# Vibration Control of Steel Wind Turbine Tower Using a Novel Tuned Mass Damper Refitted via Inner Platform

Authors: Zhenbo Lei; Jingchao Wang; Tian Peng; Huili An; Lang Shen; Tao Wang
Journal: Shock and Vibration
 (2023)

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## Abstract

Steel large-megawatts wind turbines have the light-damping and long-period properties, resulting in the adverse vibrations under the wind loads. In this paper, a novel tuned mass damper refitted via inner platform (IP-TMD) is proposed to control the excessive vibration of steel wind turbine tower (WTT). Firstly, the dynamic equation of steel WTT controlled by the IP-TMD system is established according to the principle of virtual work, and its dynamic coefficient and frequency ratio at corresponding fixed points are deduced. Then, the optimal frequency ratio and optimal stiffness and damping coefficients are obtained by the system optimization. Furthermore, a numerical simulation research is employed to analyze the frequency-response curves and resonance mitigation effect of IP-TMD under harmonic excitation. Finally, the vibration control efficiency of IP-TMD is calculated using the Wilson-θ method under the wind loadings; the results indicated that IP-TMD is able to reduce the dynamic response of steel WTT over 45% compared with the uncontrolled WTT cases.


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## Body

## 1. Introduction

As an important renewable energy source, wind turbine technology has been a significant contributor to the world energy production because of its feasible production cost, reliability, and efficiency [1]. To meet the increasing energy needs, the steel large-megawatt wind turbines have broadly built in the wind field at the higher altitudes [2]. However, the steel wind turbine tower (WTT) belongs a typical steel structure with light-damping and long-period, and this means that it will suffer from the excessive vibrations induced by the wind loads [3]. These adverse vibrations not only discount the energy generation efficiency but also will shorten the fatigue life of steel WTT [4]. Consequently, it is very important for steel WTT to control the adverse vibrations.In general, structure vibration control methods fall into three main types, including passive, active, and semiactive control technologies [5, 6]. In this respect, passive control devices have been studied actively for vibration mitigation of steel WWT in over the past decade [7, 8] since no external power is needed and the control mechanisms are simpler compared to the other two types of methods [9, 10]. Among these passive devices, the tuned mass damper (TMD) [11, 12] and tuned liquid damper (TLD) [13, 14] were extensively investigated as they are able to enhance the structural damping and can be tuned to control the structural vibration. As for the TLD system, the sloshing in the tank easily gives a rise to the irregular slope of deformed liquid, so it is not easy to predict its motion which in turn causes some difficulties in the design of the TLDs [15]. On the contrary, the TMD is generally considered as an attractive means to mitigate the adverse vibration of steel WTT because of its simple construction and high stability [16].The TMD can be divided into two categories according the structural form, including the support TMD installed in nacelle of steel wind turbine [17] and pendulum TMD suspended inside the tower [18]. As for the support TMD, Bin Zhao et al. investigated the vibration control performance of a support TMD for steel WTT under wind‐wave excitations, and the shaking table test results showed that TMD can effectively weaken the dynamic responses of steel WTT under the wind‐wave loads [19]. Xin et al. applied the bidirectional support TMD to suppress the vibration of steel WTT, and results indicated that an optimal supporting TMD could reduce substantially the fatigue load [20]. In fact, the machine parts of the wind turbine, such as the engines and gears, are located in the nacelle leaving limited space to install the support TMD in the wind turbine [21].Subsequently, the pendulum TMD suspended inside the WTT has been proposed to solve the space limitation problem. Colherinhas et al. used a pendulum TMD to control the excessive vibration of steel WTT; the numerical simulation results indicated that an optimal pendulum TMD can mitigate the dynamic responses of steel WWT under the random excitations [22]. Sun and Jahangiri developed a three-dimensional pendulum TMD with 2% mass ratio to reduce the vibration of a steel WTT; the results illustrated that it is capable to decrease the dynamic responses of nacelle by over 50% in fore-aft and side-side directions [23]. Because the frequency of the pendulum TMD can only be tuned by changing the pendulum length, the efficiency of the pendulum TMD is limited by this mass ratio.It is known that the performance of a TMD system depends on the mass ratio between control device and main structure [24], so it means that the addition mass of TMD will exceed two tons when the mass ratio is taken as 1–3%. Besides the space limitation caused by the machine parts, the extra mass can also affect the material strength and structure stability of steel WTT [25]. To solve the aforementioned problem, a novel TMD refitted via inner platform (IP-TMD) according to the structural special configuration is proposed to control the vibration of steel WTT in this paper. Meanwhile, the parameters design and system optimization of IP-TMD are performed, and the vibration control performance is analyzed to evaluate the effectiveness of IP-TMD.This paper is organized as follows. Schematic design of steel WTT controlled by the IP-TMD system is introduced, and the mathematical models of steel WTT and IP-TMD are descripted in Section2. By means of the virtual work principle, the dynamic equation of steel WTT controlled by the IP-TMD system is established, and design parameters of the IP-TMD system are performed in Section 3. In Section 4, a numerical research is used to analyze the frequency-response curve and resonance mitigation performance of IP-TMD under the harmonic excitation. Finally, the vibration control efficiency of IP-TMD is calculated by under the wind loadings in Section 5.


## 2. WTT with IP-TMD

### 2.1. Device Configuration

As shown in Figure1, a real steel wind turbine consists of four components, including three blades, nacelle, tower, and foundation. As the supporting structure of wind turbine, the steel conical tower is assembled from several segments of steel tube with variable wall thickness at the bottom and top. These segments are connected with bolts and flanges, and the mean diameter and wall thickness of each segment vary linearly with its length. Several inner platforms are installed as the pedestrian channels inside the tower.Figure 1
Schematic of steel WTT with the IP-TMD device.To avoid the space limitation caused by the machine parts and not add the too much extra mass, a novel TMD refitted via inner platform (IP-TMD) is proposed, as shown in Figure1. The IP-TMD contains a lumped mass producing the reversal inertia force, orthogonally bidirectional four springs supplying the motion stiffness, and viscous liquid dampers depleting the energy. In the vertical direction, the lumped mass is placed at a PTFE plate to decrease the sliding friction, and the PTFE plate is supported in the rigid frame. The mass block is connected to the cylindrical wall using four springs and viscous dampers in horizontal direction. When the controlled structure produces the excessive vibration induced by the wind loads, the vibration energy will be transferred to the lumped mass via the springs, and IP-TMD designed as bidirectional motion will dissipate the transmitted energy by the adjacent viscous dampers [25, 26].A conventional TMD is usually installed at the nacelle of WTT, but an unavoidable issue is that the maximum stroke of this TMD will reach to 0.75 m under the wind or wave loads [24]. This means that it has no enough space to deplete the energy in practical application since the engines and gears are located in the nacelle [27]. On the contrary, the internal available space of tower is close to 4m, so IP-TMD installed inside the tower can play a practical role in dissipating hysteresis energy. Another advantage is that the device refitted via inner platform can reduce the extra additional mass, and it is the physical mass of inner platform. Moreover, when the WTT structure need to be inspected and maintained at shutdown state of wind turbine, the IP-TMD can be locked as a temporary pedestrian platform and maintenance channel.


### 2.2. Mathematical Model

For a steel WTT, since the first vibration mode plays a dominant role in the dynamic analysis, a generalized SDOF model is developed to study the dynamic properties [25]. In addition, because the blades do not provide the lateral stiffness of nacelle motion, the tower can be simplified as a tapered tubular cantilever column, with a concentrated mass M representing the rotor and nacelle at the tower top [26], as shown in Figure 2. According to the principle of generalized displacement, the horizontal displacement x (z, t) of tower section at the height z can be expressed as follows [27]:(1)xz,t=φzx1t,where parameter φ (z) denotes the value of shape function for the first vibration mode, x1(t) is the generalized displacement of tower top, and t is the time.Figure 2
Mechanical model of WTT controlled by IP-TMD.In Figure2, x2 (t) is the generalized displacement of IP-TMD. H is the tower height, and h is the distance between upmost inner platform and tower top of steel WTT. M (z) denotes the distributional mass coefficient of the tower section at height z. k2 denotes the stiffness coefficient of springs, and c2 is the damping coefficient of viscous dampers for IP-TMD. Q denotes the relative displacement between IP-TMD and tower wall. M1, c1, k1, and Feff (t) are the generalized mass, stiffness, damping, and equivalent load of steel WTT, respectively, and they can be gained in terms of virtual work principle as follows [27, 28]:(2)m1=∫0Hmzφz2dz+MφH2,c1=∫0Hczφz2dz,k1=∫0HEIzφ″z2dz−Mg∫0Hφ′z2dz,Fefft=∫0Hpz,tφzdz+FtφH,where p (z, t) is the distributional load along with the height z, F (t) is the concentrated load of tower top, g is the gravitational acceleration, and EI is the flexural stiffness. Symbols “ and”' are the first- and second-order derivative with respect to z, respectively.


## 2.1. Device Configuration

As shown in Figure1, a real steel wind turbine consists of four components, including three blades, nacelle, tower, and foundation. As the supporting structure of wind turbine, the steel conical tower is assembled from several segments of steel tube with variable wall thickness at the bottom and top. These segments are connected with bolts and flanges, and the mean diameter and wall thickness of each segment vary linearly with its length. Several inner platforms are installed as the pedestrian channels inside the tower.Figure 1
Schematic of steel WTT with the IP-TMD device.To avoid the space limitation caused by the machine parts and not add the too much extra mass, a novel TMD refitted via inner platform (IP-TMD) is proposed, as shown in Figure1. The IP-TMD contains a lumped mass producing the reversal inertia force, orthogonally bidirectional four springs supplying the motion stiffness, and viscous liquid dampers depleting the energy. In the vertical direction, the lumped mass is placed at a PTFE plate to decrease the sliding friction, and the PTFE plate is supported in the rigid frame. The mass block is connected to the cylindrical wall using four springs and viscous dampers in horizontal direction. When the controlled structure produces the excessive vibration induced by the wind loads, the vibration energy will be transferred to the lumped mass via the springs, and IP-TMD designed as bidirectional motion will dissipate the transmitted energy by the adjacent viscous dampers [25, 26].A conventional TMD is usually installed at the nacelle of WTT, but an unavoidable issue is that the maximum stroke of this TMD will reach to 0.75 m under the wind or wave loads [24]. This means that it has no enough space to deplete the energy in practical application since the engines and gears are located in the nacelle [27]. On the contrary, the internal available space of tower is close to 4m, so IP-TMD installed inside the tower can play a practical role in dissipating hysteresis energy. Another advantage is that the device refitted via inner platform can reduce the extra additional mass, and it is the physical mass of inner platform. Moreover, when the WTT structure need to be inspected and maintained at shutdown state of wind turbine, the IP-TMD can be locked as a temporary pedestrian platform and maintenance channel.


## 2.2. Mathematical Model

For a steel WTT, since the first vibration mode plays a dominant role in the dynamic analysis, a generalized SDOF model is developed to study the dynamic properties [25]. In addition, because the blades do not provide the lateral stiffness of nacelle motion, the tower can be simplified as a tapered tubular cantilever column, with a concentrated mass M representing the rotor and nacelle at the tower top [26], as shown in Figure 2. According to the principle of generalized displacement, the horizontal displacement x (z, t) of tower section at the height z can be expressed as follows [27]:(1)xz,t=φzx1t,where parameter φ (z) denotes the value of shape function for the first vibration mode, x1(t) is the generalized displacement of tower top, and t is the time.Figure 2
Mechanical model of WTT controlled by IP-TMD.In Figure2, x2 (t) is the generalized displacement of IP-TMD. H is the tower height, and h is the distance between upmost inner platform and tower top of steel WTT. M (z) denotes the distributional mass coefficient of the tower section at height z. k2 denotes the stiffness coefficient of springs, and c2 is the damping coefficient of viscous dampers for IP-TMD. Q denotes the relative displacement between IP-TMD and tower wall. M1, c1, k1, and Feff (t) are the generalized mass, stiffness, damping, and equivalent load of steel WTT, respectively, and they can be gained in terms of virtual work principle as follows [27, 28]:(2)m1=∫0Hmzφz2dz+MφH2,c1=∫0Hczφz2dz,k1=∫0HEIzφ″z2dz−Mg∫0Hφ′z2dz,Fefft=∫0Hpz,tφzdz+FtφH,where p (z, t) is the distributional load along with the height z, F (t) is the concentrated load of tower top, g is the gravitational acceleration, and EI is the flexural stiffness. Symbols “ and”' are the first- and second-order derivative with respect to z, respectively.


## 3. Dynamic Analysis of WTT with IP-TMD

### 3.1. Motion Equation

The mechanical model of IP-TMD is a classical mass-spring-damping system, as shown in Figure2, and it is a SDOF system. Consequently, the WTT with IP-TMD is viewed as a two DOFs system, and its motion equation is established based on the principle of virtual work. In terms of the geometrical relationships in Figure 2, the total work done by external force δWE and work done by internal force δWI can be expressed as(3)δWE=−m1x¨1tδx1−m2x¨2tδx2+Fefftδx1,δWI=c1x˙1tδx1+k1x1tδxs+c2φIx˙1t−x˙2tδφIx˙1t−x˙2t+k2φIx1t−x2tδφIx1t−x2t,where t denotes the time. Symbols (˙) and (˙˙) are the first- and second-order derivative with respect to t, respectively. m2 is the additional physical mass of IP-TMD for steel WTT, and φI is the shape function value of first vibration mode at the position of IP-TMD. δx1 and δx2 are the virtual displacements corresponding to steel WTT and IP-TMD system, respectively.It is well known that the total work done by external forceδWE is equal to work done by internal force δWI. Subsequently, the balance equation can be gained as(4)m1x¨1t+c1x˙1t+k1x1t+φIc2φIx˙1t−x˙2t+φIk2φIx1t−x2t−Fefftδx1+m2x¨2t−c2φIx˙1t−x˙2t−k2φIx1t−x2tδx2=0.In (4), δxs and δxd denote the virtual displacements of steel WTT and IP-TMD, respectively, so their values are not zero. Thus, the dynamic equation of steel WTT controlled by the IP-TMD system can be expressed as(5)m1x¨1t+c1x˙1t+k1x1t+φIc2φIx˙1t−x˙2t+φIk2φIx1t−x2t=Fefft,m2x¨2t−c2φIx˙1t−x˙2t−k2φIx1t−x2t=0.Subsequently, based on (5), the matrix form of the motion equation can be rewritten as(6)m100m2x¨1x¨2+c1+φI2c2−φIc2−φIc2c2x˙1x˙2+k1+φI2k2−φIk2−φIk2k2x1x2=Fefft0,where m1, c1, k1, and Feff(t) are the generalized mass, stiffness, damping, and equivalent load of steel WTT, respectively. M2 is the physical mass of IP-TMD, and k2 is the stiffness coefficient of springs. C2 is the damping coefficient of viscous dampers, and φI is the shape function value of first vibration mode at the position of IP-TMD.


### 3.2. Dynamic Amplification Coefficient

The structural dynamic responses of steel WTT and IP-TMD are assumed as the harmonic response forms at steady state, and the motion equation of the linear WTT and IP-TMD system is solved by using the harmonic balance method (HBM) [27, 29]. Hence, the dynamic responses of WTT and IP-TMD at steady state can be obtained as(7)x1=Acosωt+Bsinωt,x˙1=Bωcosωt−Aωsinωt,x¨1=−Aω2cosωt−Bω2sinωt,x2=Dcosωt+Esinωt,x˙2=Eωcosωt−Dωsinωt,x¨2=−Dω2cosωt−Eω2sinωt,where A, B, D, and E are the intermediate coefficients and ω is the circular frequency of external excitation. x1 and x2 denote the displacement of WTT and IP-TMD, respectively. The symbols (˙) and (˙˙) are the first- and second-order derivative with respect to t, respectively.The generalized equivalent load of steel WTT is assumed as the sinusoidal form [27], and it can be expressed as(8)Fefft=P0sinωt,where Feff(t) is the external excitation, P0 is the excitation amplitude, and ω is the external excitation frequency.After substituting (7) and (8) into (6), the A, B, D, and E can be solved as(9)A=c22ωk11−m1ω2k22−m2ω2−k12k21−k22−m2ω2c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c12k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c122P0,B=k22−m2ω2k11−m1ω2k22−m2ω2−k12k21+c22ωc22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c12k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c122P0,D=c11ωk11−m1ω2k22−m2ω2−k12k21−k12c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c12k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c122P0,E=k12k11−m1ω2k22−m2ω2−k12k21+c11ωc22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c12k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c122P0,where cij and kij (i = j = 1 or 2) denote the stiffness and damping coefficients in matrix of equation (6), respectively, and they can be expressed as(10)c11=c1+φI2c2,c12=c21=φIc2,c22=c2,k11=k1+φI2k2,k12=k21=φIk2,k22=k2In terms of (7) and (8), the amplitude of dynamic displacement for steel WTT can be obtained as(11)xsmax=A2+B2=k22−m2ω22+c222ω2P02k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2−c11ωk22−m2ω2−ωk21c21+k12c122,where xsmax is the dynamic displacement amplitude.The dynamic coefficient can be defined as the ratio between dynamic displacement and static displacement amplitudes [27]; thus, the dynamic coefficient of steel WTT is obtained as(12)η=xsmaxxsta=μ2−β22+ζ2β21−β2μ2−β2−μ2α2+ζ2β21−β2+α/μ2μ2−β2−2α2,where η is the dynamic coefficient. μ denotes frequency ratio between IP-TMD frequency ω2 and WTT frequency ω1. β is the frequency ratio between load and WTT. According to (6), the IP-TMD and WWT frequencies can be defined as ω2 = (k2/m2)1/2 and ω1 = (k11/ms)1/2, respectively. ζ = c2/(m2ω1) denotes the damping tuning parameter. xsta = P0/k11 is the static displacement amplitude. α is the tuning parameter of stiffness, and it can be expressed as α = φI2k2/(k1 + φI2k2).


### 3.3. Parameters Design and Optimization

It is well known that a classical TMD system has two fixed point, which do not depend on the damping coefficient [27]; thus, the following mathematical expression can be gained in terms of (12) as(13)μ2−β221−β2μ2−β2−μ2α2=11−β2+α/μ2μ2−β2−2α2.After solving (13), two frequency ratios which are not dependent on the damping at the fixed points can be expressed as(14)βL=μ21+μ2−μ21+μ2−2α+2μ1−αα+2μ2,βR=μ21+μ2+μ21+μ2−2α+2μ1−αα+2μ2,where β is the frequency ratio of fixed point and symbols L and R of (14) are left and right frequency ratio of fixed point, respectively.In (12), if the damping coefficient c2 tends to infinity at two fixed points, their corresponding dynamic coefficients will change as(15)ηi=11−βi2+α/μ2μ2−βi2−2α,i=L,R.According to the optimization method of classical TMD [29], a TMD system is tuned to the optimal condition when the corresponding dynamic coefficients at two fixed points are equivalent [27], i.e., ηL = ηR. Hence, the mathematical expression can be obtained as(16)βL2+βR2=21−2α.Substituting (14) into (16), the optimal frequency ratio of WTT with the IP-TMD system can be deduced as(17)μopt=21−3α+21−3α2−4α22,where μopt denotes the optimal frequency ratio.After substituting (14) and (17) into (15), the maximum of dynamic amplification coefficient is obtained as(18)ηmax=α+2μ2α+μ21+μ22−2α+2μ21−α,where ηmax is the amplitude of dynamic coefficient of WTT controlled by the IP-TMD system.Because the amplitudes of dynamic amplification coefficient at two fixed points are equal, its curve slope is zero at two fixed points [27, 29], thus yielding(19)dη2dβi2i=L,R=0.By solving (19), the optimal value of damping coefficient for viscous liquid dampers is obtained as(20)copt=m22m1μopt2μopt2α+βL2−μopt22βL2k1+φI2k2μopt2+α,where copt is the optimal damping coefficient.According to equations (6), (12), and (17), the optimal stiffness coefficient of springs for IP-TMD can be deduced as(21)k2opt=k1m1m2m12+2φI2m1m2+m22,where k2opt is the optimal stiffness coefficient of springs.


## 3.1. Motion Equation

The mechanical model of IP-TMD is a classical mass-spring-damping system, as shown in Figure2, and it is a SDOF system. Consequently, the WTT with IP-TMD is viewed as a two DOFs system, and its motion equation is established based on the principle of virtual work. In terms of the geometrical relationships in Figure 2, the total work done by external force δWE and work done by internal force δWI can be expressed as(3)δWE=−m1x¨1tδx1−m2x¨2tδx2+Fefftδx1,δWI=c1x˙1tδx1+k1x1tδxs+c2φIx˙1t−x˙2tδφIx˙1t−x˙2t+k2φIx1t−x2tδφIx1t−x2t,where t denotes the time. Symbols (˙) and (˙˙) are the first- and second-order derivative with respect to t, respectively. m2 is the additional physical mass of IP-TMD for steel WTT, and φI is the shape function value of first vibration mode at the position of IP-TMD. δx1 and δx2 are the virtual displacements corresponding to steel WTT and IP-TMD system, respectively.It is well known that the total work done by external forceδWE is equal to work done by internal force δWI. Subsequently, the balance equation can be gained as(4)m1x¨1t+c1x˙1t+k1x1t+φIc2φIx˙1t−x˙2t+φIk2φIx1t−x2t−Fefftδx1+m2x¨2t−c2φIx˙1t−x˙2t−k2φIx1t−x2tδx2=0.In (4), δxs and δxd denote the virtual displacements of steel WTT and IP-TMD, respectively, so their values are not zero. Thus, the dynamic equation of steel WTT controlled by the IP-TMD system can be expressed as(5)m1x¨1t+c1x˙1t+k1x1t+φIc2φIx˙1t−x˙2t+φIk2φIx1t−x2t=Fefft,m2x¨2t−c2φIx˙1t−x˙2t−k2φIx1t−x2t=0.Subsequently, based on (5), the matrix form of the motion equation can be rewritten as(6)m100m2x¨1x¨2+c1+φI2c2−φIc2−φIc2c2x˙1x˙2+k1+φI2k2−φIk2−φIk2k2x1x2=Fefft0,where m1, c1, k1, and Feff(t) are the generalized mass, stiffness, damping, and equivalent load of steel WTT, respectively. M2 is the physical mass of IP-TMD, and k2 is the stiffness coefficient of springs. C2 is the damping coefficient of viscous dampers, and φI is the shape function value of first vibration mode at the position of IP-TMD.


## 3.2. Dynamic Amplification Coefficient

The structural dynamic responses of steel WTT and IP-TMD are assumed as the harmonic response forms at steady state, and the motion equation of the linear WTT and IP-TMD system is solved by using the harmonic balance method (HBM) [27, 29]. Hence, the dynamic responses of WTT and IP-TMD at steady state can be obtained as(7)x1=Acosωt+Bsinωt,x˙1=Bωcosωt−Aωsinωt,x¨1=−Aω2cosωt−Bω2sinωt,x2=Dcosωt+Esinωt,x˙2=Eωcosωt−Dωsinωt,x¨2=−Dω2cosωt−Eω2sinωt,where A, B, D, and E are the intermediate coefficients and ω is the circular frequency of external excitation. x1 and x2 denote the displacement of WTT and IP-TMD, respectively. The symbols (˙) and (˙˙) are the first- and second-order derivative with respect to t, respectively.The generalized equivalent load of steel WTT is assumed as the sinusoidal form [27], and it can be expressed as(8)Fefft=P0sinωt,where Feff(t) is the external excitation, P0 is the excitation amplitude, and ω is the external excitation frequency.After substituting (7) and (8) into (6), the A, B, D, and E can be solved as(9)A=c22ωk11−m1ω2k22−m2ω2−k12k21−k22−m2ω2c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c12k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c122P0,B=k22−m2ω2k11−m1ω2k22−m2ω2−k12k21+c22ωc22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c12k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c122P0,D=c11ωk11−m1ω2k22−m2ω2−k12k21−k12c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c12k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c122P0,E=k12k11−m1ω2k22−m2ω2−k12k21+c11ωc22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c12k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2+c11ωk22−m2ω2−ωk21c21+k12c122P0,where cij and kij (i = j = 1 or 2) denote the stiffness and damping coefficients in matrix of equation (6), respectively, and they can be expressed as(10)c11=c1+φI2c2,c12=c21=φIc2,c22=c2,k11=k1+φI2k2,k12=k21=φIk2,k22=k2In terms of (7) and (8), the amplitude of dynamic displacement for steel WTT can be obtained as(11)xsmax=A2+B2=k22−m2ω22+c222ω2P02k11−m1ω2k22−m2ω2−k12k212+c22ωk11−m1ω2−c11ωk22−m2ω2−ωk21c21+k12c122,where xsmax is the dynamic displacement amplitude.The dynamic coefficient can be defined as the ratio between dynamic displacement and static displacement amplitudes [27]; thus, the dynamic coefficient of steel WTT is obtained as(12)η=xsmaxxsta=μ2−β22+ζ2β21−β2μ2−β2−μ2α2+ζ2β21−β2+α/μ2μ2−β2−2α2,where η is the dynamic coefficient. μ denotes frequency ratio between IP-TMD frequency ω2 and WTT frequency ω1. β is the frequency ratio between load and WTT. According to (6), the IP-TMD and WWT frequencies can be defined as ω2 = (k2/m2)1/2 and ω1 = (k11/ms)1/2, respectively. ζ = c2/(m2ω1) denotes the damping tuning parameter. xsta = P0/k11 is the static displacement amplitude. α is the tuning parameter of stiffness, and it can be expressed as α = φI2k2/(k1 + φI2k2).


## 3.3. Parameters Design and Optimization

It is well known that a classical TMD system has two fixed point, which do not depend on the damping coefficient [27]; thus, the following mathematical expression can be gained in terms of (12) as(13)μ2−β221−β2μ2−β2−μ2α2=11−β2+α/μ2μ2−β2−2α2.After solving (13), two frequency ratios which are not dependent on the damping at the fixed points can be expressed as(14)βL=μ21+μ2−μ21+μ2−2α+2μ1−αα+2μ2,βR=μ21+μ2+μ21+μ2−2α+2μ1−αα+2μ2,where β is the frequency ratio of fixed point and symbols L and R of (14) are left and right frequency ratio of fixed point, respectively.In (12), if the damping coefficient c2 tends to infinity at two fixed points, their corresponding dynamic coefficients will change as(15)ηi=11−βi2+α/μ2μ2−βi2−2α,i=L,R.According to the optimization method of classical TMD [29], a TMD system is tuned to the optimal condition when the corresponding dynamic coefficients at two fixed points are equivalent [27], i.e., ηL = ηR. Hence, the mathematical expression can be obtained as(16)βL2+βR2=21−2α.Substituting (14) into (16), the optimal frequency ratio of WTT with the IP-TMD system can be deduced as(17)μopt=21−3α+21−3α2−4α22,where μopt denotes the optimal frequency ratio.After substituting (14) and (17) into (15), the maximum of dynamic amplification coefficient is obtained as(18)ηmax=α+2μ2α+μ21+μ22−2α+2μ21−α,where ηmax is the amplitude of dynamic coefficient of WTT controlled by the IP-TMD system.Because the amplitudes of dynamic amplification coefficient at two fixed points are equal, its curve slope is zero at two fixed points [27, 29], thus yielding(19)dη2dβi2i=L,R=0.By solving (19), the optimal value of damping coefficient for viscous liquid dampers is obtained as(20)copt=m22m1μopt2μopt2α+βL2−μopt22βL2k1+φI2k2μopt2+α,where copt is the optimal damping coefficient.According to equations (6), (12), and (17), the optimal stiffness coefficient of springs for IP-TMD can be deduced as(21)k2opt=k1m1m2m12+2φI2m1m2+m22,where k2opt is the optimal stiffness coefficient of springs.


## 4. Numerical Example

### 4.1. WTT Description

A 3.2 MW wind turbine from the National Renewable Energy Laboratory (NREL) is used to study the resonance mitigation performance of IP-TMD. The hub height of steel WTT is 86.7 m, and its detailed figure is shown in Figure3.Figure 3
Detailed figure of 3.2 MW (symbolφ denotes the tube diameter, and δ is the wall thickness).The elastic modulus of material for this WTT is 2.06 × 105 Mpa, and the density and Poisson ratio are 7850 kg/m3 and 0.3, respectively. The mean diameter of tower varies from 4.244 m at the tower base to 3.943 m at the tower top, and the corresponding thickness is from 48 mm to 15 mm. The main parameters of this steel WWT are listed in Table 1.Table 1
Main parameters of 3.2 MW steel WWT.
TypesParameterValueGross propertiesRating3.2 MWRotor orientation, configurationUpwind, 3 bladesHub height86.7 mMassBlade mass1.02 × 105 kgNacelle mass1.27 × 105 kgTower mass2.42 × 105 kgFrequency1st fore-aft mode natural frequency0.262 Hz1st side-side mode natural frequency0.265 Hz1st mode damping ratio (fore-aft and side-side)1.0%


### 4.2. Frequency-Response Curve

For the equation of the fundamental mode shape of steel WTT, there are many references which have obtained a satisfying formula [18, 23]. In terms of equation (48) listed in Ref. [23], it can be expressed as(22)φz¯=−0.6952z¯6+2.3760z¯5−3.5772z¯4+2.5337z¯3+0.3627z¯2,where z¯=z/H denotes the normalized tower height.After substituting the WTT parameters in4.1 subsection and (22) into (2), the generalized mass m1 is calculated as 2.89 × 105 kg, and stiffness k1 is 7.83 × 105 N/m. The mass ratio of IP-TMD is set at 1%, and thus the additional physical mass of IP-TMD m2 = m1 × 1% = 2.89 × 103 kg. The distance h between upmost inner platform and tower top of steel WTT is 10 m, and the optimal stiffness and damping coefficients of IP-TMD according to (20) and (21) are calculated as 7.73 × 103 N/m and 7.12 × 102 N/m/s, respectively. Meanwhile, the optimal frequency ratio of IP-TMD is calculated by using (17) as 0.99, and the stiffness tuning parameter α is 0.0064. These design parameters of the IP-TMD system are shown in Table 2.Table 2
Physical parameters of IP-TMD.
Devicek2optcoptμoptαηmaxIP-TMD7.73 × 103N7.12 × 102 N/m/s0.990.006417.60To verify the dynamic characteristics of WTT and control mechanism of IP-TMD, the frequency-response curves of WWT with and without IP-TMD are plotted in Figure4(a) at former three-order modes. It is found that the IP-TMD designed by the corresponding order frequency can effectively mitigate the resonance response at the corresponding mode. Moreover, the first vibration mode of uncontrolled WTT plays a dominant role in the dynamic response; this is because the physical mass of steel WTT mainly concentrates on the tower top. Hence, the dynamic responses induced by the first vibration mode are the main objective of vibration control. The local frequency-response curves of the first vibration mode for the uncontrolled WTT and WTT with IP-TMD are plotted in Figure 4(b).Figure 4
Frequency-response curves. (a) Multiple modes response. (b) First mode response.
(a)(b)As shown in Figure4(b), comparing with the uncontrolled WTT, the dynamic coefficient of WTT controlled by the IP-TMD system has no obvious divergence phenomenon near the natural frequency of steel WTT, and its amplitudes with the mode fission are reduced to 17.60. This is because that IP-TMD is designed to reversal vibrate with the same frequency when the external load frequency is close to the steel WTT frequency.


### 4.3. Harmonic Response

To study the resonance mitigation performance under the harmonic load, the external excitation is assumed asFeff(t) = 50sinωt kN. The uncontrolled WTT is viewed as a single DOF system with damping, and its dynamic displacement at the stability state can be expressed as(23)xU=P0k1−m1ω2sinωtk1−m1ω22+c1ω2−P0c1ωcosωtk1−m1ω22+c1ω2.In terms of (9) and (23), the dynamic responses of uncontrolled WTT and WTT with IP-TMD systems can be obtained as(24)xU=−4.790ωcosωt8352ω4−45250ω2+61310+3.915−1.445ω2sinωt8.352ω4−45.25ω2+61.31,xC=−7.191ω5+17.64ω3−0.319ωcosωt74.17ω8−179.5ω6+2886ω4−5041ω2+324.6+−1.283ω6+9.523ω4−24.50ω2+20.74sinωt7.417ω8−71.95ω6+288.6ω4−504.1ω2+32.46,(25)aU=4.790ω3sinωt8352ω4−45250ω2+61310−3.915−1.445ω2ω2cosωt8.352ω4−45.25ω2+61.31,aC=ω27.191ω5−17.64ω3+0.319ωsinωt74.17ω8−179.5ω6+2886ω4−5041ω2+324.6+ω21.283ω6−9.523ω4+24.50ω2−20.74cosωt7.417ω8−71.95ω6+288.6ω4−504.1ω2+32.46,where x denotes the dynamic displacement and a is the acceleration. Symbols U and C are the uncontrolled WTT and controlled WTT by IP-TMD, respectively.Like the classical TMD system, IP-TMD is also mainly designed to mitigate the resonance response induced by external excitations, so the structural responses of steady state are considered based on (24) and (25). Subsequently, the dynamic responses of uncontrolled WTT and WTT with IP-TMD systems can be plotted in Figures 5 and 6.Figure 5
Dynamic responses of uncontrolled WTT system. (a) Acceleration. (b) Displacement.
(a)(b)Figure 6
Dynamic responses of WTT controlled by IP-TMD system. (a) Acceleration. (b) Displacement.
(a)(b)It can be seen from Figure5 that the acceleration and displacement response amplitudes of the uncontrolled WTT system reach to 0.465 g and 1.75 m at the resonance region, respectively, since the resonance amplification phenomenon is very serious near the natural frequency of WTT. At the same time, because the IP-TMD is designed to vibrate at the resonant frequency with the same dynamic coefficient, the IP-TMD device can effectively mitigate the resonance responses. As shown in Figure 6, it can be found that the amplitudes of acceleration and displacement response for steel WTT controlled by the IP-TMD system are only 0.25 g and 0.76 m at the resonance region, respectively, and IP-TMD can reduce the acceleration and displacement responses by 46.2% and 56.6% under the harmonic excitation, respectively.


## 4.1. WTT Description

A 3.2 MW wind turbine from the National Renewable Energy Laboratory (NREL) is used to study the resonance mitigation performance of IP-TMD. The hub height of steel WTT is 86.7 m, and its detailed figure is shown in Figure3.Figure 3
Detailed figure of 3.2 MW (symbolφ denotes the tube diameter, and δ is the wall thickness).The elastic modulus of material for this WTT is 2.06 × 105 Mpa, and the density and Poisson ratio are 7850 kg/m3 and 0.3, respectively. The mean diameter of tower varies from 4.244 m at the tower base to 3.943 m at the tower top, and the corresponding thickness is from 48 mm to 15 mm. The main parameters of this steel WWT are listed in Table 1.Table 1
Main parameters of 3.2 MW steel WWT.
TypesParameterValueGross propertiesRating3.2 MWRotor orientation, configurationUpwind, 3 bladesHub height86.7 mMassBlade mass1.02 × 105 kgNacelle mass1.27 × 105 kgTower mass2.42 × 105 kgFrequency1st fore-aft mode natural frequency0.262 Hz1st side-side mode natural frequency0.265 Hz1st mode damping ratio (fore-aft and side-side)1.0%


## 4.2. Frequency-Response Curve

For the equation of the fundamental mode shape of steel WTT, there are many references which have obtained a satisfying formula [18, 23]. In terms of equation (48) listed in Ref. [23], it can be expressed as(22)φz¯=−0.6952z¯6+2.3760z¯5−3.5772z¯4+2.5337z¯3+0.3627z¯2,where z¯=z/H denotes the normalized tower height.After substituting the WTT parameters in4.1 subsection and (22) into (2), the generalized mass m1 is calculated as 2.89 × 105 kg, and stiffness k1 is 7.83 × 105 N/m. The mass ratio of IP-TMD is set at 1%, and thus the additional physical mass of IP-TMD m2 = m1 × 1% = 2.89 × 103 kg. The distance h between upmost inner platform and tower top of steel WTT is 10 m, and the optimal stiffness and damping coefficients of IP-TMD according to (20) and (21) are calculated as 7.73 × 103 N/m and 7.12 × 102 N/m/s, respectively. Meanwhile, the optimal frequency ratio of IP-TMD is calculated by using (17) as 0.99, and the stiffness tuning parameter α is 0.0064. These design parameters of the IP-TMD system are shown in Table 2.Table 2
Physical parameters of IP-TMD.
Devicek2optcoptμoptαηmaxIP-TMD7.73 × 103N7.12 × 102 N/m/s0.990.006417.60To verify the dynamic characteristics of WTT and control mechanism of IP-TMD, the frequency-response curves of WWT with and without IP-TMD are plotted in Figure4(a) at former three-order modes. It is found that the IP-TMD designed by the corresponding order frequency can effectively mitigate the resonance response at the corresponding mode. Moreover, the first vibration mode of uncontrolled WTT plays a dominant role in the dynamic response; this is because the physical mass of steel WTT mainly concentrates on the tower top. Hence, the dynamic responses induced by the first vibration mode are the main objective of vibration control. The local frequency-response curves of the first vibration mode for the uncontrolled WTT and WTT with IP-TMD are plotted in Figure 4(b).Figure 4
Frequency-response curves. (a) Multiple modes response. (b) First mode response.
(a)(b)As shown in Figure4(b), comparing with the uncontrolled WTT, the dynamic coefficient of WTT controlled by the IP-TMD system has no obvious divergence phenomenon near the natural frequency of steel WTT, and its amplitudes with the mode fission are reduced to 17.60. This is because that IP-TMD is designed to reversal vibrate with the same frequency when the external load frequency is close to the steel WTT frequency.


## 4.3. Harmonic Response

To study the resonance mitigation performance under the harmonic load, the external excitation is assumed asFeff(t) = 50sinωt kN. The uncontrolled WTT is viewed as a single DOF system with damping, and its dynamic displacement at the stability state can be expressed as(23)xU=P0k1−m1ω2sinωtk1−m1ω22+c1ω2−P0c1ωcosωtk1−m1ω22+c1ω2.In terms of (9) and (23), the dynamic responses of uncontrolled WTT and WTT with IP-TMD systems can be obtained as(24)xU=−4.790ωcosωt8352ω4−45250ω2+61310+3.915−1.445ω2sinωt8.352ω4−45.25ω2+61.31,xC=−7.191ω5+17.64ω3−0.319ωcosωt74.17ω8−179.5ω6+2886ω4−5041ω2+324.6+−1.283ω6+9.523ω4−24.50ω2+20.74sinωt7.417ω8−71.95ω6+288.6ω4−504.1ω2+32.46,(25)aU=4.790ω3sinωt8352ω4−45250ω2+61310−3.915−1.445ω2ω2cosωt8.352ω4−45.25ω2+61.31,aC=ω27.191ω5−17.64ω3+0.319ωsinωt74.17ω8−179.5ω6+2886ω4−5041ω2+324.6+ω21.283ω6−9.523ω4+24.50ω2−20.74cosωt7.417ω8−71.95ω6+288.6ω4−504.1ω2+32.46,where x denotes the dynamic displacement and a is the acceleration. Symbols U and C are the uncontrolled WTT and controlled WTT by IP-TMD, respectively.Like the classical TMD system, IP-TMD is also mainly designed to mitigate the resonance response induced by external excitations, so the structural responses of steady state are considered based on (24) and (25). Subsequently, the dynamic responses of uncontrolled WTT and WTT with IP-TMD systems can be plotted in Figures 5 and 6.Figure 5
Dynamic responses of uncontrolled WTT system. (a) Acceleration. (b) Displacement.
(a)(b)Figure 6
Dynamic responses of WTT controlled by IP-TMD system. (a) Acceleration. (b) Displacement.
(a)(b)It can be seen from Figure5 that the acceleration and displacement response amplitudes of the uncontrolled WTT system reach to 0.465 g and 1.75 m at the resonance region, respectively, since the resonance amplification phenomenon is very serious near the natural frequency of WTT. At the same time, because the IP-TMD is designed to vibrate at the resonant frequency with the same dynamic coefficient, the IP-TMD device can effectively mitigate the resonance responses. As shown in Figure 6, it can be found that the amplitudes of acceleration and displacement response for steel WTT controlled by the IP-TMD system are only 0.25 g and 0.76 m at the resonance region, respectively, and IP-TMD can reduce the acceleration and displacement responses by 46.2% and 56.6% under the harmonic excitation, respectively.


## 5. Vibration Control Induced by Wind

### 5.1. Load Design and Evaluation Index

To study the vibration control performance of IP-TMD under the wind loads, the 3.2 MW steel WTT simulation model is established using GH-Blade software. Based on the theory of blade element momentum, the horizontally thrust forces considering the wind direction, velocity, air density, and wind shear are simulated [23, 30]. Three centralized aerodynamic loads from wind are used to calculate the dynamic response as shown in Figure 7.Figure 7
Wind loads and frequency spectrums.The energy spectrums in Figure7 of these wind loads mainly distribute near the natural frequency (0.265 Hz) of steel WTT, so they can be well utilized to verify the effectiveness and rationality of IP-TMD under the wind loads.In order to evaluate the vibration mitigation efficiency, the peak indexE1 and root mean square index E2 are defined as [27](26)E1=maxxU−maxxCmaxxU,(27)E2=RMSxU−RMSxCRMSxU,where max denotes the peak response and RMS is the response of root mean square. Symbols U and C are the uncontrolled WTT and controlled WTT by the IP-TMD system, respectively.


### 5.2. Vibration Control Effect

The dynamic responses of the steel WTT with the IP-TMD system are calculated by using the Wilson-θ method under three wind loads [31]. The mass, stiffness, and damping matrix of WTT with the IP-TMD system can be calculated by using (6). Then, taking the wind load 1 as the example, the corresponding dynamic responses and power spectrum density are plotted in Figure 8.Figure 8
The dynamic responses under wind 1. (a) The response of time history. (b) The response of frequency spectrum.
(a)(b)It can be found from Figure8(a) that the dynamic responses of the WTT controlled by IP-TMD are obviously decreased than uncontrolled WTT under the wind 1 load. Since IP-TMD is designed to reversal vibrate with the same frequency near the fundamental frequency of steel WTT, it can effectively control the excessive vibration at the design frequency of steel WTT as the yellow rectangle frame in Figure 8(b). Meanwhile, the vibration control effectiveness under the three wind loads is calculated by using (26) and (27), as shown in Table 3.Table 3
Vibration control effectiveness.
CaseAcceleration (%)Velocity (%)Displacement (%)E1E2E1E2E1E2Wind 141.0552.6445.2754.1444.6853.73Wind 238.8044.2939.9046.2244.3844.82Wind 356.8568.7155.9469.1456.4567.18Mean value45.5655.2147.0456.5048.5055.24According to Table3, the IP-TMD device is noted capable to reduce the mean-peak and mean-RMS acceleration responses by 45.56% and 55.21%, respectively, under three wind loads relative to the uncontrolled cases. IP-TMD can also reduce the mean-peak and mean-RMS velocity responses by 47.04% and 56.60%, respectively, than the uncontrolled WWT. Furthermore, it can also mitigate the mean-peak and mean-RMS displacement responses by 48.50% and 55.24%, respectively, under three wind loads compared with the uncontrolled cases.


### 5.3. IP-TMD Stroke

Because the frequency of the IP-TMD device is tuned to near the fundamental frequency of the controlled steel WTT by adjusting the stiffness coefficient of springs, the vibration energy will be transferred to the mass block of IP-TMD when the controlled steel WTT structure produces the excessive vibration under wind loads. Subsequently, the IP-TMD will produce the motion stroke that is the relative displacement between mass block of IP-TMD and the cylinder wall of tower, and the viscous dampers will deplete the energy. Based on the Wilson-θ method, the stroke of IP-TMD curves is potted in Figure 9 under three wind loads.Figure 9
IP-TMD stroke.As can be seen from Figure9, the maximum of one-way stroke for the IP-TMD reaches to 0.40 m under three wind loads, so it indicated that the IP-TMD has the anticipated competence of dissipating hysteretic energy. Meanwhile, this also means that the motion space of IP-TMD will reach to 0.8 m, so the conventional TMD installed at the nacelle has no enough motion space to dissipate the energy since the engines and gears are installed inside the nacelle. On the contrary, internal available space of tower is close to 4 m; this reflects that IP-TMD placed inside the tower can play a practical role in dissipating hysteresis energy.


### 5.4. Bidirectional Vibration Control

Since the incoming wind direction of a WTT is always changing, the IP-TMD is designed as a bidirectional vibration control device. When the wind direction is different from output force, the direction misalignment effect will give a rise to the bidirectional vibration of WTT. The Wilson-θ method is applied to obtain the bidirectional displacement responses of tower top when the direction misalignment angles are taken as 0°, 10°, 20°, and 30°, respectively; these results are plotted in Figure 10.Figure 10
The absolute displacements of tower top. (a) 0° direction misalignment angle, (b) 10° direction misalignment angle, (c) 20° direction misalignment angle, and (d) 30° direction misalignment angle.
(a)(b)(c)(d)It can be found from Figure10 that the displacement response of tower top slightly descends with the increase of misalignment angle in fore-aft direction, and the displacement response ascends as the misalignment angle increases in the side-side direction. This is main because that the component of wind load decreases with the increase of misalignment angle at the range of 0°‒30° in fore-aft direction, and the corresponding component increases as the misalignment angle increases in the side-side direction. Furthermore, the displacement responses of tower top for WWT with the IP-TMD system are notably suppressed compared with the uncontrolled WWT system in fore-aft and side-side directions at different direction misalignment angles.


## 5.1. Load Design and Evaluation Index

To study the vibration control performance of IP-TMD under the wind loads, the 3.2 MW steel WTT simulation model is established using GH-Blade software. Based on the theory of blade element momentum, the horizontally thrust forces considering the wind direction, velocity, air density, and wind shear are simulated [23, 30]. Three centralized aerodynamic loads from wind are used to calculate the dynamic response as shown in Figure 7.Figure 7
Wind loads and frequency spectrums.The energy spectrums in Figure7 of these wind loads mainly distribute near the natural frequency (0.265 Hz) of steel WTT, so they can be well utilized to verify the effectiveness and rationality of IP-TMD under the wind loads.In order to evaluate the vibration mitigation efficiency, the peak indexE1 and root mean square index E2 are defined as [27](26)E1=maxxU−maxxCmaxxU,(27)E2=RMSxU−RMSxCRMSxU,where max denotes the peak response and RMS is the response of root mean square. Symbols U and C are the uncontrolled WTT and controlled WTT by the IP-TMD system, respectively.


## 5.2. Vibration Control Effect

The dynamic responses of the steel WTT with the IP-TMD system are calculated by using the Wilson-θ method under three wind loads [31]. The mass, stiffness, and damping matrix of WTT with the IP-TMD system can be calculated by using (6). Then, taking the wind load 1 as the example, the corresponding dynamic responses and power spectrum density are plotted in Figure 8.Figure 8
The dynamic responses under wind 1. (a) The response of time history. (b) The response of frequency spectrum.
(a)(b)It can be found from Figure8(a) that the dynamic responses of the WTT controlled by IP-TMD are obviously decreased than uncontrolled WTT under the wind 1 load. Since IP-TMD is designed to reversal vibrate with the same frequency near the fundamental frequency of steel WTT, it can effectively control the excessive vibration at the design frequency of steel WTT as the yellow rectangle frame in Figure 8(b). Meanwhile, the vibration control effectiveness under the three wind loads is calculated by using (26) and (27), as shown in Table 3.Table 3
Vibration control effectiveness.
CaseAcceleration (%)Velocity (%)Displacement (%)E1E2E1E2E1E2Wind 141.0552.6445.2754.1444.6853.73Wind 238.8044.2939.9046.2244.3844.82Wind 356.8568.7155.9469.1456.4567.18Mean value45.5655.2147.0456.5048.5055.24According to Table3, the IP-TMD device is noted capable to reduce the mean-peak and mean-RMS acceleration responses by 45.56% and 55.21%, respectively, under three wind loads relative to the uncontrolled cases. IP-TMD can also reduce the mean-peak and mean-RMS velocity responses by 47.04% and 56.60%, respectively, than the uncontrolled WWT. Furthermore, it can also mitigate the mean-peak and mean-RMS displacement responses by 48.50% and 55.24%, respectively, under three wind loads compared with the uncontrolled cases.


## 5.3. IP-TMD Stroke

Because the frequency of the IP-TMD device is tuned to near the fundamental frequency of the controlled steel WTT by adjusting the stiffness coefficient of springs, the vibration energy will be transferred to the mass block of IP-TMD when the controlled steel WTT structure produces the excessive vibration under wind loads. Subsequently, the IP-TMD will produce the motion stroke that is the relative displacement between mass block of IP-TMD and the cylinder wall of tower, and the viscous dampers will deplete the energy. Based on the Wilson-θ method, the stroke of IP-TMD curves is potted in Figure 9 under three wind loads.Figure 9
IP-TMD stroke.As can be seen from Figure9, the maximum of one-way stroke for the IP-TMD reaches to 0.40 m under three wind loads, so it indicated that the IP-TMD has the anticipated competence of dissipating hysteretic energy. Meanwhile, this also means that the motion space of IP-TMD will reach to 0.8 m, so the conventional TMD installed at the nacelle has no enough motion space to dissipate the energy since the engines and gears are installed inside the nacelle. On the contrary, internal available space of tower is close to 4 m; this reflects that IP-TMD placed inside the tower can play a practical role in dissipating hysteresis energy.


## 5.4. Bidirectional Vibration Control

Since the incoming wind direction of a WTT is always changing, the IP-TMD is designed as a bidirectional vibration control device. When the wind direction is different from output force, the direction misalignment effect will give a rise to the bidirectional vibration of WTT. The Wilson-θ method is applied to obtain the bidirectional displacement responses of tower top when the direction misalignment angles are taken as 0°, 10°, 20°, and 30°, respectively; these results are plotted in Figure 10.Figure 10
The absolute displacements of tower top. (a) 0° direction misalignment angle, (b) 10° direction misalignment angle, (c) 20° direction misalignment angle, and (d) 30° direction misalignment angle.
(a)(b)(c)(d)It can be found from Figure10 that the displacement response of tower top slightly descends with the increase of misalignment angle in fore-aft direction, and the displacement response ascends as the misalignment angle increases in the side-side direction. This is main because that the component of wind load decreases with the increase of misalignment angle at the range of 0°‒30° in fore-aft direction, and the corresponding component increases as the misalignment angle increases in the side-side direction. Furthermore, the displacement responses of tower top for WWT with the IP-TMD system are notably suppressed compared with the uncontrolled WWT system in fore-aft and side-side directions at different direction misalignment angles.


## 6. Conclusion

In this paper, a novel TMD refitted via inner platform (IP-TMD) is proposed to control the excessive vibration of steel WTT. Through the theoretical analysis and numerical simulation, three conclusions can be listed as follows:(1)
The dynamic amplification coefficient of WTT with the IP-TMD system has no obvious divergence phenomenon near the natural frequency of steel WTT frequency, and its amplitudes with the mode fission are reduced to 17.60.(2)
The WTT with the IP-TMD system has no obvious resonance phenomenon in the entire frequency domain, and it can reduce the acceleration and displacement responses by 46.2% and 56.6% under the harmonic excitation, respectively.(3)
Under three wind loads, the IP-TMD device is noted capable to reduce the mean-peak and mean-RMS acceleration responses by 45.56% and 55.21%, respectively, relative to the uncontrolled cases. IP-TMD can also reduce the mean-peak and mean-RMS velocity responses by 47.04% and 56.60%, respectively. Moreover, it can also mitigate the mean-peak and mean-RMS displacement responses by 48.50% and 55.24%, respectively.



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*Source: 9119331-2023-03-03.xml*