# Lightweight Design of Vibration Control Devices for Offshore Substations Based on Inerters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{in}in proportion to the relative translation accelerations between its two terminals (${\ddot{u}}_{1}$ and ${\ddot{u}}_{2}$) [26], as shown in Equation (1).

_{1}and ü

_{2}denote the displacement of the terminals. The overhead dots denote a derivation with respect to time. The inertance coefficient is b, which has the same dimension as mass. This property can be physically achieved with a mechanism (such as rack-pinion [27], ball screw [28], living hinge [29], hydraulic mechanics [30], etc.) converting the translation deflection into rotation. With a flywheel, the rotation can produce a significant inertial effect. The schematic diagram of a general simplified symbol of an inerter element is shown in Figure 2. The circular plates indicate the flywheel used to generate the inertial-like force, as presented by Equation (1). The inertial effect is quantified by the inertance coefficient, which can be valued at hundreds of times the physical mass of the inerter device. With this feature, the lightweight design of a vibration control device may be realized.

## 2. Vibration Control Mechanisms

#### 2.1. Primary Offshore Substation Structure without Control

^{2}+ Cs + K). Here, s is the Laplace complex frequency, X(s) and F(s) are the Laplace transforms of x(t) and f(t). Although the SDOF structure without control is a simple case, this approach is effective in analyzing complex cases with complex vibration control devices, as shown in the follow sections.

#### 2.2. Vibration Control with Conventional TMD

_{0}denotes the output force of the TMD. The overhead dot means a derivation with respect to time. The output force of the TMD is produced by the Kelvin–Voigt element composed of a spring and dashpot in parallel connection [11], as provided by

_{1,2,3}(s) being the characteristic functions, which are polynomial functions for single-tuned vibration control devices and are rational functions for double-tuned vibration control devices. For the most conventional TMD, they are obtained as Equation (5). Certainly, they are replaced with other forms for other complex vibration control devices.

#### 2.3. Single-Tuned Vibration Control Devices

_{1}and f

_{2}denote the output force of the two terminals, as displayed in the following. Note that Equation (6) is an extended form of Equation (2), with the two force items f

_{1}and f

_{2}for the two terminals, respectively.

#### 2.3.1. Conventional Type (TMDI)

_{1}and f

_{2}are written as

#### 2.3.2. Variant Type (V-TMDI)

_{1}and f

_{2}are written as

#### 2.4. Double-Tuned Vibration Control Devices

_{C3,4, or 6}is the output force generated by the inerter-based sub-network. For configurations C3, C4, and C6 [49,50], with inerter, damping, and stiffness coefficients of b

_{2}, c

_{2}, and k

_{2}, it is governed as follows.

#### 2.4.1. C6 Type (RIDTMD)

_{C6}in the time domain is derived as

_{1}is the displacement of the node in the C6 sub-network [49]. This dummy displacement variable can be eliminated algebraically with Laplace transforms, obtained as

_{C6}(s) is the Laplace transform of f

_{C6}(t).

#### 2.4.2. C4 Type

_{C4}is written as

_{1}is eliminated algebraically, as written by

#### 2.4.3. C3 Type

_{1}and y

_{2}. The f

_{C3}in the time domain is governed by

_{1}and y

_{2}are eliminated, as written by

#### 2.4.4. Mechanical Impedance Function

## 3. Determination of Optimal Parameters of Vibration Control Devices

#### 3.1. Parametric Optimization Method

#### 3.1.1. Dynamic Amplification Function

^{2}= −1. S

_{f}(ω) is a spectral density function of the excitation f(t), where ω denotes the frequency.

#### 3.1.2. H-Norm-Based Optimization

**π**, and underdetermined parameters are represented by a vector of

**θ**. The predetermined parameters are ones determined before parametric optimization as known parameters to be substituted into the optimization, like the mass ratio μ and the installation parameter φ. They are usually determined based on some presumptions and installation restrictions, whereas the underdetermined parameters are ones to be determined based on predetermined parameters to achieve better control performances, like the parameters (ν, ζ

_{d}) for single-tuned cases, and (ν, γ, ζ

_{d}) for double-tuned cases. The parametric optimization is to determine

**θ**(

_{opt}**π**) with an optimal target. Targeted on the norms of dynamic amplification functions, H-norm-based optimization criteria are presented.

_{∞}optimization [11] is targeted by minimizing the infinity norm (maximum value) D

_{max}of the dynamic amplification function, as per

**θ**

_{opt-∞}(

**π**) represents the optimal underdetermined parameters determined by H

_{∞}optimization.

**θ**

_{min,max}denote the lower and upper boundary of the parameter vector

**θ**. H

_{∞}optimization can be analytically approximated by the fixed-point approach, as presented in [1]. H

_{∞}optimization is targeted by minimizing the maximum possible dynamic amplification factor over the whole frequency domain. This is particularly effective for harmonically or stochastically excited structures with various frequencies.

_{2}optimization [13] is targeted on minimizing the second norm I of the dynamic amplification function, as per

**θ**

_{opt-2}(

**π**) represents the optimal parameters determined by H

_{2}optimization. The H

_{2}-norm I can be analytically obtained by the filter approach or Cauchy’s residue theorem. However, the H

_{2}optimization results can be non-analytical for complicated cases. As the H

_{2}-norm corresponds to the standard deviation value of a dynamic response subjected to a white noise with a unit intensity (S

_{f}(ω) = 1), it is particularly effective for broad-banded stochastically excitations.

#### 3.2. Single-Tuned Vibration Control Devices

**π**= {μ, β, φ}, and the underdetermined parameter vector is

**θ**= {ν, ζ

_{d}}.

#### 3.2.1. Conventional TMD

_{2}-norm for each solution is shown in the legend. Among these, the hybrid H

_{∞}–H

_{2}solution [36] combines the advantages of the fixed-point approach for the H

_{∞}-norm and filter-based approach for the H

_{2}-norm. Thus, this method is adopted in the present investigation.

_{2}-norm for the TMD with hybrid H

_{∞}–H

_{2}solutions is obtained as

#### 3.2.2. Ground Connected Single-Tuned Vibration Control Devices

_{∞}–H

_{2}optimization approach, the analytical optimal parameters are solved as Equation (28) for the conventional type and Equation (29) for the variant type.

_{2}solutions for the variant type single-tuned vibration control devices cannot exist when the inertance coefficient is sufficiently large. Thus, the H

_{∞}–H

_{2}optimization approach could provide a stable solution to adapt this situation.

#### 3.2.3. Equivalent Mass Ratio Approach to Address the Installation Location

_{∞}–H

_{2}optimization criterion [46].

_{1,2,3,4}are determined as Equation (32) for the conventional type and Equation (33) for the variant type.

_{eq}, is introduced, as defined by ${\mu}_{\mathrm{eq}}=\mu +\beta {(1-\phi )}^{2}$. The equivalent mass ratio can be used to estimate the lightweight performance of vibration control devices. The simplification Equations (34) and (35) are exactly the same as the exact solution Equations (30)–(33) when the product μβφ = 0. To be specific, when φ = 0, Equations (34) and (35) degenerate to Equations (28) and (29), respectively. When β = 0, Equation (34) degenerates to the solution of the TMD. The comparison between the exact and simplified solutions for different types and installation locations is shown in Figure 8. The simplified results are observed to be in good agreement with the exact solutions.

#### 3.3. Double-Tuned Vibration Control Devices

**π**= μ and the underdetermined parameter vector as

**θ**= {β, ν, γ, ζ

_{d}}. Different from single-tuned vibration control devices, the inerter in double-tuned vibration control devices is viewed as a sub-element of the inerter-based sub-network. Therefore, the inertance parameter β here is an underdetermined parameter rather than a predetermined parameter.

#### 3.3.1. Optimal Parameters

#### 3.3.2. Equivalent Mass Ratio

- In order to further quantify the lightweight vibration control performance, with the formant of the conventional TMD in Equation (27), an equivalent mass ratio for double-tuned vibration control devices is defined as μ
_{eq}= αμ, with α being a mass magnification effect factor. The factor α is determined by equalizing the optimal H_{2}-norms of double-tuned vibration control devices with those of the conventional TMD, as per Equation (36), via a least square technique.

- The optimal H
_{2}-norms of double-tuned vibration control devices and the conventional TMD are plotted in Figure 10. It is noticed that Equation (36) can fit well with the data, indicating the effectiveness of the equivalent mass approach. The resulting factor α is determined as 1.25, which indicates that the optimally designed double-tuned vibration control device may save 25% of the mass compared to a conventional TMD.

## 4. Vibration Control on a Practical Offshore Substation

#### 4.1. Vibration Response Analysis

#### 4.1.1. Finite Element Model of the Offshore Substation

^{3}, a passion ratio of 0.3, and a Young’s modulus of 206 GPa.

#### 4.1.2. Environmental Excitations

- (1)
- Wind load

_{z}and the turbulence intensity I

_{z}are assumed to follow a power law along the height z, with a power index of 0.12 for a marine terrain [4], as shown in Equation (37).

_{10}is a basic wind speed at a standard reference height of 10 m. The stochastic wind velocity spectrum S

_{wind}(ω) is taken as the Davenport spectrum as

_{10}) is the reduced frequency. With a spectral representation method [54], the stochastic wind speed V

_{z}(t) can be obtained. Then, the wind load on each structural node f

_{wind}(t) can be obtained as

_{p}is the wind pressure coefficient, ρ

_{air}is the air density, and A is the tributary area of the node.

- (2)
- Wave and current loads

_{x}(z, t) and a

_{x}(z, t) are expressed by

_{i}, κ

_{i}, and ε

_{i}(i = 1, 2, …, N) are the frequency, wave number, and stochastic phase of the ith wave. d is the water depth. S

_{wave}(ω) is the wave height spectrum, which is taken as the JONSWAP spectrum as per Equation (41) [55]. Δω is the frequency interval.

_{P}is the characteristic period of the wave; υ is a spectral peak factor, taken as 3.3; and σ is the spectral shape factor, taken as 0.07 for ω ≤ 2π/T

_{P}and 0.09 for ω > 2π/T

_{P}. With Morison’s formula, the wave and current loads on an underwater pipe element f

_{w&c}(t) are calculated by

_{D}is a drag coefficient, taken as 1.2; C

_{M}is an inertial coefficient, taken as 2.0; v

_{c}is the mean current velocity; ρ

_{sea}is the density of sea water; D and h are the diameter and tributary height of the pipe element, respectively; and A is the tributary area of the element.

#### 4.2. Vibration Control Performance

#### 4.2.1. Vibration Control Devices

_{eq}= 0.05. For the double-tuned vibration control device, a RIDTMD with μ

_{eq}= 0.05 is selected, which is characterized as μ = 0.04. The underdetermined parameters are calculated with simplified formulas, as presented in Section 3. The details of the calculation cases are shown in Table 2.

#### 4.2.2. Vibration Responses and Control Rates

_{dyn}and J

_{tot}is defined, as expressed by

_{UC}(t) and q

_{C}(t) are the uncontrolled and controlled responses, respectively. SD[·] denotes the standard deviation value of the dynamic responses. Max[·] denotes the maximum of the absolute value of the total responses.

#### 4.3. Comparisons of Different Vibration Control Devices

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Faraggiana, E.; Ghigo, A.; Sirigu, M.; Petracca, E.; Giorgi, G.; Mattiazzo, G.; Bracco, G. Optimal floating offshore wind farms for Mediterranean islands. Renew. Energy
**2024**, 221, 119785. [Google Scholar] [CrossRef] - Wang, H.J.; Liu, C.; Guo, Y.H.; Zhao, Y.; Li, X.Y.; Lian, J.J. Experimental and numerical research on the wet-towing of wide-shallow bucket jacket foundation for offshore substation. Ocean Eng.
**2023**, 275, 114126. [Google Scholar] [CrossRef] - Elobeid, M.; Pillai, A.C.; Tao, L.; Ingram, D.; Hanssen, J.E.; Mayorga, P. Implications of wave–current interaction on the dynamic responses of a floating offshore wind turbine. Ocean Eng.
**2024**, 292, 116571. [Google Scholar] [CrossRef] - Townsend, J.F.; Xu, G.J.; Jin, Y.J.; Yu, E.B.; Wei, H.; Han, Y. On the development of a generalized atmospheric boundary layer velocity profile for offshore engineering applications considering wind–wave interaction. Ocean Eng.
**2023**, 286, 115621. [Google Scholar] [CrossRef] - Kikuchi, Y.; Ishihara, T. Assessment of capital expenditure for fixed-bottom offshore wind farms using probabilistic engineering cost model. Appl. Energy
**2023**, 341, 120912. [Google Scholar] [CrossRef] - Sykes, V.; Collu, M.; Coraddu, A. A Review and Analysis of the Uncertainty Within Cost Models for Floating Offshore Wind Farms. Renew. Sustain. Energy Rev.
**2023**, 186, 113634. [Google Scholar] [CrossRef] - Danovaro, R.; Bianchelli, S.; Brambilla, P.; Brussa, G.; Corinaldesi, C.; Del Borghi, A.; Dell’Anno, A.; Fraschetti, S.; Greco, S.; Grosso, M.; et al. Making eco-sustainable floating offshore wind farms: Siting, mitigations, and compensations. Renew. Sustain. Energy Rev.
**2024**, 197, 114386. [Google Scholar] [CrossRef] - Machado, M.R.; Dutkiewicz, M.; Colherinhas, G.B. Metamaterial-based vibration control for offshore wind turbines operating under multiple hazard excitation forces. Renew. Energy
**2024**, 223, 120056. [Google Scholar] [CrossRef] - Rezaei, F.; Contestabile, P.; Vicinanza, D.; Azzellino, A. Towards understanding environmental and cumulative impacts of floating wind farms: Lessons learned from the fixed-bottom offshore wind farms. Ocean Coast. Manag.
**2023**, 243, 106772. [Google Scholar] [CrossRef] - Díaz-Motta, A.; Díaz-González, F.; Villa-Arrieta, M. Energy sustainability assessment of offshore wind-powered ammonia. J. Clean. Prod.
**2023**, 420, 138419. [Google Scholar] [CrossRef] - Ormondroyd, J.; Den Hartog, J.P. The theory of the dynamic vibration absorber. Trans. ASME
**1928**, 50, 9–22. [Google Scholar] [CrossRef] - Krenk, S.; Høgsberg, J. Tuned mass absorbers on damped structures under random load. Probabilistic Eng. Mech.
**2008**, 23, 408–415. [Google Scholar] [CrossRef] - Asami, T.; Nishihara, O.; Baz, A.M. Analytical solutions to H
_{∞}and H_{2}optimization of dynamic vibration absorbers attached to damped linear systems. J. Vib. Acoust.**2002**, 124, 284–295. [Google Scholar] [CrossRef] - Bisegna, P.; Caruso, G. Closed-form formulas for the optimal pole-based design of tuned mass dampers. J. Sound Vib.
**2012**, 331, 2291–2314. [Google Scholar] [CrossRef] - Krenk, S.; Høgsberg, J. Tuned mass absorber on a flexible structure. J. Sound Vib.
**2014**, 333, 1577–1595. [Google Scholar] [CrossRef] - Argenziano, M.; Faiella, D.; Carotenuto, A.R.; Mele, E.; Fraldi, M. Generalization of the Den Hartog model and rule-of-thumb formulas for optimal tuned mass dampers. J. Sound Vib.
**2022**, 538, 117213. [Google Scholar] [CrossRef] - Sun, X.; Wu, H.; Wu, Y.; Su, N. Wind-induced responses and control of a Kilometer skyscraper with mass and viscous dampers. J. Build. Eng.
**2021**, 43, 102552. [Google Scholar] [CrossRef] - Kaveh, A.; Javadi, S.M.; Moghanni, R.M. Optimal structural control of tall buildings using tuned mass dampers via chaotic optimization algorithm. Structures
**2020**, 28, 2704–2713. [Google Scholar] [CrossRef] - Domizio, M.; Garrido, H.; Ambrosini, D. Single and multiple TMD optimization to control seismic response of nonlinear structures. Eng. Struct.
**2022**, 252, 113667. [Google Scholar] [CrossRef] - Lin, Y.Y.; Cheng, C.M.; Lee, C.H. A tuned mass damper for suppressing the coupled flexural and torsional buffeting response of long-span bridges. Eng. Struct.
**2000**, 22, 1195–1204. [Google Scholar] [CrossRef] - Labbafi, S.F.; Shooshtari, A.; Mohtashami, E. Optimal design of friction tuned mass damper for seismic control of an integral bridge. Structures
**2023**, 58, 105200. [Google Scholar] [CrossRef] - Wang, J.W.; Liang, X.; Wang, L.Z.; Wang, B.X.; Wang, L.L. The influence of tuned mass dampers on vibration control of monopile offshore wind turbines under wind-wave loadings. Ocean Eng.
**2023**, 278, 114394. [Google Scholar] - Jahangiri, V.; Sun, C.; Kong, F. Study on a 3D pounding pendulum TMD for mitigating bi-directional vibration of offshore wind turbines. Eng. Struct.
**2021**, 241, 112383. [Google Scholar] [CrossRef] - Elias, S. Vibration improvement of offshore wind turbines under multiple hazards. Structures
**2024**, 59, 105800. [Google Scholar] [CrossRef] - Smith, M.C. Synthesis of mechanical networks: The inerter. IEEE Trans. Autom. Control
**2002**, 47, 1648–1662. [Google Scholar] [CrossRef] - Ma, R.S.; Bi, K.M.; Hao, H. Inerter-based structural vibration control: A state-of-the-art review. Eng. Struct.
**2021**, 243, 112655. [Google Scholar] [CrossRef] - Sun, L.; Hong, D.; Chen, L. Cables interconnected with tuned inerter damper for vibration mitigation. Eng. Struct.
**2017**, 151, 57–67. [Google Scholar] [CrossRef] - Papageorgiou, C.; Houghton, N.E.; Smith, M.C. Experimental testing and analysis of inerter devices. J. Dyn. Syst. Meas. Control
**2009**, 131, 101–116. [Google Scholar] [CrossRef] - John, E.D.A.; Wagg, D.J. Design and testing of a frictionless mechanical inerter device using living-hinges. J. Frankl. Inst.
**2019**, 356, 7650–7668. [Google Scholar] [CrossRef] - De Domenico, D.; Deastra, P.; Ricciardi, G.; Sims, N.D.; Wagg, D.J. Novel fluid inerter based tuned mass dampers for optimised structural control of base-isolated buildings. J. Frankl. Inst.
**2019**, 356, 7626–7649. [Google Scholar] [CrossRef] - Ikago, K.; Saito, K.; Inoue, N. Seismic control of single-degree-of-freedom structure using tuned viscous mass damper. Earthq. Eng. Struct. Dyn.
**2012**, 41, 453–474. [Google Scholar] [CrossRef] - Ikago, K.; Sugimura, Y.; Saito, K.; Inoue, N. Modal response characteristics of a multiple-degree-of-freedom structure incorporated with tuned viscous mass dampers. J. Asian Arch. Build. Eng.
**2012**, 11, 375–382. [Google Scholar] [CrossRef] - Zhang, R.F.; Zhao, Z.P.; Pan, C.; Ikago, K.; Xue, S.T. Damping enhancement principle of inerter system. Struct. Control Health Monit.
**2020**, 27, e2523. [Google Scholar] [CrossRef] - Su, N.; Bian, J.; Peng, S.T.; Xia, Y. Impulsive resistant optimization design of tuned viscous mass damper (TVMD) based on stability maximization. Int. J. Mech. Sci.
**2023**, 239, 107876. [Google Scholar] [CrossRef] - Mustapha, A.; Zhang, X.; Atroshchenko, E.; Jaroon, R. Vibration control of inerter-enhanced mega sub-controlled structure system (MSCSS) and the reliability analysis of the structure under seismic action. Eng. Struct.
**2024**, 241, 117508. [Google Scholar] - Marian, L.; Giaralis, A. Optimal design of a novel tuned mass-damper-inerter (TMDI) passive vibration control configuration for stochastically support-excited structural systems. Probabilistic Eng. Mech.
**2014**, 38, 156–164. [Google Scholar] [CrossRef] - Lazar, I.F.; Neild, S.A.; Wagg, D.J. Using an inerter-based device for structural vibration suppression. Earthq. Eng. Struct. Dyn.
**2014**, 43, 1129–1147. [Google Scholar] [CrossRef] - Deastra, P.; Wagg, D.; Sims, N.; Akbar, M. Tuned inerter dampers with linear hysteretic damping. Earthq. Eng. Struct. Dyn.
**2020**, 49, 1216–1235. [Google Scholar] [CrossRef] - Alotta, G.; Failla, G. Improved inerter-based vibration absorbers. Int. J. Mech. Sci.
**2021**, 192, 106087. [Google Scholar] [CrossRef] - Su, N.; Peng, S.T.; Xia, Y. Filter-based inerter location dependence analysis approach of Tuned mass damper inerter (TMDI) and optimal design. Eng. Struct.
**2022**, 250, 113459. [Google Scholar] [CrossRef] - Islam, N.U.; Jangid, R.S. Optimum parameters of tuned inerter damper for damped structures. J. Sound Vib.
**2022**, 537, 117218. [Google Scholar] [CrossRef] - Fitzgerald, B.; McAuliffe, J.; Baisthakur, S.; Sarkar, S. Enhancing the reliability of floating offshore wind turbine towers subjected to misaligned wind-wave loading using tuned mass damper inerters (TMDIs). Renew. Energy
**2023**, 221, 522–538. [Google Scholar] [CrossRef] - Ren, M.Z. A variant design of the dynamic vibration absorber. J. Sound Vib.
**2001**, 245, 762–770. [Google Scholar] [CrossRef] - Masnata, C.; Matteo, A.D.; Adam, C.; Pirrotta, A. Smart structures through nontraditional design of Tuned Mass Damper Inerter for higher control of base isolated systems. Mech. Res. Commun.
**2020**, 105, 103513. [Google Scholar] [CrossRef] - Su, N.; Bian, J.; Peng, S.T.; Xia, Y. Generic optimal design approach for inerter-based tuned mass systems. Int. J. Mech. Sci.
**2022**, 233, 107654. [Google Scholar] [CrossRef] - Su, N.; Chen, Z.Q.; Xia, Y.; Bian, J. Hybrid analytical H-norm optimization approach for dynamic vibration absorbers. Int. J. Mech. Sci.
**2024**, 264, 108796. [Google Scholar] [CrossRef] - Garrido, H.; Curadelli, O.; Ambrosini, D. Improvement of tuned mass damper by using rotational inertia through tuned viscous mass damper. Eng. Struct.
**2013**, 56, 2149–2153. [Google Scholar] [CrossRef] - Zhang, L.; Xue, S.T.; Zhang, R.F.; Xie, L.Y.; Hao, L.F. Simplified multimode control of seismic response of high-rise chimneys using distributed tuned mass inerter systems (TMIS). Eng. Struct.
**2021**, 228, 111550. [Google Scholar] [CrossRef] - Hu, Y.L.; Chen, M.Z.Q.; Shu, Z.; Huang, L.X. Analysis and optimisation for inerter-based isolators via fixed-point theory and algebraic solution. J. Sound Vib.
**2015**, 346, 17–36. [Google Scholar] [CrossRef] - Barredo, E.; Blanco, A.; Colín, J.; Penagos, V.M.; Abúndez, A.; Luis, G.V.; Meza, V.; Cruz, R.H.; Mayen, J. Closed-form solutions for the optimal design of inerter-based dynamic vibration absorbers. Int. J. Mech. Sci.
**2018**, 144, 41–53. [Google Scholar] [CrossRef] - Barredo, E.; Mendoza Larios, J.G.; Colín, J.; Mayen, J.; Flores-Hernandez, A.A.; Arias-Montiel, M. A novel high-performance passive non-traditional inerter-based dynamic vibration absorber. J. Sound Vib.
**2020**, 485, 115583. [Google Scholar] [CrossRef] - Barredo, E.; Rojas, G.L.; May´en, J.; Flores-Hernandez, A.A. Innovative negative-stiffness inerter-based mechanical networks. Int. J. Mech. Sci.
**2021**, 205, 106597. [Google Scholar] [CrossRef] - Su, N.; Peng, S.T.; Hong, N.N.; Xia, Y. Wind-induced vibration absorption using inerter-based double tuned mass dampers on slender structures. J. Build. Eng.
**2022**, 58, 104993. [Google Scholar] [CrossRef] - Melaku, A.F.; Bitsuamlak, G.T. A divergence-free inflow turbulence generator using spectral representation method for large-eddy simulation of ABL flows. J. Wind. Eng. Ind. Aerodyn
**2021**, 212, 104580. [Google Scholar] [CrossRef] - Mazzaretto, O.M.; Menéndez, M.; Lobeto, H. A global evaluation of the JONSWAP spectra suitability on coastal areas. Ocean Eng.
**2022**, 266, 112756. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagrams of offshore substations. (

**a**) Jacket-type offshore substation. (

**b**) Mono-pile-type offshore substation.

**Figure 3.**Various inerter-based single- and double- tuned vibration control devices. (

**a**) Single-tuned vibration control devices. (

**b**) Double-tuned vibration control devices.

**Figure 5.**A schematic diagram of a primary structure coupling a two-terminal-connected single-tuned vibration control device.

**Figure 6.**A schematic diagram of a primary structure coupling a one-terminal-connected vibration control device (a TMD or a double-tuned vibration control device) generically expressed by a mechanical impedance function.

**Figure 7.**Dynamic amplification functions under different optimal parameters for a conventional TMD with μ = 0.05.

**Figure 8.**The comparison of dynamic amplification functions between the exact and simplified solutions for different types and installation locations (μ = 0.01; β = 0.10; φ = 0.2, 0.4, and 0.6). (

**a**) Conventional type of TMDI; (

**b**) variant type of V-TMDI.

**Figure 9.**Dynamic amplification functions under optimal parameters for a conventional TMD and double-tuned vibration control devices with μ = 0.05.

**Figure 10.**The optimal H

_{2}-norms of double-tuned vibration control devices and the conventional TMD.

**Figure 11.**Finite element model and the fundamental mode of the offshore substation structure. (

**a**) Finite element model. (

**b**) Fundamental modal shape.

**Figure 12.**The top displacement and acceleration responses of the offshore substation for different cases. (

**a**) The top displacement responses for the uncontrolled and different controlled cases. (

**b**) The acceleration response histories for the uncontrolled and different controlled cases.

Symbol | Expression | Physical Meaning |
---|---|---|

${\omega}_{\mathrm{n}}$ | $\sqrt{K/M}$ | The natural frequency of the primary offshore substation structure. |

${\zeta}_{\mathrm{n}}$ | $\frac{C}{2\sqrt{KM}}$ | The damping ratio of the primary offshore substation structure. |

$\mu $ | $m/M$ | The tuning mass ratio of the vibration control device. |

$\beta $ | $\frac{b+{b}_{2}}{M}$ | The tuning inertance ratio of the vibration control device. Note that for a single-tuned vibration control device, b_{2} = 0, whereas for a double-tuned vibration control device, b = 0. |

${\omega}_{\mathrm{d}}$ | $\sqrt{\frac{k}{m+b}}$ | The nominal frequency of the vibration control device. Note that for a double-tuned vibration control device, b = 0. |

$\nu $ | ${\omega}_{\mathrm{d}}/{\omega}_{\mathrm{n}}$ | The tuning frequency ratio of the vibration control device. |

${\omega}_{\mathrm{d}2}$ | $\sqrt{{k}_{2}/{b}_{2}}$ | The secondary nominal frequency of the sub-network of the double-tuned vibration control device. |

$\gamma $ | ${\omega}_{\mathrm{d}2}/{\omega}_{\mathrm{d}}$ | The secondary tuning frequency ratio of the double-tuned vibration control device. |

${\zeta}_{\mathrm{d}}$ | $\frac{c}{2\sqrt{k(m+b)}}$ | The nominal damping ratio of the vibration control device. Note that for a double-tuned vibration control device, b = 0. |

$\lambda $ | $s/{\omega}_{\mathrm{n}}$ | The dimensionless Laplace complex frequency. |

Case # | Device | Predetermined Parameter | Underdetermined Parameter |
---|---|---|---|

0 | None | — | — |

1 | TMD | μ = 0.05 | ν = 0.9524, ζ_{d} = 0.1118 |

2 | TMDI | μ = 0.01, β = 0.16, φ = 0.5 | ν = 0.9524, ζ_{d} = 0.1118 |

3 | TVMD | β = 0.20, φ = 0.5 | ν = 1.0260, ζ_{d} = 0.1147 |

4 | RIDTMD | μ = 0.04 | β = 0.0046, ν = 0.9259, γ = 1.1200, ζ _{d} = 0.0234 |

Response | Case # | Device | SD[q(t)] | J_{dyn} (%) | Max[q(t)] | J_{tot} (%) |
---|---|---|---|---|---|---|

Displacement (mm) | 0 | None | 11.1 | — | 61.5 | — |

1 | TMD | 8.4 | 24.4 | 46.2 | 24.9 | |

2 | TMDI | 8.5 | 24.0 | 46.3 | 24.7 | |

3 | TVMD | 8.1 | 26.9 | 46.2 | 24.8 | |

4 | RIDTMD | 8.2 | 26.7 | 46.3 | 24.8 | |

Acceleration (m/s ^{2}) | 0 | None | 0.253 | — | 1.000 | — |

1 | TMD | 0.167 | 33.9 | 0.603 | 39.7 | |

2 | TMDI | 0.168 | 33.5 | 0.606 | 39.4 | |

3 | TVMD | 0.164 | 35.1 | 0.615 | 38.5 | |

4 | RIDTMD | 0.163 | 35.7 | 0.630 | 37.0 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Xu, C.; Yu, M.; Huang, Z.
Lightweight Design of Vibration Control Devices for Offshore Substations Based on Inerters. *Sustainability* **2024**, *16*, 3385.
https://doi.org/10.3390/su16083385

**AMA Style**

Wang Y, Xu C, Yu M, Huang Z.
Lightweight Design of Vibration Control Devices for Offshore Substations Based on Inerters. *Sustainability*. 2024; 16(8):3385.
https://doi.org/10.3390/su16083385

**Chicago/Turabian Style**

Wang, Yanfeng, Chenghao Xu, Mengze Yu, and Zhicong Huang.
2024. "Lightweight Design of Vibration Control Devices for Offshore Substations Based on Inerters" *Sustainability* 16, no. 8: 3385.
https://doi.org/10.3390/su16083385