A Compendium of Formulae for Natural Frequencies of Offshore Wind Turbine Structures

Abstract

The design of an offshore wind turbine system varies with the turbine capacity, water depth, and environmental loads. The natural frequency of the structure, considering foundation flexibility, forms an important factor in structural design, lifetime performance estimates, and cost estimates. Although nonlinear numerical analysis in the time domain is widely used in the offshore industry for detailed design, it becomes necessary for project planners to estimate the natural frequency at an earlier stage and rapidly within reasonable accuracy. This paper presents a compendium of mathematical expressions to compute the natural frequencies of offshore wind turbine (OWT) structures on various foundation types by assimilating analytical solutions for each type of OWT, obtained by a range of authors over the past decade. The calculations presented can be easily made using spreadsheets. Example calculations are also presented where the compiled solutions are compared against publicly available sources.

1. Introduction

Offshore wind power is a potential sustainable solution to the growing energy needs of the industrialized world. With rapid expansion in Europe and Asia, and reducing installation and operating costs, wind power is speculated to become the cheapest form of green energy. The huge initial capital investment cost is, however, one of the primary challenges faced by the offshore wind market. Cost estimation in offshore wind hugely relies on the design of the turbine structure, which is highly site-specific and involves multiple iterative design and analysis steps. In most cases, the turbine system and the tower structure can be identified early but the foundation system retains a degree of uncertainty owing to the site conditions and potentially unreliable design guidelines. Typically costing 16–34% of the total cost of a wind farm project, the foundation system becomes a controlling factor [1]. The choice of a foundation in turn depends on factors such as the water depth, fabrication, installation, operation and maintenance, decommissioning, and economics.

The offshore wind turbine (OWT), due to its shape and form, is a dynamically sensitive system [2]. An offshore wind turbine essentially consists of the Rotor Nacelle Assembly (RNA), tower, substructure, and the foundation. The size of OWTs has been continuously increasing since the 2000s, and larger turbines tend to have lower natural frequencies. In most cases, the dominant wave frequencies are close to the natural frequencies, where dynamic design becomes important. The target natural frequency of the chosen turbine system at a location depends on the operating range of the turbine as well as the mass and stiffness of the turbine-foundation system.

The foundation stiffness, which may have group effects, forms an important input parameter for natural frequency estimation. Several researchers have proposed closed-form expressions for the natural frequency of OWTs considering soil–structure interaction (SSI) effects. Considerable attention has been given to the vibration characteristics of wind turbines on monopile foundations [3,4,5,6,7,8,9,10]. Closed-form solutions of Eigen frequencies and design steps of monopile-supported OWTs have been reported by Arany et al. [11,12,13]. The dynamic stiffness of rigid surface foundations available from previous works [14,15,16,17,18,19] can be employed to estimate the flexible-base natural frequency of gravity-based foundation wind turbines. A solution for the first natural frequency of OWT, supported by multiple jacket foundations, was discussed by Jalbi et al. [20]. This method can be further enhanced by adopting the group-effect correction factors reported by [21,22]. Ryu et al. [23] presented a simple closed-form solution for OWTs with a tripod suction caisson foundation. For floating wind turbines, the structural responses, such as nacelle surge motion and bending moments at the tower base, are dominated by rigid body motions rather than elastic deformations [24]. Numerical methods have been largely used in studies on floating wind turbines [25,26,27,28,29,30,31,32,33,34]. Simplified expressions for spreadsheet calculation of rigid body natural frequencies of tension-leg-platform-supported OWT have been reported by [35]. For the case of spar-buoy-type wind turbines, the analytical procedure presented by Ye and Ji [36] can be used to estimate the natural frequency. Studies on semisubmersible-type wind turbines have been largely based on experimental or numerical methods [37,38,39,40].

It is in the best interest of project planners and designers that a single point reference of formulae for the foundation stiffness and natural frequencies of the various types of OWTs is readily available. Simplified analytical solutions help in choosing the tower stiffness or foundation stiffness. Such estimates can also help in understanding the output from a numerical analysis. The main objective of the present paper is to provide a compendium of formulae to estimate the undamped natural frequency of OWTs in various modes, considering SSI wherever applicable. The formulae for each type of offshore wind turbines are compiled. Example calculations are also added for clarity.

2. Offshore Wind Turbine Structure and Vibration Characteristics

2.1. The Structure of an Offshore Wind Turbine

The main components of a wind turbine include the tower, foundation, nacelle, rotor blades, and the hub as shown in Figure 1. Most offshore wind turbines are characterized by a three-bladed turbine driving a horizontally mounted generator. The nacelle houses the generator and the gear box. The tower structure supports the weight of the RNA as well as the heavy lateral wind loads. Tubular steel towers are widely used in the offshore wind power industry. The operational rotor frequency range of a turbine is called the 1P range. For example, the NREL 5 MW turbine has a 1P range of 6.9 to 12.1 rpm, essentially meaning that the turbine operates at this frequency during its entire lifecycle. The foundation of an OWT is an important design consideration that often controls the financial viability of an offshore wind farm. The foundation engineer considers factors such as seabed conditions, installation related aspects, and environmental regulations before selecting a foundation type. The foundation design can also influence the target natural frequency of the whole system, which needs to avoid the 1P range. Simplified models often idealize the tower as a beam, and the RNA as a lumped mass with or without a rotational inertia. The foundation is often idealized as translational and rotational springs at the base.

Table 1. Parameters defining the tower–RNA system.

#	Input Parameter	Symbol	Unit
1	Mass of the rotor-nacelle assembly	$m_{RNA}$	kg
2	Tower height	$L_T$	m
3	Tower top diameter	$D_t$	m
4	Tower bottom diameter	$D_b$	m
5	Average tower wall thickness	$t_T$	m
6	Tower Young’s modulus	$E_T$	N/m2
7	Tower mass	$m_T$	kg

presents the typical parameters of the tower–turbine system that are required for natural frequency estimates.

Typically, the rotor-nacelle assembly (RNA) and the tower are supplied by the turbine manufacturers as standard units, while the foundation design is site-specific. Design is often a collaborative effort between the turbine manufacturer and the substructure designer.

Table 2. Characteristics of typical wind turbine systems.

Turbine	NREL	LW	DTU	Haliade-X	Unit
Power rating	5	8	10	12	MW
Rotor diameter	126	164	178.3	218.2	m
Hub height	90	110	119	135	m
Rotor speed range	6.9–12.1	6.3–10.5	6–9.6	7.81	rpm
Cut-in, rated	3, 11.4	4, 12.5	4, 11.4	3.5	m/s
Cut-out wind speed	25	25	25	28	m/s
Nacelle mass	296.78	375	551.56	600	tonne
Blade mass	17.74	35	41.72	55	tonne
Tower mass	347.46	558	605	2500	tonne
Tower height	87.6	106.3	115.6	129.1	m
Tower top diameter	3.87	5	5.5	5.5	m
Tower bottom diameter	6	7.7	8	8	m

presents the characteristics of the NREL 5 MW, LW 8 MW, DTU 10 MW, and Haliade-X 12 MW turbines.

2.2. OWT As a Dynamical System

The OWT system can be idealized using simple structural models to estimate its dynamical characteristics. The tower–RNA system can be, for all practical purposes, idealized as a vertical beam with a lumped mass at its end. Often, an OWT falls under the category of a high-slenderness–low-stiffness dynamical system [9,41]. Simplified models simulate the flexibility of the foundation–soil system using springs representing the various degrees of freedom. The design ensures that the system frequency lies outside of the wind and wave frequencies as well as the operating range of the turbine.

The coordinate system and general rigid-body degrees of freedom used in this article are shown in Figure 2. The modes of vibration of a wind turbine structure essentially depend on the combination of foundation system and the superstructure’s stiffness and mass distribution [1]. The fundamental modes of vibration for grounded OWT systems can be mainly of two types: sway-bending mode and rocking mode. The sway-bending mode consists of flexible modes of the tower–RNA system and is dominant in cases where the foundation has sufficient axial stiffness in comparison to the tower. The rocking mode is dominant when the foundation is axially deformable (less stiff), such as in the case of an OWT supported on multiple shallow foundations. Schematic diagrams showing the modes of vibrations in shallow-embedded foundations and deep-embedded foundations are presented in Figure 3a,b, respectively. Floating OWTs, on the other hand, exhibit surge, sway, heave, roll/bending, and pitch/bending modes.

2.3. Classification of OWT Systems

Offshore wind turbines can be broadly classified based on their substructures as grounded systems and floating systems. Grounded systems, for which the structure is anchored to the seabed, are planned for water depths below 60 m. For water depths below 30 m, monopiles, gravity-based foundations, and suction caissons are currently being used. For water depths between 30 m and 60 m, jackets or seabed frame structures supported on piles or caissons are either used or planned. Suction caissons or piles can also be used in groups with the support of tripod or jacket structures. These scenarios involve group effects in addition to soil–structure interaction (SSI). Typically for water depths more than 60 m, floating systems like tension-leg platforms (TLP) and spar-buoy floating concepts are employed where the tower–RNA system is allowed to float and is anchored to the seabed by a mooring system. Figure 4 illustrates the various types of foundations used today for different depths of water.

For water depths below 30 m, monopiles, gravity-based foundations, and suction caissons are currently being used. For water depths between 30 m and 60 m, jackets or seabed frame structures supported on piles or caissons are either used or planned. Monopiles are so far the most popular foundation choice for grounded systems. A floating system is usually considered for deeper waters, typically more than 60 m. Among floating systems, the semisubmersible type has been by far the most widely adopted worldwide. Despite its poorer fundamental stability in comparison to spar-buoy and TLP concepts, the semisubmersible greatly simplifies the transportation and installation procedure, thereby reducing costs.

3. Formulae for Stiffness and Natural Frequency

3.1. Wind Turbine Supported by Gravity Based Foundation

The gravity-based foundations for OWTs are designed to avoid uplift or overturning by providing an adequate dead load. Simple macro element models of the wind turbine structure can be employed to estimate the natural frequency, provided that the foundation is modelled by three springs: lateral, rotational, and cross-coupling springs. Arany et al. [11,13] presented an analytical method based on the Timoshenko beam model to compute the flexible-base natural frequency of OWT structures. The method represents the flexible-base natural frequency in terms of the fixed-base natural frequency multiplied by flexibility factors representing the substructure and foundation. This procedure can essentially be used for any foundation types that can be represented by the three-spring model as shown in Figure 5. In this paper, we adopt the recent work equivalent framework for foundation stiffness by [19] for layered soil stratum. The parameters required for the natural frequency calculation are presented in

Table 3. Information required for estimating the natural frequency of OWT supported on a gravity-based foundation.

#	Input Parameter	Symbol	Unit
1	Foundation diameter	$D$	$\mathrm{m}$
2	Total tower height	$L_t$	$\mathrm{m}$
3	Substructure Young’s modulus	$E_s$	$\frac{\mathrm{N}}{{\mathrm{m}}^2}$
4	Substructure moment of inertia	$I_s$	${\mathrm{m}}^4$
5	Lateral stiffness of foundation	$K_L$	$\frac{\mathrm{N}}{\mathrm{m}}$
6	Rocking stiffness of foundation	$K_R$	$\frac{\mathrm{Nm}}{\mathrm{rad}}$

.

The steps involved in the estimation of natural frequency are:

• Step 1. Calculate the average tower diameter, the average wall thickness, and the average tower diameter.

The average area moment of inertia is calculated as:

$$I_T=\frac{1}{8}{(D_T-t_T)}^3{t}_T\pi$$

(1)

where $D_b$ is the tower bottom diameter, and $D_t$ is the tower top diameter. The average wall thickness and the average tower diameter are calculated as:

$$\begin{gathered} t_T=\frac{m_T}{{\rho }_T{L}_T{D}_T\pi } \\ {D}_T=\frac{D_b+D_t}{2} \end{gathered}$$

(2)

where ${\rho }_T$ is the density of the tower material (steel).

• Step 2. Calculate the non-dimensional foundation stiffness values:

The non-dimensional foundation stiffness values are computed as:

$$\begin{gathered} {\eta }_L=\frac{K_L{L}_T^3}{E{I}_{\eta }} \\ {\eta }_{LR}=\frac{K_{LR}{L}_T^2}{E{I}_{\eta }} \\ {\eta }_R=\frac{K_R{L}_T}{E{I}_{\eta }} \end{gathered}$$

(3)

where $K_L,K_{LR},K_R$ are the stiffness parameters, and $E{I}_{\eta }$ is the equivalent bending stiffness of the tower, calculated as:

$$E{I}_{\eta }=E{I}_t·f\left(q\right)$$

(4)

where:

$$\begin{gathered} f\left(q\right)=\frac{1}{3}·\frac{2q^2{\left(q-1\right)}^3}{2q^{2}lnq-3q^2+4q-1} \\ q=\frac{D_b}{D_t} \end{gathered}$$

(5)

The foundation stiffnesses are characteristics of a soil–foundation system. Simplified methods exist to estimate the foundation parameters in homogeneous and layered soils. For foundations mounted on soil surface, the cross-coupling stiffness term can be ignored for practical purposes. For strata that can be idealized as homogeneous, the formulae from various classical works collated in [42] can be adopted. For circular foundations in layered soil profiles, the recent method proposed by [19] can be adopted for a spreadsheet-based estimation. The procedure is described below:

• Identify and tabulate the shear modulus and Poisson’s ratio for a depth not less than 20 times the foundation diameter/width.
• Compute the weight distribution function as follows:

  $$\widehat{w}=\frac{a}{b}{\left(\frac{\widehat{z}}{b}\right)}^{a-1}exp\left[-{\left(\frac{\widehat{z}}{b}\right)}^a\right]$$

  (6)

  where $\widehat{z}$ is the depth normalized with the foundation diameter, and a and b are parameters defined in

  Table 4. Values of parameters a and b.

  Stiffness	a	b
  Lateral $K_L$	1.27	$0.237-0.049\nu$
  Rotational $K_R$	1.35	$0.17+5{\nu }^4$

  Here, $\nu$ is the Poisson’s ratio of the soil layer.

  .
• Evaluate the equivalent shear modulus using the expression:

  $$\frac{1}{G_{eq}}={{\displaystyle \int }}_0^{\infty }\frac{1}{G}\widehat{w}d\widehat{z}$$

  (7)
• Compute the foundation stiffnesses using the expressions:

  $$K_L=\frac{16\left(1-\nu \right)D}{7-8\nu }{G}_{eq}$$

  (8)

  $$K_R=\frac{D^3}{3\left(1-\nu \right)}{G}_{eq}$$

  (9)

• Step 3. Compute the foundation flexibility factors.

The foundation flexibility coefficients are given as:

$$\begin{gathered} C_R\left({\eta }_L,{\eta }_R,{\eta }_{LR}\right)=1-\frac{1}{1+a\left({\eta }_R-\frac{{\eta }_{LR}^2}{{\eta }_L}\right)} \\ {C}_L\left({\eta }_L,{\eta }_R,{\eta }_{LR}\right)=1-\frac{1}{1+b\left({\eta }_L-\frac{{\eta }_{LR}^2}{{\eta }_R}\right)} \end{gathered}$$

(10)

where $a\approx 0.5$ and $b\approx 0.6$ are empirical coefficients [9,10]. The substructure flexibility coefficient represents the flexibility of the substructure. The distance between the mudline and the bottom of the tower is $L_S$, and $E_s{I}_s$ is the bending stiffness of the substructure. The coefficient is expressed in terms of two dimensionless parameters, the bending stiffness ratio $\chi$ and the length ratio $\psi :$

$$C_S=\sqrt{\frac{1}{1+{\left(1+\psi \right)}^3\chi -\chi }}$$

(11)

$$\begin{gathered} \chi =\frac{E_T{I}_T}{E_s{I}_s} \\ \psi =\frac{L_S}{L_T} \end{gathered}$$

(12)

• Step 4. Compute the fixed-base natural frequency.

The fixed-base natural frequency of the tower is expressed simply with the equivalent stiffness $k_0$ and equivalent mass $m_0$ of the first mode of vibration as:

$$f_{FB}=\frac{1}{2\pi }\sqrt{\frac{k_0}{m_0}}=\frac{1}{2\pi }\sqrt{\frac{3E_T{I}_T}{L_T^3\left(m_{RNA}+\frac{33}{140}{m}_T\right)}}$$

(13)

where $E_T$ is the Young’s modulus of the tower material, $I_T$ is the average area moment of inertia of the tower, $m_T$ is the mass of the tower, $m_{RNA}$ is the mass of the rotor-nacelle assembly, and $L_T$ is the length of the tower.

• Step 5. Compute the flexible-base natural frequency.

The sway-bending mode natural frequency can be expressed as:

$$f_0=C_L{C}_R{C}_S{f}_{FB}$$

(14)

where $C_L$ and $C_R$ are the lateral and rotational foundation flexibility coefficients, $C_S$ is the substructure flexibility coefficient, and $f_{FB}$ is the fixed-base (cantilever) natural frequency of the tower.

3.2. Wind Turbine Supported by Single-Suction Caisson

Large diameter caissons are regarded as alternatives to monopiles for water depths below 30 m. A suction caisson essentially consists of a large-diameter rigid circular lid with thin skirts as shown in Figure 6. The procedure proposed by Arany et al. [9,10], presented in Section 3.1., can also be used for the single-suction caisson that can be represented by the three-spring model. The input parameters required for computing the natural frequency of a suction-caisson-supported OWT are listed in

Table 5. Information required for estimating the natural frequency of OWT supported by single suction caisson.

#	Input Parameter	Symbol	Unit
1	Platform height above mudline	$L_S$	$\mathrm{m}$
2	Substructure Young’s modulus	$E_s$	$\frac{\mathrm{N}}{{\mathrm{m}}^2}$
3	Substructure moment of inertia	$I_s$	${\mathrm{m}}^4$
4	Caisson diameter	$D$	$\mathrm{m}$
5	Caisson depth	$L$	$\mathrm{m}$
6	Initial soil Young’s modulus at 1D depth	$E_{S0}$	$\frac{\mathrm{N}}{{\mathrm{m}}^2}$
7	Soil Poisson’s ratio	${\nu }_s$
8	Lateral stiffness of foundation	$K_L$	$\frac{\mathrm{N}}{\mathrm{m}}$
9	Cross-stiffness of foundation	$K_{LR}$	$\mathrm{N}$
10	Rocking stiffness of foundation	$K_R$	$\frac{\mathrm{Nm}}{\mathrm{rad}}$

.

A list of parameters required for the estimation of natural frequency is presented in Table 5.

The steps involved in the estimation of natural frequency are the same as those described in Section 3.1. with the exception of the foundation stiffness calculations. Skirted foundations trap soil below the lid and their stiffness can resemble that of rigid embedded foundations [43]. Jalbi et al. [44] carried out numerical analysis to study the stiffness of suction caissons. The aforementioned research also found that when the aspect ratio, L/D, was greater than 2, available closed-form solutions for deep foundations reported by [45], can be used. The stiffness functions for three different soil profiles are presented in Appendix A.1:

Table A1. Impedance functions for deep foundations exhibiting rigid behavior L/D > 2.

Ground profile	$\frac{K_L}{D{E}_{SO}f\left({\upsilon }_s\right)}$	$\frac{K_{LR}}{D^2{E}_{SO}f\left({\upsilon }_s\right)}$	$\frac{K_R}{D^3{E}_{SO}f\left({\upsilon }_s\right)}$
Homogeneous	$3.2{\left(\frac{L}{D}\right)}^{0.62}$	$-1.8{\left(\frac{L}{D}\right)}^{1.56}$	$1.65{\left(\frac{L}{D}\right)}^{2.5}$
Parabolic	$2.65{\left(\frac{L}{D}\right)}^{1.07}$	$-1.8{\left(\frac{L}{D}\right)}^2$	$1.63{\left(\frac{L}{D}\right)}^3$
Linear	$2.35{\left(\frac{L}{D}\right)}^{1.53}$	$-1.8{\left(\frac{L}{D}\right)}^{2.5}$	$1.59{\left(\frac{L}{D}\right)}^{3.45}$

$f\left({\upsilon }_s\right)=1+0.6\left|0.25-{\upsilon }_s\right|$.

. For the aspect ratio in the range 0.2 < L/D < 2, solutions presented by [44] as presented in Appendix A.1:

Table A2. Impedance functions for shallow skirted foundations exhibiting rigid behavior 0.5 < L/D < 2.

Ground profile	$\frac{K_L}{D{E}_{SO}f\left({\upsilon }_s\right)}$	$\frac{K_{LR}}{D^2{E}_{SO}f\left({\upsilon }_s\right)}$	$\frac{K_R}{D^3{E}_{SO}f\left({\upsilon }_s\right)}$
Homogeneous	$2.91{\left(\frac{L}{D}\right)}^{0.56}$	$-1.87{\left(\frac{L}{D}\right)}^{1.47}$	$2.7{\left(\frac{L}{D}\right)}^{1.92}$
Parabolic	$2.7{\left(\frac{L}{D}\right)}^{0.96}$	$-1.99{\left(\frac{L}{D}\right)}^{1.89}$	$2.54{\left(\frac{L}{D}\right)}^{2.44}$
Linear	$2.53{\left(\frac{L}{D}\right)}^{1.33}$	$-2.02{\left(\frac{L}{D}\right)}^{2.29}$	$2.46{\left(\frac{L}{D}\right)}^{2.9}$

$f\left({\upsilon }_s\right)=1.1\times \left(0.096\left(\frac{L}{D}\right)+0.6\right){\upsilon }_s^2-0.7{\upsilon }_s+1.06$.

, are recommended.

3.3. Wind Turbine Supported by Monopile

The mathematical modeling of the tower–monopile system is presented in Figure 7. The natural frequency of the system can be estimated following the procedure developed in Arany et al. [11,13], discussed in the preceding sections. However, it is important to identify the pile-head stiffness based on the soil profile at the site. The calculation steps remain the same, with the exception that pile stiffness needs to be calculated in Step 2. The data required to estimate the natural frequency of a monopile-supported wind turbine are listed in

Table 6. Information required for estimating the natural frequency of an OWT supported on a monopile.

#	Input Parameter	Symbol	Unit
1	Platform height above mudline	$L_S$	$\mathrm{m}$
2	Substructure Young’s modulus	$E_s$	$\frac{\mathrm{N}}{{\mathrm{m}}^2}$
3	Substructure moment of inertia	$I_s$	${\mathrm{m}}^4$
4	Pile diameter	$D_p$	$\mathrm{m}$
5	Pile depth	$L_p$	$\mathrm{m}$
6	Pile Young’s modulus	$E_p$	$\frac{\mathrm{N}}{{\mathrm{m}}^2}$
7	Pile moment of inertia	$I_p$	${\mathrm{m}}^4$
8	Initial soil Young’s modulus at 1D depth	$E_{S0}$	$\mathrm{N}/{\mathrm{m}}^2$
9	Soil Poisson’s ratio	${\nu }_s$
10	Lateral stiffness of foundation	$K_L$	$\frac{\mathrm{N}}{\mathrm{m}}$
11	Cross-stiffness of foundation	$K_{LR}$	$\mathrm{N}$
12	Rocking stiffness of foundation	$K_R$	$\frac{\mathrm{Nm}}{\mathrm{rad}}$

.

Over the years, several authors have proposed expressions to compute the dynamic stiffness of pile foundations. The first step in the calculation of the pile-head stiffness is the classification of pile behavior, i.e., whether the monopile will behave as a long flexible pile or a short one. Rigid piles undergo rigid body rotation in soil owing to their short length, whereas slender piles undergo deflection through the formation of plastic hinge. Two commonly adopted criteria to determine whether a pile can be slender or rigid are presented in

Table 7. Criteria to determine relative pile rigidity.

Reference	Parameters Required	Criteria
Poulos and Davis [46]	$\beta =\sqrt[4]{\frac{k_h{D}_p}{4E_p{I}_p}}$	If $\beta L_p>2.5$ pile is slender If $\beta L_p<1.5$ pile is rigid
Randolph [47], Carter, and Kulhawy [48]	$E_e=\frac{E_p{I}_p}{\frac{D_p^4\pi }{64}}$ $G^{*}=G_s\left(1+\frac{3}{4}{\nu }_s\right)$	If $\frac{L_p}{D_p}\ge {\left(\frac{E_e}{G^{*}}\right)}^{\frac{2}{7}}$ pile is slender If $\frac{L_p}{D_p}\le 0.05{\left(\frac{E_e}{G^{*}}\right)}^{\frac{1}{2}}$ pile is rigid

. Formulae proposed by different researchers for K_L, K_R, and K_LR for slender piles in various soil profiles are presented in

Table A3. Formulae for stiffness by different researchers for slender piles.

Lateral stiffness $K_L$	Cross-coupling stiffness $K_{LR}$	Rotational stiffness $K_R$
Randolph (1981), slender piles, both for homogeneous and linear inhomogeneous soils
$\frac{1.67E_{S0}{D}_P}{f_{\left({\nu }_S\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.14}$	$-\frac{0.3475E_{S0}{D}_P^2}{f_{\left({\nu }_S\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.42}$	$\frac{0.1975E_{S0}{D}_P^3}{f_{\left({\nu }_S\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.7}$
Pender (1993), slender piles, homogeneous soil
$1.285E_{S0}{D}_P{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.188}$	$-0.3075E_{S0}{D}_P^2{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.47}$	$0.18125E_{S0}{D}_P^3{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.738}$
Pender (1993), slender piles, linear inhomogeneous soil
$0.85E_{S0}{D}_P{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.29}$	$-0.24E_{S0}{D}_P^2{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.53}$	$0.15E_{S0}{D}_P^3{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.77}$
Pender (1993), slender piles, parabolic inhomogeneous soil
$0.735E_{S0}{D}_P{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.33}$	$-0.27E_{S0}{D}_P^2{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.55}$	$0.1725E_{S0}{D}_P^3{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.776}$
Poulos and Davis (1980) following Barber (1953), slender pile, homogeneous soil
$\frac{k_h{D}_P}{\beta }$	$-\frac{k_h{D}_P}{{\beta }^2}$	$\frac{k_h{D}_P}{2{\beta }^3}$
Poulos and Davis (1980) following Barber (1953), slender pile, linear inhomogeneous soil
$1.074n_h^{\frac{3}{5}}{\left(E_P{I}_P\right)}^{\frac{2}{5}}$	$-0.99n_h^{\frac{2}{5}}{\left(E_P{I}_P\right)}^{\frac{3}{5}}$	$1.48n_h^{\frac{1}{5}}{\left(E_P{I}_P\right)}^{\frac{4}{5}}$
Gazetas (1984) and Eurocode 8 Part 5 (2003), slender pile, homogeneous soil
$1.08D_P{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.21}$	$-0.22D_P^2{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.50}$	$0.16D_P^3{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.75}$
Gazetas (1984) and Eurocode 8 Part 5 (2003), slender pile, linear inhomogeneous soil
$0.60D_P{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.35}$	$-0.17D_P^2{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.60}$	$0.14D_P^3{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.80}$
Gazetas (1984) and Eurocode 8 Part 5 (2003), slender pile, parabolic inhomogeneous soil
$0.79D_P{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.28}$	$-0.24D_P^2{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.53}$	$0.15D_P^3{E}_{S0}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.77}$
Shadlou and Bhattacharya (2016), slender pile, homogeneous soil
$\frac{1.45E_{S0}{D}_P}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{s0}}\right)}^{0.186}$	$-\frac{0.30E_{S0}{D}_P^2}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{s0}}\right)}^{0.50}$	$\frac{0.18E_{S0}{D}_P^3}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.73}$
Shadlou and Bhattacharya (2016), slender pile, linear inhomogeneous soil
$\frac{0.79E_{S0}{D}_P}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{s0}}\right)}^{0.34}$	$-\frac{0.26E_{S0}{D}_P^2}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.567}$	$\frac{0.17E_{S0}{D}_P^3}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.78}$
Shadlou and Bhattacharya (2016), slender pile, parabolic inhomogeneous soil
$\frac{1.02E_{S0}{D}_P}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.27}$	$-\frac{0.29E_{S0}{D}_P^2}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.52}$	$\frac{0.17E_{S0}{D}_P^3}{f_{\left({\nu }_s\right)}}{\left(\frac{E_{eq}}{E_{S0}}\right)}^{0.76}$
Parameter definitions:
$E_{eq}=\frac{E_P{I}_P}{\frac{D_P^4\pi }{64}}$ $f_{\left({\nu }_S\right)}=\frac{1+{\nu }_S}{1+0.75{\nu }_S}$ for Randolph (1981) and $f_{({\nu }_s)}=1+\left|{\nu }_s-0.25\right|$ for Shadlou and Bhattacharya (2016) $\beta =\sqrt[4]{\frac{k_h{D}_P}{E_P{I}_P}}$

.

Table A4. Formulae for stiffness by different researchers for rigid piles.

$K_L$	$K_{LR}$	$K_R$
Poulos and Davis (1980) following Barber (1953), rigid pile, homogeneous soil $\left(n=0\right)$
$k_h{D}_P{L}_P$	$-\frac{k_h{D}_P{L}_P^2}{2}$	$\frac{k_h{D}_P{L}_P^3}{3}$
Poulos and Davis (1980) following Barber (1953), rigid pile, linear inhomogeneous soil $\left(n=1\right)$
$\frac{1}{2}{L}_P^2{n}_h$	$-\frac{1}{3}{L}_P^3{n}_h$	$\frac{1}{4}{L}_P^4{n}_h$
Carter and Kulhawy (1992), rigid pile, rock
$\frac{3.15G^{*}{D}_P^{\frac{2}{3}}{L}_P^{\frac{1}{3}}}{1-0.28{\left(\frac{2L_P}{D_P}\right)}^{\frac{1}{4}}}$	$-\frac{2G^{*}{D}_P^{\frac{7}{8}}{L}_P^{\frac{9}{8}}}{1-0.28{\left(\frac{2L_P}{D_P}\right)}^{\frac{1}{4}}}$	$-\frac{4G^{*}{D}_P^{\frac{4}{3}}{L}_P^{\frac{5}{3}}}{1-0.28{\left(\frac{2L_P}{D_P}\right)}^{\frac{1}{4}}}$
Shadlou and Bhattacharya (2016), rigid pile, homogeneous soil $\left(n=0\right)$
$\frac{3.2E_{S0}{D}_P}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{0.62}$	$-\frac{1.7E_{S0}{D}_P^2}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{1.56}$	$\frac{1.65E_{S0}{D}_P^3}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{2.5}$
Shadlou and Bhattacharya (2016), rigid pile, linear inhomogeneous soil $\left(n=1\right)$
$\frac{2.35E_{S0}{D}_P}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{1.53}$	$-\frac{1.775E_{S0}{D}_P^2}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{2.5}$	$\frac{1.58E_{S0}{D}_P^3}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{3.45}$
Shadlou and Bhattacharya (2016), rigid pile, parabolic inhomogeneous soil $\left(n=1/2\right)$
$\frac{2.66E_{S0}{D}_P}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{1.07}$	$-\frac{1.8E_{S0}{D}_P^2}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{2.0}$	$\frac{1.63E_{S0}{D}_P^3}{f_{\left({\nu }_s\right)}}{\left(\frac{L_P}{D_P}\right)}^{3.0}$
Parameter definitions: $E_{eq}=\frac{E_P{I}_P}{\frac{D_P^4\pi }{64}}$ $f_{({\nu }_s)}=1+\left|{\nu }_s-0.25\right|$ $\beta =\sqrt[4]{\frac{k_h{D}_P}{E_P{I}_P}}$

presents formulae proposed by different researchers for pile-head stiffness for rigid piles in various soil profiles.

3.4. Wind Turbine Supported by Jackets on Suction Buckets

Jackets or seabed frames have been used to support OWTs in water depths ranging from 23 m to 60 m. Jackets are typically three or four legged and are supported on either piles or suction caissons. In such systems with multiple foundations, the influence of soil structure interaction and group effects becomes important [49]. A closed-form solution to estimate the natural frequency of a jacket was developed by Jalbi and Bhattacharya [20]. This approach combined with closed-form correction factors to incorporate group effects, proposed by [21] can provide a quick estimate of the natural frequency. The idealized tower jacket system is shown in Figure 8. The information required to estimate the natural frequency is presented in

Table 8. Information required for estimating the natural frequency of OWT supported by jacket on suction buckets.

#	Input Parameter	Symbol	Unit
1	Tower Young’s modulus	$E$	$\mathrm{GPa}$
2	Height of jacket	$h_J$	$\mathrm{m}$
3	Top spacing of jacket leg chords	$L_{top}$	$\mathrm{m}$
4	Bottom spacing of jacket leg chords	$L_{bottom}$	$\mathrm{m}$
5	Area of jacket leg chords	$A_c$	${\mathrm{m}}^2$
6	Distributed mass of jacket	$m_J$	$\frac{\mathrm{kg}}{\mathrm{m}}$
7	Equivalent distributed mass of tower–jacket system	$m_{eq}$	$\frac{\mathrm{kg}}{\mathrm{m}}$
8	Jacket bending stiffness	$E{I}_J$	${\mathrm{Nm}}^2$
9	Tower bending stiffness	$E{I}_T$	${\mathrm{Nm}}^2$
10	Tower–jacket system bending stiffness	$E{I}_{J-T}$	${\mathrm{Nm}}^2$
11	Number of foundations	$N$
12	Vertical stiffness of individual foundation	$K_R^f$	$\frac{\mathrm{N}}{\mathrm{m}}$
13	Rocking stiffness of individual foundation	$K_v^f$	$\frac{\mathrm{Nm}}{\mathrm{rad}}$
14	Rocking stiffness of foundation group	$K_R^{gs}$	$\frac{\mathrm{Nm}}{\mathrm{rad}}$

.

The steps involved in the calculation of natural frequency are as follows:

• Step 1. Calculate the equivalent stiffness of the jacket-truss and tower,

The equivalent flexural rigidity of the jacket and tower can be obtained using the expressions (refer to Figure 9):

$$\begin{gathered} E{I}_J=E{I}_{Jtop}·f\left(m\right) \\ {I}_{Jtop}=\frac{A_c{L}_{top}^2}{2} \\ f\left(m\right)=\frac{1}{3}·\frac{m{\left(m-1\right)}^3}{m^2-2mln\left(m\right)-1} \end{gathered}$$

(15)

where $m=\frac{L_{bottom}}{L_{top}}$

$$\begin{gathered} E{I}_{T }=E{I}_{Ttop}·f\left(m\right) \\ {I}_{Ttop}=\frac{\pi }{8}{D}_{top}^3{t}_T \\ f\left(q\right)=\frac{1}{3}·\frac{2q^2{\left(q-1\right)}^3}{q^2\left(2ln\left(q\right)-1\right)+4q-1} \end{gathered}$$

(16)

where

$$\begin{gathered} D_T=\frac{D_{top}+D_{bottom}}{2} \\ {t}_T=\frac{m_T}{\rho \pi h_T{D}_T} \\ q=\frac{D_{bottom}}{D_{top}} \end{gathered}$$

(17)

• Step 2. Calculate the equivalent bending stiffness of the tower–jacket system.

The equivalent bending stiffness, $E{I}_{T-J},$ of the tower–jacket system is given in Equation (13):

$$\begin{gathered} E{I}_{T-J}=E_T{I}_T\left(\frac{1}{1+{\left(1+\mathsf{\Psi }\right)}^3\mathsf{{\rm X}}-\mathsf{{\rm X}}}\right){\left(\frac{h_J+h_T}{h_T}\right)}^3 \\ \mathsf{{\rm X}}=\frac{E_T{I}_T}{E_J{I}_J} \\ \mathsf{\Psi }=\frac{h_J}{h_T} \end{gathered}$$

(18)

• Step 3. Calculate the equivalent mass of the tower–jacket system.

The equivalent mass can be computed by the following equation:

$$m_{eq}=\frac{\int m\left(z\right){\phi }_1^{2}dz}{\int {\phi }_1^{2}dz}=\frac{{{\displaystyle \sum }}_{i=1}^n{m}_i{{\displaystyle \int }}_{z\left(i-1\right)}^{z\left(i\right)}{\phi }_1^{2}dz}{{{\displaystyle \int }}_0^{z\left(i\right)}{\phi }_1^{2}dz}=\frac{m_J{{\displaystyle \int }}_0^{h_J}{\phi }_1^{2}dz+m_T{{\displaystyle \int }}_{h_J}^{h_J+h_T}{\phi }_1^{2}dz}{{{\displaystyle \int }}_0^{h_J+h_T}{\phi }_1^{2}dz}$$

(19)

The numerical integral can be easily computed as [20]:

$$\begin{gathered} \int {\phi }_1^{2}dz=z+\frac{1-{\beta }_1^2}{\frac{4{\lambda }_1}{L}}\times \sin \frac{2{\lambda }_1}{L}z-\frac{{\beta }_1}{\frac{2{\lambda }_1}{L}}\times \cos \frac{2{\lambda }_1}{L}z+\frac{1+{\beta }_1^2}{\frac{4{\lambda }_1}{L}}\times \sin h\frac{2{\lambda }_1}{L}z \\ +\frac{{\beta }_1^2}{\frac{2{\lambda }_1}{L}}\times \cos h\frac{2{\lambda }_1}{L}z-\frac{2{\beta }_1}{\frac{{\lambda }_1}{L}}\times \sin \frac{{\lambda }_1}{L}z\times \sin h\frac{{\lambda }_1}{L}z \\ -\frac{1+{\beta }_1^2}{\frac{{\lambda }_1}{L}}\times \sin \frac{{\lambda }_1}{L}z\times \cos h\frac{{\lambda }_1}{L}z-\frac{1-{\beta }_1^2}{\frac{{\lambda }_1}{L}}\times \cos \frac{{\lambda }_1}{L}z\times \sin h\frac{{\lambda }_1}{L}z \end{gathered}$$

(20)

The values of λ₁ and β₁ for a fixed-end cantilever beam can be taken as 1.8751 and −0.7341 respectively [50].

• Step 4. Calculate the fixed-base natural frequency:

  $$f_{FB}=\frac{1}{2\pi }\sqrt{\frac{3E{I}_{T-J}}{\left(0.243m_{eq}{h}_{total}+m_{RNA}\right){\left(h_{total}\right)}^3}}$$

  (21)

  where $h_{total}=h_t+h_J$
• Step 5. Calculate the rotational stiffness of the foundation group.

Assuming homogenous soil, and solid embedded cylindrical foundations as illustrated in Figure 10, the rotational stiffness of the foundation group, $K_R^g$ can be computed using a group interaction factor as given in Equation (17):

$$K_R^g=K_R^{gs}·\frac{1}{1+\frac{f_{1R}}{\frac{s}{D}}+\frac{f_{2R}}{{\left(\frac{s}{D}\right)}^2}}$$

$$f_{1R}=-0.67\left(1-0.13N\right)\left(1-0.53\nu \right)\left(1+0.35{\left(\frac{L}{D}\right)}^{0.49}\right)$$

$$f_{2R}=0.29\left(1-0.04N\right)\left(1-0.12\nu \right)\left(1+2.87\left(\frac{L}{D}\right)\right)$$

(22)

where s represents spacing between foundations and D represents the diameter of the cylindrical foundation element as shown in Figure 10. The rocking stiffness parameter for the foundation group, $K_R^{gs}$ is given by:

$$K_R^{gs}=N\left(K_R^f+\frac{1}{2}{r}^2{K}_V^f\right)$$

$$K_R^f=\frac{G{D}^3}{3\left(1-\nu \right)}\left[1+\left(7.5-9\nu \right)\left(\frac{L}{D}\right)+\left(10.5-7.7\nu \right){\left(\frac{L}{D}\right)}^{2.5}\right]$$

$$K_V^f=\frac{2GDln\left(3-4\nu \right)}{1-2\nu }\left[1+1.12\left(1-0.84\nu \right){\left(\frac{L}{D}\right)}^{0.84}\right]$$

(23)

• Step 6. Calculate the flexibility factor $C_J$.

The flexibility factor $C_J$ can be computed using the expression:

$$\begin{gathered} C_J=\frac{\tau }{\tau +3} \\ \tau =\frac{K_R^g{h}_{total}}{E{I}_{T-J}} \end{gathered}$$

(24)

• Step 7. Calculate the flexible-base natural frequency.

The flexible-base natural frequency can be expressed as the multiple of a flexibility parameter $C_J$ and the fixed-base natural frequency $f_{FB}$ as:

$$f_0=C_J{f}_{FB}$$

(25)

where $C_J$ is a function of the rotational stiffness of the foundation and $f_{FB}$ is the fixed-base natural frequency.

3.5. Wind Turbine Supported by Jacket on Piles

The case of jacket supported on a deep foundation can be analyzed following a similar procedure as that of a jacket on suction caissons, using the appropriate flexibility factor. The rotational stiffness of the foundation group in Equation (19) can be computed using structural models, considering both the lateral and vertical stiffness of individual piles. An approximate method proposed by Jalbi and Bhattacharya [20] leads to the following equation for rotational stiffness, $K_R^g$ for square configuration of the foundation:

$$K_R^g=K_V^p{s}^2$$

(26)

The vertical stiffness of a single pile, $K_V^p,$ can be estimated using an appropriate numerical analysis or estimation methods assuming idealized soil profiles. An approximate expression proposed by [39] is:

$$K_V^p=\frac{2\pi L_p{G}_s}{\zeta }$$

(27)

where $\zeta$ lies between 3 and 5.

Another expression proposed for the seismic design of bridges by [51] is as follows:

$$K_V^p=1.25\frac{E_{p}A}{L_p}$$

(28)

In these expressions, $L_p$ refers to the length of the pile, $G_s$ refers to the shear modulus of homogenous soil, and A represents the cross-sectional area of the pile.

3.6. Wind Turbine Supported on Tension-Leg Platform

The tension-leg platform wind turbine (TLPWT) concept design has promising application in intermediate water depths. Floating wind turbine system modeling requires simulation of complex interactions between wind- and wave-driven responses. However, following reasonable engineering assumptions and physical principles, first predictions of TLPWT behavior can be computed using simple spreadsheet calculations. Following simple linear approximations to find the added mass matrix [A], hydrostatic stiffness matrix [C], and mooring system stiffness matrix [K] discussed in previous literature [52,53], Bachynski and Moan [35] compiled the equations for quick estimation of the natural frequencies. Figure 11 presents the geometry parameters for rectangular and cylindrical pontoon designs and

Table 9. Information required for estimating the natural frequency of an OWT supported on a tension-leg platform.

#	Input Parameter	Symbol	Unit
1	Overall draft	$T$	$\mathrm{m}$
2	Number of tendons	$n_t$
3	Pretension in each cable	$T_o$	$\mathrm{N}$
4	Unstretched length of tendon	$I_o$	$\mathrm{m}$
5	Young’s modulus	$E_t$	$\mathrm{N}/{\mathrm{m}}^2$
6	Outer diameter of tendon	$d_t$	$\mathrm{m}$
7	Thickness of tendon	$t_t$	$\mathrm{m}$
8	Number of pontoons	$n_p$
9	Diameter of cylindrical pontoon	$d_p$	$\mathrm{m}$
11	Height of rectangular pontoon	$h_p$	$\mathrm{m}$
12	Width of rectangular pontoon	$w_p$	$\mathrm{m}$
13	Radius of pontoon from cylinder center	$r_p$	$\mathrm{m}$
14	Vertical location of pontoon	$z_s$	$\mathrm{m}$
15	Diameter of main cylinder of hull	$D_1$	$\mathrm{m}$
16	Diameter of base node of hull	$D_2$	$\mathrm{m}$
17	Height of main cylinder of hull	$h_1$	$\mathrm{m}$
18	Height of base node of hull	$h_2$	$\mathrm{m}$
19	Total steel mass	$M$	$\mathrm{kg}$

presents the input parameters required for the calculation.

The step-by-step procedure to arrive at the surge/sway and heave natural frequencies is as follows:

• Step 1. Calculate the mooring system stiffness factors.

The mooring system stiffness factors are calculated as:

$$\begin{gathered} K_{11}\approx {\displaystyle \sum }_{j=1}^{n_t}{k}_{11} \\ {k}_{11}=\frac{T_o}{l_o} \end{gathered}$$

(29)

$$\begin{gathered} K_{33}\approx {\displaystyle \sum }_{j=1}^{n_t}{k}_{33} \\ {k}_{33}=\frac{E_t{A}_t}{l_o} \end{gathered}$$

(30)

where $T_o$ represents the pretension in each mooring cable, $l_o$ represents the unstretched length of cable, n_t represents the number of tendons, E_t represents the elastic modulus, and A_t represents the cross-sectional area of the tendons.

• Step 2. Calculate the added-mass terms.

The added-mass terms can be calculated as:

$$A_{11}\approx a_t\left[D_1\right]\left(h_1-b_t\right)+a_t\left[D_2\right]h_2+{\displaystyle \sum }_{j=1}^{n_p}{l}_p{a}_t[h_p or d_p]{\cos }^2{\theta }_i$$

(31)

$$A_{33}\approx \rho \pi \frac{1}{12}{D}_2^3+{\displaystyle \sum }_{j=1}^{n_p}{l}_p{a}_t[w_p or d_p]$$

(32)

In these equations, $h_1$, $h_2$, $D_1$, $D_2$, $d_p$, and $w_p$ are geometry parameters as defined in Figure 5, and $b_t$ represents the distance from freeboard to tower base ($b_t=h_1+h_2-T$). The transverse added-mass per-unit lengths for a cylinder with diameter D, and a square section with side length h are given by Equations (24) and (25) respectively.

$$a_t\left[D\right]=\rho \pi D^2/4$$

(33)

$$a_t\left[h\right]=4.75\rho {\left(\frac{h}{2}\right)}^2$$

(34)

• Step 3. Calculate the hydrostatic stiffness terms.

The hydrostatic stiffness term can be computed using the equation:

$$C_{33}=\rho g{A}_{wp}$$

(35)

• Step 4. Compute the natural frequencies.

  $$f_{n1}=\frac{1}{2\pi }\sqrt{\frac{K_{11}}{M_{11}+A_{11}}}$$

  (36)

  $$f_{n3}=\frac{1}{2\pi }\sqrt{\frac{C_{33}+K_{33}}{M_{33}+A_{33}}}$$

  (37)

The parameters $M_{11}$ and $M_{33}$ represent the mass in surge and heave movements.

3.7. Wind Turbine Supported by Floating Spar Buoy

A spar-buoy-type OWT system essentially consists of a relatively deep cylindrical base providing the ballast, with a heavier lower part raising the centre of buoyancy about the centre of gravity of the system. Ye and Ji [36] idealized the spar-type OWT system into a free–free beam with two large mass components attached at its ends, which represent the RNA and the floating platform. A schematic of the idealized model of the spar-buoy OWT is presented in Figure 12. Torsion springs k₁ and k₂ are imposed at both ends of the tower beam to model the connections at the tower–floating platform and tower–nacelle joints. The angle between the longitudinal axes of the platform and the undeformed tower is represented by ${\theta }_1$ and the rotating angle at this location is represented by ${\alpha }_1$. The angle between the longitudinal axes of the nacelle and the undeformed tower is represented by ${\theta }_2,$ and the rotating angle at this location is represented by ${\alpha }_2$. The parameters required to estimate the natural frequency is presented in

Table 10. Information required for estimating the natural frequency of the spar-buoy-type OWT.

#	Input Parameter	Symbol	Unit
1	Mass of the platform	$M_1$	$\mathrm{tonne}$
2	Mass of the rotor-nacelle assembly	$M_2$	$\mathrm{tonne}$
3	Tower height	$h_T$	$\mathrm{m}$
4	Tower top diameter	$D_t$	$\mathrm{m}$
	Tower bottom diameter	$D_b$	$\mathrm{m}$
5	Average tower diameter	$D_T$	m
6	Average tower wall thickness	$t_T$	$\mathrm{m}$
7	Tower Young’s modulus	$E_T$	$\mathrm{GPa}$
8	Mass per unit length of tower	$m_T$	$\mathrm{tonne}$
9	Location of CG of platform	G1	$\mathrm{m}$
10	Location of CG of superstructure	g2	$\mathrm{m}$
11	Angle between longitudinal axes of platform and tower	${\theta }_1$	degrees
12	Angle between longitudinal axes of nacelle and tower	${\theta }_2$	degrees
13	Radius of gyration of tower section	r	$\mathrm{m}$
14	Moment of inertia of the platform about the bottom end of the tower	$I_{01}$	${\mathrm{m}}^4$
15	Moment of inertia of the nacelle about the top end of the tower	$I_{02}$	${\mathrm{m}}^4$

.

The steps to estimate the natural frequency are as follows:

• Step 1. Compute the matrix elements.

The matrix elements can be formed keeping $\lambda$ as a variable.

$$R_{11}=R_{13}=I_{01}{\omega }^2\lambda$$

(38)

$$R_{12}=-EI{\lambda }^2 \left(1-\frac{I_{01}{\omega }^2}{k_1}\right)+g_1{M}_1{\omega }^2\cos \left({\theta }_1\right)$$

$$R_{14}=EI{\lambda }^2 \left(1-\frac{I_{01}{\omega }^2}{k_1}\right)+g_1{M}_1{\omega }^2\cos \left({\theta }_1\right)$$

$$R_{21}=-EI{\lambda }^3+m{r}^2{\omega }^2\lambda -g_1{M}_1{\omega }^2\lambda \cos \left({\theta }_1\right)$$

$$R_{22}=-M_1{\omega }^2-g_1{M}_1{\omega }^2\cos \left({\theta }_1\right)\frac{EI{\lambda }^2}{k_1}$$

$$R_{23}=EI{\lambda }^3+m{r}^2{\omega }^2\lambda -g_1{M}_1{\omega }^2\lambda \cos \left({\theta }_1\right)$$

$$R_{24}=-M_1{\omega }^2+g_1{M}_1{\omega }^2\cos \left({\theta }_1\right)\frac{EI{\lambda }^2}{k_1}$$

$$R_{31}=-EI{\lambda }^{2}S-I_{01}{\omega }^2\lambda C+I_{02}{\omega }^2\frac{EI{\lambda }^2}{k_2}S-g_2{M}_2{\omega }^2\cos \left({\theta }_2\right)S$$

$$R_{32}=-EI{\lambda }^{2}C+I_{01}{\omega }^2\lambda C+I_{02}{\omega }^2\frac{EI{\lambda }^2}{k_2}C-g_2{M}_2{\omega }^2\cos \left({\theta }_2\right)C$$

$$R_{33}=EI{\lambda }^{2}SH-I_{01}{\omega }^2\lambda CH-I_{02}{\omega }^2\frac{EI{\lambda }^2}{k_2}SH-g_2{M}_2{\omega }^2\cos \left({\theta }_2\right)CH$$

$$R_{41}=-EI{\lambda }^{3}C+m{r}^2{\omega }^2\lambda C+M_2{\omega }^{2}S+g_2{M}_2{\omega }^2\cos \left({\theta }_2\right)\left(\lambda C-\frac{EI{\lambda }^2}{k_2}S\right)$$

$$R_{42}=EI{\lambda }^{3}S-m{r}^2{\omega }^2\lambda S+M_2{\omega }^{2}C+g_2{M}_2{\omega }^2\cos \left({\theta }_2\right)\left(\lambda S+\frac{EI{\lambda }^2}{k_2}C\right)$$

$$\begin{gathered} R_{43}=EI{\lambda }^{3}CH+m{r}^2{\omega }^2\lambda CH+M_2{\omega }^{2}SH+g_2{M}_2{\omega }^2\cos \left({\theta }_2\right)\left(\lambda CH+\frac{EI{\lambda }^2}{k_2}SH\right) \\ {R}_{44}=EI{\lambda }^{3}SH+m{r}^2{\omega }^2\lambda SH+M_2{\omega }^{2}CH+g_2{M}_2{\omega }^2\cos \left({\theta }_2\right)\left(\lambda SH+\frac{EI{\lambda }^2}{k_2}CH\right) \end{gathered}$$

where:

$$S=\sin \left(\lambda L_T\right), C=\cos \left(\lambda L_T\right), SH=\sin h\left(\lambda L_T\right) and CH=\cos h\left(\lambda L_T\right)$$

In these equations, M₁ is the mass of the platform, M₂ is the mass of the nacelle, I₀₁ is the moment of inertia of the platform about the bottom end of the tower, I₀₂ is the moment of inertia of the nacelle about the top end of the tower, g₁ is the location of centre of gravity of the platform, g₂ is the location of centre of gravity of the superstructure, and r is the radius of gyration of the tower section.

• Step 2. Solve the transcendental equation for $\mathsf{\lambda }$.

The transcendental equation presented in Equation (39) can now be set up. The equation can be solved numerically for $\lambda$ defined as $\lambda =\omega \sqrt{\frac{m_t}{EI}}$ where $\omega$ represents the angular frequency, $m_t$ is the mass per unit length of tower, and EI is the flexural rigidity of the tower section.

$$det\left[\begin{array}{cccc}{R}_{11}& R_{12}& R_{13}& R_{14}\\ R_{21}& R_{22}& R_{23}& R_{24}\\ R_{31}& R_{32}& R_{33}& R_{34}\\ R_{41}& R_{42}& R_{43}& R_{44}\end{array}\right]=0$$

(39)

4. Conclusions

Estimating the flexible-base natural frequency of OWTs is an essential step in the cost estimation and design stage of offshore wind projects. This article forms a single point of reference for formulae to estimate the natural frequencies of OWTs. A collection of step-by-step analytical procedures to determine the natural frequencies of OWTs considering soil–structure interaction effects is presented.

Seven different foundation options for OWTs are discussed, with procedures to estimate the foundation stiffness and natural frequency, making this paper a go-to document for rapid decision making and analyses of such turbines, along with comparison of their relative feasibility in terms of structural performance versus power production [54]. It also allows for a wide range of researchers to access and compare their results in public domain, through a common and easily interpretable set of results.

There are several additional advantages offered by these drastically simple, yet effective estimates of natural frequencies. Development of models and subsequent testing of them in scaled scenarios [55] are better understood this way, along with their variabilities and uncertainties, as evidenced from existing reviews [56, 57] and experiments [58]. On the other hand, such estimates allow for easier handling of probabilistic results, like extreme value estimates [59] and related fragility [60]. Time-dependent degradation, like scour aspects, can be considered more easily, and future re-use of sites through assessment of geotechnical fatigue can also be understood in a more direct fashion. The sensitivity of the natural frequencies with respect to various conditions make it easier for a rapid sensitivity analysis, along with better control over sensor placement strategies and non-contact measurements [61]. It is expected that these initial estimates also create baselines for system identification, control [62], model updating, and eventual digital twinning.