manufacturing tolerance of the test model. Manufacturing tolerances of hinge joints and bolted joints can degrade the effective deformation ratio of an ERC damping system in a small deformation range. However, the inter-story drift of the analytical model was transferred to deformations of the horizontal frame and the steel dampers because all elements of the ERC damping system were assumed to be connected with rigid joints and perfectly pinned joints in the analysis. Moreover, $R_{DA}$ showed that most of the deformation of the ERC damping was concentrated on the deformation of the steel damper because $k_{CH}$ was more than one thousand times higher at $κ = 0.99$ $[k_{D} = 213.8 N / mm]$ . Therefore, manufacturing tolerances that occur during the construction stage should be minimized, and/or the design value of $α_{C}$ should be increased in order to improve the seismic performance of an ERC damping system.

Figure 11.

Comparison of experimental and analytical effective deformation ratios: (a) test model C-2, (b) test model C-4, and (c) test model C-6.

The average displacement of the horizontal frames is shown in Figure 12. The designed lateral stiffness of the horizontal frame, $k_{CH}$ , was 216,064 N/mm, which was about one thousand times higher than that of the loading frame. The maximum displacement of the horizontal frames under monotonic loading was less than 0.01 mm. This indicates that the displacement of the horizontal frames may have had an insignificant effect on the damping performance of the steel damper designed for the tests. This is because the lateral stiffness of the channel-type secondary system is very high compared with that of the loading frame and the steel dampers. Considering the practical application of the ERC damping system to real structures, the lateral stiffness of the channel-type secondary system should be adjusted because a damping force in the range of 100–500 kN may produce significant displacements of the channel-type secondary system. However, as mentioned before, it is possible to increase the efficiency of an ERC damping system by adjusting the magnification factor, $α_{C}$ , whose value is determined by the dimensions of the channel-type secondary systems at the design stage.

Figure 12.

Average displacement of horizontal frames.

Equivalent damping ratio

The equivalent viscous damping ratio $(h_{eq})$ of the test models in individual inter-story drifts is summarized in Table 4. The dissipated energies, $Δ W$ , and the elastic strain energies, W, per cycle were calculated using the hysteresis loops shown in Figure 10. The results showed that because of the nonlinear restoring force characteristics of the steel damper, the damping effect was found even at loading amplitudes corresponding to ductility ratios smaller than 1. Moreover, $h_{eq}$ at $μ_{D} = 3.0$ was similar to the designed values. This indicates that the equivalent viscous damping ratios based on the elastic-perfectly plastic restoring force characteristics are almost the same as those considering the mechanical behavior of the ERC damping system.

Table 4.

Dissipated energy per cycle and equivalent viscous damping ratio.

Test model	$δ_{F} (mm)$	$μ_{D}$	$Δ W$	W	$h_{eq}$
C-2	2.0	1.0	214.2	581.5	0.029
	4.0	2.0	1293.3	2177.1	0.047
	6.0	3.0	3089.0	4800.8	0.051
	8.0	4.0	5484.9	8660.8	0.050
C-4	2.0	1.0	302.1	786.4	0.031
	4.0	2.0	2325.0	2741.4	0.065
	6.0	3.0	5861.9	6290.0	0.074
	8.0	4.0	10,454.1	11,710.8	0.071
C-6	2.0	1.0	409.0	982.0	0.033
	4.0	2.0	3078.8	3360.7	0.073
	6.0	3.0	8034.3	7243.1	0.088
	8.0	4.0	14,373.7	13,307.9	0.086

$δ_{F}$ : inter-story drift; $μ_{D}$ : ductility ratio of damper; $Δ W$ : dissipated energy per cycle; W: maximum strain energy; $h_{eq}$ : equivalent viscous damping ratio $(= Δ W / 4 π W)$ .

Vibration characteristics of full-scale test model

The amplitude time history of test model B is shown in Figure 13. The free vibration period of the test model, $T_{0}$ , was about 0.532 s, a value calculated using an average of the time taken by each cycle. The mass of the test model, m, was 1,518 kg which can be estimated using $k_{F}$ and $T_{0}$ of the test model as

\begin{matrix} m = {k_{F}}^{*} T_{0}^{2} / 4 π^{2}, m = (216 N / mm)^{*} (0.532 s)^{2} / 4 π^{2} = (216 {kg}^{*} 980 / s^{2})^{*} (0.532 s)^{2} / 4 π^{2} = 1, 518 kg \end{matrix}

(18)

m is very close to the sum of the concrete block mass and half of the mass of the loading frame (see Table 2). The structural damping ratio, $h_{0}$ , which depends on the amplitude level obtained from the amplitude time history of test model B is shown in Figure 14. The logarithmic decrement method was used to calculate $h_{0}$ . It was confirmed that the average structural damping ratio was about 0.0053.

Figure 13.

Amplitude time history of test model B.

Figure 14.

Structural damping ratio, $h_{0}$ , of test model B.

The amplitude time histories obtained from the free vibration test and the numerical analysis are compared in Figure 15. The values of m, $k_{F}$ , and $h_{0}$ obtained from the cyclic loading and free vibration tests were used in the numerical analysis considering the mechanical behavior of the ERC damping system. The analytical results showed good agreement with the experimental results. Differences between the analytical and experimental results may have been due to degradation of the steel dampers.

Figure 15.

Comparison of experimental and analytical amplitude time histories: (a) test model B, (b) test model C-2, (c) test model C-4, and (d) test model C-6.

Numerical prediction of response

A numerical analysis was conducted, with $κ$ and $α_{C}$ as the main variables, to analytically verify the effect of the ERC damping system on the response reduction of an SDOF system. In the modeling, the SDOF system had a mass of $9.67 \times 10^{3}$ $9.67 \times 10^{3}$ kg and a lateral stiffness of 10,310 kN/m, and the structural damping ratio was 0.02. In addition, the value of the relative stiffness, $ρ$ , was assumed to be 100. A time history response analysis of the structural model was performed for cases in which $κ$ ranged from 0.25 to 3.0 and $α_{C}$ was 1.0 and 2.0. The yield displacement and the initial length of the steel dampers were assumed to be 2.0 and 300.0 mm, respectively. The earthquake ground motion record for the parametric analysis was N-S component of El-Centro Earthquake, 18 May 1940, recorded in the Imperial Valley in south-eastern Southern California.

The maximum values of inter-story drift, story shear force, and acceleration response for the assumed relative stiffness are compared in Figure 16. For the $α_{C}$ of 1.0, the maximum displacement response (inter-story drift) of each structure model was reduced by about 50% $[κ = 0.5]$ and 73% $[κ = 2.0]$ . For the $R_{D}$ of 2.0, the maximum displacement response of each structure was reduced by about 64% $[κ = 0.5]$ and 80% $[κ = 2.0]$ , as shown in Figure 16(a). Figure 16(b) and (c) show that although the maximum inter-story drift of the entire structure was reduced with the increase in relative stiffness, the story shear force and the acceleration response tended to increase when the relative stiffness exceeded 0.5. The maximum acceleration response of the entire structural system with $R_{D}$ of 2.0 was about 37% higher than that of $R_{D}$ of 1.0. Analytical results showed that the seismic response does not always decrease in proportion to the magnification factor, $α_{C}$ . This means that for the practical design of the proposed damping system, an analytical review of the optimal relative stiffness and magnification factor may be required to reduce the seismic responses of building structures, even though the proposed damping system is effective at controlling the seismic response.

Figure 16.

Maximum inter-story drift, story shear force, and acceleration response for various relative stiffness: (a) maximum inter-story drift, (b) maximum story shear force, and (c) maximum acceleration response.

Conclusion

An advanced vibration control system, termed an eccentric rotational channel-type damping system (ERC damping system), has been proposed. The ERC damping system has the advantage of incorporating any type of damper, providing a wider field of view to residents compared with existing story-installation-type damping systems, as well as magnifying the efficiency by modifying the geometry of the secondary system. Furthermore, due to its own displacement mechanism, the proposed damping system is very attractive when considering buckling issues present in existing metallic dampers.

The mechanical behavior of the ERC damping system and experimental results that demonstrate the validity of a simple behavior prediction model were presented. The parametric analysis, developed over a one-story model with the ERC damping system, showed that (a) the efficiency of the ERC damping system decreases as the capacity of the steel damper increases; however, this can be increased by modifying the magnification factor, $α_{C}$ , even if the secondary system has insufficient elastic lateral stiffness, and (b) the force-displacement relation for the ERC damping system can be idealized by a bilinear model as the yield strength of steel dampers becomes larger.

Cyclic loading and free vibration tests with a full-scale test model showed that the experimental results were in good agreement with the numerical analysis. Furthermore, the ERC damping system designed in the present study effectively reduced the vibration of the test model in the target amplitude range.

The numerical analysis, conducted for an SDOF system with a hysteretic steel damper, showed the effects of the ERC damping system on seismic response reduction under earthquake ground motion. It should be noted, however, that increases in the relative stiffness or the magnification factor are not always proportional to a decrease in the seismic response of structures. Thus, further numerical simulations are needed to investigate the damping effects of various types of dampers and the relationship between seismic response reduction and existing structural variables. In order to ensure an effective deformation ratio of the ERC damping system of more than 90%, it is recommended to limit the magnification factor, $α_{C}$ , to values more than 1.25 at a value of 10 of relative stiffness, $ρ$ .

Footnotes

Appendix 1 Acknowledgements

Special thanks are extended to ACEONE Tech Co., Ltd, for technical support and advice in conducting this research.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1A6A1A03032988); the International Science and Business Belt Program through the Ministry of Science and ICT (2015-DD-RD-0068-05); and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1A2B3001656).

ORCID iD

Kil-Hee Kim

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