Introduction

The offshore wind turbine (OWT) is the most efficient turbine in generating electricity because it is far from the shore to ensure the continuity of the wind blow on the operating turbine. The monopile foundation system is the most efficient foundation system in supporting the OWT and at the same time is the most unstable one when subjected to several lateral dynamic forces (i.e., wind, sea wave, and earthquake excitations). Studying a monopile OWT with the soil–structure interaction (SSI) effect is most challenging in structural design under multiple hazards. Vibration resistance by passive tuned mass damper (TMD) is a method used to control the large range of vibrations and reduce the harmful effect of these vibrations. The key factor in the good efficiency of passive TMDs is the placement of these devices in the overall total height of the OWT tower.

Al Satari and Hussain (2008) studied the dynamic response of wind turbine towers in arctic weather conditions with the SSI effect and concluded that taking into account the soil-foundation-structure interaction yielded a more realistic estimate of the natural frequency and that the foundation system design was controlled by the natural frequency of the soil-foundation-structure system rather than by strength or serviceability considerations.

Umut et al. (2011) investigated the use of square cross-sections for wind turbine towers of different thicknesses and heights and concluded that square cross-sections might be an alternative to commonly used circular cross-sections.

Kjørlaug (2013) studied numerically the dynamic analysis of a 5-MW wind turbine in horizontal and vertical directions subject to earthquake excitation and wind-induced load and including SSI and concluded that higher-order mode shapes were excited by horizontal earthquake excitation (due to the high natural periods of the structure) and that the vertical excitation from an earthquake can produce severe vertical accelerations in the upper parts of the wind turbine (posing considerations regarding buckling in the steel tower when combined with wind, disturbance of the fine-tuned machinery in the nacelle, and the design of the connection between nacelle and turbine tower).

Kjørlaug et al. (2014) studied the dynamic response of two (65-kW and 5-MW) wind turbines with SSI effect and showed that earthquakes are not expected to govern the design for small to moderate earthquakes in stiff soils; however, for softer soils, the displacement and base moment demand from earthquakes could very well match the response from wind-induced forces.

Yu et al. (2014) studied offshore wind turbines (OWTs) on different types of foundations subjected to dynamic loads with SSI effect and concluded that the deep foundations either used as monopod or multi-pod exhibit one single dominant frequency corresponding to the bending modes of vibration, while the shallow foundation either used as monopod or multi-pod exhibit two closely spaced natural frequencies corresponding to the rocking modes of vibration in two principal axes, which has the potential to broaden the range of frequencies that can get resonant with the environmental loading, this is detrimental in the real case of design OWTs.

Kaveh et al. (2015) determined the optimum parameters of tuned mass dampers to minimize the dynamic response of multi-story building systems under seismic excitations using the Charged System Search (CSS) optimization algorithm.

Sigurðsson (2015) studied the seismic response of wind turbine structures in the near-fault region by the FEM and investigated suitable models to account for near-fault effects.

Zuo et al. (2017) proposed and numerically investigated using multiple tuned mass dampers (MTMDs) to control vibrations from the fundamental and higher modes of offshore wind turbine tower under multiple hazards, i.e., under the combined wind, sea wave, and earthquake excitations.

He et al. (2017) proposed an efficient method for restraining the large vibration displacements and loads of offshore floating wind turbines under harsh marine environment by putting tuned mass dampers in the cabin and adopted different parameter optimization methods to optimize tuned mass damper parameters by considering the standard deviation of the tower top longitudinal displacement as the objective function.

Tong et al. (2017) investigated the use of tuned mass dampers (TMDs) to suppress the vibration of a monopile wind turbine tower and the performances of the optimal TMDs in the fore–aft and side–side directions are tested through FAST simulations, which achieve substantial fatigue load reductions.

Farghaly and Kontoni (2018) studied the nonlinear analysis of a riverine platform under earthquake loads and environmental loads (water wave loads and wind loads).

Hussan et al. (2018) investigated the optimization of multiple tuned mass damper (MTMD) for multi-mode vibration reduction of offshore wind turbine under seismic excitation, where MTMD has been installed at the top and base of the turbine tower corresponding to the mode shapes of the structure, and the MTMD parameters have been optimized based on response surface methodology (RSM).

Buckley et al. (2018) studied the use of tuned liquid column damper (TLCD) to reduce the structural vibrations of a wind turbine tower with monopile foundation considering soil–structure interaction (SSI) and found that SSI plays a key role in designing TLCD for wind turbine tower vibration control and TLCDs were observed to be a viable method of vibration control.

Zhao et al. (2018) studied the vibration control effects of bidirectional tuned mass damper (TMD) on the monopile offshore wind turbine tower under wind‐wave excitations and seismic excitations, both experimentally by shaking table tests on a 1/13‐scaled test model, and numerically by the finite element method and they found that the results obtained by numerical simulation fit well to the results derived from the experimental tests.

Lian et al. (2018) studied the mitigation of the vibrations of offshore wind turbine (OWT) structures under windstorm or typhoon conditions, by using a new eddy current with tuned mass damper (EC-TMD) system install at the top of them that combines the advantages of the eddy current damper and the tuned mass damper, which showed an excellent vibration-reducing performance.

Sun and Jahangiri (2018) proposed a three-dimensional pendulum tuned mass damper (3d-PTMD) to mitigate the tower and nacelle dynamic response in the fore-aft and side-side directions and they found that the 3d-PTMD with a mass ratio of 2% can improve the mitigation of the root mean square and peak response by around 10% when compared with the dual linear TMDs in controlling the bi-directional vibration of the offshore wind turbines under the misaligned wind, wave and seismic loading.

Kaveh and Sabeti (2018a, 2018b) improved the performance and efficiency of structural optimization of jacket-supporting structures for offshore wind turbines by using colliding body optimization (CBO) and its enhanced version (ECBO), and achieved a good weight reduction effect of the jacket.

Kaveh and Sabeti (2019) presented a computerized methodology for the optimal design of monopile offshore wind turbine structures based on meta-heuristic algorithms, consisting of Colliding Bodies Optimization (CBO), Enhanced Colliding Bodies Optimization (ECBO), and Vibrating Particle System (VPS), is such that more economic upshots can be accomplished.

Kaveh and Dadras Eslamlou (2020) presented a meta-heuristic algorithm named Colliding Bodies Optimization (CBO) is employed when investigating the optimal design of jacket-supporting structures for offshore wind turbines with Frequency Constraints.

Yang et al. (2019) studied the vibration reduction for a barge-type offshore wind turbine with an optimized tuned mass damper in the floating platform and the results showed that the passive control can improve the dynamic responses of the barge-type wind turbine by placing a TMD in the floating platform, and when replacing partial ballast with a uniform mass of the tuned mass damper, a significant reduction of the dynamic response is also observed.

Tran et al. (2019) developed a new efficient graphical user interface called VC4OWT to simulate and study the seismic response of a 5-MW offshore wind turbine (OWT) analyzed by FEM, considering soil–structure interaction (SSI) and using diagonal-bracing friction damper system at the platform for the vibration control purpose.

Zuo et al. (2019) studied numerically by FEM the mitigation of tower and out-of-plane blade vibrations of offshore monopile wind turbines by using multiple tuned mass dampers (MTMDs) installed in the tower and each blade and found that MTMDs can effectively mitigate the out-of-plane vibrations of the tower and blades subjected to the combined wind and sea-wave loadings when the wind turbine is in either the operational or parked condition.

Hemmati et al. (2019) studied the vibration suppression of offshore wind turbine foundations using tuned liquid column dampers (TLCDs) and tuned mass dampers (TMDs); the TMDs were found more efficient in operational conditions, whereas the TLCDs showed better performances in parked conditions, and this highlighted the efficiency of a combined TLCD-TMD system in which the dynamic responses are minimized efficiently in a wider selection of loading conditions.

Ghassempour et al. (2019) studied the vibration mitigation via tuned mass damper in bottom-fixed, horizontal-axis offshore wind turbines and found that the tuning frequency to attain optimal reduction of structural vibrations shall be changed depending on the wind velocity in operational conditions while, in contrast, is generally equal to the natural frequency of the first support structure modes only in parked conditions.

Wang et al. (2019) studied the motion control of a bottom-fixed offshore wind turbine (OWT) under different earthquakes by tuned mass damper (TMD) and the effects of the mass ratio, location and tuned frequency of the TMD on controlling structural responses of the OWT.

Sun et al. (2019) used a three-dimensional pendulum tuned mass damper (3d-PTMD) to mitigate the nacelle structural response in the fore-aft and side-side directions under wind, wave and near-fault ground motions and showed that the 3d-PTMD responded more rapidly and provided better mitigation of the bi-directional response caused by misaligned wind, wave and near-fault ground motions, while under system variations, the 3d-PTMD was found to be more robust than the dual linear TMDs to overcome the detuning effect; moreover, the 3d-PTMD with a mass ratio of 2% could mitigate the short-term fatigue damage of the offshore wind turbine tower by up to 90%.

Jahangiri and Sun (2019) utilized a three-dimensional pendulum tuned mass damper (3d-PTMD) with a linear electromagnetic energy harvester to mitigate the bi-directional vibrations induced by misaligned wind and wave loadings, as well as harvest the kinetic energy.

Sun and Jahangiri (2019) utilized a three-dimensional pendulum tuned mass damper (3d-PTMD) attached to the offshore wind turbine (OWT) to mitigate the bi-directional vibrations resulting from wind-wave misalignment and found that the 3d-PTMD can increase the wind turbine tower fatigue life by more than 50% in comparison with the dual linear tuned mass dampers (TMDs) deployed in the side-side and fore-aft directions.

Zhang et al. (2019) studied the seismic response mitigation of a wind turbine tower using a lightweight energy dissipation device, the tuned parallel inerter mass system (TPIMS) which consists of a tuned mass, a spring, and a parallel inerter subsystem.

Kontoni and Farghaly (2019a) investigated the mitigation of the seismic response of a cable-stayed bridge with soil–structure interaction (SSI) effect subjected to four different earthquakes by using tuned mass dampers and spring dampers with different placements in four different mitigation schemes.

Kontoni and Farghaly (2019b) studied the effect of base isolation and tuned mass dampers (TMDs) on the seismic response of reinforced concrete (RC) high-rise buildings considering soil–structure interaction; a comprehensive study of the combination of TMDs with three different base-isolator types for three different soil types and under five different earthquakes was conducted to show the most suitable hybrid passive vibration control system.

Kaynia (2019) presented the state of practice in seismic design of offshore wind turbines. Zuo et al. (2020) reviewed and discussed previous studies on the vibration mitigation of wind turbines.

Kaveh et al. (2020a) investigated the optimum design of a tuned mass damper inerter (TMDI) to control a 10-story base-excited shear building by selecting the H2 and H∞ norms of three different objective functions to tune two different single and double inerter TMDI configurations using a metaheuristic technique, and concluded that the TMDI is most effective when the optimization goal is to reduce the floor accelerations.

Kaveh et al. (2020b) studied the optimum design of a tuned mass damper inerter (TMDI) in order to control a 10-story base shear building under seismic loading by minimizing the H∞ norm of the roof displacement transfer function as the objective function for robust building control, and they recommended to optimize and design the TMDI using the MDOF model and not the SDOF model.

Kaveh et al. (2020c) compared the H2 and H∞ algorithms for the optimum design of tuned mass dampers (TMDs) under near-fault and far-fault earthquake motions and concluded that the H∞ objective function is superior to the H2 objective function under NF and FF earthquake excitations.

Kontoni and Farghaly (2020) investigated the TMD effectiveness for a steel high-rise building of 15 stories under the dynamic load of wind or four different earthquakes taking into consideration the effect of soil–structure interaction (SSI) and the TMDs were found to be a successful passive resistance method.

Farghaly and Kontoni (2022) studied the mitigation of seismic pounding (due to five different earthquakes) between two adjacent RC twin high-rise buildings, founded on raft foundation supported on piles inside a liquefaction-prone soil, considering SSI. Contact pounding elements between the two buildings (distributed at all floor levels and at the raft foundation level) were used. The three mitigation methods investigated were the base isolation, the tuned mass damper (TMD) method, and the pounding tuned mass damper (PTMD) method (using PTMDs connected between the two buildings) and it was found that the PTMD method was more efficient than the other two methods in mitigating the earthquake-induced pounding risk.

Farghaly and Kontoni (2023) investigated the mitigation of the seismic pounding between two L-shape in plan high-rise buildings (HRBs) subjected to earthquake in three different direction cases, including the SSI effect, and found that the most effective technique to mitigate the seismic pounding and help in seismically protecting these adjacent HRBs is the use of a combination of pounding tuned mass dampers (PTMDs) all over the height (at the connection points) together with tuned mass dampers (TMDs) on the top of both buildings.

Kontoni and Farghaly (2023a) investigated the enhancement of the earthquake resistance of RC and steel high-rise buildings by bracings, shear walls and TMDs considering SSI.

Kontoni and Farghaly (2023b) investigated the seismic control of T–shape in plan steel high-rise buildings (HRBs) with SSI effect using six different plan arrangements of (2, 4, 8, 8, 12, 18) tuned mass dampers (TMDs) applied on the top plan of the fixed HRB model, while for the SSI HRB model, these TMDs distributions were applied on the top plan and additionally on two intermediate plans along the HRB height and this three-level TMDs distribution was required for the plan arrangement of 2 or 4 TMDs.

Zhang et al. (2023a) studied the vibration mitigation in offshore wind turbine under combined wind-wave-earthquake loads using the tuned mass damper inerter (TMDI) which combines the tuned mass damper (TMD) with a linear inerter and their results demonstrated that the TMDI outperforms the TMD in suppressing the coupled wind-wave-earthquake responses by considering smaller attached mass.

Zhang et al. (2023b) studied the vibration mitigation of an integrated structure consisting of a monopile offshore wind turbine and aquaculture cage (MOWTAC) under earthquake, wind, and wave loads. A tuned mass damper (TMD) was applied to the MOWTAC under earthquakes, and the influence of the TMD parameters on the mitigation effects was studied. Likewise, a multiple tuned mass damper (MTMD) with various tuning frequencies is applied, and comparisons with the TMD were carried out.

In this paper, a monopile offshore wind turbine (OWT) will be modeled by FEM to show the effect of service and earthquake loads on the lateral displacements in x and y directions and the internal forces of the tower and the monopile of the OWT structure. The lateral displacements and straining actions of the OWT under the effect of the service and earthquake loads will be controlled by passive TMDs arranged in a way that gives the minimum dynamic response of the tower and the monopile displacements and internal forces. New arrangements of TMDs were tested to achieve the best performance of the OWT under service and earthquake loads. Passive TMDs placed at the top of the OWT tower and at the center of gravity (CG) of the OWT tower or at the connection between the OWT tower and monopile, were investigated. Various arrangements of TMDs and also the proposed top radial TMDs arrangements have been tested to determine their validity in resisting vibrations resulting from service and earthquake loads.

Numerical model

The offshore wind turbine (OWT) under study with its main parts and dimensions is shown in Fig. 1.

Fig. 1
figure 1

The main parts of the offshore wind turbine

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The integrated model of the structure-foundation-soil system is established by SAP2000 version 17 (2015) software, including wind turbine, tower, bucket foundation, and soil.

The upper blades and the nacelle located on top of the wind tower are simplified as a 250 t mass block in modeling. The steel tower is 90 m in height, the upper steel wind tower diameter is 3.80 m, and the lower part diameter of the tower is 5.00 m, with variable shell thickness from 25 mm at the top of the tower to 35 mm at the bottom of the tower which is modeled as shell elements.

The foundation is a reinforced concrete monopile structure with a total height of 25 m, a diameter of 6 m, and a pile tube made of steel (with concrete inside the steel tube). The Mohr–Coulomb model, a simple elastic–plastic constitutive model is used to model the soil. The parameters of the soil layer are shown in Table 1. The Coulomb friction formula is used for the contact interface between the monopile and the soil.

Table 1 Parameters of the soil
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A 3D soil element representing the soil in the finite element model of the structure-foundation-soil system is shown in Fig. 2. The stiffness and damping parameters (stiffness Kx, Ky, Kz and damping Cx, Cy, Cz) of the 3D soil elements (Fig. 2) can be calculated as e.g., in Newmark and Rosenblueth (1971), Dowrick (2009), etc.

Fig. 2.
figure 2

3D soil element

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TMD optimum parameters

The optimum parameters of a TMD are given in Eq. (1) to Eq. (2) as per Zahrai and Ghannadi-Asl (2008).

$${\alpha }_{opt.}=\frac{1}{1+\mu }\sqrt{\frac{2-\mu }{2}}$$
(1)
$${\zeta }_{opt.}=\sqrt{\frac{3\mu }{8(1+\mu )}}\sqrt{\frac{2}{2-\mu }}$$
(2)
$${k}_{d}=4{\pi }^{2}\mu {\alpha }^{2}\frac{{m}_{s}}{{T}_{s}^{2}}$$
(3)
$${c}_{d}=4\pi \mu \alpha \zeta \frac{{m}_{s}}{{T}_{s}}$$
(4)

where, αopt is the optimum frequency ratio, ζ is the damping ratio, ζopt is the optimum damping, kd is the spring stiffness, cd is the damping, md is the damper mass, μ is the mass ratio μ = md/ms (ratio of damper mass md to structure mass ms). Using the values of αopt and ζopt, the optimum values of damping cd and stiffness kd of the TMD can be calculated from Eqs. (3) and (4).

Results and discussions

A 3D FEM model of an offshore wind turbine (OWT) with soil-monopile interaction was subjected to several types of environmental loads. The environmental loads which are applied on the OWT have different directions so, different straining actions occur along the tower and pile height which under the service and earthquake loads can be devastating. Mitigation schemes using passive tuned mass dampers (TMDs) which are distributed at the tower top, or at the tower top and in the CG of the tower or at the connection between the tower and the pile, where various TMDs arrangements were tested to give the best arrangements of TMDs that provide the minimum straining actions. New arrangements of TMDs were tested for the first time to achieve the best performance of the OWT under service and earthquake loads. The service loads which act on the OWT are the wind loads acting on the blades at the top and all over the height of the tower until the sea level, the current and the waves of the sea. The earthquakes act all over the monopile height. As shown in Fig. 3, three different earthquakes were used in this study: the Chi-Chi, the El Centro, and the Northridge earthquakes. The pile–soil–structure interaction (PSSI) was considered, and the 3D soil element (Fig. 2) was used to obtain the realistic effect of these service and earthquake loads on the dynamic behavior of the OWT. The OWT straining actions represented by the axial forces, shear forces, bending moments at the tower and pile heights, and displacements in horizontal (x and y) directions which judged the performance of the TMD techniques compared with the OWT without any TMD.

Fig. 3
figure 3

Three earthquake accelerograms

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Figure 4 represents the displacements and straining actions of the OWT subjected to service and earthquake loads without TMDs. Figure 4a represents the displacements in the x direction, where the maximum top displacements occur in the El Centro and Northridge earthquakes, the pile is displaced by smaller values with respect to the tower (nearly ¼ times the tower displacements) while the end of the pile is displaced less. Figure 4b represents the y-direction displacements of the tower and pile of the OWT model, where the maximum displacements occur at the top of the tower and the ratio between the maximum top displacement of the tower and the pile displacement is equal to ¼. Figure 4c represents the shear force all over the tower and pile height, where the maximum shear force occurs at about the connection between the tower and pile, then the shear force decreases until a small value for both shear forces in x and y directions (the minimum values of shear forces at the base of the pile equals nearly 0.143 times less than the maximum values at the connection between the tower and pile), at level 60m from the connection level the shear force increases from the top of the tower then it decreases (these values ratio with respect to the maximum shear values is equal to 43%), and the shear forces in both directions for all earthquake cases are almost equal. Figure 4d shows the bending moments all over the tower and the pile height, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and the base of the pile nearly are equal to zero, the bending moments increase from top tower to the maximum values then decrease until nearly zero at the base of the pile, and the bending moments in both directions for all earthquake cases are almost equal. The wind blow on the blade affects the differences between the x and y directions bending moments and also between the x and y directions shear forces. Figure 4e represents the axial forces all over the tower height and pile depth, the axial force increase from the top of the tower until the connection between the tower and pile level, then the axial force value decrease until the bearing base axial force value, the initial axial force at the top of the tower is nearly equal to the weights of the rotor, nacelle and blade, then the decrease in the axial force value is due to the friction between the pile and soil which reduces the axial force on the pile.

Fig. 4
figure 4

Displacements and straining actions all over the turbine tower and pile without TMDs

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Figure 5 represents the displacements and straining actions of the OWT subjected to service and earthquake loads with one-direction (x-direction) TMD at the tower top with a mass ratio equal to 5%. Figure 5a represents the displacements in the x direction, where the maximum top displacements occur in the El Centro and Northridge earthquakes, the pile is displaced by smaller values with respect to the tower (nearly ¼ times the tower displacements) but the end of the pile is displaced less. Figure 5b represents the y-direction displacements of the tower and pile of the OWT model, where the maximum displacements occur at the top of the tower, and the ratio between the maximum top displacement of the tower and the pile displacement is equal to 1/3 times. Figure 5c represents the shear force all over the tower and pile height, where the maximum shear force occurs at about the connection between the tower and the pile, then the shear force decreases until small values for both shear force in x and y directions (the minimum values of shear forces at the end of the pile equals nearly 0.143 times less than the maximum values at about the connection between the tower and pile), at level 60m from the connection level the shear force increases from the top of the tower then it decreases (these value ratios with respect to the maximum shear values are equal to 57%), the shear forces in both directions for all earthquake cases are almost equal. Figure 5d shows the bending moments at the tower and the pile height, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and the base of the pile are nearly equal to zero, the bending moments increase from the top of the tower to the maximum values then decrease until nearly zero in the base of the pile, and the bending moments in both directions for all earthquake cases are almost equal. Figure 5e represents the axial forces all over the tower and pile lengths, the axial force increase from the top of the tower until about the connection between the tower and pile level, then the values of the axial force decrease until the bearing base axial force value, the ratio between the bearing base value and the maximum axial force value is equal to 0.25, the initial axial force at the top of the tower is nearly equal to the values of the weights of the rotor, nacelle and blade, then the decrease in the axial force value is due to the friction between the pile and soil which reduces the axial force on the pile.

Fig. 5
figure 5

Displacements and straining actions all over the turbine tower and pile with 1 top tower TMD (µ = 5%)

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Figure 6 represents the displacements and straining actions of the OWT subjected to service and earthquake loads with 1 top tower multiple TMD (MTMD) with a mass ratio of 5% (bi-directional TMD in x and y directions). Figure 6a represents the displacements in x direction, the maximum top displacements occur in the El Centro and Northridge earthquakes, the pile is displaced by a small value with respect to the tower (nearly 2/5 times the tower displacements) but the end of the pile is displaced less. Figure 6b represents the y-direction displacements of the tower and pile of the OWT model, the maximum displacements occur at the top of the tower, and the ratio between the maximum top displacement of the tower and the pile displacement is equal to 1/2.5 times. Figure 6c represents the shear force all over the tower and pile height, the maximum shear force occurs at about the connection between the tower and pile, then the shear force decreases until a small value for both shear forces in x and y directions, where the minimum values of the shear forces at the end of the pile equals nearly 0.13 times less than the maximum values at about the connection between the tower and pile; at top and 60m level from the connection level, the shear force decreases by 1.5 times than the maximum shear force acting at about the connection between the tower and pile. Figure 6d shows the bending moments all over the tower and the pile lengths, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and base of the pile are nearly equal to zero, and the bending moments increase from the top tower to the maximum values, then decrease until nearly zero at the base of the pile. Figure 6e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until the connection between the tower and pile level, then the axial force values decrease until the base bearing base axial forces value, the ratio between the bearing base value and the maximum axial force value is equal to 0.25, the initial axial force at the top of the tower is nearly equal to the values of weights of the rotor, nacelle and blade, then the decrease in axial force value is due to the friction between the pile and soil which reduces the axial force on the pile.

Fig. 6
figure 6

Displacements and straining actions all over the turbine tower and pile with 1 top tower MTMD (µ = 5%)

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Figure 7 shows the displacements and straining actions of the OWT subjected to service and earthquake loads with one TMD with a mass ratio of 5% at the tower top and also one TMD at the connection between the tower and pile with a mass ratio of 10%. Figure 7a represents the displacements in the x direction, the values of the displacements are close for all earthquakes and the pile displaces by a smaller value with respect to the tower (nearly 1/4 times the tower displacements). Figure 7b represents the y-direction displacements where their behavior is like that at the x-direction. Figure 7c represents the shear force all over the tower and pile heights, the maximum shear force occurs at about the connection between the tower and pile, then the shear forces decrease until small values for both shear force in the x and y directions (the minimum values of the shear forces at the end of the pile equals nearly 0.13 times less than the maximum values at about the connection between the tower and pile), and the shear force all over the height of the tower is nearly constant with ratio with respect to the maximum shear equal to nearly 30%. Figure 7d shows the bending moments all over the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and base of the pile are nearly equal to zero, and the bending moments increase from the tower top to the maximum values then decrease until nearly zero at the base of the pile. Figure 7e represents the axial forces all over the tower height and pile depth, the axial force increases from the top of the tower until about the connection between the tower and pile level, and then the values of the axial force decrease until the base bearing base axial forces value, the ratio between the bearing base value and the maximum axial force value is equal to 0.13, the initial axial force at the top of the tower is nearly equal to the value of the weights of the rotor, nacelle and blade, then the decrease in the axial force value is due to the friction between the pile and soil which reduces the axial force on the pile.

Fig. 7
figure 7

Displacements and straining actions all over the turbine tower and pile with 1 top tower TMD (µ = 5%) and 1 TMD (µ = 10%) at the connection between the tower and the pile

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Figure 8 illustrates the displacements and straining actions of the OWT subjected to service and earthquake loads with one TMD at the top of the tower and one TMD at the connection between the tower and pile, both with a mass ratio of 10%. Figure 8a represents the displacements in the x direction, the values of the displacements are close for all earthquakes and the pile displaces by smaller values with respect to the tower (nearly ¼ times the tower displacements). Figure 8b represents the y-direction displacements, where the values of the displacements are close for all earthquakes and the pile displaces by smaller values with respect to the tower (nearly 1/3 times the tower displacements). Figure 8c represents the shear force all over the tower and pile heights, the maximum shear force occurs at about the connection between the tower and pile, then the shear force decreases until small values for both the shear force in the x and y directions, the minimum values of the shear forces at the end of the pile are nearly equal 0.1 times less than the maximum values at about the connection between the tower and pile, and the shear force all over the height of the tower is nearly constant with respect to the maximum shear and nearly equal to 16.67%. Figure 8d shows the bending moments all over the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and base of the pile are nearly equal to zero, and the bending moments increase from the top tower to the maximum values and then decrease until nearly zero at the base of the pile. Figure 8e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until about the connection between the tower and pile level, and then the values of the axial force decrease until the base bearing base axial forces value, the ratio between the bearing base value and the maximum axial force value is equal to 0.13, the initial axial force at the top of the tower is nearly equal to the value of the weights of the rotor, nacelle and blade, then the decrease in the axial force value is due to the friction between the pile and soil which reduces the axial force on the pile.

Fig. 8
figure 8

Displacements and straining actions all over the turbine tower and pile with 1 top tower TMD (µ = 10%) and 1 TMD at the connection between the tower and the pile (µ = 10%)

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Figure 9 illustrates the displacements and straining actions of the OWT subjected to service and earthquake loads with 1 TMD at the top and 1 at the CG of the tower, both with a mass ratio of 10%. Figure 9a represents the displacements in the x direction, where the values of the displacements are close for all earthquakes and the pile displaces by a small value with respect to the tower (nearly ½ times the tower displacement). Figure 9b represents the y-direction displacements, where the values of the displacements are close for all earthquakes and the pile displaces by a small value with respect to the tower (nearly ½ times the tower displacement). Figure 9c represents the shear force all over the tower and pile heights, the maximum shear force occurs at about the connection between the tower and pile. Figure 9d shows the bending moments at the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the tower top and the base of the pile are nearly equal to zero, and the bending moments increase from the tower top to the maximum values and then decrease until nearly zero at the base of the pile. Figure 9e represents the axial forces all over the tower height and pile depth, the axial force increases from the top of the tower until about the connection between the tower and pile level, and then the value of the axial force decreases until the base bearing base axial force value, the ratio between the bearing base value and the maximum axial force value equals to 0.56, the initial axial force at the top of the tower is nearly equal to the value of the weights of the rotor, nacelle and blade, then the decrease in the axial force value is due to the friction between the pile and soil which reduces the axial force on the pile.

Fig. 9
figure 9

Displacements and straining actions all over the turbine tower and pile with 1 top tower TMD and 1 tower CG TMD (µ = 10%)

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Figure 10 represents the displacements and straining actions of the OWT subjected to service and earthquake loads with four TMDs at the top of the tower, with a mass ratio of 5%, Fig. 10a represents the displacements in the x direction, the values of the displacements are close for all earthquakes and pile displace by small values with respect to the tower (nearly 1/3 times tower displacements). Figure 10b represents the y-direction displacements, the values of the displacements are close for all earthquakes, and the monopile is displaced by small values with respect to the tower (nearly 1/3 times tower displacements). Figure 10c represents the shear force all over the tower and pile heights, the maximum shear force occurs at about the connection between the tower and pile and top of the tower, where at level 60 m the ratio between the shear and the maximum shear is equal to nearly 0.43. Figure 10d shows the bending moments at the tower and the pile heights, the maximum bending moments occur about at the connection level between the tower and the pile, and the bending moments at the top tower and base of the pile are nearly equal to zero, and the bending moments increase from top tower to the maximum values then decrease until nearly zero in the base of the pile. Figure 10e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until about the connection between the tower and pile level then the values of axial force values decrease until the base bearing base axial forces values, the ratios between the bearing base values and the maximum axial force values equals to nearly 0.56, the initial axial force at the top of the tower is nearly equal to the value of the weights of the rotor, nacelle and blade, then the decrease in the axial force value is due to the friction between the pile and soil which reduces the axial force on the pile.

Fig. 10
figure 10

Displacements and straining actions all over the turbine tower and pile with the top tower 4 TMDs (µ = 5%)

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Figure 11 shows the displacements and straining actions of the OWT subjected to service and earthquake loads with four TMDs at the top of the tower, with a mass ratio of 10%, Fig. 11a shows the displacements in the x direction, the values of the displacements are close for all earthquakes, and the pile displaced by small values with respect to the tower (nearly 0.2 times tower displacements). Figure 11b represents the y-direction displacements, the values of the displacements are close for all earthquakes, and the pile displaced by small values with respect to the tower (nearly ¼ times tower displacements). Figure 11c represents the shear force all over the tower and pile heights, the maximum shear force occurs at about the connection between the tower and pile, for both shear force in the x and y directions are nearly identical, and the maximum shear forces occur at the Northridge earthquake. Figure 11d shows the bending moments at the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and the base of the pile are nearly equal to zero, the bending moments increase from top tower to the maximum values then decrease until nearly zero in the base of the pile, and the maximum bending moments occur at the Northridge earthquake. Figure 11e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until about the connection between the tower and pile level, then the values of axial force values decrease until the base bearing base axial forces values, the ratio between the bearing base value and the maximum axial force value equals to 0.56, the initial axial force at the top of the tower is nearly equal to the value of the rotor, nacelle and blade weights, then the decrease in axial force value is due to the friction between the pile and soil which reduces the axial force on the pile.

Fig. 11
figure 11

Displacements and straining actions all over the turbine tower and pile with the top tower 4 TMDs (µ = 10%)

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Figure 12 shows the displacements and straining actions of the OWT subjected to service and earthquake loads with 1 TMD at the top of the tower and 1 TMD at the connection between the tower and pile, both with a mass ratio of 5%. Figure 12a shows the displacements in x direction, the values of the displacements are close for all earthquakes except in pile displacements. Figure 12b represents the y-direction displacements, where the values of the displacements are close for all earthquakes except in pile displacements. Figure 12c represents the shear force all over the tower and pile heights, the maximum shear force occurs at about the connection between tower and pile, and the shear forces in the x and y directions are nearly close. Figure 12d shows the bending moments at the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and the base of the pile are nearly equal to zero, the bending moments increase from top tower to the maximum values then decrease until nearly zero in the base of the pile, and maximum bending moments occur at the Northridge earthquake. Figure 12e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until the connection between the tower and pile level then the values of axial force values decrease until the base bearing base axial force value, the ratio between the bearing base value and the maximum axial force values equals to nearly 0.1875, the initial axial force at the top of the tower nearly equals to the values of the weights of the rotor, nacelle and blade weights, the decrease in axial force values is due to friction between pile and soil which reduce the axial force on the pile.

Fig. 12
figure 12

Displacements and straining actions all over the turbine tower and pile with top tower 1 TMD (µ = 5%) and 1 TMD at the connection between the tower and the pile (µ = 5%)

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Figure 13 shows the displacements and straining actions of the OWT subjected to service and earthquake loads with six TMDs at the top of the tower, with a mass ratio of 10%. Figure 13a shows the displacements in the x direction, and the pile displacements are small with respect to the values of the tower. Figure 13b represents the y-direction displacements, and the pile displacements are small with respect to the values of the tower. Figure 13c represents the shear force all over the tower and pile heights, the shear forces that occur on the tower are constant all over the tower, while the maximum shear force at about the connection between the tower and the pile equals about five times the shear force in the tower, and the shear forces in x and y directions are nearly close. Figure 13d shows the bending moments at the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and the base of the pile are nearly equal to zero, the bending moments increase from top tower to the maximum values then decrease until nearly zero in the base of the pile, and the maximum bending moments occur at the Chi-Chi earthquake. Figure 13e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until about the connection level between the tower and the pile, then the values of axial force values decrease until the base bearing base axial forces values, and the ratio between the bearing base value and the maximum axial force value is equal to nearly 0.64.

Fig. 13
figure 13

Displacements and straining actions all over the turbine tower and the pile with the top tower 6 TMDs (µ = 10%)

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Figure 14 shows the displacements and straining actions of the OWT subjected to service and earthquake loads with six TMDs at the top of the tower, with a mass ratio of 5%. Figure 14a shows the displacements in the x direction, where the pile displacements are small with respect to the displacements of the tower. Figure 14b represents the y-direction displacements, where the pile displacements are small with respect to the displacements of the tower. Figure 14c represents the shear force all over the tower and pile heights, where the shear forces in the tower are constant all over the tower, the shear force at about the connection between the tower and pile is equal to about six times the shear force in the tower, and both shear forces in x and y directions are nearly identical. Figure 14d shows the bending moments all over the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and the base of the pile are nearly equal to zero, the bending moments increase from top tower to the maximum values then decrease until nearly zero in the base of the pile, and the maximum bending moments occur at the Chi-Chi earthquake. Figure 14e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until about the connection between the tower and pile level, then the values of axial force values decrease until the base bearing base axial forces values, the ratio between the bearing base value and the maximum axial force value is equal to nearly 0.56.

Fig. 14
figure 14

Displacements and straining actions all over the turbine tower and the pile with the top tower 6 TMDs (µ = 5%)

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Figure 15 shows the displacements and straining actions of the OWT subjected to service and earthquake loads with twelve TMDs at the top of the tower with a mass ratio of 10%. Figure 15a shows the displacements in the x direction, where the pile displacements are small with respect to the displacements of the tower. Figure 15b represents the y-direction displacements, and the pile displacements are small with respect to the values of the tower displacements. Figure 15c represents the shear force all over the tower and pile heights, the shear forces in the tower are constant all over the tower, and the shear force at about the connection between the tower and the pile is equal to nearly six times the shear force in the tower, where both shear forces in x and y directions are nearly close. Figure 15d shows the bending moments all over the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the tower top and the base of the pile are nearly equal to zero, the bending moments increase from tower top to the maximum values and then decrease until nearly zero in the base of the pile, and the maximum bending moments occur at the Chi-Chi earthquake. Figure 15e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until about the connection level between the tower and the pile and then the axial force values decrease until the base bearing base axial force value, where the ratio between the bearing base values and the maximum axial force values is equal to 0.56.

Fig. 15
figure 15

Displacements and straining actions all over the turbine tower and the pile with the top tower 12 TMDs (µ = 10%)

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Figure 16 shows the displacements and straining actions of the OWT subjected to service and earthquake loads with twelve TMDs at the top of the tower, with a mass ratio of 5%. Figure 16a shows the displacements in the x direction, and the pile displacements are small with respect to the values of the tower where all are displaced on one side. Figure 16b represents the y-direction displacements, and the pile displacements are small with respect to the values of the tower displacements where all are displaced on one side. Figure 16c represents the shear force all over the tower and pile heights, the shear forces that occur on the tower are constant all over the tower, and the maximum shear force at about the connection between the tower and the pile equals nearly 5.5 times the shear force in the tower, where both shear forces in x and y directions are close. Figure 16d shows the bending moments at the tower and the pile heights, the maximum bending moments occur at about the connection level between the tower and the pile, the bending moments at the top tower and the base of the pile are equal to zero, the bending moments increase from top tower to the maximum values then decrease until nearly zero in the base of the pile, and the maximum bending moments occur at the Chi-Chi earthquake. Figure 16e represents the axial forces all over the tower height and pile depth, the axial forces increase from the top of the tower until about the connection between the tower and pile level then the values of axial force values decrease until the base bearing base axial force value, and the ratio between the bearing base value and the maximum axial force value equals to nearly 0.56 times.

Fig. 16
figure 16

Displacements and straining actions all over the turbine tower and the pile with the top tower 12 TMDs (µ = 5%)

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Figure 17 summarizes the maximum displacements and the maximum straining actions of the tower and the monopile of the OWT under the service and earthquake loads with the effect of the aforementioned different arrangements of TMDs. Figure 17a represents the maximum displacements occurring at the top of the tower and at the end of the pile, where the lowest “maximum displacements” (at the tower top and for pile end) occur at the 12 TMDs with 5% mass ratio case, while the next lowest “maximum displacements” are recorded at the 6 TMDs with 5% mass ratio case. Figure 17b represents the maximum shear forces all over the height of the tower, where the lowest “maximum shear force” that occurs in the tower of the OWT is when using 6 TMDs with a 5% mass ratio. Figure 17c represents the “maximum bending moments” all over the height of the tower, where the lowest “maximum bending moment” that occurs in the tower of the OWT is also when using 6 TMDs with a 5% mass ratio. Finally, Fig. 17d represents the maximum axial forces all over the height of the tower, where the lowest “maximum axial force” that occurs in the tower of the OWT is also when using 6 TMDs with a 5% mass ratio.

Fig. 17
figure 17

The maximum displacements and straining actions of the monopile OWT for different TMD cases

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Conclusions

The monopile offshore wind turbine (OWT) is one of the most used electricity generators in the world because of its effectiveness under wind blowing. The special construction of an OTW as a slender structure with vertical mass irregularity and foundation system containing a monopile (under the turbine tower), requires a special control system to mitigate the harmful effect of the vibrations generated by the lateral dynamic loads (wind, sea waves and earthquake wave in x and y directions) on the structure. Herein passive tuned mass dampers (TMDs) were used as are not consuming energy for operation and are easy to be installed. One key factor of TMDs' good performance is their arrangement on the tower of the OWT which has a big effect on the total performance of the OWT. Many different arrangements of TMDs were tested to mitigate the service and seismic loads affecting an offshore wind turbine (OWT) including the pile–soil–structure interaction (PSSI) effect. The lateral displacements, shear forces and bending moments in both horizontal directions and the axial forces all over the OWT tower and monopile heights were recorded to compare the performance of each mitigation scheme of TMDs. The following conclusions can be drawn:

  • A monopile OWT subjected to service and earthquake loads with pile–soil–structure interaction (PSSI) requires a critical design and special procedure to be analyzed.

  • The behavior of the OWT monopile with respect to displacements under service and earthquake loads shows lower values compared to the OWT tower displacements.

  • Different schemes of passive TMDs, placed at the top of the OWT tower or also at the center of gravity (CG) of the OWT tower or at the connection between the OWT tower and monopile, were tested.

  • Various arrangements of TMDs including the proposed herein radial TMDs arrangements at the top of the tower have been investigated to determine their validity in resisting vibrations resulting from service and earthquake loads.

  • Most of the tested TMDs arrangements mitigated the harmful effect of the vibrations resulting from dynamic lateral loads with respect to the model without TMDs.

  • The addition of TMDs at the connection between the OWT tower and the monopile was not effective.

  • The most effective TMDs placement was only at the top of the OWT tower.

  • The most effective TMDs arrangement occurred when using 6 TMDs in a radial arrangement (every 60 degrees) at the top of the OWT tower, as the comparison results showed that the top radial 6 TMDs arrangement was the most effective mitigation scheme for all straining actions in the tower and the monopile of the OWT subjected to service and earthquake loads.