# Equivalent Static Wind Load for Structures with Inerter-Based Vibration Absorbers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Inerter-Based Vibration Absorbers

#### 2.1. Generic Equations of Motion

_{0}is the mean up-crossing rate that can be approximated by ${\omega}_{\mathrm{n}}/2\pi $, and γ is the Euler constant taken as 0.5772.

_{0}and z

_{1}, as shown in Figure 1, the Ritz-Galerkin method is adopted, as referred to [29,39,43,44], assuming that u(z, t) = Φ(z)·x(t). According to the principle of visual work, the equations of motion are rewritten as Equation (4). In the equation, ${\phi}_{0}=\Phi ({z}_{0})$ and ${\phi}_{1}=\Phi ({z}_{1})$ are the location parameters of the IVA. f

_{0}and f

_{1}are the control force generated by the IVA at installation locations z

_{0}and z

_{1}, respectively. y represents the absolute displacement of the IVA.

_{0}and f

_{1}are written in Equation (5). Note that when b = 0, it become a Tuned Mass Damper (TMD). Whereas, when the mass can be ignored (m = 0), it has a similar configuration with a Tuned Inerter Damper (TID). Thus, Equation (5) is applicable for all of the above-mentioned situations. If the dashpot of the TMDI is connected to the inerter side, a variant design of TMDI is formed, namely VTMDI. In this case, f

_{0}and f

_{1}are expressed as Equation (6). When the inerter is absent, it forms a variant design of TMD (VTMD). When the mass becomes absent, it has a similar configuration with the Tuned Viscous Mass Damper (TVMD [21]), which is also denoted as TID2 in [37]. Equation (6) is applicable for these variants. Also note that, when φ

_{1}= 0, the IVAs are connected to the ground, known as grounded IVAs. Conventional IVAs usually takes φ

_{0}= 1 and φ

_{1}= 0 regardless of the installation locations. They are assumed to be connected between the tip of the building and the ground.

_{j}(j = 0, 1, 2, 3, 4). The numerator polynomial $\Theta (s)={\displaystyle \sum _{j=0}^{2}{\theta}_{j}{(s/{\omega}_{\mathrm{n}})}^{j}}$ is quadratic, with dimensionless coefficients θ

_{j}(j = 0, 1, 2).

#### 2.2. Analytical Optimal Design Based on Fixed-Point Approach

_{opt}, ζ

_{dopt}} with respect to the input parameters {μ, β, φ

_{0}, φ

_{1}}. The optimization can be performed based on different performance targets, e.g., the maximum (infinity norm) or 2nd norm of the dynamic amplification function (H

_{∞}, H

_{2}optimization). Among them, the most basic optimal design method is the Fixed-point approach (FPA), which can lead to analytical results.

_{n}, as shown in Equation (8), where λ = ω/ω

_{n}is the normalized frequency. The polynomials A

_{1}(λ), A

_{2}(λ), B

_{1}(λ), and B

_{2}(λ) are determined by substituting Equation (7) into Equation (8), as shown in Equation (9). In the Equation, ${\tilde{\gamma}}_{1}=\frac{{\left.{\gamma}_{1}\right|}_{{\zeta}_{\mathrm{n}}=0}}{{\zeta}_{\mathrm{d}}}$, ${\tilde{\gamma}}_{2}={\left.{\gamma}_{2}\right|}_{{\zeta}_{\mathrm{n}}=0}$, ${\tilde{\gamma}}_{3}=\frac{{\left.{\gamma}_{3}\right|}_{{\zeta}_{\mathrm{n}}=0}}{{\zeta}_{\mathrm{d}}}$, ${\tilde{\theta}}_{1}={\theta}_{1}/{\zeta}_{\mathrm{d}}$. For TMDI and VTMDI, the coefficients are shown in Table 2.

_{1,2}can be solved by letting ζ

_{d}be 0 and infinity, i.e., $\frac{{A}_{1}(\lambda )}{{A}_{2}(\lambda )}=\pm \frac{{B}_{1}(\lambda )}{{B}_{2}(\lambda )}$. At the optimal tuning frequency ratio ν

_{∞}, the DAF values of the two fixed points are equal, i.e., D(λ

_{1}) = D(λ

_{2}) = D

_{opt}. Consequently, we obtain the following equation:

_{∞}can be obtained by solving Equation (11) analytically. The coordinates of the optimal fixed points (λ

_{1, 2}, D

_{opt}) are given by Equation (12).

_{d∞}can be determined by assigning the fixed points to the peaks of the DAF, i.e., $\frac{\mathrm{d}{D}_{{\zeta}_{\mathrm{d}}={\zeta}_{\mathrm{d}\infty}}^{2}({\lambda}_{1,2})}{\mathrm{d}\lambda}=0$, as shown in Figure 2b. Based on the quotient rule of derivation, the following equation may be obtained:

_{d∞1,2}, corresponding to the fixed points at λ

_{1}and λ

_{2}, are given by Equation (14).

_{d∞}is taken as ${\zeta}_{\mathrm{d}\infty}=\sqrt{\frac{{\zeta}_{\mathrm{d}\infty 1}^{2}+{\zeta}_{\mathrm{d}\infty 2}^{2}}{2}}$. Substituting Equations (9) and (12) into Equation (14), ζ

_{d∞}is analytically obtained, as shown in Equation (15).

**, ζ**

_{∞}_{d∞}} for various IVAs are obtained, as shown in Table 3. As the solution of a general VTMDI is complex, it is given in Appendix A.

_{∞}optimization, which aims at minimizing the maximum of the DAF. The H

_{2}optimization is targeted for minimizing the frequency domain integration, which corresponds to the variance of the response based on the stochastic vibration theory. In the presented paper, we adopted the FPA for an H

_{∞}optimal solution considering the two reasons. Firstly, the optimal results of the H

_{∞}and H

_{2}optimization are similar for stationary stochastic vibration responses according to previous studies [29,39]. Secondly, based on the fixed-point approach, the closed form solution can be derived, providing a feasible formula for practical design.

_{∞}and ζ

_{d∞}) of Equations (11), (12) and (15) are only based on the coefficients of the transfer function. No extra assumption is introduced. Therefore, the derivation can be applied for DVAs that follows a transfer function with a quadratic numerator polynomial and a quartic denominator polynomial. A linear DVA with a single DOF usually follows this characteristic. The abovementioned equations can be applied to such DVAs other than the IVAs investigated in this paper.

_{1}= 0 (i.e., absent mass μ = 0 or grounded IVA φ

_{1}= 0), the optimal parameters of the TMDI (or VTMDI) can be formulated with an equivalent mass ratio μ

_{eq}compared to the corresponding conventional TMD (or VTMD). The equivalent mass ratio μ

_{eq}is formulated in Equation (16), revealing the influence of installation locations. This equivalent mass ratio approach can be extended to IVAs for approximating the optimal parameters neglecting the higher order items, as displayed in Equations (17) and (18).

## 3. Wind-Induced Response Estimation

#### 3.1. Filter-Based Wind Load Spectrum

_{c}approaches 1. Otherwise, as λ

_{c}tends to infinity, the result converges to the along-wind spectrum. This model agrees well with the experimental data for various slender structures reported in the literatures, as shown in Figure 4b.

#### 3.2. Closed Form Solutions to Wind-Induced Responses Based on Filter Approach

_{j}(j = 0, 1, 2, …, n) being the dimensionless filter coefficients. The numerator polynomial of the even degree is less than the 2(n–1)th order, written as, $\Xi ({\omega}^{2})={\displaystyle \sum _{j=0}^{n-1}{\xi}_{j}{(\omega /{\omega}_{\mathrm{n}})}^{2j}}$, with ξ

_{j}(j = 0, 1, 2, …, n–1) being the dimensionless numerator coefficients.

_{j}. Whereas, N is the numerator determinant, similar to D, merely replacing the first row with the numerator coefficients ξ

_{j}. With this approach, the closed form solutions are obtained in this section.

#### 3.2.1. Uncontrolled Response

_{d0}denotes the ratio between the dynamic response and quasi-static (also known as “background”) response.

_{0}and D

_{0}are calculated through Equations (A5) and (A6), as shown in Equation (24). In the equation, coefficients ψ

_{1}, ψ

_{2}, and ψ

_{12}for along-wind and cross-wind (denoted by superscripts “a” and “c”) are given by Equations (25) and (26), respectively. Note that, for cross-wind response, the aerodynamic damping ratio ζ

_{a}should be considered. Consequently, the dimensionless responses are obtained and generically described in Equation (24).

#### 3.2.2. Controlled Response

_{4}κ

_{0}for controlled responses. The dimensionless responses are obtained with the filter approach in Equation (22). The determinants N and D are calculated from Equation (A5) and (A6). Moreover, the aerodynamic damping ratio ζ

_{a}should be considered in the cross-wind situation.

## 4. Equivalent Static Wind Load

_{eq}targeting the top displacement of the building is formulated in Equation (28).

#### 4.1. Gust Response Factor for Along-Wind ESWL

_{u}being the turbulence intensity, and r being a modification factor), the gust response factor G may be provided by Equation (29). Furthermore, the equivalent static wind pressure p

_{eq}(z) may be provided by Equation (30), where $\overline{p}(z)$ is the time-averaged wind pressure of p(z, t).

_{d}is a factor that considers the dynamic effect. For an uncontrolled structure, it is taken as β

_{d0}in Equation (23). While controlled with IVA, it should be calculated by Equation (27). In order to make this process explicit, a control factor is defined as the ratio between controlled β

_{d}and uncontrolled β

_{d0}, as calculated by Equation (31). In this manner, the relationship between the controlled G and the uncontrolled G

_{0}is provided by Equation (32).

#### 4.2. Cross-Wind ESWL

_{eq,c}(z), may be given by Equation (34) in proportion to the mean along-wind load ${\overline{p}}_{\mathrm{a}}(z)$.

_{eq,c}(z), is in proportion to the modal function Φ(z).

_{d}. The relationship between the controlled load p

_{eq,c}and the uncontrolled load p

_{eq,c0}is provided by Equation (36). The control ratio J is estimated from Equation (32) using the cross-wind spectrum.

## 5. Case Study

_{n}= 2.48 rad/s. The corresponding modal function Φ(z) is also presented in Table 6. The modal mass is M = 4588 t. The critical wind velocity of the chimney is determined as U

_{Cr}= 50.2 m/s. The critical damping ratio of the chimney is ζ

_{n}= 1.5%. According to the wind tunnel tests on an aeroelastic model [43], the aerodynamic damping ratio at U

_{Cr}is ζ

_{a}= –0.96%. In the numerical case, the most unfavorable case is considered, as the design wind velocity is equal to the critical wind velocity.

_{0}= 260 m (φ

_{0}= 0.933) and z

_{1}= 210 m (φ

_{1}= 0.605). Using Equations (16)–(18), the optimal parameters of the IVAs are calculated, as shown in Table 7. Here, the TMDI and VTMDI cases are considered. The modulus of the frequency response functions |X(iω)/F(iω)| of uncontrolled and controlled chimney cases are shown in Figure 5. The theoretical curves (using the optimal parameters in Table 3) and the proposed ones (obtained with parameters in Table 7, obtained by Equations (17) and (18)) are compared in the figures. The results have demonstrated good agreements between the proposed formulas and the theoretical curves on the chimney case, indicating the effectiveness of the proposed optimal design method.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Analytical Solutions of {ν_{∞}, ζ_{d∞}} for VTMDI

**, ζ**

_{∞}_{d∞}} of VTMDI based on FPA are shown in Equations (A1) and (A2), where the polynomials Ψ

_{1}and Ψ

_{2}are shown in Equations (A3) and (A4).

## Appendix B. The Determinants of D and N for Filter Approach

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**Figure 2.**The DAF curves D(λ) with different ν and ζ

_{d}. (

**a**) D(λ) with different ν; the fixed points are equal when ν = ν

_{∞}. (

**b**) D(λ) at ν

_{∞}with different ζ

_{d}; the fixed points are approximately assigned to the equal peaks when ζ

_{d}= ζ

_{d∞}.

**Figure 3.**A flow chart for determining the analytical optimal parameters via FPA (the procedures that output optimal parameters are highlighted).

**Figure 5.**The modulus of frequency response functions |X(iω)/F(iω)| of uncontrolled and controlled chimney cases. (

**a**) TMDI. (

**b**) VTMDI.

**Figure 6.**The time-history of wind-induced top displacement responses of uncontrolled and controlled chimney cases. (

**a**) Along-wind. (

**b**) Cross-wind.

Tuning Parameter | Symbol | Definition |
---|---|---|

Tuning mass ratio | μ | m/M |

Tuning inertance ratio | β | b/M |

Nominal frequency | ω_{d} | $\sqrt{k/(m+b)}$ |

Nominal damping ratio | ν | $c/2\sqrt{k(m+b)}$ |

Tuning frequency ratio | ζ_{d} | ω_{d}/ω_{n} |

Coefficient | TMDI | VTMDI |
---|---|---|

γ_{0} | ${\nu}^{2}$ | ${\nu}^{2}$ |

γ_{1} | $2{\zeta}_{\mathrm{n}}{\nu}^{2}+2{\zeta}_{\mathrm{d}}\nu $ | $2{\zeta}_{\mathrm{n}}{\nu}^{2}+2{\zeta}_{\mathrm{d}}\nu \left[1+{\nu}^{2}(\mu +\beta ){({\phi}_{0}-{\phi}_{1})}^{2}\right]$ |

γ_{2} | $1+4{\zeta}_{\mathrm{n}}{\zeta}_{\mathrm{d}}\nu +{\nu}^{2}\left[1+\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}\right]$ | $1+4{\zeta}_{\mathrm{n}}{\zeta}_{\mathrm{d}}\nu +{\nu}^{2}\left[1+\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}\right]$ |

γ_{3} | $2{\zeta}_{\mathrm{n}}+2{\zeta}_{\mathrm{d}}\nu \left[1+\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}\right]$ | $2{\zeta}_{\mathrm{n}}+2{\zeta}_{\mathrm{d}}\nu (1+\mu {\phi}_{1}^{2})$ |

γ_{4} | $1+{\scriptscriptstyle \frac{\mu}{\mu +\beta}}\beta {\phi}_{1}^{2}$ | $1+{\scriptscriptstyle \frac{\mu}{\mu +\beta}}\beta {\phi}_{1}^{2}$ |

${\tilde{\gamma}}_{1}$ | $2\nu $ | $2\nu \left[1+{\nu}^{2}(\mu +\beta ){({\phi}_{0}-{\phi}_{1})}^{2}\right]$ |

${\tilde{\gamma}}_{2}$ | $1+{\nu}^{2}\left[1+\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}\right]$ | $1+{\nu}^{2}\left[1+\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}\right]$ |

${\tilde{\gamma}}_{3}$ | $2\nu \left[1+\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}\right]$ | $2\nu (1+\mu {\phi}_{1}^{2})$ |

θ_{0} | ${\nu}^{2}$ | ${\nu}^{2}$ |

θ_{1} | $2{\zeta}_{\mathrm{d}}\nu $ | $2{\zeta}_{\mathrm{d}}\nu $ |

θ_{2} | 1 | 1 |

${\tilde{\theta}}_{1}$ | $2\nu $ | $2\nu $ |

**Table 3.**Analytical solutions to the optimal parameters {ν

**, ζ**

_{∞}_{d∞}} of various IVAs obtained based on FPA.

IVA | Parameter | ν_{∞} | ζ_{d∞} |
---|---|---|---|

TMD [18,19] | β = 0, φ_{0} = 1, φ_{1} = 0 | $\frac{1}{1+\mu}$ | $\sqrt{\frac{3\mu}{8(1+\mu )}}$ |

TMDI [28,39] | Arbitrary {μ, β, φ_{0}, φ_{1}} | $\frac{\sqrt{1+{\scriptscriptstyle \frac{\mu}{\mu +\beta}}\beta {\phi}_{1}^{2}}}{1+\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}}$ | $\sqrt{\frac{3}{8}\frac{\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}-{\scriptscriptstyle \frac{\mu}{\mu +\beta}}\beta {\phi}_{1}^{2}}{1+\mu {\phi}_{0}^{2}+\beta {({\phi}_{0}-{\phi}_{1})}^{2}}}$ |

TMDI | μφ_{1} = 0 (TID [32,33], or grounded TMDI [25,26,27]) | $\frac{1}{1+{\mu}_{\mathrm{eq}}}$ | $\sqrt{\frac{3{\mu}_{\mathrm{eq}}}{8(1+{\mu}_{\mathrm{eq}})}}$ |

VTMD [34,35] | β = 0, φ_{0} = 1, φ_{1} = 0 | $\sqrt{\frac{1}{1-\mu}}$ | $\sqrt{\frac{3\mu}{4(2-\mu )}}$ |

VTMDI [28,39] | Arbitrary {μ, β, φ_{0}, φ_{1}} | Equation (A1) | Equation (A2) |

VTMDI | φ_{1} = 0 (grounded VTMDI [36]) | $\sqrt{\frac{1}{1-(\mu +\beta ){\phi}_{0}^{2}}}$ | $\sqrt{\frac{3(\mu +\beta ){\phi}_{0}^{2}}{4\left[2-(\mu +\beta ){\phi}_{0}^{2}\right]}}$ |

VTMDI | μ = 0 (TVMD [21], TID2 [37], or VTID [39]) | $\sqrt{\frac{1}{1-\beta {({\phi}_{0}-{\phi}_{1})}^{2}}}$ | $\sqrt{\frac{3\beta {({\phi}_{0}-{\phi}_{1})}^{2}}{4\left[2-\beta {({\phi}_{0}-{\phi}_{1})}^{2}\right]}}$ |

VTMDI | μφ_{1} = 0 | $\sqrt{\frac{1}{1-{\mu}_{\mathrm{eq}}}}$ | $\sqrt{\frac{3{\mu}_{\mathrm{eq}}}{4(2-{\mu}_{\mathrm{eq}})}}$ |

Coefficient | Along-Wind | Cross-Wind |
---|---|---|

χ_{0} | 1 | ${\lambda}_{\mathrm{c}}^{2}$ |

χ_{1} | ${\Omega}_{\mathrm{a}}+2{\zeta}_{\mathrm{n}}$ | ${\rho}_{\mathrm{c}}{\Omega}_{\mathrm{c}}+2({\zeta}_{\mathrm{n}}+{\zeta}_{\mathrm{a}}){\lambda}_{\mathrm{c}}^{2}$ |

χ_{2} | $2{\zeta}_{\mathrm{n}}{\Omega}_{\mathrm{a}}+1$ | ${\Omega}_{\mathrm{c}}^{2}+2({\zeta}_{\mathrm{n}}+{\zeta}_{\mathrm{a}}){\rho}_{\mathrm{c}}{\Omega}_{\mathrm{c}}+{\lambda}_{\mathrm{c}}^{2}$ |

χ_{3} | ${\Omega}_{\mathrm{a}}$ | $2({\zeta}_{\mathrm{n}}+{\zeta}_{\mathrm{a}}){\Omega}_{\mathrm{c}}^{2}+{\rho}_{\mathrm{c}}{\Omega}_{\mathrm{c}}$ |

χ_{4} | — | ${\Omega}_{\mathrm{c}}^{2}$ |

ξ_{0} | 1 | 1 |

κ_{0} | 1 | ${\rho}_{\mathrm{c}}{\lambda}_{\mathrm{c}}^{2}/{\Omega}_{\mathrm{c}}$ |

Coefficient | Along-Wind | Cross-Wind |
---|---|---|

χ_{0} | ${\gamma}_{0}$ | ${\gamma}_{0}{\lambda}_{\mathrm{c}}^{2}$ |

χ_{1} | ${\gamma}_{0}{\Omega}_{\mathrm{a}}+{\gamma}_{1}$ | ${\gamma}_{0}{\rho}_{\mathrm{c}}{\Omega}_{\mathrm{c}}+{\gamma}_{1}{\lambda}_{\mathrm{c}}^{2}$ |

χ_{2} | ${\gamma}_{1}{\Omega}_{\mathrm{a}}+{\gamma}_{2}$ | ${\gamma}_{0}{\Omega}_{\mathrm{c}}^{2}+{\gamma}_{1}{\rho}_{\mathrm{c}}{\Omega}_{\mathrm{c}}+{\gamma}_{2}{\lambda}_{\mathrm{c}}^{2}$ |

χ_{3} | ${\gamma}_{2}{\Omega}_{\mathrm{a}}+{\gamma}_{3}$ | ${\gamma}_{1}{\Omega}_{\mathrm{c}}^{2}+{\gamma}_{2}{\rho}_{\mathrm{c}}{\Omega}_{\mathrm{c}}+{\gamma}_{3}{\lambda}_{\mathrm{c}}^{2}$ |

χ_{4} | ${\gamma}_{3}{\Omega}_{\mathrm{a}}+{\gamma}_{4}$ | ${\gamma}_{2}{\Omega}_{\mathrm{c}}^{2}+{\gamma}_{3}{\rho}_{\mathrm{c}}{\Omega}_{\mathrm{c}}+{\gamma}_{4}{\lambda}_{\mathrm{c}}^{2}$ |

χ_{5} | ${\gamma}_{4}{\Omega}_{\mathrm{a}}$ | ${\gamma}_{3}{\Omega}_{\mathrm{c}}^{2}+{\gamma}_{4}{\rho}_{\mathrm{c}}{\Omega}_{\mathrm{c}}$ |

χ_{6} | — | ${\gamma}_{4}{\Omega}_{\mathrm{c}}^{2}$ |

ξ_{0} | ${\theta}_{0}^{2}$ | ${\theta}_{0}^{2}$ |

ξ_{1} | ${\theta}_{1}^{2}-2{\theta}_{0}{\theta}_{2}$ | ${\theta}_{1}^{2}-2{\theta}_{0}{\theta}_{2}$ |

ξ_{2} | ${\theta}_{2}^{2}$ | ${\theta}_{2}^{2}$ |

κ | ${\gamma}_{4}$ | ${\gamma}_{4}{\rho}_{\mathrm{c}}{\lambda}_{\mathrm{c}}^{2}/{\Omega}_{\mathrm{c}}$ |

Height (m) | Outer Diameter (m) | Thickness (m) | Modal Function Φ(z) |
---|---|---|---|

270 | 16.9 | 0.45 | 1.000 |

260 | 17.3 | 0.45 | 0.933 |

250 | 17.7 | 0.45 | 0.866 |

240 | 18.1 | 0.50 | 0.799 |

230 | 18.5 | 0.50 | 0.733 |

220 | 18.9 | 0.50 | 0.668 |

210 | 19.3 | 0.50 | 0.605 |

200 | 19.7 | 0.55 | 0.543 |

180 | 20.5 | 0.55 | 0.429 |

160 | 22.1 | 0.60 | 0.328 |

140 | 23.7 | 0.65 | 0.240 |

120 | 25.3 | 0.65 | 0.168 |

100 | 26.9 | 0.70 | 0.110 |

80 | 28.9 | 0.80 | 0.066 |

40 | 33.7 | 0.90 | 0.015 |

0 | 38.5 | 1.00 | 0.000 |

Parameter | TMDI | VTMDI |
---|---|---|

μ_{eq} (%) | 3.15 | 3.15 |

ν_{opt} | 0.969 | 1.016 |

ζ_{dopt} (%) | 10.71 | 10.96 |

m (ton) | 45.88 | 45.88 |

c (kN·s/m) | 496 | 532 |

k (kN/m) | 5569 | 6119 |

b (kN·s^{2}/m) | 917.6 | 917.6 |

Direction | Parameter | Uncontrolled | TMDI | VTMDI |
---|---|---|---|---|

Along-wind | Top displacement (m) | 0.245 | 0.207 | 0.206 |

Base shear force (MN) | 17.72 | 15.50 | 15.46 | |

Base bending moment (GN·m) | 2.793 | 2.388 | 2.381 | |

Control ratio J (time domain method) | — | 0.676 | 0.670 | |

Gust Response Factor G (time domain method) | 1.93 | 1.63 | 1.62 | |

Control ratio J (proposed method) | — | 0.692 | 0.680 | |

Gust Response Factor G (proposed method) | 1.95 | 1.66 | 1.64 | |

Cross-wind | Top displacement (m) | 0.422 | 0.197 | 0.194 |

Base shear force (MN) | 19.36 | 9.14 | 8.95 | |

Base bending moment (GN·m) | 4.033 | 1.901 | 1.868 | |

Control ratio J (time domain method) | — | 0.467 | 0.461 | |

Gust Response Factor G’ (time domain method) | 3.32 | 1.55 | 1.53 | |

Control ratio J (proposed method) | — | 0.476 | 0.469 | |

Gust Response Factor G’ (proposed method) | 3.38 | 1.61 | 1.59 |

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## Share and Cite

**MDPI and ACS Style**

Su, N.; Peng, S.; Chen, Z.; Hong, N.; Uematsu, Y.
Equivalent Static Wind Load for Structures with Inerter-Based Vibration Absorbers. *Wind* **2022**, *2*, 766-783.
https://doi.org/10.3390/wind2040040

**AMA Style**

Su N, Peng S, Chen Z, Hong N, Uematsu Y.
Equivalent Static Wind Load for Structures with Inerter-Based Vibration Absorbers. *Wind*. 2022; 2(4):766-783.
https://doi.org/10.3390/wind2040040

**Chicago/Turabian Style**

Su, Ning, Shitao Peng, Zhaoqing Chen, Ningning Hong, and Yasushi Uematsu.
2022. "Equivalent Static Wind Load for Structures with Inerter-Based Vibration Absorbers" *Wind* 2, no. 4: 766-783.
https://doi.org/10.3390/wind2040040