*2.1. Mathematical Model of Transmission Lines*

The nonlinear equivalent method can be used to compute the dynamic responses of transmission lines using the Hamilton variational method and the Lagrange equation. As displayed in Figure 1, a transmission line is simulated by many masses and elements. Following the Hamilton variational principle, the line can be described as a series of generalized coordinates, namely the difference of the angle *θ* and element length *l*. If the line vibrates in the in-plane direction, the kinetic energy *Tline* is [27]:

$$T\_{\rm line} = \sum\_{i=1}^{4} \frac{1}{2} m\_i (\dot{\mathbf{x}}\_i^2 + \dot{y}\_i^2) \ = \ T\_{\rm line} (\dot{\xi}\_{2\prime} \dot{\xi}\_{3\prime} \dot{\xi}\_{4\prime} \delta l\_{1\prime} \delta l\_{2\prime} \delta \dot{l}\_{3\prime} \delta \dot{l}\_{4\prime} \delta \dot{l}\_{5}) \tag{1}$$

where *ξ<sup>i</sup>* (*i* = 2,3,4) and *δ<sup>i</sup>* (*i* = 1,2,3,4,5) are the structural generalized coordinates related to the *θ* and *l*, respectively [27]; *x<sup>i</sup>* and *y<sup>i</sup>* are the horizontal and vertical displacement of the *i*th mass, respectively.

*i*th mass, respectively.

*s*

/ *U <sup>i</sup>* ∂ ∂ξ

Similarly, structural potential energy *Uline* is:

where *ξi* (*i* = 2,3,4) and *δ<sup>i</sup>* (*i* = 1,2,3,4,5) are the structural generalized coordinates related to the *θ* and *l*, respectively [27]; *xi* and *yi* are the horizontal and vertical displacement of the

*U m gy l l*

where *E* is Young's modulus of the line; *A* is the cross-sectional area of the line; <sup>0</sup>

1 1

= =

*line i i*

*<sup>j</sup> l* are the initial and deformation length of the *j*th element.

2 2 4 5

*i j j j*

0 0

*<sup>j</sup> l* and

( ) ( ) <sup>2</sup> *s s jj j*

δ

*EA ll l*

+ =+ − (2)

**Figure 1.** Analytical model of a transmission line. (**a**) In-plane vibration; (**b**) out-of-plane vibration. **Figure 1.** Analytical model of a transmission line. (**a**) In-plane vibration; (**b**) out-of-plane vibration.

The equation of motion is derived based on Hamilton's equation in terms of a set of Similarly, structural potential energy *Uline* is:

generalized coordinates *qi*, namely *ξ* and *δ*:

$$\mathcal{U}\_{line} = \sum\_{i=1}^{4} m\_i \text{g} y\_i + \sum\_{j=1}^{5} \frac{EA}{2} (\frac{l\_j^s + \delta l\_j)^2}{l\_j^0} - \frac{l\_j^{s2}}{l\_j^0}) \tag{2}$$

In which *Wline*(*t*) denotes the virtual work. Then, computing the variation yields into Lagrange's equation: where *E* is Young's modulus of the line; *A* is the cross-sectional area of the line; *l* 0 *j* and *l s j* are the initial and deformation length of the *j*th element.

*line line line i <sup>d</sup> T TU <sup>Q</sup> dt q q q* ∂ ∂∂ −+ = ∂ ∂∂ (4) The equation of motion is derived based on Hamilton's equation in terms of a set of generalized coordinates *q<sup>i</sup>* , namely *ξ* and *δ*:

$$\int\_{t\_1}^{t\_2} \delta[T\_{line}(t) - \mathcal{U}\_{line}(t)]dt + \int\_{t\_1}^{t\_2} \delta\mathcal{W}\_{line}(t)dt = \mathbf{0} \tag{3}$$

If the line vibrates in the in-plane direction, the stiffness matrix *in* **K***l* is established In which *Wline*(*t*) denotes the virtual work.

δ

by calculating the partial differential of the *Uline* to the generalized displacement and / *U <sup>i</sup>* ∂ ∂Then, computing the variation yields into Lagrange's equation:

$$\frac{d}{dt}\left(\frac{\partial T\_{line}}{\partial \dot{q}\_i}\right) - \frac{\partial T\_{line}}{\partial q\_i} + \frac{\partial L\_{line}}{\partial q\_i} = \dot{Q}\_i \tag{4}$$

pendulum, as shown in Figure 1b. The system matrices **M***<sup>l</sup>* out and **K***<sup>l</sup>* where *Q<sup>i</sup>* is the generalized forcing function of the line.

duced similarly, as follows: 1 2 *out l m m* <sup>=</sup> **M** (5) If the line vibrates in the in-plane direction, the stiffness matrix **K***in l* is established by calculating the partial differential of the *Uline* to the generalized displacement *∂U*/*∂ξ<sup>i</sup>* and *∂U*/*∂δ<sup>i</sup>* . Similarly, the mass matrix **M***in l* is established by calculating the partial differential of the *Tline* to the generalized velocity *∂T*/*∂* . *ξi* and *∂T*/*∂* . *δi* . If the line vibrates in the out-ofplane direction, the line can be simulated as a suspended pendulum, as shown in Figure 1b. The system matrices **M***<sup>l</sup> out* and **K***<sup>l</sup> out* of the line are deduced similarly, as follows:

$$\mathbf{M}\_l^{out} = \begin{bmatrix} m\_1 \\ & m\_2 \end{bmatrix} \tag{5}$$

out of the line are de-

$$\mathbf{K}\_{l}^{out} = \begin{bmatrix} \frac{m\_{1\mathcal{S}}}{l\_1} & -\frac{m\_{1\mathcal{S}}}{l\_1} \\ -\frac{m\_{1\mathcal{S}}}{l\_1} & \frac{m\_{1\mathcal{S}}}{l\_1} + \frac{(m\_1 + m\_2)\mathcal{X}}{l\_2} \end{bmatrix} \tag{6}$$

#### *2.2. Mathematical Model of Tower-Line Coupled System*

The three-dimensional (3D) model of a real transmission tower in China is constructed using ANSYS, as shown in Figure 2a. If a 3D finite element (FE) model is used for the large-scale tower and is incorporated with SMA dampers subjected to wind excitations, the step-by-step dynamic computation will be unbearably time-consuming. This may make the parametric study tedious and impractical. In addition, the numerical simulation of wind loading of the 3D FE model is commonly carried out using the spectral representation method, which requires enormous series calculus. In practice, a lumped mass model is commonly adopted for vibration control and parametric studies, as shown in Figure 2b.

shown in Figure 2b.

1 1 1 1

The three-dimensional (3D) model of a real transmission tower in China is constructed using ANSYS, as shown in Figure 2a. If a 3D finite element (FE) model is used for the large-scale tower and is incorporated with SMA dampers subjected to wind excitations, the step-by-step dynamic computation will be unbearably time-consuming. This may make the parametric study tedious and impractical. In addition, the numerical simulation of wind loading of the 3D FE model is commonly carried out using the spectral

<sup>=</sup> <sup>+</sup> − +

*mg mg l l*

*out l*

*2.2. Mathematical Model of Tower-Line Coupled System* 

1 1 12 11 2

<sup>−</sup>

*mg mg m m g ll l*

( )

**K** (6)

**Figure 2.** Analytical model of a large transmission tower. (**a**) 3D FE model; (**b**) 2D dynamic model. **Figure 2.** Analytical model of a large transmission tower. (**a**) 3D FE model; (**b**) 2D dynamic model.

A transmission tower-line coupled system is a complex continuous system consisting of many towers and lines. It is impossible and unnecessary to establish the system model considering all the lines and towers. Thus, the single tower associated with connected lines can be adopted in the dynamic analysis, as shown in Figure 3. The kinetic energy *T* and potential energy *U* of the tower-line system are expressed as follows: A transmission tower-line coupled system is a complex continuous system consisting of many towers and lines. It is impossible and unnecessary to establish the system model considering all the lines and towers. Thus, the single tower associated with connected lines can be adopted in the dynamic analysis, as shown in Figure 3. The kinetic energy *T* and potential energy *U* of the tower-line system are expressed as follows:

$$T = \ \ T\_t + \sum\_{j=1}^{nl} \ T\_l^{(j)} \tag{7}$$

$$\mathcal{U}\_{\!\!\!I} = \mathcal{U}\_{\!\!\!I} + \sum\_{j=1}^{n!} \mathcal{U}\_{\!\!\!I}^{(j)} \tag{8}$$

where *T<sup>t</sup>* and *U<sup>t</sup>* are the kinetic and potential energy of the single tower; *T* (*j*) *l* and *U* (*j*) *l* are the kinetic and potential energy of the jth line; *nl* is the number of all the lines.

Then, Equations (7) and (8) can be substituted into the Lagrange equation. Similar to the computation of the transmission line expressed in Equation (4), the stiffness matrix of the entire coupled system **K***in* can be established in line with *∂U*/*∂ξ<sup>i</sup>* and *∂U*/*∂δ<sup>i</sup>* . The mass matrix **M***in* can be established in line with *∂T*/*∂* . *ξi* and *∂T*/*∂* . *δi* . For the out-of-plane vibration, the stiffness matrix **K***out* and mass matrix **M***out* of the entire coupled system are determined by combing the system matrices of the tower and lines.

**Figure 3.** Analytical model of a transmission tower-line system. **Figure 3.** Analytical model of a transmission tower-line system. **Figure 3.** Analytical model of a transmission tower-line system.

#### **3. Mathematical Model of SMA Dampers 3. Mathematical Model of SMA Dampers 3. Mathematical Model of SMA Dampers**

SMA wires have excellent inherited properties and can be used to fabricate smart energy-dissipating dampers. The SMA material can be described by the widely-used constitutive model [36–38]. The 2D and 3D configuration of an SMA damper is displayed in Figure 4. The SMA damper consists of the outer tube, inner tube, and circular plates. The SMA wires are incorporated in tension to dissipate energy during its reciprocal movement in vibration. SMA wires have excellent inherited properties and can be used to fabricate smart energy-dissipating dampers. The SMA material can be described by the widely-used constitutive model [36–38]. The 2D and 3D configuration of an SMA damper is displayed in Figure 4. The SMA damper consists of the outer tube, inner tube, and circular plates. The SMA wires are incorporated in tension to dissipate energy during its reciprocal movement in vibration. SMA wires have excellent inherited properties and can be used to fabricate smart energy-dissipating dampers. The SMA material can be described by the widely-used constitutive model [36–38]. The 2D and 3D configuration of an SMA damper is displayed in Figure 4. The SMA damper consists of the outer tube, inner tube, and circular plates. The SMA wires are incorporated in tension to dissipate energy during its reciprocal movement in vibration.

( )

( )

*j*

*j*

( )

( )

*j*

*j*

= + (7)

= + (7)

= + (8)

= + (8)

ξ*<sup>i</sup>* ∂ ∂ and / *<sup>T</sup>*

ξ*<sup>i</sup>* ∂ ∂ and / *<sup>T</sup>*

ξ

ξ

and / *U*

δ*<sup>i</sup>* ∂ ∂ . For the

δ*<sup>i</sup>* ∂ ∂ . For the

and / *U*

δ*<sup>i</sup>* ∂ ∂ .

δ*<sup>i</sup>* ∂ ∂ .

1

1

=

=

1

*nl*

*t l j*

=

1

Then, Equations (7) and (8) can be substituted into the Lagrange equation. Similar to the computation of the transmission line expressed in Equation (4), the stiffness matrix

Then, Equations (7) and (8) can be substituted into the Lagrange equation. Similar to the computation of the transmission line expressed in Equation (4), the stiffness matrix

*UU U*

*UU U*

where *Tt* and *Ut* are the kinetic and potential energy of the single tower; ( )*<sup>j</sup> Tl* and ( )*<sup>j</sup> Ul*

where *Tt* and *Ut* are the kinetic and potential energy of the single tower; ( )*<sup>j</sup> Tl* and ( )*<sup>j</sup> Ul*

out-of-plane vibration, the stiffness matrix *out* **K** and mass matrix **M***out* of the entire coupled system are determined by combing the system matrices of the tower and lines.

out-of-plane vibration, the stiffness matrix *out* **K** and mass matrix **M***out* of the entire coupled system are determined by combing the system matrices of the tower and lines.

are the kinetic and potential energy of the jth line; *nl* is the number of all the lines.

are the kinetic and potential energy of the jth line; *nl* is the number of all the lines.

of the entire coupled system *in* **K** can be established in line with / *U <sup>i</sup>* ∂ ∂

of the entire coupled system *in* **K** can be established in line with / *U <sup>i</sup>* ∂ ∂

The mass matrix **M***in* can be established in line with / *T*

The mass matrix **M***in* can be established in line with / *T*

*Materials* **2022**, *15*, x FOR PEER REVIEW 5 of 25

*nl*

*t l j TT T*

=

*nl*

*t l j*

*t l j TT T*

*nl*

**Figure 4.** Configuration of an SMA damper. **Figure 4.** Configuration of an SMA damper.

phases, respectively.

A

σ

σ

*Af* σ

*EA*

*Ms* σ

*M <sup>f</sup>* σ

Figure 5 shows the hysteretic model of the widely-used Ni-Ti SMA material. In the figure, *Mf* and *Ms* are the martensite finish and start temperature, respectively. *Af* and *As* are the austenite finish and start temperature, respectively; *Ms* σ and *Ms* ε are the critical stress and strain at martensite start temperature, respectively; *<sup>M</sup> <sup>f</sup>* σ and *<sup>M</sup> <sup>f</sup>* ε are the critical stress and strain at martensite finish temperature, respectively; *As* σ and *As* ε are the critical stress and strain at austenite start temperature, respectively; *Af* σ and *Af* ε are Figure 5 shows the hysteretic model of the widely-used Ni-Ti SMA material. In the figure, *M<sup>f</sup>* and *M<sup>s</sup>* are the martensite finish and start temperature, respectively. *A<sup>f</sup>* and *A<sup>s</sup>* are the austenite finish and start temperature, respectively; *σM<sup>s</sup>* and *εM<sup>s</sup>* are the critical stress and strain at martensite start temperature, respectively; *σM<sup>f</sup>* and *εM<sup>f</sup>* are the critical stress and strain at martensite finish temperature, respectively; *σA<sup>s</sup>* and *εA<sup>s</sup>* are the critical stress and strain at austenite start temperature, respectively; *σA<sup>f</sup>* and *εA<sup>f</sup>* are the critical stress and strain at austenite finish temperature, respectively; *ε<sup>L</sup>* is the maximum residual strain; *E<sup>A</sup>* and *E<sup>M</sup>* are Young's moduli at the austenite and martensite phases, respectively.

the critical stress and strain at austenite finish temperature, respectively; *ε<sup>L</sup>* is the maximum residual strain; *EA* and *EM* are Young's moduli at the austenite and martensite

C

*EM*

B

The relationships of strain and stress of the Ni-Ti SMA material shown in Figure 5

ε

The elastic stages (Paths O-A and E-O) are the full austenite stage. The damper force

() ( () ) *<sup>A</sup>*

The forward transformation stage (Path A-B) is the loading stage and the damper

*w E A ut dt l*

where *u*(*t*) is damper force; *d*(*t*) is the damper length with deformation; *A* is the

*w*

*<sup>l</sup>* = − (9)

*EA*

*As* ε

*<sup>M</sup> <sup>f</sup> ε* ε

*L*

cross-sectional area of a wire; *lw* is the original length of a wire.

**Figure 5.** Hysteretic model of an SMA wire.

*Af*

*As* D

can be described based on different paths.

is given by:

O

ε *Ms* ε

E

force *u*(*t*) is given by:

**Figure 4.** Configuration of an SMA damper.

phases, respectively.

are the austenite finish and start temperature, respectively; *Ms*

stress and strain at martensite start temperature, respectively; *<sup>M</sup> <sup>f</sup>*

critical stress and strain at martensite finish temperature, respectively; *As*

the critical stress and strain at austenite start temperature, respectively; *Af*

**Figure 5.** Hysteretic model of an SMA wire. **Figure 5.** Hysteretic model of an SMA wire.

The relationships of strain and stress of the Ni-Ti SMA material shown in Figure 5 can be described based on different paths. The relationships of strain and stress of the Ni-Ti SMA material shown in Figure 5 can be described based on different paths.

Figure 5 shows the hysteretic model of the widely-used Ni-Ti SMA material. In the figure, *Mf* and *Ms* are the martensite finish and start temperature, respectively. *Af* and *As*

the critical stress and strain at austenite finish temperature, respectively; *ε<sup>L</sup>* is the maximum residual strain; *EA* and *EM* are Young's moduli at the austenite and martensite

σ

 and *Ms* ε

σ

 and *<sup>M</sup> <sup>f</sup>* ε

σ

σ

are the critical

 and *As* εare

 and *Af* εare

are the

The elastic stages (Paths O-A and E-O) are the full austenite stage. The damper force is given by: The elastic stages (Paths O-A and E-O) are the full austenite stage. The damper force is given by:

$$u(t) = \frac{E\_A A}{l\_w} (d(t) - l\_w) \tag{9}$$

*w w <sup>l</sup>* = − (9) where *u*(*t*) is damper force; *d*(*t*) is the damper length with deformation; *A* is the crosssectional area of a wire; *l<sup>w</sup>* is the original length of a wire.

where *u*(*t*) is damper force; *d*(*t*) is the damper length with deformation; *A* is the cross-sectional area of a wire; *lw* is the original length of a wire. The forward transformation stage (Path A-B) is the loading stage and the damper force *u*(*t*) is given by:

$$u(t) = \left[\sigma\_{M\_\delta} + \frac{\sigma\_{M\_f} - \sigma\_{M\_\delta}}{\varepsilon\_{M\_f} - \varepsilon\_{M\_\delta}}(\varepsilon(t) - \varepsilon\_{M\_\delta})\right]A\tag{10}$$

For the full martensite stage (Path B-C), the elastic deformation is observed and the damper force is:

$$u(t) = \frac{E\_M A}{l\_w} (d(t) - l\_w) \tag{11}$$

For the full martensite stage (Path B-D), the unloading process is observed and the damper force is:

$$u(t) = \sigma\_{M\_f} A + \mathbb{E}\_M \Big[ (d(t) - l\_w) - \varepsilon\_{M\_f} \Big] A \tag{12}$$

The reverse transformation stage (Path D-E) is the unloading stage and the damper force *u*(*t*) is given by:

$$u(t) = \left[\sigma\_{\mathcal{M}\_s} + \frac{\sigma\_{\mathcal{M}\_s} - \sigma\_{A\_f}}{\varepsilon\_{A\_s} - \varepsilon\_{A\_f}}(\varepsilon(t) - \varepsilon\_{A\_s})\right]A\tag{13}$$

in which *ε*(*t*) is the stress of a wire.

force *u*(*t*) is given by:

#### **4. Equation of Motion of the Controlled Structure**

The equation of motion of the tower-line system equipped with SMA dampers is:

$$\mathbf{M}\ddot{\mathbf{x}}(t) + \mathbf{C}\dot{\mathbf{x}}(t) + \mathbf{K}\mathbf{x}(t) \ = \mathbf{W}(t) + \mathbf{H}\mathbf{u}(t) \tag{14}$$

$$\mathbf{M} = \begin{bmatrix} \mathbf{M}^{in} & \mathbf{0} \\ \mathbf{0} & \mathbf{M}^{out} \end{bmatrix} \tag{15}$$

$$\mathbf{C} = \begin{bmatrix} \mathbf{C}^{in} & 0\\ 0 & \mathbf{C}^{out} \end{bmatrix} \tag{16}$$

$$\mathbf{K} = \begin{bmatrix} \mathbf{K}^{in} & \mathbf{0} \\ \mathbf{0} & \mathbf{K}^{out} \end{bmatrix} \tag{17}$$

$$\mathbf{W}(t) = \begin{bmatrix} \mathbf{W}^{in}(t); & \mathbf{W}^{out}(t) \end{bmatrix} \tag{18}$$

$$\mathbf{u}(t) \;= \begin{bmatrix} u\_1 & u\_2 & \cdots & u\_n \end{bmatrix}^T \tag{19}$$

where **x**(*t*), **. <sup>x</sup>**(*t*), and **.. x**(*t*) are the displacement, velocity, and acceleration of the towerline system, respectively; **M**, and **K** are the mass and stiffness matrices of the system, respectively; **C** is the Rayleigh damping matrix; **W**(*t*) is the wind-loading vector; **W***in*(*t*) and **W***out*(*t*) are the wind-loading vectors in the in-plane and out-of-plane direction, respectively; **u**(*t*) is the control force vector; **H** is the position matrix of **u**(*t*); *n* is the damper number.

The wind excitations acting on the structural system are simulated using the spectral representation method. The vibration of a tower-line coupled system can also be illustrated using energy responses. The energy equations of the entire coupled system without and with SMA dampers are formed by integrating Equation (15). The total input energy from wind loading to the structural system *E<sup>W</sup>* is the sum of the kinetic energy *EK*, the strain energy *ES*, the energy dissipated by structural damping *ED*, and the energy dissipated by SMA dampers *EC*.

#### **5. Case Study**
