Performance Evaluation of Inerter-Based Dynamic Vibration Absorbers for Wind-Induced Vibration Control of a Desulfurization Tower

Abstract

High-rise flue gas desulfurization towers are susceptible to wind loads, which can cause instability and failure in the along-wind and across-wind directions. The tuned mass damper (TMD) has been widely applied in the wind-induced vibration control of high-rise structures. To enhance the control performance and reduce the auxiliary mass of TMD, this study focuses on inerter-based dynamic vibration absorbers (IDVAs) for controlling the vibration response of a desulfurization tower. The dynamical equations of the tower–IDVA systems are established under wind loads, and a parameter optimization strategy for IDVAs is proposed by using the genetic algorithm. The performance of the traditional TMD and six IDVAs in the vibration control of the tower are systematically compared. Numerical simulations demonstrate that both the TMD and IDVAs can substantially mitigate the vibration response of the tower. However, compared to the TMD with the same response mitigation ratio, more than 34% of the auxiliary mass can be reduced by two optimal IDVAs. In addition, the energy dissipation enhancement and lightweight effect of the two IDVAs are explained through parametric studies.

1. Introduction

Large-scale flue gas desulfurization towers, characterized as high-rise thin-walled steel structures, are critical devices used by petrochemical enterprises. Such structures are notably sensitive to wind loads due to their high flexibility and low damping [1,2]. At the bottom of the desulfurization tower, industrial exhaust gases undergo desulfurization, and a slender steel chimney is set up on the upper sections to raise the height of the smoke discharge. The variation in the sections weakens the stiffness and carrying capacity of the upper parts of the structure. As the high-rise structure with a circular cross-section, the desulfurization tower can easily suffer severe across-wind vortex-induced vibrations at certain wind speeds in addition to the along-wind fluctuating response [3,4,5]. Therefore, it is imperative to conduct research on the wind-induced vibration control of the tower.

Passive measures have been recognized as an effective approach [6,7]. As a typical dynamic vibration absorber, the tuned mass damper (TMD) has been widely adopted in engineering structures [8,9,10]. The optimal parameter design approach, which disregards the damping of the main structure proposed by Den Hartog [11], serves as the foundation for TMD parameter optimization. Brownjohn et al. [12] installed a monitoring system for TMDs, which allows the real-time display of the resulting vibration status and modal damping values. This system facilitates the adjustment of the TMD parameters, consequently enhancing its control performance. Elias et al. [13] designed a distributed multiple tuned mass damper, which achieves multimodal dynamic response control of the along-wind response of the chimney. Xiang et al. [14] performed a seismic design of the hysteretic damping tuned mass damper using the H_∞/H₂ optimization for undamped and damped structures, providing valuable references for designers. Theoretical analysis and experimental verification have demonstrated that the application of TMDs in reducing chatter during the turning of the thin-walled cylinder also shows a good control performance [15]. Vibration control analyses of a high-rise structure indicate the effectiveness of linearizing the equations of motion to obtain optimal control device parameters [16]. Moreover, the pendulum TMDs installed at the top of Shanghai Tower and Taipei 101 significantly mitigate the wind-induced vibrations, thereby improving the comfort levels of residents [17]. Testing research on the TMD system in the Shanghai tower indicates that it is highly suitable for controlling vibrations of structures with relatively small damping ratios [18]. However, the control performance of TMD relies considerably on its tuned mass, which may have adverse effects on the structure. Additionally, the large displacement of its mass block and the narrow frequency band limit the development of TMD [19]. Especially for high-rise structures with limited installation space, like desulfurization towers, the large auxiliary mass of the TMD increases the difficulty of its application.

In 2002, based on the force–current analogy, Smith [20] introduced a two-terminal mechanical device dubbed an inerter. This groundbreaking innovation has allowed new avenues for the vibration control strategies of high-rise structures [21,22]. Combined with a TMD, the inerter element can effectively reduce the auxiliary mass and improve the control performance of traditional dynamic vibration absorbers [23,24,25]. Alotta et al. [26] included a standard inerter-based device within a rhombus truss, which can tune the frequency of the vibration absorber by changing only the geometry of the rhombus truss. In terms of the vibration control of high-rise structures, Liang et al. [27] seismically controlled a mega-substructure system with a tuned inerter damper (TID) and showed an enhanced effect under optimal conditions relative to the traditional viscous damper and the TMD. With reference to a multiple tuned mass damper system, Chen et al. [28] developed a multiple tuned inerter-based damper system for controlling the seismic response, which demonstrated significant advantages. Marian et al. [29] proposed a tuned mass damper–inerter (TMDI) by combining a TMD with an inerter element in series. By leveraging the mass amplification effect of the inerter, the TMDI can perform better than the traditional TMD. Based on a constrained multi-objective evolutionary algorithm, Wang et al. [30] proposed an efficient inerter-based absorber parameter optimization scheme for the wind-induced vibration control of high-rise structures, and they applied it to the TMDI/TID in a 340-meter-high building. However, TID and TMDI require connections at both ends of the structure. Their control performance depends on the connection length of the inerter device, posing installation difficulties in high-rise tubular towers.

Different from TID and TMDI, inerter-based dynamic absorbers (IDVAs) connected to one end of the primary structure are expected to solve the connection problem of the inerter device. Replacing the viscous damper in a TMD with a tuned viscous mass damper [31], Garrido et al. [32] proposed a rotational inertia double tuned mass damper and demonstrated the advantages of it with respect to the TMD. Zhang et al. [33] installed a rotational inertia double tuned mass damper on wind turbines to control their in-plane vibration, establishing comprehensive numerical models and optimization algorithms. Building on this basis, Zhang et al. [34,35,36] combined a TMD with an inerter subsystem, leading to the development of a generalized tuned mass inerter system. Their findings suggested that the tuned mass inerter system could reduce the auxiliary mass and improve the control performance compared to TMD. Wen et al. [37], Su et al. [38], and Hu et al. [39] compared the performance levels of different combinations of IDVAs, as shown in Figure 1. These studies have verified the advantages of IDVAs over the traditional TMD, which were manifested in energy dissipation enhancement, a lightweight effect, and a wider effective frequency band.

Given the limited space at the top of the tower, it is necessary to investigate the lightweight effects of different vibration absorbers. However, the specific effectiveness of different IDVAs in controlling the wind-induced vibration of the desulfurization towers are not clear. Additionally, in the parameter design of the vibration absorbers, the distinct wind-induced response characteristics of cylindrical structures in the along-wind and across-wind directions should be considered. The present study aims to provide an optimal solution for the wind-induced vibration control of the desulfurization tower and investigate the control performances of different IDVAs.

To this end, the control performances of six IDVAs on the wind-induced vibration of a desulfurization tower are systematically evaluated and compared. The organization of this paper is as follows. First, a numerical model of the tower is established, and the transfer functions are derived. Second, an optimal parameter design strategy for the IDVAs is established. Third, the control performances of a TMD and six IDVAs on the tower are compared. Finally, a parametric study is carried out to further analyze the control performance enhancement and lightweight effect of two optimal IDVAs.

2. Tower–IDVA Systems

2.1. Model of the Desulfurization Tower

The desulfurization tower is 85 meters tall and divided into five sections. The tower is mainly made of Q235 steel, with an elastic modulus of 2.06 × 10⁵ MPa, a density of 7.85 g/cm³, and a Poisson’s ratio of 0.3. The thickness of the tower body decreases with increasing height, ranging from 1.6 cm to 2 cm. The dimensions of the segmental structures of the tower are shown in

Table 1. Inner diameter and thickness values of each section of the tower.

Segment	I	II	III	IV	V
Inside diameter/m	7.700	7.700/5.760	5.760	2.700	2.700
Thickness/m	0.020	0.020/0.016	0.016	0.016	0.016

and Figure 2a.

The desulfurization tower is regarded as a cantilever beam structure with the base fully constrained. A finite element model, comprising 32 elements, is established using the Beam188 element in ANSYS, as shown in Figure 2b. To validate the accuracy of the simplified beam element model, a solid model is established using the Solid95 element, as shown in Figure 2c. The parameters of dynamic characteristics and mode shapes of the two models are compared in

Table 2. Natural vibration frequencies and periods of the tower.

Mode	Frequency (Beam199/Solid95)	Period (Beam188/Solid95)	Error
1st	0.908/0.888	1.101/1.126	2.3%
2nd	4.192/4.160	0.239/0.241	0.8%
3rd	9.156/9.050	0.109/1.111	1.2%

Note: Error = (Beam model frequency – Solid model frequency)/Solid model frequency × 100%.

and Figure 3, respectively. The comparison reveals minimal discrepancies between the beam element model and the solid model. Conservatively, the first two modal damping ratios for the desulfurization tower are set as 0.5% [13,40]. To enhance computational efficiency, the structural mass matrix, stiffness matrix, and damping matrix of the beam element model are extracted to construct a numerical model of the tower for subsequent analysis.

2.2. Equations of the Tower–IDVA System

The desulfurization tower is simplified into a generalized system:

$$m={\mathbf{\Phi }}^{\mathrm{T}}\mathit{M}\mathbf{\Phi }; c={\mathbf{\Phi }}^{\mathrm{T}}\mathit{C}\mathbf{\Phi }; k={\mathbf{\Phi }}^{\mathrm{T}}\mathit{K}\mathbf{\Phi },$$

(1)

where M, C, and K represent the mass matrices, damping matrices, and stiffness matrices, respectively; and Φ represents the mode shape function of the desulfurization tower. In this study, only the first mode is considered in wind-induced vibration control.

The analytical model of the desulfurization tower–IDVA system under wind loads is shown in Figure 4. Cn may represent any inerter subsystem of the different configurations C1~C6, as shown in Figure 1. If Cn is replaced by a damping element c_t, it represents the TMD system. The dynamical equation for the tower–TMD system is Equation (2), and the equations for the six tower–IDVA systems are represented as Equations (3)–(8):

$$\left\{\begin{array}{c}m\ddot{x}+c\dot{x}+c_{\mathrm{t}}(\dot{x}-{\dot{x}}_{\mathrm{t}})+kx+k_{\mathrm{t}}(x-x_{\mathrm{t}})=f(t)\\ m_{\mathrm{t}}{\ddot{x}}_{\mathrm{t}}+c_{\mathrm{t}}({\dot{x}}_{\mathrm{t}}-\dot{x})+k_{\mathrm{t}}(x_{\mathrm{t}}-x)=0\end{array}\right.;$$

(2)

$$\left\{\begin{array}{c}m\ddot{x}+c\dot{x}+kx+k_{\mathrm{t}}(x-x_{\mathrm{t}})+k_{\mathrm{in}}(x-x_{\mathrm{in}})=f(t)\\ m_{\mathrm{t}}{\ddot{x}}_{\mathrm{t}}+m_{\mathrm{in}}({\ddot{x}}_{\mathrm{t}}-{\ddot{x}}_{\mathrm{in}})+c_{\mathrm{in}}({\dot{x}}_{\mathrm{t}}-{\dot{x}}_{\mathrm{in}})+k_{\mathrm{t}}(x_{\mathrm{t}}-x)=0\\ m_{\mathrm{in}}({\ddot{x}}_{\mathrm{in}}-{\ddot{x}}_{\mathrm{t}})+c_{\mathrm{in}}({\dot{x}}_{\mathrm{in}}-{\dot{x}}_{\mathrm{t}})+k_{\mathrm{in}}(x_{\mathrm{in}}-x)=0\end{array}\right.;$$

(3)

$$\left\{\begin{array}{c}m\ddot{x}+m_{\mathrm{in}}(\ddot{x}-{\ddot{x}}_{\mathrm{in}})+c\dot{x}+kx+k_{\mathrm{t}}(x-x_{\mathrm{t}})=f(t)\\ m_{\mathrm{t}}{\ddot{x}}_{\mathrm{t}}+c_{\mathrm{in}}({\dot{x}}_{\mathrm{t}}-{\dot{x}}_{\mathrm{in}})+k_{\mathrm{t}}(x_{\mathrm{t}}-x)+k_{\mathrm{in}}(x_{\mathrm{t}}-x_{\mathrm{in}})=0\\ m_{\mathrm{in}}({\ddot{x}}_{\mathrm{in}}-\ddot{x})+c_{\mathrm{in}}({\dot{x}}_{\mathrm{in}}-{\dot{x}}_{\mathrm{t}})+k_{\mathrm{in}}(x_{\mathrm{in}}-x_{\mathrm{t}})=0\end{array}\right.;$$

(4)

$$\left\{\begin{array}{c}m\ddot{x}+c\dot{x}+c_{\mathrm{in}}(\dot{x}-{\dot{x}}_{\mathrm{in}})+kx+k_{\mathrm{t}}(x-x_{\mathrm{t}})=f(t)\\ m_{\mathrm{t}}{\ddot{x}}_{\mathrm{t}}+m_{\mathrm{in}}({\ddot{x}}_{\mathrm{t}}-{\ddot{x}}_{\mathrm{in}})+k_{\mathrm{t}}(x_{\mathrm{t}}-x)+k_{\mathrm{in}}(x_{\mathrm{t}}-x_{\mathrm{in}})=0\\ m_{\mathrm{in}}({\ddot{x}}_{\mathrm{in}}-{\ddot{x}}_{\mathrm{t}})+c_{\mathrm{in}}({\dot{x}}_{\mathrm{in}}-\dot{x})+k_{\mathrm{in}}(x_{\mathrm{in}}-x_{\mathrm{t}})=0\end{array}\right.;$$

(5)

$$\left\{\begin{array}{l}m\ddot{x}+m_{\mathrm{in}}(\ddot{x}-{\ddot{x}}_{\mathrm{t}})+c\dot{x}+c_{\mathrm{t}}(\dot{x}-{\dot{x}}_{\mathrm{t}})+kx+k_{\mathrm{t}}(x-x_{\mathrm{t}})=f(t)\\ m_{\mathrm{t}}{\ddot{x}}_{\mathrm{t}}+m_{\mathrm{in}}({\ddot{x}}_{\mathrm{t}}-\ddot{x})+c_{\mathrm{t}}(x_{\mathrm{t}}-x)+k_{\mathrm{t}}(x_{\mathrm{t}}-x)=0\end{array}\right.;$$

(6)

$$\left\{\begin{array}{c}m\ddot{x}+c\dot{x}+kx+k_{\mathrm{t}}(x-x_{\mathrm{t}})+k_{\mathrm{in}}(x-x_1)=f(t)\\ m_{\mathrm{t}}{\ddot{x}}_{\mathrm{t}}+c_{\mathrm{in}}({\dot{x}}_{\mathrm{t}}-{\dot{x}}_2)+k_{\mathrm{t}}(x_{\mathrm{t}}-x)=0\\ m_{\mathrm{in}}({\ddot{x}}_2-{\ddot{x}}_1)+c_{\mathrm{in}}({\dot{x}}_2-{\dot{x}}_{\mathrm{t}})=0\\ m_{\mathrm{in}}({\ddot{x}}_1-{\ddot{x}}_2)+k_{\mathrm{in}}(x_1-x)=0\end{array}\right.;$$

(7)

$$\left\{\begin{array}{c}m\ddot{x}+c\dot{x}+c_{\mathrm{in}}(\dot{x}-{\dot{x}}_{\mathrm{in}})+kx+k_{\mathrm{t}}(x-x_{\mathrm{t}})=f(t)\\ m_{\mathrm{t}}{\ddot{x}}_{\mathrm{t}}+m_{\mathrm{in}}({\ddot{x}}_{\mathrm{t}}-{\ddot{x}}_{\mathrm{in}})+k_{\mathrm{t}}(x_{\mathrm{t}}-x)=0\\ m_{\mathrm{in}}({\ddot{x}}_{\mathrm{in}}-{\ddot{x}}_{\mathrm{t}})+c_{\mathrm{in}}({\dot{x}}_{\mathrm{in}}-\dot{x})=0\end{array}\right.,$$

(8)

where x, x_t, and x_in represent the displacements of the tower, the tuned mass, and the inerter element, respectively, and f(t) represents the generalized wind load.

As previous research on TMDs is relatively mature and the derivation processes of various configurations are similar, only the configuration C1 is taken as an example to demonstrate the derivation of the transfer functions and the frequency domain responses of the tower–IDVA systems.

First, the following dimensionless parameters are defined:

$${\mu }_{\mathrm{t}}=m_{\mathrm{t}}/m; {\mu }_{\mathrm{in}}=m_{\mathrm{in}}/m_{\mathrm{t}}; {\xi }_{\mathrm{in}}=c_{\mathrm{in}}/(2\sqrt{m_{\mathrm{t}}{k}_{\mathrm{t}}}); {\upsilon }_{\mathrm{t}}={\omega }_{\mathrm{t}}/{\omega }_0; {\upsilon }_{\mathrm{in}}=k_{\mathrm{in}}/k_{\mathrm{t}},$$

(9)

where μ_t and μ_in represent the tuned mass ratio and inertance–mass ratio, respectively; ξ_in represents the nominal damping ratio of the inerter subsystem; υ_t is the frequency ratio of C1, where ω_t and ω₀ represent the circular frequency of C1 and the desulfurization tower, respectively; and υ_in is the nominal stiffness ratio of spring k_in in the inerter subsystem.

Accordingly, a substitution can be made for the mass, stiffness, and damping parameters in Equation (3):

$$m_{\mathrm{t}}={\mu }_{\mathrm{t}}m; m_{\mathrm{in}}={\mu }_{\mathrm{in}}{\mu }_{\mathrm{t}}m; c=2m{\omega }_0{\xi }_0; c_{\mathrm{in}}=2m{\xi }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}{\omega }_0; k=m{\omega }_0^2; k_{\mathrm{t}}={\mu }_{\mathrm{t}}m{({\omega }_0{\upsilon }_{\mathrm{t}})}^2; k_{\mathrm{in}}={\upsilon }_{\mathrm{in}}{\mu }_{\mathrm{t}}m{({\omega }_0{\upsilon }_{\mathrm{t}})}^2,$$

(10)

where ξ₀ represents the first-order damping ratio of the desulfurization tower.

$${\mathit{M}}_1\ddot{\mathit{X}}+{\mathit{C}}_1\dot{\mathit{X}}+{\mathit{K}}_1\mathit{X}=\mathit{F}(t),$$

(11)

where X = [x, x_t, x_in]^T, F(t) = [f(t)/m, 0, 0]^T. The first modal mass matrices M₁, damping matrices C₁, and stiffness matrices K₁ of C1 are represented as follows:

$$\begin{array}{l}{\mathit{M}}_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\mu }_{\mathrm{t}}+{\mu }_{\mathrm{t}}{\mu }_{\mathrm{in}}& -{\mu }_{\mathrm{t}}{\mu }_{\mathrm{in}}\\ 0& -{\mu }_{\mathrm{t}}{\mu }_{\mathrm{in}}& {\mu }_{\mathrm{t}}{\mu }_{\mathrm{in}}\end{array}\right]; {\mathit{C}}_1=\left[\begin{array}{ccc}2{\xi }_0{\omega }_0& 0& 0\\ 0& 2{\xi }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}{\omega }_0& -2{\xi }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}{\omega }_0\\ 0& -2{\xi }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}{\omega }_0& 2{\xi }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}{\omega }_0\end{array}\right]; \\ {\mathit{K}}_1=\left[\begin{array}{ccc}{\omega }_0^2(1+{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}^2+{\upsilon }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}^2)& -{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2& -{\upsilon }_{\mathrm{in}}{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2\\ -{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2& {\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2& 0\\ -{\upsilon }_{\mathrm{in}}{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2& 0& {\upsilon }_{\mathrm{in}}{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2\end{array}\right]\end{array}$$

(12)

Upon substituting the variables in Equation (10), the mass, stiffness, and damping matrices of C2~C6 can be found in Appendix A. If f(t) = e^iωt, X = H(iω)·e^iωt, where H(iω) = [H₀(iω), H_t(iω), H_in(iω)]^T, H₀(iω), H_t(iω), and H_in(iω) represent the transfer functions of the main structure, the mass block. and the inerter element of the tower–C1 system, respectively. By substitution into Equation (11), the following equation is derived:

$$\begin{array}{l}\mathit{H}(\mathrm{i}\omega )=\frac{1}{m}\cdot {[-{\omega }^2{\mathit{M}}_1+\mathrm{i}\omega {\mathit{C}}_1+{\mathit{K}}_1]}^{-1}\cdot {[1,0,0]}^{\mathrm{T}}={\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]}^{-1}\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right] \\ \left\{\begin{array}{c}\begin{array}{c}{A}_{11}=-{\omega }^2+\mathrm{i}2\omega {\omega }_0{\xi }_0+{\omega }_0^2(1+{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}^2+{\upsilon }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}^2)\\ A_{12}=-{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2; A_{21}=A_{12}\end{array}\\ \begin{array}{c}{A}_{13}=-{\upsilon }_{\mathrm{in}}{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2; A_{31}=A_{13}\\ A_{22}=-{\omega }^2({\mu }_{\mathrm{in}}+{\mu }_{\mathrm{t}}{\mu }_{\mathrm{in}})+\mathrm{i}2\omega {\xi }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}{\omega }_0+{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2\end{array}\\ \begin{array}{c}{A}_{23}={\omega }^2{\mu }_{\mathrm{t}}{\mu }_{\mathrm{in}}-\mathrm{i}2\omega {\xi }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}{\omega }_0; A_{32}=A_{23}\\ A_{33}=-{\omega }^2{\mu }_{\mathrm{t}}{\mu }_{\mathrm{in}}+\mathrm{i}2\omega {\xi }_{\mathrm{in}}{\mu }_{\mathrm{t}}{\upsilon }_{\mathrm{t}}{\omega }_0+{\upsilon }_{\mathrm{in}}{\mu }_{\mathrm{t}}{({\omega }_0{\upsilon }_{\mathrm{t}})}^2\end{array}\end{array}\right.\end{array}$$

(13)

The root mean square of the displacement response of the tower–C1 system under wind load can be represented as follows:

$${\sigma }_x=\sqrt{{\displaystyle {\int }_{{\omega }_1}^{{\omega }_2}{S}_x(\mathrm{i}\omega )\mathrm{d}\omega }}=\sqrt{{\displaystyle {\int }_{{\omega }_1}^{{\omega }_2}{\left|H_0(\mathrm{i}\omega )\right|}^2{S}_F(\mathrm{i}\omega )\mathrm{d}\omega }},$$

(14)

where ω₁~ω₂ represent the range of the integral frequency, and the integration is performed near the first-order frequency of the structure; S_x(iω) represents the response spectrum of the structure, and it is resolved using the pseudo excitation method [41], which has a relatively high computational efficiency; and S_F(iω) represents the load spectrum of the structure, which can be derived from the wind speed spectrum.

2.3. Optimal Design

For the along-wind vibration control, the minimum root-mean-square displacement response σ_x of the tower obtained from the frequency domain analysis or the minimum tuned mass ratio μ_t are taken as the optimization objectives. The genetic algorithm, a widely used computational method for optimizing damper parameters in design [10,42], is renowned for its global search capabilities and reduced likelihood of entrapment in local optima. Its ability to process multiple variables simultaneously enhances the computational efficiency, making it suitable for optimizing multiple computational parameters under a single objective. Therefore, the genetic algorithm is then used to optimize the IDVA parameters μ_in, ξ_in, υ_t, and υ_in under the specific μ_t,lim or σ_x,lim. The control performances of IDVAs can also be compared under varying μ_t. The control parameter settings of the genetic algorithm are as follows: the maximum evolution generation is set to 150 generations, the population size is 900, the crossover probability is 0.8, the mutation probability is 0.05, the proportion of the newly generated population is 0.25, and the calculation accuracy is 1 × 10⁻⁹.

H_∞ control theory is renowned for high control accuracy and strong robustness, and it is suitable for the control of the single-order modal vibration of the structure [14,43]. The desulfurization tower is a cylindrical structure, and its vibration in the across-wind direction is dominated by single-order modal vortex resonance.

As a cylindrical high-rise structure, the across-wind response of the desulfurization tower is predominantly governed by vortex-induced resonance at a single modal frequency. It is feasible to simulate the vortex excitation by employing sinusoidal excitation (refer to Appendix B). The H_∞ optimization theory is typically employed for structural peak response suppression under harmonic/sinusoidal excitation [14,43,44]. Therefore, for the across-wind vibration control of the tower, based on the transfer function of the tower–IDVA system and the dimensionless parameters defined by Equation (9), the parameters of the IDVAs are optimized using the genetic algorithm and H_∞ performance index:

$$G(\mathrm{i}\omega )=\underset{\omega }{\sup }{\tau }_{\max }[H_0(\mathrm{i}\omega )],$$

(15)

where G(iω) represents the H_∞ norm of the six typical tower–IDVA systems, and τ_max represents the maximum singular value.

In summary, the parameter optimization of IDVAs for vibration control in the along-wind and across-wind directions of the desulfurization tower obtained from the above analysis is shown in Figure 5.

3. Evaluation of Wind-Induced Vibration Control

3.1. Comparative Analysis of H_∞ Performance

By setting the tuned mass ratios to μ_t = 2% and μ_t = 5%, the genetic algorithm is used to search and optimize the four parameters of IDVAs, specifically μ_in, υ_t, υ_in, and ξ_in. The results, including the optimized parameters for the TMD calculated using the fixed point theory [9], are collated in

Table 3. Optimal H_∞ performance parameters of the TMD and IDVAs.

	μt (%)	μin	υt	υin	ξin
TMD	2	/	0.9794	/	0.0874 (ξt)
	5	/	0.9524	/	0.1336 (ξt)
C1	2	0.0646	0.9547	0.0738	0.0149
	5	0.1322	0.9060	0.1937	0.0548
C2	2	0.0683	0.9990	0.0621	0.0136
	5	0.0950	0.9705	0.0866	0.0176
C3	2	0.0038	0.9031	0.4170	0.1619
	5	0.0037	0.9013	0.8104	0.1520
C4	2	→0	0.9796	/	0.0862
	5	→0	0.9511	/	0.1346
C5	2	0.0413	0.9596	0.0438	0.0764
	5	0.1123	0.8978	0.1333	0.1156
C6	2	→+∞	1.0136	/	0.0975
	5	→+∞	0.9511	/	0.1359

. The frequency response curves of the TMD and six optimally designed IDVAs are performed in Figure 6 and Figure 7, and the tuned mass ratios μ_t of the vibration absorbers are 2% and 5%, respectively.

Based on the comparison results of the frequency response curves and the optimal parameters of IDVAs, C1 and C5 exhibit the best H_∞ performance. When the mass ratio is 2%, the peak values of the frequency response curves of C1 and C5 can be reduced to below 80.02% and 80.43% of the traditional TMD, respectively. In addition, the H_∞ performance of C2 is also significantly superior to that of the TMD. The frequency response curve of C3 resides between that of C2 and TMD, resulting in a limited enhancement to the control performance. This indicates that in the inerter subsystem, the spring element is suitable for series connection with the inerter element and damping element. Furthermore, the inerter makes the effective frequency band of IDVAs wider than that of the TMD.

According to the results of parameter optimization, when the mass ratio μ_t is determined, C1, C2, C3, and C5 can obtain the optimal inertance–mass ratio μ_in within the set range. However, the optimal inertance–mass ratio μ_in of C4 is infinitely close to zero, while the optimal μ_in of C6 is close to positive infinity. Therefore, an additional analysis of the H_∞ performance of C4 is performed. The tuned mass ratio μ_t is set as 2%, and the inertance–mass ratio μ_in is varied. For each μ_in, the genetic algorithm is employed to find the optimal υ_t, υ_in, and ξ_in of C4. Figure 8a shows that the H_∞ performance under the control of C4 is optimal and equivalent to that of the TMD when μ_in is zero. As μ_in gradually increases from 0 to 2, the peak of the frequency response curve continues to increase. This indicates that the inerter does not have a significant effect in C4, and C4 does not enhance the control performance of the TMD.

Similarly, the H_∞ performance analysis of C6 is performed, as shown in Figure 8b. The frequency response curves of C6 under various inertance–mass ratios μ_in are compared when μ_t = 2%. Contrary to C4, the control performance of C6 improves with increasing μ_in. The optimal frequency ratios υ_t and the damping ratios ξ_in of both C4 and C6 are similar to the optimal parameters of the TMD. By comparing C4, C6, and the traditional TMD, it can be determined that introducing only an inerter element, whether it is in parallel or series, cannot enhance the control performance of the TMD. For the IDVAs connected to one end of the primary structure, it is necessary to introduce another spring element within the inerter subsystem.

The subsequent vibration control analysis and the parametric study are no longer focused on C4 and C6. The variation trends of the optimal parameters of C1~C3 and C5 under different tuned mass ratios μ_t are shown in Figure 9. For an intuitive comparison, the dimensionless coefficients ξ_in, μ_in, υ_t, and υ_in of the IDVAs are calculated into c_in, m_in, k_t, and k_in, which can be used directly for engineering design through Equation (10). The range of the tuned mass ratio μ_t is set from 0.5% to 10%, and nine sets of H_∞ performance optimal parameters of C1, C2, C3, and C5 are further optimized.

As shown in Figure 9a,b, the damping coefficients c_in of C1 and C2 are relatively small, but their inertance mass m_in is the largest. This demonstrates that in the inerter subsystems of C1 and C2, the connection modes of the elements make the inerter more involved in vibration control. Moreover, these configurations achieve greater energy dissipation of the main structure because of their small damping coefficients. C3 has the largest damping coefficient and a small inertance mass, which indicates that in C3, the inerter element has not participated significantly in vibration control. This phenomenon is also the reason for the control performance of C3 compared to the other three configurations. The tuned mass m_in and damping coefficient c_in of C5 are relatively large, demonstrating that it achieves energy dissipation enhancement through its large damping by connecting the three elements in series in the inerter subsystem. This mechanism of energy dissipation is different from that of C1 and C2.

As shown in Figure 9c,d, the optimal stiffnesses k_t of the four IDVAs and TMD are relatively close, demonstrating that the introduction of the inerter subsystem has a minor impact on the stiffness of the TMD. In the inerter subsystems, the spring stiffness k_in of C3 is relatively large, while the k_in values of the remaining three IDVAs are fairly similar and significantly small.

3.2. Along-Wind Vibration Control Analysis

The desulfurization tower is in the deltaic plain of the Yangtze River, where the basic wind pressure data corresponding to a 50-year return period are 0.40 kPa at a height of 10 meters. The along-wind load is represented by the Davenport spectrum [45], which is defined by Equation (16).

$$S_u(n)=\frac{2}{3}\frac{{\sigma }_u^2}{n}\frac{y}{{(1+y^2)}^{4/3}},$$

(16)

where S_u(n) represents the power spectrum of fluctuating wind; n represents the frequency; u represents the across-wind fluctuating wind speed; y represents the dimensionless frequency, $y=1200n/{\overline{u}}_{10}$; ${\overline{u}}_{10}=24.9$ m/s, representing the mean wind speed at an elevation of 10 m on the tower, which is derived through basic wind-pressure; and σ_u represents the root mean square of the fluctuating wind speed.

The optimal parameters of C1, C2, C3, and C5 can be obtained under the principle of minimizing the frequency domain displacement response σ_x in the along-wind direction. The IDVAs are then coupled with the numerical model of the tower for a comparative analysis of the control performance in the frequency domain and for verification in the time domain. To compare the control performances of the TMD and different IDVAs, the displacement response mitigation ratio γ is defined as:

$$\gamma =(1-{\sigma }_{x,\mathrm{controlled}}/{\sigma }_{x,\mathrm{original}})\times 100\%,$$

(17)

where σ_x,original and σ_x,controlled are the root-mean-square displacements of the tower in the frequency domain before and after the installation of the vibration absorber, respectively.

The relationships between the root-mean-square displacement response and the vibration mitigation ratio γ with respect to the mass ratio μ_t for the four typical IDVAs and TMD in application to the desulfurization tower are compared in Figure 10. The control performance of IDVAs on the along-wind vibration response of the desulfurization tower agrees with their H_∞ performance, demonstrating their advantages over the TMD. The control efficiency of the vibration absorbers is progressively improved with the increase in the mass ratio μ_t. However, once μ_t is larger than 5%, the increase rates of the vibration mitigation ratios progressively slow down. Therefore, in the subsequent time-domain vibration control analysis, the tuned mass ratios μ_t are all capped at 5%. The above results show that the H_∞ performance of C5 is superior to that of C1, but the advantage is not obvious. In the response control of the structure, the control ratios of the two configurations are even close. Therefore, subsequent analyses are carried out for both C1 and C5, and other IDVAs are no longer considered.

The simulation of the pulsating wind speed and wind load time is performed in accordance with the Davenport spectrum (Equation (16)) using the harmonic superposition method [13]. The along-wind speed curves of Elements 30, 31, and 32 at the top of the tower are shown in Figure 11, which demonstrate the spatial correlation of wind speed time history curves at these three locations. A comparison between the simulated wind speed spectrum and the target wind speed spectrum on the tower is further conducted in Figure 12, verifying the reliability of the simulated wind load.

A time-domain response analysis of wind-induced vibration in the along-wind direction of the tower is conducted, comparing the displacement and acceleration vibration control performance of C1, C5, and the TMD with μ_t = 2%. Figure 13 presents the time history response curves at the top (Element 32) of the tower. The analysis results also validate that the control performances of C1 and C5 are better than those of the TMD.

In addition to the enhancement of the control performance, the more significant advantage of C1 and C5 is that their auxiliary masses are smaller than that of the TMD. As depicted in the frequency domain analysis in Figure 10, C1 and C5 possess much smaller mass ratios than the TMD under the same displacement response mitigation ratio γ. To further validate the lightweight effects of C1 and C5, the frequency domain response mitigation ratio (γ = 59.21%) of the TMD (μ_t = 5%) is adopted as the objective, and the minimum mass ratios μ_t and other optimal parameters of C1 and C5 are designed again using the genetic algorithm. The results reveal that when μ_t = 3.3%, C1 and C5 can meet the vibration mitigation ratio, yielding an equivalent control performance with TMD when μ_t = 5%. The time domain response curves in Figure 14 corroborate this finding. For the identical control objective, C1 and C5 can achieve a 34% reduction in the auxiliary mass compared to the TMD.

3.3. Across-Wind Vibration Control Analysis

The across-wind loads on high-rise structures are mainly composed of vortex excitation and across-wind turbulence. When vortex-induced resonance occurs, the impact of turbulence on the structural vibration can be neglected [5]; thus, only the vortex force is taken into account as the across-wind load. Taking into account the dimensions of various sections of the desulfurization tower, the Reynolds number, and the critical wind speed parameters, we refer to numerous standards [46,47,48,49,50,51] for the characteristics and modalities involved in vortex-induced resonance of the desulfurization tower. The detailed analysis process can be found in Appendix B. The results reveal that the upper part of the tower (Section V, Figure 1a) will be subject to vortex-induced resonant forces, leading to severe vortex-induced resonance, at its first-order critical wind speed. When vortex-induced resonance occurs, the wind speed at a height of 10 m is 11.7 m/s. The sinusoidal force proposed by Rumman [52] is applied to the upper section of the tower:

$$F_v(\mathrm{t})=\frac{1}{2}\rho {\overline{v}}^{2}A{\mu }_{\mathrm{L}}\sin (2\mathsf{\pi }{n}_{\mathrm{s}}\mathrm{t}),$$

(18)

where F_v(t) represents the vortex force; $\overline{v}$ represents the mean wind speed; n_s represents the vortex shedding frequency, which is taken as the first-order frequency of the tower; and μ_L represents the lift coefficient, which is set at 0.25 for a circular cross-section structure.

Time history response analysis is carried out to evaluate the performance of across-wind vibration control on the tower using a TMD (μ_t = 5%). The TMD yields a maximum vibration mitigation ratio of 93.51% for the top of the tower, resulting in a peak displacement response of 0.035 m at the tower top. Starting from 5%, the tuned mass ratios μ_t of C1 and C5 are gradually reduced in increments of 0.1%, and other parameters are optimally designed with the H_∞ performance index. When the tuned mass ratios of C1 and C5 are limited to 2.3%, the results of the time domain analysis show that the mitigation ratios for the largest across-wind displacement responses reach 93.46% and 93.52%, respectively. The control performance is approximately equivalent to that of the TMD under μ_t = 5%.

A comparative analysis in the time domain of TMD (μ_t = 2.3%, μ_t = 5%), C1 (μ_t = 2.3%), and C5 (μ_t = 2.3%) on the vibration control is carried out. Figure 15 shows the displacement responses at the top (Element 32) of the tower with different control devices installed. It verifies that the equivalent C1 and C5 can exhibit a consistent control performance relative to the TMD (μ_t = 5%), which is significantly superior to that of the TMD (μ_t = 2.3%). This finding indicates that when the tower experiences across-wind vortex-induced resonance, C1 and C5 based on H_∞ optimization exhibit superior vibration control performance. Compared to the along-wind vibration control, C1 and C5 reveal better results regarding the response mitigation ratios and the lightweight effects in the across-wind direction. Compared to the TMD under μ_t = 5%, C1 and C5 can reduce the auxiliary mass of the vibration absorber by 54%.

4. Parametric Study

Both the control devices C1 and C5 demonstrate a remarkable superiority in mitigating the along-wind and across-wind vibrations in the desulfurization tower. To further explore the principle of the control performance enhancement and lightweight effects of the inerter system, Figure 16 presents the hysteresis curves for the whole vibration absorber and damping elements of TMD, C1, and C5, all set at an identical mass ratio (μ_t = 2%). The hysteresis curves of C1 and C5 are observed to closely parallel those of the TMD. However, the hysteresis curve for the damping element in C1 shows a significant difference from those in C5 and the TMD. This difference is attributed to the inerter subsystem in C1, which can effectively amplify the displacement of the vibration absorber, leading to a significant enhanced energy dissipation. Additionally, the hysteresis curve for the damping element in C5 closely mirrors that of the TMD. In combination with the earlier parameter analysis results, it is evident that the inertance–mass ratio of C5 is smaller than that of C1, whereas its damping coefficient aligns closely with that of the TMD. This observation suggests that the energy dissipation mechanism of C5 aligns closely with that of the TMD. The difference is that the inerter element of C5 can increase the apparent mass through a series connection with a damping element, thereby realizing an enhancement in energy dissipation efficiency.

Based on the optimized parameters of the control devices C1 and C5, a sensitivity analysis is conducted to evaluate the influence of various parameters including μ_in, μ_t, υ_in, and ξ_in on the displacement response mitigation ratio γ. In both Case 1 and Case 2, as depicted in Figure 17, Figure 18, Figure 19 and Figure 20, the tuned mass ratios μ_t are set at 2% and 5%, respectively, with other primary optimal parameters referenced from Table 3. Figure 17 illustrates the influences of variations in μ_in and μ_t on the γ of C1. A comparative analysis of Figure 17a,b reveals that an increase in μ_t leads to a diminished sensitivity of C1 to changes in μ_in. This finding indicates that when m_t of C1 is relatively large, the role of the inerter system in vibration control gradually decreases. The increment of γ progressively decelerates with higher values of μ_t, consistent with the results obtained from Figure 10. Figure 18 presents the influences of υ_in and ξ_in on γ of C1 with μ_t values set at 2% and 5%, respectively. The outcomes demonstrate that the displacement response mitigation rate of C1 is insensitive to variations in the tuned stiffness, implying strong control robustness. This observation is consistent with the frequency response curve comparisons previously conducted.

As depicted in Figure 19, compared to C1, the control performance of C5 exhibits greater sensitivity to variations in μ_in. This finding indicates that when the μ_t of C5 is relatively small, an excessively low μ_in may even lead to an adverse effect on the response mitigation capability. When the value of μ_in slightly exceeds its optimal value, its impact on the control performance of C5 is relatively insignificant. Figure 20 indicates a similar pattern of C5 and C1 regarding the influence of υ_in and ξ_in on γ. The control performance of the two control devices exhibits greater sensitivity to variations in ξ_in but insensitivity to the variations in frequency within a certain range.

5. Conclusions

A study of IDVAs for the wind-induced vibration control of high-rise desulfurization towers is carried out. The transfer functions and displacement responses in the frequency domain of desulfurization tower–IDVA systems are derived and solved. Based on the vibration characteristics of the tower in both the along-wind and across-wind directions, the optimal design strategy for the IDVA parameters is established by using the genetic algorithm. The H_∞ performances and control efficiencies of the traditional TMD and six typical IDVAs are systematically compared. The principles of the energy dissipation enhancement and lightweight effects of the inerter in two optimal IDVAs are further elucidated through parametric analyses. The main conclusions are as follows:

• C4 and C6, which only introduced an inerter element, cannot enhance the control performance of the TMD. It is necessary to add another spring element. The configurations C1, C2, C3, and C5 with three components—the spring, damper, and inerter—can enhance the H_∞ performance and expand the effective frequency band of the TMD when parallel with the tuning spring component. The configurations C1 and C5, where the spring element is in series with other components, have a relatively close control performance and are significantly superior to other configurations.
• Under the along-wind loads, the application of C1 and C5 can reduce the tuned mass ratio of the TMD from 5% to below 3.3%, maintaining an equivalent vibration control performance for the tower. Under vortex excitation simulated by sinusoidal excitation in the across-wind direction, C1 and C5 demonstrate an even higher vibration mitigation ratio and greater lightweight effect. With mass ratios of 2.3%, C1 and C5 match the control performance of the TMD with a 5% tuned mass ratio, and their vibration mitigation ratio of the displacement response reaches 93.5%.
• C1 features a high optimal inertance–mass ratio and a low damping ratio, and it achieves further energy dissipation through an increased displacement response of the damping element by paralleling it with the inerter element. C5 adopts a series connection of the inerter, damping, and stiffness elements, with a larger inertance–mass ratio and an optimal damping coefficient close to that of the TMD, achieving further energy dissipation through the apparent mass enhancement of the inerter.