A novel energy dissipation outrigger system with rotational inertia damper

Abstract

Rotational inertia damper, a novel damper, possessing the advantage of displacement amplification, has been employed in outrigger system for seismic mitigation. The equivalent analysis model composed by a uniform cantilever beam and an equivalent spring was proposed to simulate the rotational inertia damper outrigger system, by which the corresponding dynamic characteristic equation was derived based on numerical assembly technique. To gain the response of the damped system, finite element method and state space method have been utilized. Finally, the results show that the pseudo-undamped natural frequency ratios and system modal damping ratios are significantly influenced by stiffness parameter of the exterior column, while the mass parameter of the rotational inertia damper has little effect on them. The optimal damping ratio can be acquired for one mode, but it may be worse for the other mode in the same position equipping rotational inertia damper. Furthermore, numerical simulation results for the typical earthquake records have verified that the rotational inertia damper outrigger has excellent control performance in displacement as well as acceleration. A good agreement between damping force and equivalent force also suggests that the damping force of rotational inertia damper is predominant and the inertial force has no significant effect on the structure.

Keywords

dynamic characteristics numerical assembly technique optimal position and damping rotational inertia damper outrigger seismic mitigation

Introduction

The structural safety, especially for tall building, greatly affects the society’s stability, human lives, and property. Therefore, suppressing the excess vibration for tall building in a reasonable range is every engineer’s mission. However, it has been known that the assumed damping for many high-rise buildings is overestimated (Satake et al., 2003; Soong and Spencer, 2002; Smith and Willford, 2007), which makes it even more important to enhance energy dissipation for the structure. Fortunately, many approaches such as passive control and active control are gradually introduced into the building improving the damping performance for vibration suppression.

Recently, damped outrigger system has been proposed by Smith and Willford (2007) to reduce the excess dynamic response. Practical engineering has proved that it can provide higher damping ratio for the structure (Willford et al., 2008). In order to explore the damping characteristics, Chen et al. (2010) assumed that the core is a cantilever beam with infinite stiffness for peripheral column and presented relatively accurate approximations for the pseudo-undamped natural frequency and damping ratio of the first mode. The semi-active control device, such as magnetorheological (MR) dampers, was as well adopted to replace the traditional dampers installed between peripheral columns and outriggers, which suggests that MR damper can be a good alternative to control (Chang et al., 2013; Wang et al., 2010). To verify the previous theoretical studies, a real-time hybrid simulation test has been conducted which demonstrates the efficacy of the smart outrigger damping system (Asai et al., 2013). It should be noted that the stiffness of the perimeter column is not taken into consideration, while Tan et al. (2014) pointed out that it has a significant effect on modal damping ratios.

The beam with additional dampers to lower the unwanted vibration has been investigated before. The complex modal analysis was adopted by Oliveto et al. (1997) to develop the supported beam with two rotational viscous dampers attached at two ends, and the relation to optimal damping and impulse response of the continuous beams was illustrated (Krenk, 2004). The frequency equation of the Bernoulli–Euler beam with attachments such as concentrated masses, springs, or dampers was derived to investigate the dynamic characteristics (Gürgöze, 1996; Gürgöze and Erol, 2002). In addition to that, Timoshenko beam carrying masses and translational and rotational springs was investigated by analytical-and-numerical-combined method (ANCM), and it turned out that the method can be used for the determination of natural frequency and its mode shapes (Wu and Huang, 1995).

Inerter device was first proposed by Smith (2002) and then the corresponding mechanical property has been tested and analyzed (Papageorgiou and Smith, 2005). Inerter devices, in many cases, are utilized to be a mass amplifier with heavy flywheel; for example, tuned mass damper (TMD) with an additional inerter improves its damping effect (Marian and Giaralis, 2014), meanwhile tuned viscous mass damper (TVMD; Ikago et al., 2012) and tuned inerter damper (TID) (Lazar et al., 2013) become the alternative approaches instead of TMD due to larger mass amplification for the inerter. On the other hand, researchers expected to magnify deformation for effective energy dissipation with a smaller inerter, just as the case for the rotational inertial damper (RID) having a ball screw amplifying deformation without the flywheel (Hwang et al., 2007). In this article, RID will be introduced to participate in energy dissipation for an outrigger system.

This article presents the vibration equation for the RID outrigger system which can be simplified as a cantilever beam with damping equivalent springs based on Bernoulli–Euler beam theory. With numerical assembly technique (Wu and Chou, 1999), a complex dynamic characteristic equation satisfying the boundary conditions is formulated to obtain the pseudo-undamped natural frequencies and modal damping ratios. Furthermore, a finite element model for the RID outrigger system is built to verify the proposed method and solve the response, global optimal damping, and position. Various performance parameters of the RID outrigger system were investigated to identify their influences and provide some suggestions for engineering application.

RID

The basic schematic diagram of RID is shown in Figure 1. The detailed configuration will not be given here as different RIDs can be described in the same mechanism. From Figure 1, we can see that a linear motion is transformed to rotational movement due to the ball screw; meanwhile, the small linear displacement is amplified so that more energy from the system can be dissipated in the viscous material with a relatively small damping coefficient.

Figure 1.

RID model.

An RID can be represented as shown in Figure 2 and the force applied to it can be determined using the following expression

F = b (\overset{\cdot\cdot}{u} - {\overset{\cdot\cdot}{u}}_{c}) + c_{b} (\overset{\cdot}{u} - {\overset{\cdot}{u}}_{c})

(1)

where $F$ is the equivalent force caused by RID, $b$ is the nominal mass which is equivalent amplified mass because of rotational motion, $c_{b}$ is the equivalent amplified damping coefficient of the RID (Hwang et al., 2007), and $u$ and $u_{c}$ are the displacements of the upside and downside of the RID, respectively.

Figure 2.

Simplified model of RID.

The force–deformation curves in Figure 3 show the hysteretic behavior of viscous, inertia, and rotational inertia dampers. It can be seen that the RID has the characteristics of general viscous damper and pseudo-negative stiffness caused by inertia.

Figure 3.

Force–deformation relation: (a) viscous damper, (b) inertia damper, and (c) rotational inertia damper.

Modeling of the outrigger system with RID

A new outrigger system with RID can be developed considering the attractive characteristics of RIDs. Figure 4(a) presents a simplified model of a high-rise building with the proposed system and an equivalent model in which the outrigger system is represented by a rotational spring (Figure 5). The RIDs are symmetrically equipped in the outriggers. Assume that the bending stiffness of the outrigger is infinite and the length of each outrigger is r. The bending stiffness of the core is EI and m represents the mass per unit length of the core. E_cA_c is the axial stiffness of the peripheral column, H is the length of the core, and $α H$ is the location of the RID outrigger.

Figure 4.

Model of a building with an RID outrigger system: (a) simplified model and (b) simplified model of RID with peripheral column.

Figure 5.

Equivalent model of a building with an RID outrigger system.

As ignoring deformation of the exterior columns in the analysis of the outrigger system may result in numerical errors (Tan et al., 2014), the peripheral columns will be included in the models in this study. A mechanical model of the RID with a peripheral column is presented in Figure 4(b), where the model is drawn in fuchsia dot-dash line box. Then, the governing equation of this model can be expressed as

{\begin{matrix} F_{d} = b (\overset{\cdot\cdot}{u} - {\overset{\cdot\cdot}{u}}_{c}) + c_{b} (\overset{\cdot}{u} - {\overset{\cdot}{u}}_{c}) \\ b ({\overset{\cdot\cdot}{u}}_{c} - \overset{\cdot\cdot}{u}) + c_{b} ({\overset{\cdot}{u}}_{c} - \overset{\cdot}{u}) + k_{e} u_{c} = 0 \end{matrix}

(2)

where $F_{d}$ is the equivalent force caused by RID and $k_{e} = E_{c} A_{c} / α H$ represents the axial stiffness of the outer column.

With Laplace transform, equation (2) can be expressed as

F_{d} = Z_{u} u

(3)

where

Z_{u} = \frac{k_{e} (c_{b} s + b s^{2})}{k_{e} + c_{b} s + b s^{2}}

(4)

with $s = i ω$ . Considering that $u = θ r$ , the resistant moment of the RID outrigger is

M_{θ} = - 2 F_{d} r = - k_{θ} θ

(5)

where $k_{θ}$ is the equivalent resistant rotational stiffness of the RID with the outer column

\begin{matrix} k_{θ} & = \frac{2 k_{e} (c_{b} s + b s^{2}) r^{2}}{k_{e} + c_{b} s + b s^{2}} \\ = \frac{EIk}{H} (\frac{2 kci - 2 k^{3} η^{2}}{1 + 2 k^{2} c α β i - 2 k^{4} α β η^{2}}) \end{matrix}

(6)

where

{\begin{matrix} η = \sqrt{\frac{b r^{2}}{m H^{3}}} = \frac{r}{H} \sqrt{μ_{b}} \\ k^{4} = \frac{m H^{4}}{EI} ω^{2} \\ c = \frac{c_{b} r^{2}}{H \sqrt{mEI}} \\ β = \frac{EI}{2 E_{c} A_{c} r^{2}} \end{matrix}

(7)

with $μ_{b}$ being a nominal mass ratio for the RID

μ_{b} = \frac{b}{mH}

(8)

Thus, with the equivalent model shown in Figure 5, the new outrigger system can be analyzed.

Dynamic characteristics of the RID outrigger system

Dynamic equation formulation

The motion equation of a vibrating Bernoulli–Euler beam with no damping can be expressed as

EI \frac{\partial^{4} y (x, t)}{\partial x^{4}} + m \frac{\partial^{2} y (x, t)}{\partial t^{2}} = 0

(9)

where $y (x, t)$ represents the transverse displacement. The solution for equation (9) is assumed to be in the form

y (x, t) = Y (x) e^{i ω t}

(10)

Nondimensionalizing by $\tilde{Y} = Y / H$ , $\tilde{x} = x / H$ , and separating variables for equation (9) yields

\tilde{Y} ″ ″ + k^{4} \tilde{Y} = 0

(11)

The general solution of equation (11) can be expressed as

\begin{matrix} \tilde{Y} (\tilde{x}) = F_{1} \cosh (k \tilde{x}) + F_{2} \sinh (k \tilde{x}) \\ + F_{3} \cos (k \tilde{x}) + F_{4} \sin (k \tilde{x}) \end{matrix}

(12)

where $F_{υ} (υ = 1, 2, 3, 4)$ are unknown constants that can be determined using boundary conditions.

In this study, only one pair of RID outriggers is used, and therefore the cantilever beam shown in Figure 4(a) is divided into two segments, with the lower part as the first segment. Let the solutions for the segments be expressed as

\begin{matrix} {\tilde{Y}}_{j} (\tilde{x}) = F_{j 1} \cosh (k \tilde{x}) + F_{j 2} \sinh (k \tilde{x}) \\ + F_{j 3} \cos (k \tilde{x}) + F_{j 4} \sin (k \tilde{x}) \end{matrix}

(13)

where $F_{j υ} (j = 1, 2; υ = 1, 2, 3, 4)$ are unknown for the two segments. In order to develop the characteristics of the RID outrigger system, the numerical assembly technique (Wu and Chou, 1999) is employed to derive dynamic characteristic equations. The continuity conditions at the connection point $\tilde{x} = α$ are

{\tilde{Y}}_{1} (α) = {\tilde{Y}}_{2} (α)

(14)

{\tilde{Y}}_{1}^{'} (α^{-}) = {\tilde{Y}}_{2}^{'} (α^{+})

(15)

{\tilde{Y}}_{1}^{''} (α^{-}) + \frac{k_{θ} H}{EIk} {\tilde{Y}}_{1}^{'} (α^{-}) - {\tilde{Y}}_{2}^{''} (α^{+}) = 0

(16)

{\tilde{Y}}_{1}^{'''} (α^{-}) = \tilde{Y} 2^{'''} (α^{+})

(17)

where the superscripts “–” and “+” of $α$ indicate the left and right derivatives, respectively. Substituting equation (13) into equations (14) to (17) yields

[\begin{matrix} D_{1} & D_{2} \end{matrix}] {\begin{matrix} F_{1} \\ F_{2} \end{matrix}} = 0

(18)

where

F_{1} = {[\begin{matrix} F_{11} & F_{12} & F_{13} & F_{14} \end{matrix}]}^{T}

(19)

F_{2} = {[\begin{matrix} F_{21} & F_{22} & F_{23} & F_{24} \end{matrix}]}^{T}

(20)

D_{1} = [\begin{matrix} \cosh (k α) & \sinh (k α) & \cos (k α) & \sin (k α) \\ \sinh (k α) & \cosh (k α) & - \sin (k α) & \cos (k α) \\ β_{1} & β_{2} & β_{3} & β_{4} \\ \sinh (k α) & \cosh (k α) & \sin (k α) & - \cos (k α) \end{matrix}]

(21)

D_{2} = [\begin{matrix} - \cosh (k α) & - \sinh (k α) & - \cos (k α) & - \sin (k α) \\ - \sinh (k α) & - \cosh (k α) & \sin (k α) & - \cos (k α) \\ - \cosh (k α) & - \sinh (k α) & \cos (k α) & \sin (k α) \\ - \sinh (k α) & - \cosh (k α) & - \sin (k α) & \cos (k α) \end{matrix}]

(22)

with

β_{1} = \cosh (k α) + k_{p} \sinh (k α)

(23)

β_{2} = \sinh (k α) + k_{p} \cosh (k α)

(24)

β_{3} = - \cos (k α) - k_{p} \sin (k α)

(25)

β_{4} = - \sin (k α) + k_{p} \cos (k α)

(26)

k_{p} = \frac{k_{θ} H}{kEI}

(27)

The boundary conditions for the fixed end of the left of the first segment, namely $\tilde{x} = 0$ , are

{\begin{matrix} {\tilde{Y}}_{1} (0) = 0 \\ {\tilde{Y}}_{1}^{'} (0^{+}) = 0 \end{matrix}

(28)

substituting equation (13) into which yields

D_{α 0} F_{1} = 0

(29)

where

D_{α 0} = [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}]

(30)

The boundary conditions for the free end of the right of the second segment $\tilde{x} = 1$ are

{\begin{matrix} {\tilde{Y}}_{2}^{''} (1^{-}) = 0 \\ {\tilde{Y}}_{2}^{'''} (1^{-}) = 0 \end{matrix}

(31)

substituting equation (13) into which yields

D_{α 1} F_{2} = 0

(32)

where

D_{α 1} = [\begin{matrix} \cosh (k) & \sinh (k) & - \cos (k) & - \sin (k) \\ \sinh (k) & \cosh (k) & \sin (k) & - \cos (k) \end{matrix}]

(33)

Integrating equations (18), (29), and (32) into a characteristic equation, one can get

DF = 0

(34)

where

D = [\begin{matrix} D_{α 0} & 0_{2 \times 4} \\ D_{1} & D_{2} \\ 0_{2 \times 4} & D_{α 1} \end{matrix}]

(35)

F = {\begin{matrix} F_{1} \\ F_{2} \end{matrix}}

(36)

To obtain the nontrivial solution for equation (34), the determinant of the coefficient matrix must vanish

| D | = 0

(37)

After rearrangement of equation (37), a new equation is obtained as follows

\begin{matrix} k_{p} [\cos (k - 2 k α) \sinh (k) + \cosh (k - 2 k α) \sin (k) \\ + 2 \cos (k α) \sinh (k α) + 2 \cosh (k α) \sin (k α) \\ + \cos (k) \sinh (k) + \cosh (k) \sin (k) \\ - 2 \cos (k - k α) \sinh (k - k α) \\ - 2 \cosh (k - k α) \sin (k - k α)] \\ + 4 [\cos (k) \cosh (k) + 1] = 0 \end{matrix}

(38)

When $η = 0$ , the above is in agreement with the results obtained by Tan et al. (2014)

\begin{matrix} ick {\cos (k - 2 k α) \sinh (k) + \cosh (k - 2 k α) \sin (k) \\ + 2 \cos (k α) \sinh (k α) + 2 \cosh (k α) \sin (k α) \\ + \cos (k) \sinh (k) + \cosh (k) \sin (k) \\ - 2 \cos (k - k α) \sinh (k - k α) \\ - 2 \cosh (k - k α) \sin (k - k α) \\ + 4 k β α [\cos (k) \cosh (k) + 1]} \\ + 2 [\cos (k) \cosh (k) + 1] = 0 \end{matrix}

(39)

However, if the axial stiffness of the column is simultaneously viewed as infinite, like $β = 0$ , one obtains another equation

\begin{array}{l} 2 c k i [\cos (k - 2 k α) \sinh (k) + \cosh (k - 2 k α) \sin (k) \\ + 2 \cos (k α) \sinh (k α) + 2 \cosh (k α) \sin (k α) \\ + \cos (k) \sinh (k) + \cosh (k) \sin (k) \\ - 2 \cos (k - k α) \sinh (k - k α) \\ - 2 \cosh (k - k α) \sin (k - k α)] \\ + 4 [\cos (k) \cosh (k) + 1] = 0 \end{array}

(40)

Equation (40) is the characteristic equation of the structure with damped outriggers (Chen et al., 2010). Based on equation (38), the complex frequency of the RID outrigger system can be expressed in the following form

i ω = - ξ ω_{0} + i \sqrt{1 - ξ^{2}} ω_{0}

(41)

where $ω_{0}$ is the pseudo-undamped natural circular frequency and $ξ$ is the damping ratio for the system, which are calculated by

ξ = - \frac{Re (i ω)}{| i ω |}

(42)

ω_{0} = | i ω |

(43)

Dynamic characteristic analysis

Structural safety is commonly related to dynamic characteristics. In order to avoid or reduce unwanted vibration, it is common practice to keep the natural frequencies of the structure away from the dominant frequency for earthquake. One of the other methods is to enhance the structural damping ratio. The use of the RID outrigger system belongs to the latter.

In order to verify the proposed analysis method, Figure 6 shows the comparisons between the finite element method (FEM) and the analysis method with $β = 1, η = 0.002$ , where “FEM” stands for finite element method, “Analysis*” represents the proposed method, and the “*” represents the corresponding mode. Obviously, the curves for frequency and damping ratio versus $c$ or $α$ indicate that the results obtained by FEM and the proposed method agree very well.

Figure 6.

Comparisons between the FEM method and the analysis method $(β = 1, η = 0.002)$ : (a) frequency versus $c$ $(α = 0.5)$ , (b) damping ratio versus $c$ $(α = 0.5)$ , (c) frequency versus $α$ $(c = 0.2)$ , and (d) damping ratio versus $α$ $(c = 0.2)$ .

The position of the RID outriggers influences the performance for structure vibration attenuation. Figure 7 illustrates the changes of modal frequency ratios and modal damping ratios with the damping parameter c when RID is installed at 0.55 H. It is to be noted that, as similar conclusions can be drawn when RID is installed at other positions, the corresponding results will not be presented here. Figure 7(a), (c), and (e) shows the pseudo-undamped natural frequency ratio of the RID outrigger system in the first, second, and third modes, respectively, where $ω_{j}, j = 1, 2, 3,$ are the first three natural frequencies for the undamped cantilever beam. Apparently, each frequency ratio increases with the damping parameter c from 0 to 1 and gradually tends to be stable, from which it can be explained that RID behaves as a rigid bar to that of $c \to \infty$ . This is also in accordance with the results of Tan et al. (2014). Interestingly, pseudo-undamped natural frequency ratios in the first mode totally acquire the highest value, while the counterparts in the third mode take the second place and the lowest at the second mode. This phenomenon suggests that the RID outriggers installed at 0.55 H causes less effect on the frequency in the second mode. In effect, these conclusions are appropriate for damping ratios analyzed later as well. Furthermore, the larger the $β$ value is, the lower the frequency ratio that is acquired. It is worth noting that the case $β = 0$ representing infinite stiffness of the perimeter column can give the largest frequency ratio. Figure 7(a) also shows that the mass parameter $η$ for RID hardly plays a role in the frequency ratio except that a quite slight difference in the third mode as shown in Figure 7(c), which indicates that pseudo-negative stiffness caused by rotational inertia of the RID can be ignored even if $η = 0.01$ selected is far larger than its actual physical mass. This information can also be easily validated in the thumbnails for $η = 0$ , $η = 0.0001$ , $η = 0.001$ , and $η = 0.01$ , respectively, when $β = 0$ . It should be known that considering the space and convenience, other situations for $β = 0.5$ , $β = 1$ , and $β = 4$ of the thumbnails have not been presented here (Figure 7(a)), and also the same for other diagrams in Figures 7 and 8.

Figure 7.

The first three frequency ratios and damping ratios for $α = 0.55$ : (a) first frequency ratio versus $c$ , (b) first damping ratio versus $c$ , (c) second frequency ratio versus $c$ , (d) second damping ratio versus $c$ , (e) third frequency ratio versus $c$ , and (f) third damping ratio versus $c$ .

Figure 8.

The first three frequency ratios and damping ratios for $c = 0.2$ : (a) first frequency ratio versus $α$ , (b) first damping ratio versus $α$ , (c) second frequency ratio versus $α$ , (d) second damping ratio versus $α$ , (e) third frequency ratio versus $α$ , and (f) third damping ratio versus $α$ .

Figure 7(b), (d), and (f) presents the modal damping ratios varying with the damping parameter $c$ . An optimal damping ratio can be obtained; however, these ratios reveal the discrepancies for different modes, which show that there are no optimal damping ratios for all modes. In addition to that, the lower the mode is, the larger the damping ratio is, but a slight distinction is found for the second mode damping ratio because the outrigger located at 0.55 H is rightly close to the modal shape node of the undamped cantilever beam resulting in much less effect on damping. It can also be observed that the stiffness ratio of the perimeter column has a significant effect on the damping ratio, for example, the largest damping ratio for $β = 0$ and the smallest one for $β = 4$ . Similarly, one can find that the different line types such as dashed line for $η = 0.01$ , dash-dotted line for $η = 0.001$ , dotted line for $η = 0.0001$ , and solid line for $η = 0$ in Figure 7(a) overlap each other except the quite small difference for the third mode, which indicates that the mass parameters for the RID have no essential effect on the modal damping ratio.

Further detailed information about the pseudo-undamped natural frequency ratios and modal damping ratios versus the position parameter $α$ for the RID outrigger with $c = 0.2$ is shown in Figure 8. Similarly, thumbnails based on the consideration of the space and convenience are plotted for $β = 0$ in Figure 8(a), but no repetition for $β = 0.5, 1, 4$ and for Figure 8(b) to (f). It is interesting that one, two, and three peaks for the pseudo-undamped natural frequency ratio vary with the position $α$ increasing for the first, second, and third modes, respectively, and the analogous phenomena can be observed for damping ratios. Obviously, the optimal damping ratio may be acquired in one site for the certain mode, while poor results may be obtained for the other modes in the same position from the analysis of Figure 8(b), (d), and (f). As shown in Figure 8(a), the pseudo-undamped natural frequency ratio in the first mode increases with $α$ ranging from 0 to 1. It can be found that the frequency ratio can reach a relatively larger value for $β = 0$ with $α \to 1$ , whereas $β = 0.5$ , $β = 1$ , and $β = 4$ can reach their lower maximum in different peaks mentioned above. It can also be observed that the curves for $β = 0$ under different mass parameters such as $η = 0$ , $η = 0.0001$ , $η = 0.001$ , $η = 0.01$ always coincide with each other, as well as the same cases for $β = 0.5$ , $β = 1$ , $β = 4$ in Figure 8(a) and for Figure 8(b) to (f). This also demonstrates that mass parameters $η$ hardly play a role on the frequency ratios and damping ratios varying versus position of the RID outrigger.

FEM

To determine the response of the outrigger structure with RID under seismic excitations, FEM is utilized. The dynamic equation for the system can be expressed as

M \overset{\cdot\cdot}{u} + C \dot{u} + K \dot{u} = - M I {\overset{\cdot\cdot}{u}}_{g} + f

(44)

where $M$ , $C$ , $K$ are the mass matrix, damping matrix, and stiffness matrix, respectively, $u$ is the displacement vector, ${\overset{\cdot\cdot}{u}}_{g}$ is the acceleration of horizontal ground motion, $I$ is the unit vector, and $f$ is the equivalent load vector for the forces generated by the RID outrigger.

The high-rise building is modeled by conventional Bernoulli–Euler beam element. The mass matrix for one element can be written as

M_{e} = \frac{mL}{420} [\begin{matrix} 156 & 22 L & 54 & - 13 L \\ 22 L & 4 L^{2} & 13 L & - 3 L^{2} \\ 54 & 13 L & 156 & - 22 L \\ - 13 L & - 3 L^{2} & - 22 L & 4 L^{2} \end{matrix}]

(45)

where $L$ stands for the element length. Similarly, the stiffness matrix for one element is

K_{e} = \frac{EI}{L^{3}} [\begin{matrix} 12 & 6 L & - 12 & 6 L \\ 6 L & 4 L^{2} & - 6 L & 2 L^{2} \\ - 12 & - 6 L & 12 & - 6 L \\ 6 L & 2 L^{2} & - 6 L & 4 L^{2} \end{matrix}]

(46)

The inertial force caused by earthquake on one element is given by

F_{* e} = \frac{- m {\overset{\cdot\cdot}{u}}_{g} L}{12} {[\begin{matrix} 6 & L & 6 & - L \end{matrix}]}^{T}

(47)

Assume that the building is modeled with n beam elements and the displacement vector contains displacements for nodes $1, 2, \dots, n$ and can be expressed as

u = {u_{1}, θ_{1}, u_{2}, θ_{2}, \dots, u_{n}, θ_{n}}

(48)

Furthermore, assume that the RID outrigger is installed at node j. The displacement $u_{c}$ in the RID is added to the displacement vector and the displacement components at node j will be $u_{j}, θ_{j}, u_{c}$ . Thus, the displacement vector to be considered will be

u_{f} = {u_{1}, θ_{1}, \dots, u_{j - 1}, θ_{j - 1}, u_{j}, θ_{j}, u_{c}, u_{j + 1}, θ_{j + 1}, \dots, u_{n}, θ_{n}}

(49)

The contributions of the outrigger to the global matrices will be

{\begin{matrix} m_{r} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 2 b r^{2} & - 2 br \\ 0 & - 2 br & 2 b \end{matrix}] \\ c_{r} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 2 c_{b} r^{2} & - 2 c_{b} r \\ 0 & - 2 c_{b} r & 2 c_{b} \end{matrix}] \\ k_{r} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 2 k_{e} \end{matrix}] \end{matrix}

(50)

By assembling the beam elements using the above equations, one can obtain the dynamic equation for the whole system

M_{f} {\overset{\cdot\cdot}{u}}_{f} + C_{f} {\overset{\cdot}{u}}_{f} + K_{f} u_{f} = Γ {\overset{\cdot\cdot}{u}}_{g}

(51)

where $M_{f}$ , $C_{f}$ , $K_{f}$ are the global mass matrix, global damping matrix and global stiffness matrix, and $Γ {\overset{\cdot\cdot}{u}}_{g}$ made by equation (51) is the earthquake action vector in which the location for RID outrigger is set to be 0.

Introducing the state variable in

z = {\begin{matrix} u \\ \overset{\cdot}{u} \end{matrix}}

(52)

equation (52) can be written in the following form

\overset{\cdot}{z} = Az + B {\overset{\cdot\cdot}{u}}_{g}

(53)

where

A = [\begin{matrix} 0 & I \\ - M_{f}^{- 1} K_{f} & - M_{f}^{- 1} C_{f} \end{matrix}]

(54)

B = [\begin{matrix} 0 \\ - M_{f}^{- 1} Γ \end{matrix}]

(55)

In this study, the Kanai–Tajimi filtered white noise ground motion model (Housner, 1955), which is utilized to obtain the optimum parameters, can be characterized by the following form

S_{{\overset{\cdot\cdot}{u}}_{g}} = \frac{ω_{g}^{4} + 4 ξ_{g}^{2} ω_{g}^{2} ω^{2}}{{(ω_{g}^{2} - ω^{2})}^{2} + 4 ξ_{g}^{2} ω_{g}^{2} ω^{2}} S_{0}

(56)

where $S_{0}$ is the input power spectrum intensity of the earthquake and $ω_{g}$ and $ξ_{g}$ are the natural frequency and damping ratio of the site soil, respectively. Let equation (56) be a single degree of system and extend equation (53) with it; thus, the extended state equation can be expressed as

{\overset{\cdot}{z}}_{T} = A_{T} z_{T} + B_{T} w_{0} (t)

(57)

where

A_{T} = [\begin{matrix} A & B C_{w} \\ 0 & A_{w} \end{matrix}]

(58)

B_{T} = [\begin{matrix} 0 \\ B_{w} \end{matrix}]

(59)

z_{T} = {\begin{matrix} z \\ z_{w} \end{matrix}}

(60)

A_{w} = [\begin{matrix} 0 & 1 \\ - ω_{g}^{2} & - 2 ξ_{g} ω_{g} \end{matrix}]

(61)

B_{w} = [\begin{matrix} 0 \\ - 1 \end{matrix}]

(62)

Let $E_{z_{T}} = E (z_{T} z_{T}^{T})$ and $E_{w} = B_{T} 2 π S_{0} B_{T}^{T}$ . Then, the Lyapunov equation can be written as

\frac{d E_{z_{T}}}{dt} = A_{T} E_{z_{T}} + E_{z_{T}} A_{T}^{T} + E_{w}

(63)

For the stationary response of the structure subjected to stationary random excitation, equation (63) reduces to a simpler form

A_{T} E_{Y_{T}} + E_{Y_{T}} A_{T}^{T} + E_{w} = 0

(64)

Therefore, the mean square response can be determined by solving the last equation.

Simulation analysis

A numerical example for the RID outrigger system is provided to investigate its parameters and performance of the seismic mitigation. The main model parameters are as follows: $EI = 1.47 \times 10^{13} N m^{2}$ , $m = 1.08 \times 10^{5} kg / m$ , $H = 200 m$ , $r = 15 m$ , and $ξ_{0} = 0.02$ . The idea seeking the optimum damping ratios for different modes in section “Dynamic characteristics of the RID outrigger system” may be a good solution for structural mitigation; however, it is not a global approach. In this article, top displacement variance is minimized to find the global optimal parameters and the corresponding minimal variance based on Kanai–Tajimi filtered white noise earthquake motion model with power density $S_{0} = 0.0048 m^{2} / rad s^{3}$ , $ω_{g} = 10.55 rad / s$ , and $ξ_{g} = 0.5$ for mid-firm soil (Lin and Yan, 1987; Zhu and Xu, 2005) is employed.

Figure 9 illustrates the displacement variance at the top of the structure versus $c$ and $α$ for the given values of $β$ and $η$ . When comparing Figure 9(a) and (c) as well as Figure 9(b) and (d), the wider areas with the lower $σ^{2}$ value can be observed for larger $α$ and appropriate lower $c$ values. In addition, it also suggests that optimal parameters such as $c$ and $α$ are slightly larger for $β = 0.5$ (the perimeter column tending to be rigid) when compared with $β = 4$ (the perimeter column tending to be soft). Furthermore, as shown in Figure 9(a) to (d), one can find that $η$ has less impact on $σ^{2}$ .

Figure 9.

Top displacement variance versus $c$ and $α$ : (a) $β = 0.5, η = 0.0001$ , (b) $β = 0.5, η = 0.1$ , (c) $β = 4, η = 0.0001$ , and (d) $β = 4, η = 0.1$ .

Figure 10 shows the optimal position parameter $α_{opt}$ versus the damping parameter $c$ and the optimal damping parameter $c_{opt}$ versus the position parameter $α$ . Figure 10(a) shows that $α_{opt}$ is close to 1 for very small values of the damping parameter $c$ , and it monotonically decreases with increasing $c$ ; but generally it will be smaller when the stiffness parameter $β$ increases. On the other hand, the mass parameter $η$ has a relatively small effect on $α_{opt}$ . A similar analysis can be carried out on the relationship between $c_{opt}$ and the position parameter $α$ and the related information will not be presented again.

Figure 10.

Optimum parameters: (a) $α_{opt}$ versus $c$ and (b) $c_{opt}$ versus $α$ .

Two typical seismic records are applied to investigate the performance of the RID outrigger system. The two records are for the North and South component of El Centro earthquake in 1940 and Kobe earthquake in 1995, both of which are adjusted to the maximum ground acceleration of 3 m/s². The known parameters of the RID are listed in Table 1 and the corresponding optimal position $α_{opt}$ is obtained by minimizing top displacement variance under Kanai–Tajimi filtered white noise earthquake excitation. Mass information of RID can be deduced from Table 2 with the determined $η$ .

Table 1.

Optimal position of the outrigger.

$c$	$β$	$η$	$α_{opt}$
0.2	0.5	0.002	0.65
0.2	0.5	0.01	0.65
0.2	0.5	0.1	0.65

Table 2.

Mass parameters.

$η$	$μ_{b}$	$b$ (kg)
0.002	0.0007	1.54 × 10⁴
0.01	0.0178	3.84 × 10⁵
0.1	1.78	3.84 × 10⁷

From Table 2, one can observe that $η = 0.1$ may not be practical because the corresponding nominal mass of RID is far too large. It is also not quite reasonable for $η = 0.01$ with the reason that the nominal mass $b = 3.84 \times 10^{5} kg$ of RID means hundreds of times than its real physical mass of which is generally several tons. Commonly, it is not necessary for RID using flywheel to create the large inertial force (amplified mass); contrarily, the amplified mass for RID should be designed to be much smaller, for instance several times or 10 times the physical mass. However, $η = 0.002$ with the nominal mass $b = 1.54 \times 10^{4} kg$ is suitable for RID with several tons of physical mass.

The responses at the top of the structure under earthquake excitations for $c = 0.2$ , $β = 0.5$ , $η = 0.002$ , and $α_{opt} = 0.65$ are shown in Figures 11 and 12. The numerical simulation results show that the structure with RID outriggers has excellent control performance for displacement and acceleration. Reductions in peak responses are 70.42% in displacement and 81.82% in acceleration for Kobe wave, and 67.59% in displacement and 90.10% in acceleration for El Centro wave.

Figure 11.

Response under Kobe excitation: (a) displacement and (b) absolute acceleration.

Figure 12.

Response under El Centro excitation: (a) displacement and (b) absolute acceleration.

The above analysis results indicate that the use of RID is effective for suppressing the vibration. Figure 13 presents the force–displacement relationship of the structure with RID outriggers subjected to El Centro excitation. Curves for the total equivalent force of RID and just damping force are included. Note that the total equivalent force includes both damping force and inertial force caused by RID. It can easily be found that, for the two cases of $η = 0.002$ and $η = 0.01$ , the damping force of RID is predominant and the inertial force is negligible. For the third case, where the RID has an excessively large mass, histories of the total equivalent force of RID and damping force are significantly different, indicating that the inertial force of RID is important for this unpractical case.

Figure 13.

Force–deformation relationship for El Centro excitation: (a) $η = 0.002$ , (b) $η = 0.01$ , and (c) $η = 0.1$ .

Conclusion

RID was introduced into the outrigger system to suppress excessive vibrations of high-rise buildings. Based on Bernoulli–Euler beam theory, an equivalent analysis model was proposed to simulate the structure with an RID outrigger. The dynamic characteristic equation of the damped system was derived using numerical assembly technique, and the influence of the RID on the first three modes was investigated. Finite element analysis was also conducted on seismic responses of the high-rise building model to investigate the performance of the new outrigger system. The following conclusions can be drawn:

The use of an RID outrigger in high-rise buildings can be effective for earthquake mitigation. Numerical simulation shows that the RID outrigger system has an excellent control performance. The reductions in peak responses are 70.42% in displacement and 80.82% in acceleration for Kobe wave, and 67.59% in displacement and 90.10% in acceleration for El Centro wave.

The relative bending stiffness of the peripheral columns has a significant influence on the pseudo-undamped natural frequency ratios and modal damping ratios of the structure, while the mass parameter of RID has little effect. The optimal outrigger positions for maximum modal damping ratios are different for different modes, and the optimal position for one mode may not be appropriate for another mode. A global analysis of the structure subjected to the filtered white noise earthquake excitation has been conducted and it has been found that $α_{opt}$ and $c_{opt}$ optimized by minimal top variance decrease with the stiffness parameter $β$ increasing, but they have hardly been influenced by mass parameters.

The damping force in RID is predominant and the inertial force is insignificant. It may have significant influence on the damped system for larger $η$ values due to the large inertial force, but there is actually no possibility for that case.

Generally, the detailed research considering bending stiffness and shear stiffness should be further conducted to provide a complete understanding.

Footnotes

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 Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT13057) and the National Natural Science Foundation of China (Grant No. 51478129), and also in part by Guangdong Special Program (Grant No. 2014TX01C141) and Yangcheng Scholars Program (Grant No. 1201541630)

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