A new reduced model of damped moment frame for rapid optimization of viscous damper placement

A new model reduction method is presented for moment-resisting frames with viscous dampers. The proposed method considers (1) the interactive effect of added damping between the horizontal-vertical directions and (2) the direct effect with regard to the vertical direction different from the conventional static condensation method. When brace-type dampers are treated, these effects cannot be neglected originally. The proposed method gives a better correspondence of the fundamental eigenmode with that of the non-reduced moment frame than the static condensation method. In addition, the proposed reduced model can accurately evaluate the maximum interstory drifts of the non-reduced moment frame. The proposed model reduction method is formulated in the frequency domain. The frequency dependency of the stiffness matrix and the damping matrix of the reduced model is investigated. The inﬂuences of the added damping matrix by viscous dampers on the natural circular frequencies, the damping ratios and the participation vectors are also investigated. Finally, it is demonstrated through a numerical example that the displacement-based optimization of the damper allocation can be accurately and rapidly conducted by using the reduced models throughout the optimization process.


Introduction
Viscous dampers have been widely used for the vibration control of high-rise buildings since both of deformations and floor accelerations can be reduced. The optimal allocation of viscous dampers has been extensively investigated. [1][2][3][4][5] Takewaki 6 proposed an optimality criterion-based approach to the minimization of the sum of transfer function amplitudes evaluated at the undamped fundamental natural frequency of a structural system. Aydin et al. 7 extended the method of Takewaki 6 to the minimization of the transfer function amplitude of the base shear. Trombetti and Silvestri 8 investigated the efficiency of the mass proportional type damping systems. Nomura et al. 9 used the evolutionary structural optimization method to optimize the damper placement. Terazawa and Takeuchi 10 used the generalized response spectrum analysis in the damper optimization for saving the computational load. Harada and Yoshitomi 11 proposed an optimization method to determine the allocation of buckling restrained braces. Some researchers tackled the damper optimization problems for elastic-plastic structures. [12][13][14][15][16][17][18] When the optimization is conducted for large-scale structures, the computational efficiency should be considered seriously. Some researchers effectively use mathematical techniques (like as domain decomposition), [19][20][21][22][23][24] limitation or assumption of the solution (like symmetrical planning), 17,18,25 and reduced-order models 17,18,26 to improve the computational efficiency. It is also pointed out that the static condensation is a well-known method to reduce the behavioral order of moment frames. However, the static condensation method does not consider (1) the interactive effect of added damping between the horizontal-vertical directions and (2) the direct effect with regard to the vertical direction. Therefore, the static condensation method cannot determine the efficient distribution of brace-type dampers among different bays, although it can be used for the design of the damper distribution along the building height.
In this paper, a new model reduction method is presented for moment frames with viscous dampers. In Section 2, the proposed method and the conventional static condensation method are explained for showing the difference between the two methods. In Section 3, the frequency dependency of the stiffness matrix and the damping matrix of the reduced model is investigated. The influences of the added damping matrix by viscous dampers on the natural circular frequencies, the damping ratios and the participation vectors are also investigated. In addition, the time-history response analyses are conducted to compare the responses of the original moment frames and the reduced models. In Section 4, a numerical example is presented for the displacement-based optimization of the viscous damper placement. It is demonstrated that the only use of the proposed reduced models throughout the optimization process leads to an accurate and rapid search of the solutions.

Reduction Model of Moment Frame with Viscous Damper
In this section, a new model reduction method is presented for moment frames with viscous dampers. The conventional static condensation method is also explained to show the difference between these two methods.
In the conventional static condensation method, the stiffness matrix K eq of the reduced model is obtained by neglecting the damping and inertial forces and moments related to _ θ and € θ (see Figure 1). Then K eq is expressed as When the model has only the structural damping, the stiffness-proportional damping matrix C eq may be given by where h 1 , ω 1 denote the fundamental damping ratio by the structural damping and the undamped fundamental natural circular frequency (see Figure 1). When dampers are added, C eq may be express as where C add,11 denotes the submatrix related to _ u in the added damping matrix by viscous dampers. Equation (3b) means that the influences of the remaining damping submatrices C add,12 , C add,21 , C add,22 related to added dampers on the responses are not considered. In other words, the interactive effect of added damping between the horizontal-vertical directions and the direct effect with regard to the vertical direction are not considered. When brace-type dampers are treated, these effects cannot be neglected originally.

Proposed method
A new model reduction method is presented for moment frames with viscous dampers.
The neglect of the inertial force and moment due to J (see Figure 1) and the Fourier transformation of Equation (1) lead to The lower half of Equation (4) can be reduced to By substituting Equation (5) into the upper half of Equation (4), the following equation is obtained.
Equation (6) can be rewritten in the following form.
where e K eq ¼ Re It is noted that e K eq , e C eq depend on ω. It is also noted that C 11 , C 12 , C 21 , C 22 include both the structural damping and the added damping (see Figure 1), although the static condensation method treats them separately. Figure 1 shows the different treatment among the original frame model, the static condensation model and the proposed condensation model. By substituting ω ¼ ω eq (ω eq : a selected particular frequency), the stiffness matrix K eq and the damping matrix C eq of the proposed reduced model can be obtained as K eq ¼ e K eq ω eq ð Þ, C eq ¼ e C eq ω eq ð Þ: As indicated in Equation (5), the relation between U, Θ depends on ω. Equation (9) holds for all the frequency range. It is pointed out that the relation between U, Θ of undamped models is independent of ω, and C eq is not defined. It is also noted that this independent relation with regard to ω is common in the case of proportionally damped models (see Appendix 1). Assuming that the relation between U, Θ is quasi-static ω eq ! 0 ð Þ , K eq , C eq are obtained as The detailed formulation is shown in Appendix 2. K eq , C eq under ω eq ! ∞ are also formulated in Appendix 2. In the case of ω eq ! 0, K eq totally corresponds to the stiffness matrix by the static condensation. On the contrary, C eq in Equation (10b) considers the interactive effect of added damping between the horizontal-vertical directions and the direct effect with regard to the vertical direction. Since the relation between U, Θ is quasi-static and is not "totally" static, these effects are transferred through the stiffness matrix. These effects are not considered in the static condensation since it is a totally static method. In Section 3, it will be demonstrated that the settings of ω eq ! 0 (Equation (10)) is suitable for the calculation of K eq , C eq , and the reduced model under ω eq ! 0 evaluates the displacement and acceleration responses more accurately than that by the static condensation.

Accuracy of Proposed Reduction Method
In this section, the accuracy of the proposed reduction method is investigated. In Sections 3.1 and 3.2, the influences of ω eq on the natural circular frequencies, the damping ratios, and the participation vectors are investigated. In Section 3.3, the time-history response analyses are conducted to compare the responses of the original moment frames and the reduced models.
3.1 Reduction of one-story, one-span moment frame and onestory, 5-span moment frame Consider a one-story, one-span moment frame and two onestory, 5-span moment frames (see Figure 2). The crosssections of the beams and columns are common in the three frames. The structural damping ratio is 0.02 (stiffness proportional type). V-type linear viscous dampers are treated. The values of the mass and the added damping coefficient c of the dampers are set so that the natural circular frequencies and the damping ratios of the three frames become almost identical. Two patterns of the damper allocation are considered for the one-story, 5-span moment frames. The one-story, 5-span frame with damper only at the center span is designated as one-story, 5-span frame-A, and the one-story, 5-span frame with dampers at all the spans is designated as one-story, 5-span frame-B. Figures 3-5 show the natural circular frequencies ω nr and the damping ratios h nr of the non-reduced frames, and the natural circular frequencies ω r and the damping ratios h r of their reduced models. ω eq in the figures is the frequency which is substituted into Equation (9). It can be observed from Figures 3-5 that ω nr , h nr and ω r , h r correspond well in the range of ω eq ≤ ω nr . With the increase of ω eq , the value of ω r increases, and the value of h r decreases. Then, these converge on constant values, respectively, in the case of ω eq ! ∞. It is noted that, when the value of c becomes larger, the values of ω r , h r vary more largely with the increase of ω eq . These mean that the viscous dampers apparently add the stiffness to the models depending on ω eq and the amount of added damping. It is also noted that this tendency can be seen markedly in the case of Figure 4. This is because the nonproportionality of the damping matrix of the one-story, 5-span frame-A is stronger than the other frames. As explained in Section 2.2, e K eq , e C eq for the proportional damping do not change.

Reduction of 20-story, 3-span moment frame
Consider next a 20-story, 3-span moment frame. The details of the frame are shown in Figure 6. V-type linear viscous dampers are treated again. Figure 7 shows the 1-3th pseudoundamped natural circular frequencies ω nrÃ 1 , ω nrÃ 2 , ω nrÃ 3 and the damping ratios h nrÃ 1 , h nrÃ 2 , h nrÃ 3 of the frame by the complex modal analysis. The pseudo-undamped natural circular frequencies ω rÃ 1 , ω rÃ 2 , ω rÃ 3 and the damping ratios h rÃ 1 , h rÃ 2 , h rÃ 3 of the reduced model are also shown. In the range of ω eq ≤ ω nrÃ 1 , ω nrÃ 1 , h nrÃ 1 , and ω rÃ 1 , h rÃ 1 correspond well, and ω nrÃ 2 , ω nrÃ 3 , h nrÃ 2 , h nrÃ 3 differ from ω rÃ 2 , ω rÃ 3 , h rÃ 2 , h rÃ 3 by 13% at most, respectively. As in the cases of Section 3.1, the value of ω r increases and the value of h r decreases with the increase of ω eq . When ω eq ! 0 is adopted in the calculation of K eq , C eq , the fundamental modes of the non-reduced frames and the reduced models correspond well, and K eq , C eq are simply expressed by Equation (10). Furthermore, all the eigenmodes of the non-reduced frames and those of the reduced models correspond well when the amount of added damping is small. This is because K eq is equal to the stiffness matrix by the static condensation. Therefore, the settings of ω eq ! 0 is suitable for the calculation of K eq , C eq .

Comparison of original moment frame and reduced model through time-history response analysis
The 20-story, 3-span moment frame shown in Figure 6 is treated again. Four patterns of the distribution of the added A B C   On the contrary, the proposed method underestimates the maximum floor acceleration in the middle floors. This tendency can be seen remarkably in the cases where dampers are not allocated in the upper stories and at side spans. In other words, when the higher-mode responses are not well-controlled or the higher modes of the reduced model do not correspond well to those of the original frame, the maximum floor acceleration tends to be underestimated. Figures 13-16 demonstrate the participation vectors of the original frame, the reduced model and the static condensation model by the complex modal analysis. As in Figure 8, complex participation vector components (horizontal direction) at every floor are drawn. It is noted that the proposed method gives a better correspondence of the fundamental eigenmodes than the static condensation method. This can be seen especially in Figures 15 and 16. In those cases, the dampers are allocated at the side spans and the fundamental damping ratio by the static condensation is smaller than that of the proposed reduced model and the non-reduced original frame. The elongation (or shortening) of the outer columns is larger than that of the inner columns, and then it leads to the ineffective working of brace-type dampers with respect to horizontal deformation. It should be noted that, as stated in Sections 2.1 and 2.2, the static condensation method does not consider the

Displacement-Based Optimization for Viscous Damper Placement
In this section, a numerical example is presented for the displacement-based optimization for the viscous damper placement. Only the proposed reduced models are used throughout the optimization process. Since non-reduced original frames are not treated in the optimization process, the optimization is rapidly conducted. The 20-story, 3-span moment frame shown in Figure 6 is treated again. El Centro NS component during the Imperial Valley earthquake (1940) is employed as the input ground motion. The peak ground velocity (PGV) is adjusted to 0.5 m/s. The sensitivity-based algorithm is adopted as the optimization algorithm. 16 In the sensitivity-based algorithm, a sufficient amount of added damping coefficients is given to all the probable locations at first. Then, a small amount of added damping coefficients Δc is sequentially removed based on the sensitivity by the finite difference of the maximum interstory drift. This procedure is repeated until the maximum interstory drift attains the target value d target . In this numerical example, the added damping coefficient 5=3 ð ÞÂ10 7 Ns=m is given to each span in each story at first, and Δc ¼ 5=120 ð ÞÂ10 7 Ns=m and d target ¼ 4=200 m are adopted. The value of d target corresponds to 1/200 of the interstory drift ratio. In addition, the damper allocation keeps to be symmetric.  Figure 17 shows the optimization results. The maximum interstory drifts of the frame without dampers are also plotted. It can be observed that the dampers effectively reduce the maximum interstory drifts. In more detail, all the interstory drifts are almost uniform except for the upper and lower stories, and those stories' interstory drifts are smaller than those of the middle stories. Furthermore, the maximum interstory drifts of the reduced model correspond well to those of the original frame.
It is noted that it took about 780 s for the optimization. A PC whose clock frequency is 3.6 GHz was used without parallel computing. It is also noted that the time-history response analysis of the non-reduced 20-story, 3-span frame requires the computational time about 35 times longer than that of the reduced model. Therefore, it will take about 7.5 h to use the non-reduced frame throughout the optimization process.
It is also noted that the proposed method cannot treat nonlinear models. However, an optimal design for a linear model works as a good initial design for the local search-based optimization for the corresponding nonlinear model. 17,18 Therefore, the optimization for nonlinear models can be efficiently conducted by using the proposed method in obtaining the initial design.

Conclusions
A new model reduction method was presented for moment frames with viscous dampers. The main conclusions can be summarized as follows.
1 A new model reduction method was presented for moment frames with viscous dampers. The conventional static condensation method does not consider the interactive effect of added damping between the horizontal-vertical directions and the direct effect with regard to the vertical direction. When brace-type dampers are treated, these effects cannot be neglected originally. On the contrary, the proposed method provides a better correspondence of the fundamental eigenmode with that of the non-reduced moment frame than the static condensation method because the proposed method considers these effects. 2 The proposed reduction method was formulated in the frequency domain and the stiffness matrix K eq and the damping matrix C eq of the reduced model depend on a selected particular frequency ω eq . The added dampers apparently add the stiffness to the reduced models depending on ω eq and the amount of added damping. Furthermore, nonproportionality of the damping matrix strengthens this frequency dependency. On the contrary, K eq , C eq for the proportional damping do not change with respect to frequency. Finally, it was demonstrated that ω eq ! 0 is suitable for the calculation of K eq , C eq . 3 Regardless of damper allocation, the proposed reduced model can accurately evaluate the maximum interstory drifts of the non-reduced moment frames. On the contrary, the reduced models may underestimate the maximum floor acceleration slightly, depending on the damper allocation. This can be seen in the cases where the higher-mode responses are not wellcontrolled or the higher modes of the reduced model do not correspond well to those of the non-reduced frame. 4 A displacement-based optimization for viscous damper placement can be accurately and efficiently conducted by using the reduced model. In the numerical example of a 20story, 3-span frame, the optimization was conducted about 35 times faster than the case where the non-reduced frame was used throughout the optimization process.
Equation (A3) means that e K eq , e C eq are independent of ω. When C is proportional to the mass matrix and J is assumed to be 0, C 12 ¼ C T 21 ¼ 0, C 22 ¼ 0. By substituting this into Equation (8), e K eq ¼ K 11 ÀK 12 K À1 22 K 21 , e C eq ¼ C 11 are obtained.